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566
Quantifying social group evolution
 Nature
, 2007
"... The rich set of interactions between individuals in the society [1,2,3,4,5,6,7] results in complex community structure, capturing highly connected circles of friends, families, or professional cliques in a social network [3,7,8,9,10]. Thanks to frequent changes in the activity and communication patt ..."
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Cited by 126 (3 self)
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The rich set of interactions between individuals in the society [1,2,3,4,5,6,7] results in complex community structure, capturing highly connected circles of friends, families, or professional cliques in a social network [3,7,8,9,10]. Thanks to frequent changes in the activity and communication patterns of individuals, the associated social and communication network is subject to constant evolution [7,11,12,13,14,15,16]. Our knowledge of the mechanisms governing the underlying community dynamics is limited, but is essential for a deeper understanding of the development and selfoptimisation of the society as a whole [17,18,19,20,21,22]. We have developed a new algorithm based on clique percolation [23,24], that allows, for the first time, to investigate the time dependence of overlapping communities on a large scale and as such, to uncover basic relationships characterising community evolution. Our focus is on networks capturing the collaboration between scientists and the calls between mobile phone users. We find that large groups persist longer if they are capable of dynamically altering their membership, suggesting that an ability to change the composition results in better adaptability. The behaviour of small groups displays the opposite tendency, the condition
The Internet ASLevel Topology: Three Data Sources and One Definitive Metric
"... We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the W ..."
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Cited by 108 (15 self)
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We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. Among the widely considered metrics, we find that the joint degree distribution appears to fundamentally characterize Internet AS topologies as well as narrowly define values for other important metrics. We discuss the interplay between the specifics of the three data collection mechanisms and the resulting topology views. In particular, we show how the data collection peculiarities explain differences in the resulting joint degree distributions of the respective topologies. Finally, we release to the community the input topology datasets, along with the scripts and output of our calculations. This supplement should enable researchers to validate their models against real data and to make more informed selection of topology data sources for their specific needs.
Collecting the Internet ASlevel Topology
 ACM SIGCOMM Computer Communications Review (CCR
, 2005
"... At the interdomain level, the Internet topology can be represented by a graph with Autonomous Systems (ASes) as nodes and AS peerings as links. This ASlevel topology graph has been widely used in a variety of research efforts. Conventionally this topology graph is derived from routing tables colle ..."
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Cited by 107 (12 self)
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At the interdomain level, the Internet topology can be represented by a graph with Autonomous Systems (ASes) as nodes and AS peerings as links. This ASlevel topology graph has been widely used in a variety of research efforts. Conventionally this topology graph is derived from routing tables collected by RouteViews or RIPE RIS. In this work, we assemble the most complete ASlevel topology by extending the conventional method along two dimensions. First, in addition to using data from RouteViews and RIPE RIS, we also collect data from many other sources, including route servers, looking glasses, and routing registries. Second, in addition to using routing tables, we also accumulate topological information from routing updates over time. The resulting topology graph on a recent day contains 44 % more links and 3 % more nodes than that from using RouteViews routing tables alone. Our data collection and topology generation process have been automated, and we publish the latest topology on the web on a daily basis. 1.
Systematic topology analysis and generation using degree correlations
 In SIGCOMM
"... Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph’s resilience to failure or its routing efficiency. Knowledge of appropriate metric values may inf ..."
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Cited by 94 (7 self)
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Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph’s resilience to failure or its routing efficiency. Knowledge of appropriate metric values may influence the engineering of future topologies, repair strategies in the face of failure, and understanding of fundamental properties of existing networks. Unfortunately, there are typically no algorithms to generate graphs matching one or more proposed metrics and there is little understanding of the relationships among individual metrics or their applicability to different settings. We present a new, systematic approach for analyzing network topologies. We first introduce the dKseries of probability distributions specifying all degree correlations within dsized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dKseries implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d =0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d = 2 case is sufficient for most practical purposes, while d = 3 essentially reconstructs the Internet AS and routerlevel topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.
