We describe a simple and yet very scalable implementation of static functions (VFunc) and of static filters (VFilter) based on hypergraphs. We introduce the idea of ε-cost sharding, which allows us to build structures that can manage trillions of keys, at the same time increasing memory locality in hypergraph-based constructions. Contrarily to the commonly used HEM sharding method, ε-cost sharding does not require to store of additional information, and does not introduce dependencies in the computation chain; its only cost is that of few arithmetical instructions, and of a relative increase ε in space usage. We apply ε-cost sharding to the classical MWHC construction, but we obtain the best result by combining Dietzfelbinger and Walzer's fuse graphs for large shards with lazy Gaussian elimination for small shards. We obtain large structures with an overhead of 10.5% with respect to the information-theoretical lower bound and with a query time that is a few nanoseconds away from the query time of the non-sharded version, which is the fastest currently available within the same space bounds. Besides comparing our structures with a non-sharded version, we contrast its tradeoffs with bumped ribbon constructions, a space-saving alternative to hypergraph-based static functions and filters, which provide optimum space consumption but slow construction and query time (though construction can be parallelized very efficiently). We build offline a trillion-key filter using commodity hardware in just 60 ns/key.

Download from arXiv; Rust implementations.

When the Mersenne Twister made his first appearance in 1997 it was a powerful example of how linear maps on F₂ could be used to generate pseudorandom numbers. In particular, the easiness with which generators with long periods could be defined gave the Mersenne Twister a large following, in spite of the fact that such long periods are not a measure of quality, and they require a large amount of memory. Even at the time of its publication, several defects of the Mersenne Twister were predictable, but they were somewhat obscured by other interesting properties. Today the Mersenne Twister is the default generator in C compilers, the Python language, the Maple mathematical computation system, and in many other environments. Nonetheless, knowledge accumulated in the last 20 years suggests that the Mersenne Twister has, in fact, severe defects, and should never be used as a general-purpose pseudorandom number generator. Many of these results are folklore, or are scattered through very specialized literature. This paper surveys these results for the non-specialist, providing new, simple, understandable examples, and it is intended as a guide for the final user, or for language implementors, so that they can take an informed decision about whether to use the Mersenne Twister or not.

Download from arXiv.

Among the properties describing the behavior of centrality measures with respect to network modifications, score monotonicity means that adding an arc increases the centrality score of the target of the arc; rank monotonicity means that adding an arc improves the importance of the target with respect to the remaining nodes. It is known that score and rank monotonicity hold in directed graphs for almost all the classical centrality measures. In undirected graphs one expects that the corresponding properties hold when adding a new edge—in this case, both endpoints of the new edge should enjoy the increase in score/rank. However, recent results have shown that this is not true: for many centrality measures, it is possible to find situations in which adding an edge reduces the rank of one of its two endpoints. In this paper we introduce a weaker property for undirected networks, semi-monotonicity, in which just one of the two endpoints of a new edge is required to enjoy score or rank monotonicity. We show that this property is satisfied by closeness centrality, by a large class of distance-based centralities, and (somehow surprisingly) by betweenness centrality. In the last two cases, we prove in fact a stronger property, basin dominance, which is of independent interest.

doi:10.1007/s13278-024-01285-y.

Is it always beneficial to create a new relationship (have a new follower/friend) in a social network? This question can be formally stated as a property of the centrality measure that defines the importance of the actors of the network. Score monotonicity means that adding an arc increases the centrality score of the target of the arc; rank monotonicity means that adding an arc improves the importance of the target of the arc relatively to the remaining nodes. It is known that most centralities are both score and rank monotone on directed, strongly connected graphs. In this paper, we study the problem of score and rank monotonicity for classical centrality measures in the case of undirected networks: in this case, we require that score, or relative importance, improve at both endpoints of the new edge. We show that, surprisingly, the situation in the undirected case is very different, and in particular that closeness, harmonic centrality, betweenness, eigenvector centrality, Seeley's index, Katz's index, and PageRank are not rank monotone; betweenness and PageRank are not even score monotone. In other words, while it is always a good thing to get a new follower, it is not always beneficial to get a new friend.

doi:10.1017/nws.2022.42; Download from arXiv.

We describe a new statistical test for pseudorandom number generators (PRNGs). Our test can find bias induced by dependencies among the Hamming weights of the outputs of a PRNG, even for PRNGs that pass state-of-the-art tests of the same kind from the literature, and in particular for generators based on F₂-linear transformations such as the dSFMT, xoroshiro1024+, and WELL512.

doi:10.1145/3527582; PDF version; more data can be found here.

In 2014, Steele, Lea, and Flood presented SplitMix, an object-oriented pseudorandom number generator (prng) that is quite fast (9 64-bit arithmetic/logical operations per 64 bits generated) and also splittable. A conventional prng object provides a generate method that returns one pseudorandom value and updates the state of the prng; a splittable prng object also has a second operation, split, that replaces the original prng object with two (seemingly) independent prng objects, by creating and returning a new such object and updating the state of the original object. Splittable prng objects make it easy to organize the use of pseudorandom numbers in multithreaded programs structured using fork-join parallelism. This overall strategy still appears to be sound, but the specific arithmetic calculation used for generate in the SplitMix algorithm has some detectable weaknesses, and the period of any one generator is limited to 2⁶⁴.

Here we present the LXM family of prng algorithms. The idea is an old one: combine the outputs of two independent prng algorithms, then (optionally) feed the result to a mixing function. An LXM algorithm uses a linear congruential subgenerator and an F₂-linear subgenerator; the examples studied in this paper use a linear congruential generator (LCG) of period 2¹⁶, 2³², 2⁶⁴, or 2¹²⁸ with one of the multipliers recommended by L'Ecuyer or by Steele and Vigna, and an F₂-linear xor-based generator (XBG) of the xoshiro family or xoroshiro family as described by Blackman and Vigna. For mixing functions we study the MurmurHash3 finalizer function; variants by David Stafford, Doug Lea, and degski; and the null (identity) mixing function.

Like SplitMix, LXM provides both a generate operation and a split operation. Also like SplitMix, LXM requires no locking or other synchronization (other than the usual memory fence after instance initialization), and is suitable for use with simd instruction sets because it has no branches or loops.

We analyze the period and equidistribution properties of LXM generators, and present the results of thorough testing of specific members of this family, using the TestU01 and PractRand test suites, not only on single instances of the algorithm but also for collections of instances, used in parallel, ranging in size from 2 to 2²⁴. Single instances of LXM that include a strong mixing function appear to have no major weaknesses, and LXM is significantly more robust than SplitMix against accidental correlation in a multithreaded setting. We believe that LXM, like SplitMix, is suitable for “everyday” scientific and machine-learning applications (but not cryptographic applications), especially when concurrent threads or distributed processes are involved.

doi:10.1145/3485525; code can be found in the java.util.random package (Java 17+).

F₂-linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer from linear artifacts that show as failures in linearity-related statistical tests such as the binary-rank and the linear-complexity test. In this article, we give two new contributions. First, we introduce two new F₂-linear transformations that have been handcrafted to have good statistical properties and at the same time to be programmable very efficiently on superscalar processors, or even directly in hardware. Then, we describe some scramblers, that is, nonlinear functions applied to the state array that reduce or delete the linear artifacts, and propose combinations of linear transformations and scramblers that give extremely fast pseudorandom number generators of high quality. A novelty in our approach is that we use ideas from the theory of filtered linear-feedback shift registers to prove some properties of our scramblers, rather than relying purely on heuristics. In the end, we provide simple, extremely fast generators that use a few hundred bits of memory, have provable properties, and pass strong statistical tests.

doi:10.1145/3460772; PDF version; more data can be found at the xoshiro / xoroshiro generators and the PRNG shootout page; xoroshiro128++ and xoshiro256++ are now part of the java.util.random package (Java 17+).

Congruential pseudorandom number generators rely on good multipliers, that is, integers that have good performance with respect to the spectral test. We provide lists of multipliers with a good lattice structure up to dimension eight and up to lag eight for generators with typical power-of-two moduli, analyzing in detail multipliers close to the square root of the modulus, whose product can be computed quickly.

doi:10.1002/spe.3030; code and data can be found on GitHub.

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

Nature Methods link.

Recent advances in the analysis of random linear systems on finite fields have paved the way for the construction of constant-time data structures representing static functions and minimal perfect hash functions using less space with respect to existing techniques. The main obstacle for any practical application of these results is the time required to solve such linear systems: despite they can be made very small, the computation is still too slow to be feasible. In this paper, we describe in detail a number of heuristics and programming techniques to speed up the solution of these systems by orders of magnitude, making the overall construction competitive with the standard and widely used MWHC technique, which is based on hypergraph peeling. In particular, we introduce broadword programming techniques for fast equation manipulation and a lazy Gaussian elimination algorithm. We also describe a number of technical improvements to the data structure which further reduce space usage and improve lookup speed. Our implementation of these techniques yields a minimal perfect hash function data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based ones, and a static function data structure which reduces the multiplicative overhead from 1.23 to 1.024. For functions whose output has low entropy, we are able to implement feasibly for the first time the Hreinsson–Krøyer–Pagh approach, which makes it possible, for example, to store a function with an output of 10⁶ values distributed following a power law of exponent 2 in just 2.76 bits per key instead of 20.

doi:10.1016/j.ic.2020.104517. The algorithms described in the paper are implemented in Sux4J.

