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The Average Distance in a Random Graph with Given Expected Degrees
"... Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d wher ..."
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Cited by 188 (13 self)
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Random graph theory is used to examine the “smallworld phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d where ˜ d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1/k β for some fixed exponent β. For the case of β> 3, we prove that the average distance of the power law graphs is almost surely of order log n / log ˜ d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having n c / log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.
Statistical properties of community structure in large social and information networks
"... A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structur ..."
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Cited by 120 (10 self)
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A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse realworld networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large realworld networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “communitylike.” This behavior is not explained, even at a qualitative level, by any of the commonlyused network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are wellembeddable in a lowdimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.
Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters
, 2008
"... A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins wit ..."
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Cited by 78 (6 self)
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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large realworld networks, ranging from traditional and online social networks, to technological and information networks and
ON THE COVERINGS OF GRAPHS
, 1980
"... Let p(n) denote the smallest integer with the property that any graph with n vertices can be covered by p(n) complete bipartite subgraphs. We prove a conjecture of J.C. Bermond by showing p(n) = n + o(n 11’14+c) for any positive E. ..."
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Cited by 69 (6 self)
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Let p(n) denote the smallest integer with the property that any graph with n vertices can be covered by p(n) complete bipartite subgraphs. We prove a conjecture of J.C. Bermond by showing p(n) = n + o(n 11’14+c) for any positive E.
Sic Transit Gloria Telae: Towards an Understanding of the Web's Decay
 In Proceedings of the 13th conference on World Wide Web
, 2004
"... The rapid growth of the web has been noted and tracked extensively. Recent studies have however documented the dual phenomenon: web pages have small half lives, and thus the web exhibits rapid death as well. Consequently, page creators are faced with an increasingly burdensome task of keeping links ..."
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Cited by 52 (0 self)
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The rapid growth of the web has been noted and tracked extensively. Recent studies have however documented the dual phenomenon: web pages have small half lives, and thus the web exhibits rapid death as well. Consequently, page creators are faced with an increasingly burdensome task of keeping links uptodate, and many are falling behind. In addition to just individual pages, collections of pages or even entire neighborhoods of the web exhibit significant decay, rendering them less e#ective as information resources. Such neighborhoods are identified only by frustrated searchers, seeking a way out of these stale neighborhoods, back to more uptodate sections of the web; measuring the decay of a page purely on the basis of dead links on the page is too naive to reflect this frustration. In this paper we formalize a strong notion of a decay measure and present algorithms for computing it e#ciently. We explore this measure by presenting a number of validations, and use it to identify interesting artifacts on today's web. We then describe a number of applications of such a measure to search engines, web page maintainers, ontologists, and individual users.
The Markov Chain Simulation Method for Generating Connected Power Law Random Graphs
 In Proc. 5th Workshop on Algorithm Engineering and Experiments (ALENEX). SIAM
, 2003
"... Graph models for realworld complex networks such as the Internet, the WWW and biological networks are necessary for analytic and simulationbased studies of network protocols, algorithms, engineering and evolution. To date, all available data for such networks suggest heavy tailed statistics, most ..."
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Cited by 32 (6 self)
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Graph models for realworld complex networks such as the Internet, the WWW and biological networks are necessary for analytic and simulationbased studies of network protocols, algorithms, engineering and evolution. To date, all available data for such networks suggest heavy tailed statistics, most notably on the degrees of the underlying graphs. A practical way to generate network topologies that meet the observed data is the following degreedriven approach: First predict the degrees of the graph by extrapolation from the available data, and then construct a graph meeting the degree sequence and additional constraints, such as connectivity and randomness. Within the networking community, this is currently accepted as the most successful approach for modeling the interdomain topology of the Internet.
privacy in social networks
 ICDE 2008 Poster
"... We consider a privacy threat to a social network in which the goal of an attacker is to obtain knowledge of a significant fraction of the links in the network. We formalize the typical social network interface and the information about links that it provides to its users in terms of lookahead. We co ..."
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Cited by 23 (2 self)
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We consider a privacy threat to a social network in which the goal of an attacker is to obtain knowledge of a significant fraction of the links in the network. We formalize the typical social network interface and the information about links that it provides to its users in terms of lookahead. We consider a particular threat where an attacker subverts user accounts to get information about local neighborhoods in the network and pieces them together in order to get a global picture. We analyze, both experimentally and theoretically, the number of user accounts an attacker would need to subvert for a successful attack, as a function of his strategy for choosing users whose accounts to subvert and a function of lookahead provided by the network. We conclude that such an attack is feasible in practice, and thus any social network that wishes to protect the link privacy of its users should take great care in choosing the lookahead of its interface, limiting it to 1 or 2, whenever possible.
A sequential importance sampling algorithm for generating random graphs with prescribed degrees
, 2006
"... Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a ra ..."
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Cited by 22 (0 self)
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Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random
Distances in random graphs with finite variance degrees
, 2008
"... In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the socalled configuration model, and includes heavy tail degree ..."
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Cited by 21 (11 self)
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In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the socalled configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N, when the base of the logarithm equals ν = E[Dj(Dj − 1)]/E[Dj]> 1. This confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.
Distances in random graphs with infinite mean degrees
, 2004
"... We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ (1, 2). Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal nu ..."
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Cited by 19 (13 self)
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We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ (1, 2). Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with N nodes is investigated when N → ∞. The paper is part of a sequel of three papers. The other two papers study the case where τ ∈ (2, 3), and τ ∈ (3, ∞), respectively. The main result of this paper is that the graph distance converges for τ ∈ (1, 2) to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most N α for some α> 0. For τ −1 < α < (τ −1) −1, the hopcount converges to 3 in probability, while for α> (τ − 1) −1, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.