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Unformatted text preview: Electronic copy available at: Communities in Networks Mason A. Porter, Jukka-Pekka Onnela, and Peter J. Mucha Introduction: Networks and Communities “But although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments quite outside of thermodynamics.” — Josiah Willard Gibbs, Elementary Principles in Statistical Mechanics, 1902 [47] F rom an abstract perspective , the term network is used as a synonym for a math- ematical graph . However, to scientists across a variety of fields, this label means so much more [13,20,44,83,88,120,124]. In sociology, each node (or vertex) of a network represents an agent , and a pair of nodes can be connected by a link (or edge) that signifies some social interaction or tie between them (see Figure 1 Mason A. Porter, Oxford Centre for Industrial and Ap- plied Mathematics, Mathematical Institute, University of Oxford, and CABDyN Complexity Centre, University of Oxford. His email address is [email protected] . Jukka-Pekka Onnela, Harvard Kennedy School, Harvard University; Department of Physics, University of Oxford; CABDyN Complexity Centre, University of Oxford; and Department of Biomedical Engineering and Computa- tional Science, Helsinki University of Technology. His email address is [email protected] . Peter J. Mucha, Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, and Institute for Advanced Materials, Nanoscience and Technology, University of North Carolina. His email address is [email protected] . for an example). Each node has a degree given by the number of edges connected to it and a strength given by the total weight of those edges. Graphs can represent either man-made or natural con- structs, such as the World Wide Web or neuronal synaptic networks in the brain. Agents in such networked systems are like particles in traditional statistical mechanics that we all know and (pre- sumably) love, and the structure of interactions between agents reflects the microscopic rules that govern their behavior. The simplest types of links are binary pairwise connections, in which one only cares about the presence or absence of a tie. How- ever, in many situations, links can also be assigned a direction and a (positive or negative) weight to designate different interaction strengths. Traditional statistical physics is concerned with the dynamics of ensembles of interacting and noninteracting particles. Rather than tracking the motionofalloftheparticlessimultaneously,which isanimpossibletaskduetotheirtremendousnum- ber, one averages (in some appropriate manner) the microscopic rules that govern the dynamics of individual particles to make precise statements of macroscopic observables such as temperature...
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This note was uploaded on 10/24/2011 for the course SCIENCE PHY 453 taught by Professor Barnard during the Winter '11 term at BYU.

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