We analyze in detail the probability that sequences of equal length generated by a pseudorandom number generator starting from random points of the state space overlap, providing for the first time an exact result and manageable bounds. While the computation of the probability is almost elementary, the value has been reported erroneously several times in the literature.

The combinatorial part of this paper has actually been proved in 1968 by Joseph I. Naus (thanks to Samuel Neves for reporting this reference).

doi:10.1016/j.ipl.2020.105939; PDF version.

The Fenwick tree is a classical implicit data structure that stores an array in such a way that modifying an element, accessing an element, computing a prefix sum and performing a predecessor search on prefix sums all take logarithmic time. We introduce a number of variants which improve the classical implementation of the tree: in particular, we can reduce its size when an upper bound on the array element is known, and we can perform much faster predecessor searches. Our aim is to use our variants to implement an efficient dynamic bit vector: our structure is able to perform updates, ranking and selection in logarithmic time, with a space overhead in the order of a few percents, outperforming existing data structures with the same purpose. Along the way, we highlight the pernicious interplay between the arithmetic behind the Fenwick tree and the structure of current CPU caches, suggesting simple solutions that improve performance significantly.

Download from arXiv; the algorithms described in the paper are implemented in Sux.

Although web crawlers have been around for twenty years by now, there is virtually no freely available, open-source crawling software that guarantees high throughput, overcomes the limits of single-machine systems, and, at the same time, scales linearly with the amount of resources available. This article aims at filling this gap, through the description of BUbiNG, our next-generation web crawler built upon the authors' experience with UbiCrawler and on the last ten years of research on the topic. BUbiNG is an open-source Java fully distributed crawler; a single BUbiNG agent, using sizeable hardware, can crawl several thousand pages per second respecting strict politeness constraints, both host- and IP-based. Unlike existing open-source distributed crawlers that rely on batch techniques (like MapReduce), BUbiNG job distribution is based on modern high-speed protocols to achieve very high throughput.

PDF version; online version; BUbiNG can be downloaded from the LAW web site or from Maven.

Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant ∧-representation by ∧-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in details properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.

doi:10.1007/s11083-016-9418-8; PDF version.

A measure of centrality is rank monotone if after adding an arc x → y, all nodes with a score smaller than (or equal to) y have still a score smaller than (or equal to) y. If in particular all nodes with a score smaller than or equal to y get a score smaller than y (i.e., all ties with y are broken in favor of y) the measure is called strictly rank monotone. We prove that harmonic centrality is strictly rank monotone, whereas closeness is just rank monotone on strongly connected graphs, and that some other measures, including betweenness, are not rank monotone at all (sometimes not even on strongly connected graphs). Among spectral measures, damped scores such as Katz's index and PageRank are strictly rank monotone on all graphs, whereas the dominant eigenvector is strictly monotone on strongly connected graphs only.

PDF version; doi:10.1017/nws.2017.21.

We sketch the history of spectral ranking—a general umbrella name for techniques that apply the theory of linear maps (in particular, eigenvalues and eigenvectors) to matrices that do not represent geometric transformations, but rather some kind of relationship between entities. Albeit recently made famous by the ample press coverage of Google's PageRank algorithm, spectral ranking was devised more than a century ago, and has been studied in psychology, social sciences, bibliometrics, economy, and choice theory. We describe the contribution given by previous scholars in precise and modern mathematical terms: Along the way, we show how to express in a general way damped rankings, such as Katz's index, as dominant eigenvectors of perturbed matrices, and then use results on the Drazin inverse to go back to the dominant eigenvectors by a limit process. The result suggests a regularized definition of spectral ranking that yields for a general matrix a unique vector depending on a boundary condition.

Current version; Network Science version.

xorshift* generators are a variant of Marsaglia's xorshift generators that eliminate linear artifacts typical of generators based on Z/2Z-linear operations using multiplication by a suitable constant. Shortly after high-dimensional xorshift* generators were introduced, Saito and Matsumoto suggested a different way to eliminate linear artifacts based on addition in Z/2³²Z, leading to the XSadd generator. Starting from the observation that the lower bits of XSadd are very weak, as its reverse fails systematically several statistical tests, we explore variants of XSadd using 64-bit operations, and describe in detail xorshift128+, an extremely fast generator that passes strong statistical tests using only three shifts, four xors and an addition.

PDF version; doi:10.1016/j.cam.2016.11.006; more data can be found at the xoshiro / xoroshiro generators and the PRNG shootout page.

Minimal-interval semantics associates with each query over a document a set of intervals, called witnesses, that are incomparable with respect to inclusion (i.e., they form an antichain): witnesses define the minimal regions of the document satisfying the query. Minimal-interval semantics makes it easy to define and compute several sophisticated proximity operators, provides snippets for user presentation, and can be used to rank documents. In this paper we provide algorithms for computing conjunction and disjunction that are linear in the number of intervals and logarithmic in the number of operands; for additional operators, such as ordered conjunction and Brouwerian difference, we provide linear algorithms. In all cases, space is linear in the number of operands. More importantly, we define a formal notion of optimal laziness, and either prove it, or prove its impossibility, for each algorithm. We cast our results in a general framework of antichains of intervals on total orders, making our algorithms directly applicable to other domains.

PDF version; doi:10.1016/j.tcs.2016.07.036; the algorithms described in the paper are implemented in MG4J.

Knowledge about the general graph structure of the World Wide Web is important for understanding the social mechanisms that govern its growth, for designing ranking methods, for devising better crawling algorithms, and for creating accurate models of its structure. In this paper, we analyze a large web graph. The graph was extracted from a large publicly accessible web crawl that was gathered by the Common Crawl Foundation in 2012. The graph covers over 3:5 billion web pages and 128:7 billion hyperlinks. We analyze and compare, among other features, degree distributions, connectivity, average distances, and the structure of weakly/strongly connected components. We conduct our analysis on three different levels of aggregation: page, host, and pay-level domain (PLD) (one “dot level” above public suffixes). Our analysis shows that, as evidenced by previous research (Serrano et al., 2007), some of the features previously observed by Broder et al., 2000 are very dependent on artifacts of the crawling process, whereas other appear to be more structural. We confirm the existence of a giant strongly connected component; we however find, as observed by other researchers (Donato et al., 2005; Boldi et al., 2002; Baeza-Yates and Poblete, 2003), very different proportions of nodes that can reach or that can be reached from the giant component, suggesting that the “bow-tie structure” as described by Broder et al. is strongly dependent on the crawling process, and to the best of our current knowledge is not a structural property of the Web. More importantly, statistical testing and visual inspection of size-rank plots show that the distributions of indegree, outdegree and sizes of strongly connected components of the page and host graph are not power laws, contrarily to what was previously reported for much smaller crawls, although they might be heavy tailed. If we aggregate at pay-level domain, however, a power law emerges. We also provide for the first time accurate measurement of distance-based features, using recently introduced algorithms that scale to the size of our crawl (Boldi and Vigna, 2013).

PDF version; doi:10.1561/106.00000003.

Marsaglia proposed recently xorshift generators as a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L'Ecuyer has lowered the expectations raised by Marsaglia's paper, showing several weaknesses of such generators, verified experimentally using the TestU01 suite. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a non-linear operation (as originally suggested by Marsaglia). In this paper we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 equispaced points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia's paper, finding choices of parameters providing periods of length 2¹⁰²⁴ − 1 and 2⁴⁰⁹⁶ − 1. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition and one multiplication by a constant.

PDF Version; doi.org:10.1145/2845077; more data can be found at the xoshiro / xoroshiro generators and the PRNG shootout page.

Wikipedia is a huge global repository of human knowledge that can be leveraged to investigate interwinements between cultures. With this aim, we apply methods of Markov chains and Google matrix for the analysis of the hyperlink networks of 24 Wikipedia language editions, and rank all their articles by PageRank, 2DRank and CheiRank algorithms. Using automatic extraction of people names, we obtain the top 100 historical figures, for each edition and for each algorithm. We investigate their spatial, temporal, and gender distributions in dependence of their cultural origins. Our study demonstrates not only the existence of skewness with local figures, mainly recognized only in their own cultures, but also the existence of global historical figures appearing in a large number of editions. By determining the birth time and place of these persons, we perform an analysis of the evolution of such figures through 35 centuries of human history for each language, thus recovering interactions and entanglement of cultures over time. We also obtain the distributions of historical figures over world countries, highlighting geographical aspects of cross-cultural links. Considering historical figures who appear in multiple editions as interactions between cultures, we construct a network of cultures and identify the most influential cultures according to this network.

Online version.Some related Wikipedia datasets are available from the LAW.

Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this paper, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed for. Our axioms suggest some simple, basic properties that a centrality measure should exhibit.

Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas's classic closeness centrality designed to take unreachable nodes into account in a natural way.

As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the effectiveness of the various indices in locating web pages that are relevant to a query. While there are some examples of such comparisons in the literature, here for the first time we also take into consideration centrality measures based on distances, such as closeness, in an information-retrieval setting. The results closely match the data we gathered using our axiomatic approach.

