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43
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1415 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Evolution of networks
 Adv. Phys
, 2002
"... We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence rece ..."
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Cited by 269 (2 self)
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We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scalefree and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect. We discuss how growing networks selforganize into scalefree structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems
Comparing community structure identification
 Journal of Statistical Mechanics: Theory and Experiment
, 2005
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Efficient semistreaming algorithms for local triangle counting in massive graphs
 in KDD’08, 2008
"... In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph ha ..."
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Cited by 41 (4 self)
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In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph has been considered before, but to our knowledge this is the first paper that addresses the problem of local triangle counting with a focus on the efficiency issues arising in massive graphs. The distribution of the local number of triangles and the related local clustering coefficient can be used in many interesting applications. For example, we show that the measures we compute can help to detect the presence of spamming activity in largescale Web graphs, as well as to provide useful features to assess content quality in social networks. For computing the local number of triangles we propose two approximation algorithms, which are based on the idea of minwise independent permutations (Broder et al. 1998). Our algorithms operate in a semistreaming fashion, using O(V ) space in main memory and performing O(log V ) sequential scans over the edges of the graph. The first algorithm we describe in this paper also uses O(E) space in external memory during computation, while the second algorithm uses only main memory. We present the theoretical analysis as well as experimental results in massive graphs demonstrating the practical efficiency of our approach. Luca Becchetti was partially supported by EU Integrated
Fast Counting of Triangles in Large Real Networks: Algorithms and Laws
"... How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straightforward and even approximate counting algorithms ..."
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Cited by 39 (9 self)
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How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straightforward and even approximate counting algorithms can be slow, trying to execute or approximate the equivalent of a 3way database join. In this paper, we provide two algorithms, the EigenTriangle for counting the total number of triangles in a graph, and the EigenTriangleLocal algorithm that gives the count of triangles that contain a desired node. Additional contributions include the following: (a) We show that both algorithms achieve excellent accuracy, with up to ≈ 1000x faster execution time, on several, real graphs and (b) we discover two new power laws ( DegreeTriangle and TriangleParticipation laws) with surprising properties. Figure 1. Speedup ratio versus accuracy for the Wikipedia web graph ( ≈ 3, 1M nodes, ≈ 37M edges). Proposed method achieves 1021x faster time, for 97.4 % accuracy, compared to a typical competitor, the Node Iterator method. 1
Doulion: Counting Triangles in Massive Graphs with a Coin
 PROCEEDINGS OF ACM KDD,
, 2009
"... Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a ..."
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Cited by 31 (14 self)
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Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a triangle counting algorithm. Furthermore, several interesting graph mining applications rely on computing the number of triangles in the graph of interest. In this paper, we focus on the problem of counting triangles in a graph. We propose a practical method, out of which all triangle counting algorithms can potentially benefit. Using a straightforward triangle counting algorithm as a black box, we performed 166 experiments on realworld networks and on synthetic datasets as well, where we show that our method works with high accuracy, typically more than 99 % and gives significant speedups, resulting in even ≈ 130 times faster performance.
Using curvature and markov clustering in graphs for lexical acquisition and word sense discrimination
 In Workshop MEANING2005
, 2004
"... We introduce two different approaches for clustering semantically similar words. We accommodate ambiguity by allowing a word to belong to several clusters. Both methods use a graphtheoretic representation of words and their paradigmatic relationships. The first approach is based on the concept of c ..."
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Cited by 14 (0 self)
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We introduce two different approaches for clustering semantically similar words. We accommodate ambiguity by allowing a word to belong to several clusters. Both methods use a graphtheoretic representation of words and their paradigmatic relationships. The first approach is based on the concept of curvature and divides the word graph into classes of similar words by removing words of low curvature which connect several dispersed clusters. The second method, instead of clustering the nodes, clusters the links in our graph. These contain more specific contextual information than nodes representing just words. In so doing, we naturally accommodate ambiguity by allowing multiple class membership. Both methods are evaluated on a lexical acquisition task, using clustering to add nouns to the WordNet taxonomy. The most effective method is link clustering. 1
Efficient Triangle Counting in Large Graphs via Degreebased Vertex Partitioning
"... The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering t ..."
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Cited by 9 (2 self)
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The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering the hidden thematic structures in the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting approximation algorithm which can be adapted to the semistreaming model [23]. The key idea of our algorithm is to combine the sampling algorithm of [51,52] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [5], treating each set appropriately. From a mathematical perspective, we show a simplified proof of [52] which uses the powerful KimVu concentration inequality [31] based on the HajnalSzemerédi theorem [25]. Furthermore, we improve bounds of existing triple sampling ( techniques based on a theorem of Ahlswede and Katona [3]. We obtain a running time O m + m3/2 log n tɛ2) and an (1 ± ɛ)
Navigability of strong ties: Small worlds, tie strength and network topology
 Complexity
, 2002
"... We examine data on and models of small world properties and parameters of social networks. Our focus, on tiestrength, multilevel networks and searchability in strongtie social networks, allows us to extend some of the questions and findings of recent research and the fit of small world models to s ..."
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Cited by 9 (2 self)
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We examine data on and models of small world properties and parameters of social networks. Our focus, on tiestrength, multilevel networks and searchability in strongtie social networks, allows us to extend some of the questions and findings of recent research and the fit of small world models to sociological and anthropological data on human communities. We offer a ‘navigability of strong ties ’ hypothesis about network topologies tested with data from kinship systems, and potentially applicable to corporate cultures and business networks. Small Worlds A small world (SW) is a (large) graph with both local clustering and, on average, short distances between nodes [44,45]. Short distances promote accessibility, while local clustering and redundancy of edges, as some research suggests [38,48], promotes robustness to disconnection and, through multiple independent pathways, reliable accessibility as well. For paths to transmit materials and information via network traversal, a small world also requires navigability. This was the property investigated in the first small world experiment by Travers and Milgram [42]: Could people randomly selected in Omaha, Nebraska, successfully send letters to a predetermined target in Boston, when asked to direct their letters to single acquaintances who are asked in turn