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242
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 2600 (7 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.
Connected Components in Random Graphs with Given Expected Degree Sequences
 ANNALS OF COMBINATORICS
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Heuristically optimized tradeoffs: a new paradigm for power laws in the internet
, 2002
"... Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: "last mile " connection costs, and transmissi ..."
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Cited by 178 (1 self)
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Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: &quot;last mile &quot; connection costs, and transmission delays measured in hops. We also point out a similar phenomenon, anticipated in [2], in the distribution of file sizes. Our results seem to suggest that power laws tend to arise as a result of complex, multiobjective optimization.
A General Model of Web Graphs
, 2003
"... We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web. ..."
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Cited by 118 (6 self)
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We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.
Using PageRank to Characterize Web Structure
"... Recent work on modeling the web graph has dwelt on capturing the degree distributions observed on the web. Pointing out that this represents a heavy reliance on “local” properties of the web graph, we study the distribution of PageRank values on the web. Our measurements suggest that PageRank value ..."
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Cited by 114 (0 self)
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Recent work on modeling the web graph has dwelt on capturing the degree distributions observed on the web. Pointing out that this represents a heavy reliance on “local” properties of the web graph, we study the distribution of PageRank values on the web. Our measurements suggest that PageRank values on the web follow a power law. We then develop generative models for the web graph that explain this observation and moreover remain faithful to previously studied degree distributions. We analyze these models and compare the analysis to both snapshots from the web and to graphs generated by simulations on the new models. To our knowledge this represents the first modeling of the web that goes beyond fitting degree distributions on the web.
Random Evolution in Massive Graphs
, 2001
"... Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for ge ..."
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Cited by 107 (7 self)
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Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for generating power law graphs by adding one node/edge at a time. We will show that for any given edge density and desired distributions for indegrees and outdegrees (not necessarily the same, but adhered to certain general conditions), the resulting graph will almost surely satisfy the power law and the in/outdegree conditions. We will show that our most general directed and undirected models include nearly all known models as special cases. In addition, we consider another crucial aspects of massive graphs that is called "scalefree" in the sense that the f requency of sampling (w.r.t. the growth rate) is independent of the parameter of the resulting power law graphs. We will show that our evolution models generate scalefree power law graphs. 1
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
, 2006
"... We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into Gn; “right convergence” defined in terms of the densities of homomorphisms from Gn into small graphs ..."
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Cited by 100 (6 self)
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We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into Gn; “right convergence” defined in terms of the densities of homomorphisms from Gn into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs Gn, and for graphs Gn with nodeweights and edgeweights. One of the main steps here is the introduction of a cutdistance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural
Triangulation and Embedding using Small Sets of Beacons
, 2008
"... Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance ..."
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Cited by 96 (11 self)
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Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into lowdimensional Euclidean space. There is a fundamental distinction, however, between the theoretical approaches to the embedding problem and this recent Internetrelated work: in addition to computational limitations, Internet measurement algorithms operate under the constraint that it is only feasible to measure distances for a linear (or nearlinear) number of node pairs, and typically in a highly structured way. Indeed, the most common framework for Internet measurements of this type is a beaconbased approach: one chooses uniformly at random a constant number of nodes (‘beacons’) in the network, each node measures its distance to all beacons, and one then has access to only these measurements for the remainder of the algorithm. Moreover, beaconbased algorithms are often designed not for embedding but for the more basic problem of triangulation, in which one uses the triangle inequality to infer the distances that have not been measured. Here we give algorithms with provable performance guarantees for beaconbased triangulation and
On Certain Connectivity Properties of the Internet Topology
 IN PROC. 35TH ACM SYMP. ON THEORY OF COMPUTING
, 2003
"... We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the rando ..."
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Cited by 85 (3 self)
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We show that random graphs in the preferential connectivity model have constant conductance, and hence have worstcase routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the VickreyClarkeGroves mechanism for shortest paths) of a random graph is bounded by a small constant.
On Compressing Social Networks
"... Motivated by structural properties of the Web graph that support efficient data structures for in memory adjacency queries, we study the extent to which a large network can be compressed. Boldi and Vigna (WWW 2004), showed that Web graphs can be compressed down to three bits of storage per edge; we ..."
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Cited by 83 (2 self)
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Motivated by structural properties of the Web graph that support efficient data structures for in memory adjacency queries, we study the extent to which a large network can be compressed. Boldi and Vigna (WWW 2004), showed that Web graphs can be compressed down to three bits of storage per edge; we study the compressibility of social networks where again adjacency queries are a fundamental primitive. To this end, we propose simple combinatorial formulations that encapsulate efficient compressibility of graphs. We show that some of the problems are NPhard yet admit effective heuristics, some of which can exploit properties of social networks such as link reciprocity. Our extensive experiments show that social networks and the Web graph exhibit vastly different compressibility characteristics.