Our results suggest that centrality measures based on distances, which in the last years have been neglected in information retrieval in favor of spectral centrality measures, do provide high-quality signals; moreover, harmonic centrality pops up as an excellent general-purpose centrality index for arbitrary directed graphs.

PDF version.

Given a social network, which of its nodes have a stronger impact in determining its structure? More precisely: which node-removal order has the greatest impact on the network structure? We approach this well-known problem for the first time in a setting that combines both web graphs and social networks. Our experiments are performed on datasets that are orders of magnitude larger than those appearing in the previous literature: this is possible thanks to some recently developed algorithms and software tools that approximate accurately the number of reachable pairs and the distribution of distances in large graphs. Our experiments highlight deep differences in the structure of social networks and web graphs, show significant limitations of previous experimental results; at the same time, they reveal clustering by label propagation as a new and very effective way of locating nodes that are important from a structural viewpoint.

Public datasets are available at the LAW site; Java™ implementations are available as free software at the WebGraph home page.

It is well known that in a binary tree the external path length minus the internal path length is exactly 2n − 2, where n is the number of external nodes. We show that a generalization of the formula holds for compacted tries, replacing the role of paths with the notion of extent, and the value 2n − 2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.

PDF version; doi.org:10.1016/j.ipl.2010.12.012.

Minimal perfect hash functions have been shown to be useful to compress data in several data management tasks. In particular, order-preserving minimal perfect hash functions have been used to retrieve the position of a key in a given list of keys: however, the ability to preserve any given order leads to an unavoidable Ω(n log n) lower bound on the number of bits required to store the function. Recently, it was observed that very frequently the keys to be hashed are sorted in their intrinsic (i.e., lexicographical) order. This is typically the case of dictionaries of search engines, list of URLs of web graphs, etc. We refer to this restricted version of the problem as monotone minimal perfect hashing. We analyse experimentally the data structures proposed in our paper “Monotone Minimal Perfect Hashing: Searching a Sorted Table with O(1) Accesses”, and along our way we propose some new methods that, albeit asymptotically equivalent or worse, perform very well in practice, and provide a balance between access speed, ease of construction, and space usage.

PDF version; doi.org:10.1145/1963190.2025378; the algorithms described in the paper are implemented in Sux4J.

doi:10.1145/1953122.1953154.

Understanding query reformulation patterns is a key task towards next generation web search engines. If we can do that, then we can build systems able to understand and possibly predict user intent, providing the needed assistance at the right time, and thus helping users locate information more effectively and improving their web-search experience. As a step in this direction, we build a very accurate model for classifying user query reformulations into broad classes (generalization, specialization, error correction or parallel move), achieving 92% accuracy. We then apply the model to automatically label two very large query logs sampled from different geographic areas, and containing a total of approximately 17 million query reformulations. We study the resulting reformulation patterns, matching some results from previous studies performed on smaller manually annotated datasets, and discovering new interesting reformulation patterns, including connections between reformulation types and topical categories. We annotate two large query-flow graphs with reformulation type information, and run several graph-characterization experiments on these graphs, extracting new insights about the relationships between the different query reformulation types. Finally we study query recommendations based on short random walks on the query-flow graphs. Our experiments show that these methods can match in precision, and often improve, recommendations based on query-click graphs, without the need of users' clicks. Our experiments also show that it is important to consider transition-type labels on edges for having recommendations of good quality.

Since the first investigations on web graph compression, it has been clear that the ordering of the nodes of the graph has a fundamental influence on the compression rate (usually expressed as the number of bits per link). The authors of the LINK database, for instance, investigated three different approaches: an extrinsic ordering (URL ordering) and two intrinsic (or coordinate-free) orderings based on the rows of the adjacency matrix (lexicographic and Gray code); they concluded that URL ordering has many advantages in spite of a small penalty in compression. In this paper we approach this issue in a more systematic way, testing some old orderings and proposing some new ones. Our experiments are made in the WebGraph framework, and show that the compression technique and the structure of the graph can produce significantly different results. In particular, we show that for the transpose web graph URL ordering is significantly less effective, and that some new orderings combining host information and Gray/lexicographic orderings outperform all previous methods. In particular, in some large transposed graphs they yield the quite incredible compression rate of 1 bit per link.

PDF version; datasets and Java™ implementations are available as free software at the LAW site.

PageRank is defined as the stationary state of a Markov chain. The chain is obtained by perturbing the transition matrix induced by a web graph with a damping factor α that spreads uniformly part of the rank. The choice of α is eminently empirical, and in most cases the original suggestion α=0.85 by Brin and Page is still used. In this paper, we give a mathematical analysis of PageRank when α changes. In particular, we show that, contrarily to popular belief, for real-world graphs values of α close to 1 do not give a more meaningful ranking. Then, we give closed-form formulae for PageRank derivatives of any order, and by proving that the k-th iteration of the Power Method gives exactly the value obtained by truncating the PageRank power series at degree k, we show how to obtain an approximation of the derivatives. Finally, we view PageRank as a linear operator acting on the preference vector and show a tight connection between iterated computation and derivation.

PDF version; doi.org:10.1145/1629096.1629097.

You are back from that very long, marvelous journey. You have a thousand pictures, but your friends and relatives will stand just a few dozens. Choosing is a painful process, in particular when you cannot decide between the silent vastity of that desert and the idyllic picture of that tranquil, majestic lake. We are going to help.

PDF version; doi.org:10.1007/s00224-008-9113-2.

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

PDF version; doi.org:10.1051/ita:2006004; Java™ classes for computing a minimum base are available as free software at the LAW site.

Deciding which kind of visit accumulates high-quality pages more quickly is one of the most often debated issue in the design of web crawlers. It is known that breadth-first visits work well, as they tend to discover pages with high PageRank early on in the crawl. Indeed, this visit order is much better than depth first, which is in turn even worse than a random visit; nevertheless, breadth-first can be superseded using an omniscient visit that chooses, at every step, the node of highest PageRank in the frontier.

This paper discusses a related, and previously overlooked, measure of effectivity for crawl strategies: whether the graph obtained after a partial visit is in some sense representative of the underlying web graph as far as the computation of PageRank is concerned. More precisely, we are interested in determining how rapidly the computation of PageRank over the visited subgraph yields relative ranks that agree with the ones the nodes have in the complete graph; ranks are compared using Kendall's τ.

We describe a number of large-scale experiments that show the following paradoxical effect: visits that gather PageRank more quickly (e.g., highest-quality-first) are also those that tend to miscalculate PageRank. Finally, we perform the same kind of experimental analysis on some synthetic random graphs, generated using well-known web-graph models: the results are almost opposite to those obtained on real web graphs.

PDF version; doi.org:10.1080/15427951.2005.10129106; Java™ classes for computing Kendall's τ efficiently are available as free software at the LAW site.

We introduce a new family of simple, complete instantaneous codes for positive integers, called ζ codes, which are suitable for integers distributed as a power law with small exponent (smaller than 2). The main motivation for the introduction of ζ codes comes from web-graph compression: if nodes are numbered according to URL lexicographical order, gaps in successor lists are distributed according to a power law with small exponent. We give estimates of the expected length of ζ codes against power-law distributions, and compare the results with analogous estimates for the more classical γ, δ and variable-length block codes.

PDF version; doi.org:10.1080/15427951.2005.10129113; to download software and data sets, please look at the WebGraph home page.

The Java string classes, String and StringBuffer, lie at the extremes of a spectrum (immutable, reference-based and mutable, content-based). Analogously, available text-search methods on string classes are implemented either as trivial, brute-force double loops, or as very sophisticated and resource-consuming regular-expression search methods. Motivated by our experience in data-intensive text applications, we propose a new string class, MutableString, which tries to get the right balance between extremes in both cases. Mutable strings can be in one of two states, compact and loose, in which they behave more like String and StringBuffer, respectively. Moreover, they support a wide range sophisticated text-search algorithms with a very low resource usage and setup time, using a new, very simple randomised data structure (a generalisation of Bloom filters) that stores an approximation from above of a lattice-valued function. Computing the function value requires a constant number of steps, and the error probability can be balanced with space usage. As a result, we obtain practical implementations of Boyer-Moore type algorithms that can be used with very large alphabets, such as Unicode collation elements. The techniques we develop are very general and amenable to a wide range of applications.

We report our experience in implementing UbiCrawler, a scalable distributed web crawler, using the Java programming language. The main features of UbiCrawler are platform independence, linear scalability, graceful degradation in the presence of faults, a very effective assignment function (based on consistent hashing) for partitioning the domain to crawl, and more in general the complete decentralization of every task. The necessity of handling very large sets of data has highlighted some limitation of the Java APIs, which prompted the authors to partially reimplement them.

A graph G with n vertices and maximum degree Δ_G cannot be given weak sense of direction using less than Δ_G colours. It is known that n colours are always sufficient, and it was conjectured that just Δ_G+1 are really needed, that is, one more colour is sufficient. Nonetheless, it has just been shown that for sufficiently large n there are graphs requiring ω(n/log n) more colours than Δ_G. In this paper, using recent results in asymptotic graph enumeration, we show not only that (somehow surprisingly) the same bound holds for regular graphs, but also that it can be improved to Ω(n log log n/log n) We also show that Ω(d_G(log log d_G)^1/2 colours are necessary, where d_G is the degree of G.

