Results 1  10
of
76
Graph mining: laws, generators, and algorithms
 ACM COMPUT SURV (CSUR
, 2006
"... How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M: N relation in ..."
Abstract

Cited by 132 (7 self)
 Add to MetaCart
How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M: N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: “How can we generate synthetic but realistic graphs? ” To answer this, we must first understand what patterns are common in realworld graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.
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 ..."
Abstract

Cited by 100 (6 self)
 Add to MetaCart
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
Conductance and Congestion in Power Law Graphs
, 2003
"... It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size ..."
Abstract

Cited by 69 (6 self)
 Add to MetaCart
It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size of the network? We study routing in families of sparse random graphs whose degrees follow heavy tailed distributions. Instantiations of such random graphs have been proposed as models for the topology of the Internet at the level of Autonomous Systems as well as at the level of routers. Let n be the number of nodes. Suppose that for each pair of nodes with degrees du and dv we have O(dudv ) units of demand. Thus the total demand is O(n ). We argue analytically and experimentally that in the considered random graph model such demand patterns can be routed so that the flow through each link is at most O . This is to be compared with a bound # that holds for arbitrary graphs. Similar results were previously known for sparse random regular graphs, a.k.a. "expander graphs." The significance is that Internetlike topologies, which grow in a dynamic, decentralized fashion and appear highly inhomogeneous, can support routing with performance characteristics comparable to those of their regular counterparts, at least under the assumption of uniform demand and capacities. Our proof uses approximation algorithms for multicommodity flow and establishes strong bounds of a generalization of "expansion," namely "conductance." Besides routing, our bounds on conductance have further implications, most notably on the gap between first and second eigenvalues of the stochastic normalization of the adjacency matrix of the graph.
On the spread of viruses on the internet
 In SODA
, 2005
"... We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a nonvanishing chance of becoming epidemic. Quantita ..."
Abstract

Cited by 66 (6 self)
 Add to MetaCart
(Show Context)
We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a nonvanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy: for a virus with effective spread rate λ, if the infection starts at a typical vertex, then it develops log(1/λ) into an epidemic with probability λ Θ ( log log(1/λ)), but on average the epidemic probability is λΘ(1). 1
Counting graph homomorphisms
 IN:TOPICS IN DISCRETE MATH
, 2006
"... Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this paper we survey recent developments in the study of ..."
Abstract

Cited by 58 (16 self)
 Add to MetaCart
(Show Context)
Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this paper we survey recent developments in the study of homomorphism numbers, including the characterization of the homomorphism numbers in terms of the semidefiniteness of “connection matrices”, and some applications of this fact in extremal graph theory. We define a distance of two graphs in terms of similarity of their global structure, which also reflects the closeness of (appropriately scaled) homomorphism numbers into the two graphs. We use homomorphism numbers to define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. The convergence can also be characterized in terms of mappings of the graphs into fixed small graphs, which is strongly connected to important parameters like ground state energy in statistical physics, and to weighted maximum cut problems in computer science.
Network structure and systemic risk in banking systems
, 2012
"... We present a quantitative methodology for analyzing the potential for contagion and systemic risk in a network of interlinked financial institutions, using a metric for the systemic importance of institutions: the Contagion Index. We apply this methodology to a data set of mutual exposures and capi ..."
Abstract

Cited by 44 (3 self)
 Add to MetaCart
We present a quantitative methodology for analyzing the potential for contagion and systemic risk in a network of interlinked financial institutions, using a metric for the systemic importance of institutions: the Contagion Index. We apply this methodology to a data set of mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, and analyze the role of balance sheet size and network structure in each institution’s contribution to systemic risk. Our results emphasize the contribution of heterogeneity in network structure and concentration of counterparty exposures to a given institution in explaining its systemic importance. These observations plead for capital requirements which depend on exposures, rather than aggregate balance sheet size, and which
Distances in random graphs with finite mean and infinite variance degrees.
 Electron. J. Probab.,
, 2007
"... Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph wi ..."
Abstract

Cited by 38 (13 self)
 Add to MetaCart
(Show Context)
Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with N nodes is investigated when N → ∞. When τ ∈ (2, 3), this graph distance grows like 2 log log N  log(τ −2) . In different papers, the cases τ > 3 and τ ∈ (1, 2) have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.
Multilayer networks
 TOOL FOR MULTILAYER ANALYSIS AND VISUALIZATION OF NETWORKS 17 OF 18
, 2014
"... In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is impo ..."
Abstract

Cited by 34 (7 self)
 Add to MetaCart
(Show Context)
In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such “multilayer” features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize “traditional ” network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary
Link Privacy in Social Networks
 ICDE 2008 POSTER
, 2008
"... 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 ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 29 (12 self)
 Add to MetaCart
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.