Evolutionary dynamics of social dilemmas in structured heterogeneous populations
 Proc. Natl. Acad. Sci. USA
, 2006
"... Abbreviations frequently used: T – Payoff for defecting on a cooperator R – Payoff for mutual cooperation P – Payoff for mutual defection S – Payoff for cooperating with a defector NoC – Network of Contacts SI – Supporting Information Real populations have been shown to be heterogeneous, in which so ..."
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Cited by 93 (22 self)
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Abbreviations frequently used: T – Payoff for defecting on a cooperator R – Payoff for mutual cooperation P – Payoff for mutual defection S – Payoff for cooperating with a defector NoC – Network of Contacts SI – Supporting Information Real populations have been shown to be heterogeneous, in which some individuals have many more contacts than others. This fact contrasts with the traditional homogeneous setting used in studies of evolutionary game dynamics. We incorporate heterogeneity in the population by studying games on graphs, in which heterogeneity ranges from singlescale graphs, where it is small and associated degree distributions exhibit a Gaussian tail, to scalefree graphs, where it is large with degreedistributions exhibiting a powerlaw behavior. We study the evolution of cooperation, modeled in terms of the most popular dilemmas of cooperation. We show that, for all dilemmas, increasing heterogeneity favors the emergence of cooperation, such that longterm cooperative behavior easily resists shortterm noncooperative behavior. Moreover, we show how cooperation depends on the intricate ties between individuals in scalefree populations. accepted 15dec2005 in the Proceedings of the National Academy of Sciences; published online 16feb2006 2 1.
Spatial networks
 PHYSICS REPORTS
, 2010
"... Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topolo ..."
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Cited by 93 (5 self)
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Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information. Characterizing and understanding
The role of the airline transportation network in the prediction and predictability of global epidemics.
 Proc. Natl Acad. Sci. USA 103,
, 2006
"... The systematic study of largescale networks has unveiled the ubiquitous presence of connectivity patterns characterized by largescale heterogeneities and unbounded statistical fluctuations. These features affect dramatically the behavior of the diffusion processes occurring on networks, determini ..."
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Cited by 86 (7 self)
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The systematic study of largescale networks has unveiled the ubiquitous presence of connectivity patterns characterized by largescale heterogeneities and unbounded statistical fluctuations. These features affect dramatically the behavior of the diffusion processes occurring on networks, determining the ensuing statistical properties of their evolution pattern and dynamics. In this article, we present a stochastic computational framework for the forecast of global epidemics that considers the complete worldwide air travel infrastructure complemented with census population data. We address two basic issues in global epidemic modeling: (i) we study the role of the large scale properties of the airline transportation network in determining the global diffusion pattern of emerging diseases; and (ii) we evaluate the reliability of forecasts and outbreak scenarios with respect to the intrinsic stochasticity of disease transmission and traffic flows. To address these issues we define a set of quantitative measures able to characterize the level of heterogeneity and predictability of the epidemic pattern. These measures may be used for the analysis of containment policies and epidemic risk assessment. complex systems ͉ epidemiology ͉ networks T he mathematical modeling of epidemics has often dealt with the predictions and predictability of outbreaks in real populations with complicated social and spatial structures and with heterogeneous patterns in the contact network (18). All these factors have led to sophisticate modeling approaches including disease realism, metapopulation grouping, and stochasticity, and more recently to agentbased numerical simulations that recreate entire populations and their dynamics at the scale of the single individual (9, 10). In many instances, however, the introduction of the inherent complex features and emerging properties (1113) of the network in which epidemics occur implies the breakdown of standard homogeneous approaches (5, 6) and calls for a systematic investigation of the impact of the detailed system's characteristics in the evolution of the epidemic outbreak. These considerations are particularly relevant in the study of the geographical spread of epidemics where the various longrange heterogeneous connections typical of modern transportation networks naturally give rise to a very complicated evolution of epidemics characterized by heterogeneous and seemingly erratic outbreaks (14, 15), as recently documented in the severe acute respiratory syndrome case (www.who.int͞csr͞ sars͞en). In this context, airtransportation represents a major channel of epidemic propagation, as pointed out in the modeling approach to global epidemic diffusion of Rvachev and Longini (16) capitalizing on previous studies on the Russian airline network (17). Similar modeling approaches, even if limited by a partial knowledge of the worldwide transportation network, have been used to study specific outbreaks such as pandemic influenza Results and Discussion The AirTransportationNetwork Heterogeneity. The International Air Transport Association database contains the world list of airport pairs connected by direct flights and the number of available seats on any given connection for the year 2002. The resulting worldwide airtransportation network (WAN) is therefore a weighted graph comprising V ϭ 3,880 vertices denoting airports and E ϭ 18,810 weighted edges whose weight w jl accounts for the passenger flow between the airports j and l. This data set has been complemented by the population N j of the large metropolitan area served by the airport as obtained by different sources. The final network data set contains the 3,100 largest airports, 17,182 edges (accounting for 99% of the worldwide traffic), and the respective urban population data. The obtained network is highly heterogeneous both in the connectivity pattern and the traffic capacities (see
The “robust yet fragile” nature of the Internet
 Proceedings of the National Academy of Sciences
, 2005
"... The search for unifying properties of complex networks is popular, challenging, and important. For modeling approaches that focus on robustness and fragility as unifying concepts, the Internet is an especially attractive case study, mainly because its applications are ubiquitous and pervasive, and w ..."