A graph with n and maximum degree Δ cannot be given weak sense of direction using less than Δ colours. It is known that n colours are always sufficient, but it has been conjectured that just Δ+1 are really needed. On the contrary, we show that for sufficiently large n there are graphs requiring Δ+ω(n/log n) colours. We also give simple examples of small graphs requiring Δ+2 colours, which have been verified mechanically.

We prove the existence of a "universal" synchronous self-stabilizing protocol, that is, a protocol that allows a distributed system to stabilize to a desired nonreactive behaviour (as long as a protocol stabilizing to that behaviour exists). Previous proposals required drastic increases in asymmetry and knowledge to work, whereas our protocol does not use any additional knowledge, and does not require more symmetry-breaking conditions than available; thus, it is also stabilizing with respect to dynamic changes in the topology. We prove an optimal quiescence time n+D for a synchronous network of n processors and diameter D; the protocol can be made finite state with a negligible loss in quiescence time. Moreover, an optimal D+1 protocol is given for the case of unique identifiers. As a consequence, we provide an effective proof technique that allows to show whether self-stabilization to a certain behaviour is possible under a wide range of models.

PDF version.

In this paper we prove that deciding whether a distributed system (represented as a coloured digraph with n nodes) has weak sense of direction is in AC¹ (using n⁶ processors). Moreover, we show that deciding sense of direction is in P. Our algorithms can also be used to decide in AC¹ whether a coloured graph is a Cayley colour graph.

A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.

PDF version; see also the Graph-fibrations home page; Java™ classes for computing a minimum base are available as free software at the LAW site.

A δ-uniform BSS machine is a standard BSS machine which does not rely on exact equality tests. We prove that, for any real closed archimedean field R, a set is δ-uniformly semi-decidable iff it is open and semi-decidable by a BSS machine which is locally time bounded; we also prove that the local time bound is nontrivial. This entails a number of results about BSS machines, in particular the existence of decidable sets whose interior (closure) is not even semi-decidable without adding constants. Finally, we show that the sets semi-decidable by Turing machines are the sets semi-decidable by δ-uniform machines with coefficients in Q or T, the field of Turing computable numbers.

We define a notion of degree of unsolvability for subsets of Rⁿ (where R is a real closed Archimedean field) and prove that, in contrast to Type 2 computability, the presence of exact equality in the BSS model forces exactly one jump of the unsolvability degree of decidable sets.

This paper presents an equivalence result between computability in the BSS model and in a suitable distributive category. It is proved that the class of functions R^m→Rⁿ (with n,m finite and R a commutative, ordered ring) computable in the BSS model, and the functions distributively computable over a natural distributive graph based on the operations of R coincide. Using this result, a new structural characterization, based on iteration, of the same functions is given.

We study the jug problem in its most general form: given a set of jugs of fixed capacities, find out which quantities are measurable, and provide upper and lower bounds on the number of steps necessary for measurements.

A BSS machine is δ-uniform if it does not use exact tests; such machines are equivalent (modulo parameters) to Type 2 Turing machines. We define a notion of closure related to Turing machines for archimedean fields, and show that such fields admit nontrivial δ-uniformly decidable sets iff they are not Turing closed. Then, the partially ordered set of Turing closed fields is proved isomorphic to the ideal completion of unsolvability degrees.

Sense of direction is a property of the labelling of (possibly anonymous) networks which allows to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. A graph has minimal sense of direction whenever it has sense of direction and the number of colours equals its maximum outdegree. We prove that an outregular digraph with minimal weak sense of direction is a Cayley colour graph (in the general sense, i.e., we do not require connectedness). Since Cayley colour graphs are known to possess minimal transitive sense of direction, we obtain a characterization of outregular graphs with minimal (weak,transitive) sense of direction. As a consequence, deciding whether a coloured graph is a Cayley colour graph reduces to deciding whether it has weak sense of direction, which can be done in AC¹.

We show that computing (and even approximating) maximum clique and minimum graph coloring for circulant graphs is essentially as hard as in the general case. In contrast, we show that, under additional constraints, e.g., prime order and/or spareness, graph isomorphism and minimum graph coloring become easier in the circulant case, and we take advantage of spectral techniques for their efficient computation.

Sense of direction is a property of labelled networks (i.e., arc-coloured graphs) that allows one to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. We prove that (weak) sense of direction is preserved by the construction of regular coverings (i.e., coverings induced by voltage assignments in a group) whose voltage assignment depends only on colours. Moreover, this construction preserves minimality.

The Open Wikipedia Ranking is an open dataset published yearly, containing the ranking of Wikipedia pages with respect to centrality measures computed on the whole Wikipedia graph for that year. In this paper, ten years after its start, we report some details, results and anecdotal observations on this dataset. The goal of the Open Wikipedia Ranking is to provide a completely open and reproducible ranking of Wikipedia pages based on indegree, PageRank, harmonic centrality, and page views; the Wikipedia graphs themselves are also made available by the Laboratory of Web Algorithmics. What characterizes the Open Wikipedia Ranking is that the whole graph construction and ranking process are meticulously documented and reproducible. All computations are based on open-source Java software and algorithms from the literature. Thus, the reason of the centrality score of pages can be exactly traced back to structural graph properties.

doi:10.1145/3701716.3715510.

We report the results of a yearlong effort at the Laboratory for Web Algorithmics and Inria to port the WebGraph framework from Java to Rust. For two decades WebGraph has been instrumental in the analysis and distribution of large graphs for the research community of TheWebConf, but the intrinsic limitations of the Java Virtual Machine had become a bottleneck for very large use cases, such as the Software Heritage Merkle graph with its half a trillion arcs. As part of this clean-slate implementation of WebGraph in Rust, we developed a few ancillary projects bringing to the Rust ecosystem some missing features of independent interest, such as easy, consistent and zero-cost memory mapping of data structures. WebGraph in Rust offers impressive performance improvements over the previous implementation, enabling open-source graph analytics on very large datasets like Common Crawl, on top of a modern systems programming language.

doi:10.1145/3589335.3651581.

In the study of the behavior of centrality measures with respect to network modifications, score monotonicity means that adding an arc increases the centrality score of the target of the arc; rank monotonicity means that adding an arc improves the importance of the target of the arc relative to the remaining nodes. It is known that score and rank monotonicity hold in directed graphs for almost all the classical centrality measures. In undirected graphs one expects that the corresponding properties (where both endpoints of the new edge enjoy the increase in score/rank) hold when adding a new edge. However, recent results have shown that in undirected networks this is not true: for many centrality measures, it is possible to find situations where adding an edge reduces the rank of one of its two endpoints. In this paper we introduce a weaker condition for undirected networks, semi-monotonicity, in which just one of the endpoints of a new edge is required to enjoy score or rank monotonicity. We show that this condition is satisfied by closeness and betweenness centrality, and that harmonic centrality satisfies it in an even stronger sense.

PDF version; doi:10.1007/978-3-031-53472-0_9.

Progress in High-Performance Computing in general, and High-Performance Graph Processing in particular, is highly dependent on the availability of publicly-accessible, relevant, and realistic data sets. To ensure continuation of this progress, we (i) investigate and optimize the process of generating large sequence similarity graphs as an HPC challenge and (ii) demonstrate this process in creating MS-BioGraphs, a new family of publicly available real-world edge-weighted graph datasets with up to 2.5 trillion edges, that is, 6.6 times greater than the largest graph published recently. The largest graph is created by matching (i.e., all-toall similarity aligning) 1.7 billion protein sequences. The MSBioGraphs family includes also seven subgraphs with different sizes and direction types. We describe two main challenges we faced in generating large graph datasets and our solutions, that are, (i) optimizing data structures and algorithms for this multi-step process and (ii) WebGraph parallel compression technique. The datasets are available online on https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs.

doi:10.1109/BigData59044.2023.10386309.

We study the problem of score and rank monotonicity for spectral ranking methods, such as eigenvector centrality and PageRank, in the case of undirected networks. Score monotonicity means that adding an edge increases the score at both ends of the edge. Rank monotonicity means that adding an edge improves the relative position of both ends of the edge with respect to the remaining nodes. It is known that common spectral rankings are both score and rank monotone on directed, strongly connected graphs. We show that, surprisingly, the situation is very different for undirected graphs, and in particular that PageRank is neither score nor rank monotone.

PDF version; doi:10.1007/978-3-030-93409-5_20; you can get the Sage code used in the paper.

We consider the problem of mining the development history—as captured by modern version control systems—of ultra-large-scale software archives (e.g., tens of millions software repositories corresponding).

We show that graph compression techniques can be applied to the problem, dramatically reducing the hardware resources needed to mine similarly-sized corpus. As a concrete use case we compress the full Software Heritage archive, consisting of 5 billion unique source code files and 1 billion unique commits, harvested from more than 80 million software projects— encompassing a full mirror of GitHub.

