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58
The effect of network topology on the spread of epidemics
 IN IEEE INFOCOM
, 2005
"... Many network phenomena are well modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. Many types of information dissemination can also be modeled as spreads of epidemics. In this paper we address the question ..."
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Cited by 115 (8 self)
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Many network phenomena are well modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. Many types of information dissemination can also be modeled as spreads of epidemics. In this paper we address the question of what makes an epidemic either weak or potent. More precisely, we identify topological properties of the graph that determine the persistence of epidemics. In particular, we show that if the ratio of cure to infection rates is smaller than the spectral radius of the graph, then the mean epidemic lifetime is of order log n, where n is the number of nodes. Conversely, if this ratio is bigger than a generalization of the isoperimetric constant of the graph, then the mean epidemic lifetime is of order � Ò�, for a positive constant �. We apply these results to several network topologies including the hypercube, which is a representative connectivity graph for a distributed hash table, the complete graph, which is an important connectivity graph for BGP, and the power law graph, of which the ASlevel Internet graph is a prime example. We also study the star topology and the ErdősRényi graph as their epidemic spreading behaviors determine the spreading behavior of power law graphs.
Complex Networks and Decentralized Search Algorithms
 In Proceedings of the International Congress of Mathematicians (ICM
, 2006
"... The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that ..."
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Cited by 71 (1 self)
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The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in largescale network data; such models have the potential to help us reason, at a general level, about the ways in which realworld networks are organized. We survey one particular line of network research, concerned with smallworld phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a wellknown experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graphtheoretic questions, and how they are supported by recent empirical studies of large social networks.
Effects of missing data in social networks
 Social Networks
, 2003
"... We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three pr ..."
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Cited by 28 (1 self)
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We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three principal missing data mechanisms: network boundary specification (noninclusion of actors or affiliations), survey nonresponse, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos Eprint Archive as well as random bipartite graphs. The simulation results show that network boundary specification and fixed choice designs can dramatically alter estimates of networklevel statistics. The observed clustering and assortativity coefficients are overestimated via omission of affiliations or fixed choice thereof, and underestimated via actor nonresponse, which results in inflated measurement error. We also find that social networks with multiple interaction contexts may have certain interesting properties due to the presence of overlapping cliques. In particular, assortativity by degree does not necessarily improve network robustness to random omission of nodes as predicted by current theory.
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.
Rescuing TitforTat with Source Coding
 7th IEEE International Conference on PeertoPeer Computing (P2P
, 2007
"... Titfortat is widely believed to be the most effective strategy to enforce collaboration among selfish users. However, it has been shown that its usefulness for decentralized and dynamic environments such as peertopeer networks is marginal, as peers can rapidly end up in a deadlock situation. Many ..."
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Cited by 14 (4 self)
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Titfortat is widely believed to be the most effective strategy to enforce collaboration among selfish users. However, it has been shown that its usefulness for decentralized and dynamic environments such as peertopeer networks is marginal, as peers can rapidly end up in a deadlock situation. Many proposed solutions to this problem are either less resilient to freeloading behavior or induce a computational overhead that cannot be sustained by regular peers. In contrast, we retain titfortat, but enhance the system with a novel form of source coding and an effective scheme to prevent peers from freeloading from seeding peers. We show that our system performs well without the risk of peer starvation and without sacrificing fairness. The proposed solution has a reasonably low overhead, and may hence be suitable for fully distributed content distribution applications in real networks. 1
On the Chromatic Number of Random Graphs
, 2007
"... In this paper we consider the classical ErdősRényi model of random graphs Gn,p. We show that for p = p(n) ≤ n−3/4−δ, for any fixed δ>0, the chromatic number χ(Gn,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1) log(ℓ−1) ≤ p(n−1). ..."
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Cited by 14 (0 self)
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In this paper we consider the classical ErdősRényi model of random graphs Gn,p. We show that for p = p(n) ≤ n−3/4−δ, for any fixed δ>0, the chromatic number χ(Gn,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1) log(ℓ−1) ≤ p(n−1).
Universal Rewriting in Constrained Memories
"... A constrained memory is a storage device whose elements change their states under some constraints. A typical example is flash memories, in which cell levels are easy to increase but hard to decrease. In a general rewriting model, the stored data changes with some pattern determined by the applicati ..."
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Cited by 13 (10 self)
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A constrained memory is a storage device whose elements change their states under some constraints. A typical example is flash memories, in which cell levels are easy to increase but hard to decrease. In a general rewriting model, the stored data changes with some pattern determined by the application. In a constrained memory, an appropriate representation is needed for the stored data to enable efficient rewriting. In this paper, we define the general rewriting problem using a graph model. This model generalizes many known rewriting models such as floating codes, WOM codes, buffer codes, etc. We present a novel rewriting scheme for the flashmemory model and prove it is asymptotically optimal in a wide range of scenarios. We further study randomization and probability distributions to data rewriting and study the expected performance. We present a randomized code for all rewriting sequences and a deterministic code for rewriting following any i.i.d. distribution. Both codes are shown to be optimal asymptotically.
Robustness in LargeScale Random Networks
, 2003
"... We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to reliability. In particular, we take the case of random regular networks for simplici ..."
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Cited by 12 (0 self)
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We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to reliability. In particular, we take the case of random regular networks for simplicity and consider the length of primary and backup paths in terms of the number of hops. First, for a randomly picked pair of nodes, we derive a lower bound on the average distance between the pair and discuss the tightness of the bound. In addition, noting that primary and protection paths form cycles, we obtain a lower bound on the average length of the shortest cycle around the pair. Finally, we show that the protected connections of a given maximum finite length are rare. We then generalize our network model so that different degrees are allowed according to some arbitrary distribution, and show that the second moment of degree over the first moment is an important shorthand for behavior of a network. Notably, we show that most of the results in regular networks carry over with minor modifications, which significantly broadens the scope of networks to which our approach applies. We present as an example the case of networks with a powerlaw degree distribution.