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Cited by 72 (14 self)
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The search for unifying properties of complex networks is popular, challenging, and important. For modeling approaches that focus on robustness and fragility as unifying concepts, the Internet is an especially attractive case study, mainly because its applications are ubiquitous and pervasive, and widely available expositions exist at every level of detail. Nevertheless, alternative approaches to modeling the Internet often make extremely different assumptions and derive opposite conclusions about fundamental properties of one and the same system. Fortunately, a detailed understanding of Internet technology combined with a unique ability to measure the network means that these differences can be thoroughly understood and unambiguously resolved. This paper aims to make recent results of this process accessible beyond Internet specialists to the broader scientific community, and to clarify several sources of basic methodological differences that are relevant beyond either the Internet or the two specific approaches focused on here; i.e., scalefree networks and highly optimized tolerance networks. A popular case study for complex networks has been the Internet, with a central issue the extent to which its design and evolution have made it “robust yet fragile ” (RYF)—that is, unaffected by random component failures
Compact routing on Internetlike graphs
 In Proc. IEEE INFOCOM
, 2004
"... Abstract — The ThorupZwick (TZ) compact routing scheme is the first generic stretch3 routing scheme delivering a nearly optimal pernode memory upper bound. Using both direct analysis and simulation, we derive the stretch distribution of this routing scheme on Internetlike interdomain topologies. ..."
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Cited by 64 (7 self)
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Abstract — The ThorupZwick (TZ) compact routing scheme is the first generic stretch3 routing scheme delivering a nearly optimal pernode memory upper bound. Using both direct analysis and simulation, we derive the stretch distribution of this routing scheme on Internetlike interdomain topologies. By investigating the TZ scheme on random graphs with powerlaw node degree distributions, Pk � k −γ, we find that the average TZ stretch is quite low and virtually independent of γ. In particular, for the Internet interdomain graph with γ � 2.1, the average TZ stretch is around 1.1, with up to 70 % of all pairwise paths being stretch1 (shortest possible). As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for 10 4node networks. Furthermore, we find that both the average shortest path length (i.e. distance) d and width of the distance distribution σ observed in the real Internet interAS graph have values that are very close to the minimums of the average stretch in the d and σdirections. This leads us to the discovery of a unique critical point of the average TZ stretch as a function of d and σ. The Internet distance distribution is located in a close neighborhood of this point. This is remarkable given the fact that the Internet interdomain topology has evolved without any direct attention paid to properties of the stretch distribution. It suggests the average stretch function may be an indirect indicator of the optimization criteria influencing the Internet’s interdomain topology evolution.
Nonequilibrium phase transition in the coevolution of networks and opinions
 Growth dynamics of the WorldWide Web,” Nature: VOL 401: 9 SEPTEMBER
, 2006
"... Models of the convergence of opinion in social systems have been the subject of a considerable amount of recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of per ..."
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Cited by 63 (2 self)
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Models of the convergence of opinion in social systems have been the subject of a considerable amount of recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion. We characterize the static and dynamical properties of this transition.