The resulting compressed graph fits in less than 100GB of RAM, corresponding to a hardware cost of less than 300 U.S. dollars. We show that the compressed in-memory representation of the full corpus can be accessed with excellent performances, with edge lookup times close to memory random access. As a sample exploitation experiment we show that the compressed graph can be used to conduct clone detection at this scale, benefiting from main memory access speed.

PDF version.

Since the seminal work of Litvak and van der Hofstad, it has been known that Newman's assortativity, being based on Pearson's correlation, is subject to a pernicious size effect which makes large networks with heavy-tailed degree distributions always unassortative. Usage of Spearman's ρ, or even Kendall's τ was suggested as a replacement but the treatment of ties was problematic for both measures. In this paper we first argue analytically that the tie-aware version of τ solves the problems observed, and we show that Newman's assortativity is heavily influenced by tightly knit communities. Then, we perform for the first time a set of large-scale computational experiments on a variety of networks, comparing assortativity based on Kendall's τ and assortativity based on Pearson's correlation, showing that the pernicious effect of size is indeed very strong on real-world large networks, whereas the tie-aware Kendall's τ can be a practical, principled alternative.

doi:10.1007/978-3-030-36683-4_24; PDF version.

A minimal perfect hash function bijectively maps a key set S out of a universe U into the first |S| natural numbers. Minimal perfect hash functions are used, for example, to map irregularly-shaped keys, such as strings, in a compact space so that metadata can then be simply stored in an array. While it is known that just 1.44 bits per key are necessary to store a minimal perfect hash function, no published technique can go below 2 bits per key in practice. We propose a new technique for storing minimal perfect hash functions with expected linear construction time and expected constant lookup time that makes it possible to build for the first time, for example, structures which need 1.56 bits per key, that is, within 8.3% of the lower bound, in less than 2ms per key. We show that instances of our construction are able to simultaneously beat the construction time, space usage and lookup time of the state-of-the-art data structure reaching 2 bits per key. Moreover, we provide parameter choices giving structures which are competitive with alternative, larger-size data structures in terms of space and lookup time. The construction of our data structures can be easily parallelized or mapped on distributed computational units (e.g., within the MapReduce framework), and structures larger than the available RAM can be directly built in mass storage.

Download from arXiv. The algorithms described in the paper are implemented in Sux.

Once upon a time, a king had a very, very long list of names of his subjects. The king was also a bit obsessed with name days: every day he would ask his servants to look the list for all persons having their name day. Reading every day the whole list was taking an enormous amount of time to the king's servants. One day, the chancellor had a magnificent idea: he wrote a book with instructions. The number of pages in the book was equal to the number of names, but following the instructions one could find all people having their name day by looking at only a few pages—in fact, as many pages as the length of the name—and just glimpsing at the list. Everybody was happy, but in time the king's servants got lazy: when the name was very long they would find excuses to avoid looking at so many pages, and some name days were skipped. Desperate, the king made a call through its reign, and a fat sorceress answered. There was a way to look at much, much fewer pages using an additional magic book. But sometimes, very rarely, it would not work (magic does not always work). The king accepted the offer, and name days parties restarted. Only, once every a few thousand years, the magic book fails, and the assistants have to go by the chancellor book. So the parties start a bit later. But they start anyway.

PDF version. The algorithms described in the paper are implemented in Sux4J.

Recent advances in the compact representation of static functions (with constant access time) have made it possible to fully exploit constructions based on random linear system. Such constructions, albeit theoretically appealing, were previously too slow to be usable. In this paper, we extend such techniques to the problem of storing compressed static functions, in the sense that the space used per key should be close to the entropy of the list of values. From a theoretical viewpoint, we are inspired by the approach of Hreinsson, Krøyer and Pagh. Values are represented using a near-optimal instantaneous code. Then, a bit array is created so that by XOR'ing its content at a fixed number of positions depending on the key one obtains the value, represented by its associated codeword. In the construction phase, every bit of the array is associated with an equation on Z/2Z, and solving the associated system provides the desired representation. Thus, we pass from one equation per key (the non-compressed case) to one equation per bit: the size of the system is thus approximately multiplied by the empirical entropy of the values, making the problem much more challenging. We show that by carefully engineering the value representation we can obtain a practical data structure. For example, we can store a function with geometrically distributed output in just 2.28 bits per key, independently of the key set, with a construction time double with respect to that of a state-of-the-art non-compressed function, which requires ≈log log n bits per key, where n is the number of keys, and slightly improved lookup time. We can also store a function with an output of 10⁶ values distributed following a power law of exponent 2 in just 2.75 bits per key, whereas a non-compressed function would require more than 20, with a threefold increase in construction time and significantly faster lookups.

PDF version. The algorithms described in the paper are implemented in Sux4J.

Recent advances in random linear systems on finite fields have paved the way for the construction of constant-time data structures representing static functions and minimal perfect hash functions using less space with respect to existing techniques. The main obstruction for any practical application of these results is the cubic-time Gaussian elimination required to solve these linear systems: despite they can be made very small, the computation is still too slow to be feasible.

In this paper we describe in detail a number of heuristics and programming techniques to speed up the resolution of these systems by several orders of magnitude, making the overall construction competitive with the standard and widely used MWHC technique, which is based on hypergraph peeling. In particular, we introduce broadword programming techniques for fast equation manipulation and a lazy Gaussian elimination algorithm. We also describe a number of technical improvements to the data structure which further reduce space usage and improve lookup speed.

Our implementation of these techniques yields a minimal perfect hash function data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based ones, and a static function data structure which reduces the multiplicative overhead from 1.23 to 1.03.

The Open-Source IR Reproducibility Challenge brought together developers of open-source search engines to provide reproducible baselines of their systems in a common environment on Amazon EC2. The product is a repository that contains all code necessary to generate competitive ad hoc retrieval baselines, such that with a single script, anyone with a copy of the collection can reproduce the submitted runs. Our vision is that these results would serve as widely accessible points of comparison in future IR research. This project represents an ongoing effort, but we describe the first phase of the challenge that was organized as part of a workshop at SIGIR 2015. We have succeeded modestly so far, achieving our main goals on the Gov2 collection with seven open-source search engines. In this paper, we describe our methodology, share experimental results, and discuss lessons learned as well as next steps.

Github repository.

Understanding the correlation between two different scores for the same set of items is a common problem in graph analysis and information retrieval. The most commonly used statistics that quantifies this correlation is Kendall's τ however, the standard definition fails to capture that discordances between items with high rank are more important than those between items with low rank. Recently, a new measure of correlation based on average precision has been proposed to solve this problem, but like many alternative proposals in the literature it assumes that there are no ties in the scores. This is a major deficiency in a number of contexts, and in particular when comparing centrality scores on large graphs, as the obvious baseline, indegree, has a very large number of ties in social networks and web graphs. We propose to extend Kendall's definition in a natural way to take into account weights in the presence of ties. We prove a number of interesting mathematical properties of our generalization and describe an O(n log n) algorithm for its computation. We also validate the usefulness of our weighted measure of correlation using experimental data on social networks and web graphs.

PDF version.

Knowledge about the general graph structure of the World Wide Web is important for understanding the social mechanisms that govern its growth, for designing ranking methods, for devising better crawling algorithms, and for creating accurate models of its structure. In this paper, we describe and analyse a large, publicly accessible crawl of the web that was gathered by the Common Crawl Foundation in 2012 and that contains over 3.5 billion web pages and 128.7 billion links. This crawl makes it possible to observe the evolution of the underlying structure of the World Wide Web within the last 10 years: we analyse and compare, among other features, degree distributions, connectivity, average distances, and the structure of weakly/strongly connected components.

Our analysis shows that, as evidenced by previous research, some of the features previously observed by Broder et al. are very dependent on artefacts of the crawling process, whereas other appear to be more structural. We confirm the existence of a giant strongly connected component; we however find, as observed by other researchers, very different proportions of nodes that can reach or that can be reached from the giant component, suggesting that the “bow-tie structure” as described by Broder et al. is strongly dependent on the crawling process, and to the best of our current knowledge is not a structural property of the web.

More importantly, statistical testing and visual inspection of size-rank plots show that the distributions of indegree, outdegree and sizes of strongly connected components are not power laws, contrarily to what was previously reported for much smaller crawls, although they might be heavy tailed. We also provide for the first time accurate measurement of distance-based features, using recently introduced algorithms that scale to the size of our crawl.

Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this paper, we approach the problem of computing geometric centralities, such as closeness and harmonic centrality, on very large graphs; traditionally this task requires an all-pairs shortest-path computation in the exact case, or a number of breadth-first traversals for approximated computations, but these techniques yield very weak statistical guarantees on highly disconnected graphs. We rather assume that the graph is accessed in a semi-streaming fashion, that is, that adjacency lists are scanned almost sequentially, and that a very small amount of memory (in the order of a dozen bytes) per node is available in core memory. We leverage the newly discovered algorithms based on HyperLogLog counters, making it possible to approximate a number of geometric centralities at a very high speed and with high accuracy. While the application of similar algorithms for the approximation of closeness was attempted in the MapReduce framework, our exploitation of HyperLogLog counters reduces exponentially the memory footprint, paving the way for in-core processing of networks with a hundred billion nodes using “just” 2TiB of RAM. Moreover, the computations we describe are inherently parallelizable, and scale linearly with the number of available cores.

