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We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection method in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity.(More)
— We discuss an old distributed algorithm for reaching consensus that has received a fair amount of recent attention. In this algorithm, a number of agents exchange their values asynchronously and form weighted averages with (possibly outdated) values possessed by their neighbors. We overview existing convergence results, and establish some new ones, for(More)
We study fifteen months of human mobility data for one and a half million individuals and find that human mobility traces are highly unique. In fact, in a dataset where the location of an individual is specified hourly, and with a spatial resolution equal to that given by the carrier's antennas, four spatio-temporal points are enough to uniquely identify(More)
We introduce a concept of similarity between vertices of directed graphs. Let G A and G B be two directed graphs with, respectively, n A and n B vertices. We define an n B × n A similarity matrix S whose real entry s ij expresses how similar vertex j (in G A) is to vertex i (in G B): we say that s ij is their similarity score. The similarity matrix can be(More)
The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of(More)
We prove that there exist (infinitely many) values of the real parameters a and b for which the matrices a 1 1 0 1 and b 1 0 1 1 have the following property: all infinite periodic products of the two matrices converge to zero, but there exists a nonperiodic product that doesn't. Our proof is self-contained and fairly elementary; it uses only elementary(More)
—We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents in the same cluster holding the same opinion. We then introduce a(More)
In this paper, we analyze statistical properties of a communication network constructed from the records of a mobile phone company. The network consists of 2.5 million customers that have placed 810 million communications (phone calls and text messages) over a period of 6 months and for whom we have geographical home localization information. It is shown(More)
Social, technological and information systems can often be described in terms of complex networks that have a topology of interconnected nodes that combines organization and randomness [1, 2, 3, 4, 5]. The typical size of large networks such as social network services, mobile phone networks or the web now counts in millions when not billions of nodes and(More)
Citation Tsitsiklis, John N. et al. "Weighted Gossip: Distributed Averaging using non-doubly stochastic matrices. Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this(More)