PDF version.

The computation of a peeling order in a randomly generated hypergraph is the most time-consuming step in a number of constructions, such as perfect hashing schemes, random r-SAT solvers, error-correcting codes, and approximate set encodings. While there exists a straightforward linear time algorithm, its poor I/O performance makes it impractical for hypergraphs whose size exceeds the available internal memory.

We show how to reduce the computation of a peeling order to a small number of sequential scans and sorts, and analyze its I/O complexity in the cache-oblivious model. The resulting algorithm requires O(sort(n)) I/Os and O(n log n) time to peel a random hypergraph with n edges.

We experimentally evaluate the performance of our implementation of this algorithm in a real-world scenario by using the construction of minimal perfect hash functions (MPHF) as our test case: our algorithm builds a MPHF of 7.6 billion keys in less than 21 hours on a single machine. The resulting data structure is both more space-efficient and faster than that obtained with the current state-of-the-art MPHF construction for large-scale key sets.

Download from arXiv.

We recently measured the average distance of users in the Facebook graph, spurring comments in the scientific community as well as in the general press (“Four Degrees of Separation”). A number of interesting criticisms have been made about the meaningfulness, methods and consequences of the experiment we performed. In this paper we want to discuss some methodological aspects that we deem important to underline in the form of answers to the questions we have read in newspapers, magazines, blogs, or heard from colleagues. We indulge in some reflections on the actual meaning of “average distance” and make a number of side observations showing that, yes, 3.74 “degrees of separation” are really few.

PDF version.

Compressed inverted indices in use today are based on the idea of gap compression: documents pointers are stored in increasing order, and the gaps between successive document pointers are stored using suitable codes which represent smaller gaps using less bits. Additional data such as counts and positions is stored using similar techniques. A large body of research has been built in the last 30 years around gap compression, including theoretical modeling of the gap distribution, specialized instantaneous codes suitable for gap encoding, and ad hoc document reorderings which increase the efficiency of instantaneous codes. This paper proposes to represent an index using a different architecture based on quasi-succinct representation of monotone sequences. We show that, besides being theoretically elegant and simple, the new index provides expected constant-time operations, space savings, and, in practice, significant performance improvements on conjunctive, phrasal and proximity queries.

PDF version. Quasi-succinct indices are now the default indices in MG4J.

Frigyes Karinthy, in his 1929 short story “Láancszemek” (“Chains”) suggested that any two persons are distanced by at most six friendship links. (The exact wording of the story is slightly ambiguous: “He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual […]”. It is not completely clear whether the selected individual is part of the five, so this could actually allude to distance five or six in the language of graph theory, but the “six degrees of separation” phrase stuck after John Guare's 1990 eponymous play. Following Milgram's definition and Guare's interpretation, we will assume that “degrees of separation” is the same as “distance minus one”, where “distance” is the usual path length—the number of arcs in the path.) Stanley Milgram in his famous experiment challenged people to route postcards to a fixed recipient by passing them only through direct acquaintances. The average number of intermediaries on the path of the postcards lay between 4.4 and 5.7, depending on the sample of people chosen.

We report the results of the first world-scale social-network graph-distance computations, using the entire Facebook network of active users (≈721 million users, ≈69 billion friendship links). The average distance we observe is 4.74, corresponding to 3.74 intermediaries or “degrees of separation”, showing that the world is even smaller than we expected, and prompting the title of this paper. More generally, we study the distance distribution of Facebook and of some interesting geographic subgraphs, looking also at their evolution over time.

The networks we are able to explore are almost two orders of magnitude larger than those analysed in the previous literature. We report detailed statistical metadata showing that our measurements (which rely on probabilistic algorithms) are very accurate.

PDF version; Java™ implementations are available as free software at the WebGraph home page.You can download the property files, HyperBall runs and degree distributions presented in the paper. The graphs are also described on the LAW dataset page.

Given a social network, which of its nodes have a stronger impact in determining its structure? More formally: which node-removal order has the greatest impact on the network structure? We approach this well-known problem for the first time in a setting that combines both web graphs and social networks, using datasets that are orders of magnitude larger than those appearing in the previous literature, thanks to some recently developed algorithms and software tools that make it possible to approximate accurately the number of reachable pairs and the distribution of distances in a graph. Our experiments highlight deep differences in the structure of social networks and web graphs, show significant limitations of previous experimental results, and at the same time reveal clustering by label propagation as a new and very effective way of locating nodes that are important from a structural viewpoint.

The neighbourhood function N_G(t) of a graph G gives, for each t, the number of pairs of nodes <x, y> such that y is reachable from x in less that t hops. The neighbourhood function provides a wealth of information about the graph (e.g., it easily allows one to compute its diameter), but it is very expensive to compute it exactly. Recently, the ANF algorithm (approximate neighbourhood function) has been proposed with the purpose of approximating N_G(t) on large graphs. We describe a breakthrough improvement over ANF in terms of speed and scalability. Our algorithm, called HyperANF, uses the new HyperLogLog counters and combines them efficiently through broadword programming; our implementation uses decomposition to exploit multi-core parallelism. With HyperANF, for the first time we can compute in a few hours the neighbourhood function of graphs with billions of nodes with a small error and good confidence using a standard workstation. Then, we turn to the study of the distribution of the distances between reachable nodes (that can be efficiently approximated by means of HyperANF), and discover the surprising fact that its index of dispersion provides a clear-cut characterisation of proper social networks vs. web graphs. We thus propose the spid (Shortest-Paths Index of Dispersion) of a graph as a new, informative statistics that is able to discriminate between the above two types of graphs. We believe this is the first proposal of a significant new non-local structural index for complex networks whose computation is highly scalable.

Implementations are available as free software at the WebGraph home page; public datasets are available at the LAW site.

We continue the line of research on graph compression started with WebGraph, but we move our focus to the compression of social networks in a proper sense (e.g., LiveJournal): the approaches that have been used for a long time to compress web graphs rely on a specific ordering of the nodes (lexicographical URL ordering) whose extension to general social networks is not trivial. In this paper, we propose a solution that mixes clusterings and orders, and devise a new algorithm, called Layered Label Propagation, that builds on previous work on scalable clustering and can be used to reorder very large graphs (billions of nodes). Our implementation uses decomposition to perform aggressively on multi-core architecture, making it possible to reorder graphs of more than 600 millions nodes in a few hours. Experiments performed on a wide array of web graphs and social networks show that combining the order produced by the proposed algorithm with the WebGraph compression framework provides a major increase in compression with respect to all currently known techniques, both on web graphs and on social networks. These improvements make it possible to analyse in main memory significantly larger graphs.

PDF version; public datasets and Java™ implementations are available as free software at the LAW site.

Understanding query reformulation patterns is a key step towards next generation web search engines: it can help improving users' web-search experience by predicting their intent, and thus helping them to locate information more effectively.

As a step in this direction, we build an accurate model for classifying user query reformulations into broad classes (generalization, specialization, error correction or parallel move), achieving 92% accuracy. We apply the model to automatically label two large query logs, creating annotated query-flow graphs. We study the resulting reformulation patterns, finding results consistent with previous studies done on smaller manually annotated datasets, and discovering new interesting patterns, including connections between reformulation types and topical categories.

Finally, applying our findings to a third query log that is publicly available for research purposes, we demonstrate that our reformulation classifier leads to improved recommendations in a query recommendation system.

A voting system is a set of rules that a community adopts to take collective decisions. In this paper we study voting systems for a particular kind of community: electronically mediated social networks. In particular, we focus on delegative democracy (a.k.a. proxy voting) that has recently received increased interest for its ability to combine the benefits of direct and representative systems, and that seems also perfectly suited for electronically mediated social networks. In such a context, we consider a voting system in which users can only express their preference for one among the people they are explicitly connected with, and this preference can be propagated transitively, using an attenuation factor.

We present this system and we study its properties. We also take into consideration the problem of missing votes, which is particularly relevant in online networks, as some recent case shows. Our experiments on real-world networks provide interesting insight into the significance and stability of the results obtained with the suggested voting system.

A prefix search returns the strings out of a given collection S that start with a given prefix. Traditionally, prefix search is solved by data structures that are also dictionaries, that is, they actually contain the strings in S. For very large collections stored in slow-access memory, we propose extremely compact data structures that solve weak prefix searches—they return the correct result only if some string in S starts with the given prefix. Our data structures for weak prefix search use O(|S |log ℓ) bits in the worst case, where ℓ is the average string length, as opposed to O(|S| ℓ) bits for a dictionary. We show a lower bound implying that this space usage is optimal.

The query-flow graph is an aggregated representation of the latent querying behavior contained in a query log. Intuitively, in the query-flow graph a directed edge from query q_i to query q_j means that the two queries are likely to be part of the same search mission. Any path over the query-flow graph may be seen as a possible search task, whose likelihood is given by the strength of the edges along the path. An edge (q_i, q_j) is also labelled with some information: e.g., the probability that user moves from q_i to q_j, or the type of the transition, for instance, the fact that q_j is a specialization of q_i.

In this paper we propose, and experimentally study, query recommendations based on short random walks on the query-flow graph. Our experiments show that these methods can match in precision, and often improve, recommendations based on query-click graphs, without using users' clicks. Our experiments also show that it is important to consider transition-type labels on edges for having good quality recommendations.

Finally, one feature that we had in mind while devising our methods was that of providing diverse sets of recommendations: the experimentation that we conducted provides encouraging results in this sense.

We describe a dynamic version of the z-fast trie, a new data structure inspired by the research started by the van Emde Boas trees and followed by the development of y-fast tries. The dynamic z-fast trie is a very simple, uniform data structure: given a set S of n variable-length strings, it is formed by a standard compacted trie on S (with two additional pointers per node), endowed with a dictionary of size n−1. With this simple setup, the dynamic z-fast trie provides predecessors/successors in time O(log max{ |x|, |x⁺|, |x^-| }) (x^± is the successor/predecessor of x in S) for strings of length linear in the machine-word size w. Prefix queries are answered in time O(log|x|+k), and range queries in time O(log max{ |x|, |y|, |x^-|, |y⁺| }+k), where k is the number of elements in the output and x (and y) represent the input of the prefix (range) queries. Updates are performed within the same bounds in expectation (or with high probability using an appropriate dictionary). We then show a simple modification that makes it possible to handle strings of length up to 2^w; in this case, predecessor/successor queries and updates are supported in O(|x|/w + log max{ |x|, |x⁺|, |x^-| }) time, (and O(|x|/B + log max{ |x|, |x⁺|, |x^-| }) I/Os in the cache-oblivious model) with high probability. The space occupied by a dynamic z-fast trie, beside that necessary to store S, is just of 12n pointers, n integers and, in the “long string” case, O(n) signatures of O(w) bits each.

A minimal perfect hash function maps a set S of n keys into the set {0,1,…, n − 1} bijectively. Classical results state that minimal perfect hashing is possible in constant time using a structure occupying space close to the lower bound of log e bits per element. Here we consider the problem of monotone minimal perfect hashing, in which the bijection is required to preserve the lexicographical ordering of the keys. A monotone minimal perfect hash function can be seen as a very weak form of index that provides ranking just on the set S (and answers randomly outside of S). Our goal is to minimise the description size of the hash function: we show that, for a set S of n elements out of a universe of 2^w elements, O(n log log w) bits are sufficient to hash monotonically with evaluation time O(log w). Alternatively, we can get space O(n log w) bits with O(1) query time. Both of these data structures improve a straightforward construction with O(n log w) space and O(log w) query time. As a consequence, it is possible to search a sorted table with O(1) accesses to the table (using additional O(n log log w) bits). Our results are based on a structure (of independent interest) that represents a trie in a very compact way, but admits errors. As a further application of the same structure, we show how to compute the predecessor (in the sorted order of S) of an arbitrary element, using O(1) accesses in expectation and an index of O(n log w) bits, improving the trivial result of O(nw) bits. This implies an efficient index for searching a blocked memory.

PDF version. The algorithms described in the paper are implemented in Sux4J.

Query logs record the queries and the actions of the users of search engines, and as such they contain valuable information about the interests, the preferences, and the behavior of the users, as well as their implicit feedback to search-engine results. Mining the wealth of information available in the query logs has many important applications including query-log analysis, user profiling and personalization, advertising, query recommendation, and more.

In this paper we introduce the query-flow graph, a graph representation of the interesting knowledge about latent querying behavior. Intuitively, in the query-flow graph a directed edge from query q_i to query q_j means that the two queries are likely to be part of the same “search mission”. Any path over the query-flow graph may be seen as a searching behavior, whose likelihood is given by the strength of the edges along the path.

The query-flow graph is an outcome of query-log mining and, at the same time, a useful tool for it. We propose a methodology that builds such a graph by mining time and textual information as well as aggregating queries from different users. Using this approach we build a real-world query-flow graph from a large-scale query log and we demonstrate its utility in concrete applications, namely, finding logical sessions, and query recommendation. We believe, however, that the usefulness of the query-flow graph goes beyond these two applications.

doi:10.4230/DagSemProc.07071.15.

Since the first investigations on web graph compression, it has been clear that the ordering of the nodes of the graph has a fundamental influence on the compression rate (usually expressed as the number of bits per link). The authors of the LINK database, for instance, investigated three different approaches: an extrinsic ordering (URL ordering) and two intrinsic (or coordinate-free) orderings based on the rows of the adjacency matrix (lexicographic and Gray code); they concluded that URL ordering has many advantages in spite of a small penalty in compression. In this paper we approach this issue in a more systematic way, testing some old orderings and proposing some new ones. Our experiments are made in the WebGraph framework, and show that the compression technique and the structure of the graph can produce significantly different results. In particular, we show that for the transpose web graph URL ordering is significantly less effective, and that some new orderings combining host information and Gray/lexicographic orderings outperform all previous methods. In particular, in some large transposed graphs they yield the quite incredible compression rate of 1 bit per link.

Recently, a new temporal dataset has been made public: it is made of a series of twelve 100M pages snapshots of the .uk domain. The Web graphs of the twelve snapshots have been merged into a single time-aware graph that provide constant-time access to temporal information. In this paper we present the first statistical analysis performed on this graph, with the goal of checking whether the information contained in the graph is reliable (i.e., whether it depends essentially on appearance and disappearance of pages and links, or on the crawler behaviour). We perform a number of tests that show that the graph is actually reliable, and provide the first public data on the evolution of the Web that use a large scale and a significant diversity in the sites considered.

Latent semantic analysis is a well-known technique to extrapolate concepts from a set of documents; it discards noise by reducing the rank of (a variant of) the term/document matrix of a document collection by singular value decomposition. The latter is performed by solving an equivalent symmetric eigenvector problem on a related matrix. Scaling to large set of documents, however, is problematic because every vector-matrix multiplication required by iterative solvers requires a number of multiplications equal to twice the number of postings of the collection. We show how to combine standard search-engine algorithmic tools in such a way to compute (reasonably) quickly the cooccurrence matrix C of a large document collection, and solve directly the associated symmetric eigenvector problem. Albeit the size of C is quadratic in the number of terms, we can distribute its computation among any number of computational unit without increasing the overall number of multiplications. Moreover, our approach is advantageous when the document collection is large, because the number of terms over which latent semantic analysis has to be performed is inherently limited by the size of a language lexicon. We present experiments over a collection with 3.6 billions of postings—two orders of magnitudes larger than any published experiment in the literature.

Research on succinct data structures (data structures occupying space close to the information-theoretical lower bound, but achieving speed similar to their standard counterparts) has steadily increased in the last few years. However, many theoretical constructions providing asymptotically optimal bounds are unusable in practice because of the very large constants involved. The study of practical implementations of the basic building blocks of such data structures is thus fundamental to obtain practical applications. In this paper we argue that 64-bit and wider architectures are particularly suited to very efficient implementations of rank (counting the number of ones up to a given position) and select (finding the position of the i-th bit set), two essential building blocks of all succinct data structures. Contrarily to typical 32-bit approaches, involving precomputed tables, we use pervasively broadword (a.k.a. SWAR—“SIMD in A Register”) programming, which compensates the constant burden associated to succinct structures by solving problems in parallel in a register. We provide an implementation named rank9 that addresses 2⁶⁴ bits, consumes less space and is significantly faster then current state-of-the-art 32-bit implementations, and a companion select9 structure that selects in nearly constant time using only access to aligned data. For sparsely populated arrays, we provide a simple broadword implementation of the Elias–Fano representation of monotone sequences. In doing so, we develop broadword algorithms to perform selection in a word or in a sequence of words that are of independent interest.

PDF version; a complete C++ and Java™ implementation is available at the Sux project home page.

We discuss a number of issues in the definition, computation and comparison of PageRank values that have been addressed sparsely in the literature, often with contradictory approaches. We study the difference between weakly and strongly preferential PageRank, which patch the dangling nodes with different distributions, extending analytical formulae known for the strongly preferential case, and corroborating our results with experiments on a snapshot of 100 millions of pages of the .uk domain. The experiments show that the two PageRank versions are poorly correlated, and results about each one cannot be blindly applied to the other; moreover, our computations highlight some new concerns about the usage of exchange-based correlation indices (such as Kendall's τ) on approximated rankings.

We describe the WEBSPAM-UK2006 collection, a large set of Web pages that have been manually annotated with labels indicating if the hosts are include Web spam aspects or not. This is the first publicly available Web spam collection that includes page contents and links, and that has been labelled by a large and diverse set of judges.

Large inverted indices are by now common in the construction of web-scale search engines. For faster access, inverted indices are indexed internally so that it is possible to skip quickly over unnecessary documents. The classical approach to skipping dictates that a skip should be positioned every f^1/2 document pointers, where f is the overall number of documents where the term appears. We argue that due to the growing size of the web more refined techniques are necessary, and describe how to embed a compressed perfect skip list in an inverted list. We provide statistical models that explain the empirical distribution of the skip data we observe in our experiments, and use them to devise good compression techniques that allow us to limit the waste in space, so that the resulting data structure increases the overall index size by just a few percents, still making it possible to index pointers with a rather fine granularity.

PDF version of the extended draft.

PageRank is defined as the stationary state of a Markov chain depending on a damping factor α that spreads uniformly part of the rank. The choice of α is eminently empirical, and in most cases the original suggestion α = 0.85 by Brin and Page is still used. It is common belief that values of α closer to 1 give a “truer to the web” PageRank, but a small α accelerates convergence. Recently, however, it has been shown that when α = 1 all pages in the core component are very likely to have rank 0. This behaviour makes it difficult to understand PageRank when α ≈ 1, as it converges to a meaningless value for most pages. We propose a simple and natural modification to the standard preprocessing performed on the adjacency matrix of the graph, resulting in a ranking scheme we call TruRank. TruRank ranks the web with principles almost identical to PageRank, but it gives meaningful values also when α ≈ 1.

Minimal-interval semantics associates with each query over a document a set of intervals, called witnesses, that are incomparable with respect to inclusion (i.e., they form an antichain): witnesses define the minimal regions of the document satisfying the query. Minimal-interval semantics makes it easy to define and compute several sophisticated proximity operators, provides snippets for user presentation, and can be used to rank documents: thus, computing efficiently the antichains obtained by operations such as logic conjunction and disjunction is a basic issue. In this paper we provide the first algorithms for computing such operators that are linear in the number of intervals and logarithmic in the number of input antichains. The space used is linear in the number of antichains. Moreover, the algorithms are lazy—they do not assume random access to the input antichains. These properties make the usage of our algorithms feasible in large-scale web search engines.

The algorithms described in this paper are by now obsolete. Please refer to our new paper.

PageRank is defined as the stationary state of a Markov chain. The chain is obtained by perturbing the transition matrix induced by a web graph with a damping factor α that spreads uniformly part of the rank. The choice of α is eminently empirical, and in most cases the original suggestion α=0.85 by Brin and Page is still used. Recently, however, the behaviour of PageRank with respect to changes in α was discovered to be useful in link-spam detection. Moreover, an analytical justification of the value chosen for α is still missing. In this paper, we give the first mathematical analysis of PageRank when α changes. In particular, we show that, contrarily to popular belief, for real-world graphs values of α close to 1 do not give a more meaningful ranking. Then, we give closed-form formulae for PageRank derivatives of any order, and an extension of the Power Method that approximates them with convergence O(t^k α^t) for the k-th derivative. Finally, we show a tight connection between iterated computation and analytical behaviour by proving that the k-th iteration of the Power Method gives exactly the PageRank value obtained using a Maclaurin polynomial of degree k. The latter result paves the way towards the application of analytical methods to the study of PageRank.

PDF version.

Recent developments in reification of ER schemata include automatic generation of web-based database administration systems. These systems enforce the schema cardinality constraints, but, beyond unsatisfiable schemata, this feature may create unreachable instances. We prove sound and complete characterisations of schemata whose instances satisfy suitable reachability properties; these theorems translate into linear algorithms that can be used to prevent the administrator from reifying schemata with unreachable instances.

Studying web graphs is often difficult due to their large size. Recently, several proposals have been published about various techniques that allow to store a web graph in memory in a limited space, exploiting the inner redundancies of the web. The WebGraph framework is a suite of codes, algorithms and tools that aims at making it easy to manipulate large web graphs. This papers presents the compression techniques used in WebGraph, which are centred around referentiation and intervalisation (which in turn are dual to each other). WebGraph can compress the WebBase graph (118 Mnodes, 1 Glinks) in as little as 3.08 bits per link, and its transposed version in as little as 2.89 bits per link.

PDF version; to download software and data sets, please look at the WebGraph home page.

ERW is an innovative open-source system for handling complex databases using a web browser. Once the details of an enhanced entity-relationship schema have been specified in XML, ERW generates a complete application that lets the user interact with the database. Then, specification percolation makes it possible to customise heavily the application while maintaining the flexibility of a model-driven approach.

The Java string classes, String and StringBuffer, lie at the extremes of a spectrum (immutable, reference-based and mutable, content-based). Motivated by data-intensive text applications, we propose a new string class, MutableString, which tries to embody the best of both approaches.

This paper describes a multirelation semantics for a fragment of the Extended Entity-Relationship schemata formalism based on the bicategorical definition of multirelations. The approach we follow is elementary—we try to introduce as few notions as possible. We claim that bicategorical algebra handles gracefully multirelations and their operations, and that multirelations are essential in a number of applications; moreover, the bicategorical composition of multirelations turns out to correspond to natural joins. From the formal semantics we derive an algorithm that can establish statically the possibility of building parallel ownership paths of weak entities. The ideas described in this paper have been implemented in a free tool, ERW, which lets users edit sets and multirelations instantiating an EER schema via a sophisticated web interface.

We report on the implementation of UbiCrawler, a scalable distributed web crawler, analyze its performance, and describe its fault tolerance.

ERW is an innovative system for handling complex databases using a web browser. It uses the most recent standards endorsed by the W3C to offer to the user a sophisticated environment, similar to a dedicated client. Moreover, the user interface is generated in a completely automatic way starting from a conceptual description of the database by means of an XML-based language for entity-relationship schemata.

In this poster we illustrate some data about the African web. These data have been collected using UbiCrawler, a distributed Web crawler designed and developed by the authors.

Trovatore is an ongoing project aimed at realizing an efficient distributed and highly scalable web crawler. This poster illustrates the main ideas behind its design.

It is known that computations of anonymous networks can be reduced to the construction of a certain graph, the minimum base of the network. The crucial step of this construction is the inference of the minimum base from a finite tree that each processor can build (its truncated view). We isolate those trees that make this inference possible, and call them holographic. Intuitively, a tree is holographic if it is enough self-similar to be uniquely extendible to an infinite tree. This possibility depends on a size function for the class of graphs under examination, which we call a holographic bound for the class. Holographic bounds give immediately, for instance, bounds for the quiescence time of self-stabilizing protocols. In this paper we give weakly tight holographic bounds for some classes of graphs.

We provide effective (i.e., recursive) characterizations of the relations that can be computed on networks where all processors use the same algorithm, start from the same state, and know at least a bound on the network size. Three activation models are considered (synchronous, asynchronous, interleaved).

PDF version. doi.org:10.1007/3-540-45414-4_3.

In this paper we study several notions of approximability of functions in the framework of the BSS model. Denoting with φ_M^δ the function computed by a BSS machine M when its comparisons are against δ rather than 0, we study classes of functions f for which φ_M^δ→f in some sense (pointwise, uniformly, etc.). The main equivalence results show that this notion coincides with Type 2 computability when the convergence speed is recursively bounded. Finally, we study the possibility of extending these results to computations over Archimedean fields.

We provide characterizations of the relations that can be computed with arbitrary knowledge on networks where all processors use the same algorithm and start from the same state (in particular, we do not assume that a bound on the network size is known). Three activation models are considered (synchronous, asynchronous, interleaved).

The push algorithm was proposed first by Jeh and Widom in the context of personalized PageRank computations (albeit the name “push algorithm” was actually used by Andersen, Chung and Lang in a subsequent paper). In this note we describe the algorithm at a level of generality that make the computation of the spectral ranking of any nonnegative matrix possible. Actually, the main contribution of this note is that the description is very simple (almost trivial), and it requires only a few elementary linear-algebra computations. Along the way, we give new precise ways of estimating the convergence of the algorithm, and describe some of the contribution of the existing literature, which again turn out to be immediate when recast in our framework.

PDF version.

This note argues that when dot-plotting distributions typically found in papers about web and social networks (degree distributions, component-size distributions, etc.), and more generally distributions that have high variability in their tail, an exponentially binned version should always be plotted, too, and suggests Fibonacci binning as a visually appealing, easy-to-use and practical choice.

PDF version; a Ruby script implementing Fibonacci binning.

We describe the techniques developed to gather and distribute in a highly compressed, yet accessible, form a series of twelve snapshot of the .uk web domain. Ad hoc compression techniques made it possible to store the twelve snapshots using just 1.9 bits per link, with constant-time access to temporal information. Our collection makes it possible to study the temporal evolution link-based scores (e.g., PageRank), the growth of online communities, and in general time-dependent phenomena related to the link structure.

Datasets and Java™ implementations are available as free software at the LAW site.

Collections are a fundamental tool for reproducible evaluation of information retrieval techniques. We describe a new method for distributing the document lengths and term counts (a.k.a. within-document frequencies) of a web snapshot in a highly compressed and nonetheless quickly accessible form. Our main application is reproducibility of the behaviour of focused crawlers: by coupling our collection with the corresponding web graph compressed with WebGraph we make it possible to apply text-based machine learning tools to the collection, while keeping the data set footprint small. Finally, we describe a collection based on a crawl of 100Mpages of the .uk domain, publicly available in bundle with a Java open-source implementation of our techniques.

In this paper we survey the fundamental constructions of a presheaf topos in the case of the elementary topos of graphs. We prove that the transition graphs of nondeterministic automata (a.k.a. labelled transition systems) are the separated presheaves for the double negation topology, and obtain as an application that their category is a quasitopos.

PDF version.