Random Walks on Stochastic Temporal Networks
In the study of dynamical processes on networks, there has been intense focus on network structure—i.e., the arrangement of edges and their associated weights—but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we
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Abstract In the study of dynamical processes on networks, there has been intense focus on network structure—i.e., the arrangement of edges and their associated weights—but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we develop a mathematical framework for random walks on temporal networks using an approach that provides a compromise between abstract but unrealistic models and data-driven but non-mathematical approaches. To do this, we introduce a stochastic model for temporal networks in which we summarize the temporal and structural organization of a system using a matrix of waiting-time distributions. We show that random walks on stochastic temporal networks can be described exactly by an integro-differential master equation and derive an analytical expression for its asymptotic steady state. We also discuss how our work might be useful to help build centrality measures for temporal networks.
1 Introduction A wide variety of systems are composed of interacting elements (e.g., nodes connected by edges) and can be represented as networks. Important examples include the Internet, highways and other transportation systems, and many social
T. Hoffmann Department of Physics, University of Oxford, Oxford, OX1 3RH, UK e-mail: [email protected] M.A. Porter Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute and CABDyN Complexity Centre, University of Oxford, Oxford, OX1 3LB, UK e-mail: [email protected] R. Lambiotte () Department of Mathematics, University of Namur, Namur, Belgium e-mail: [email protected] P. Holme and J. Saram¨aki (eds.), Temporal Networks, Understanding Complex Systems, DOI 10.1007/978-3-642-36461-7 15, © Springer-Verlag Berlin Heidelberg 2013
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Fig. 1 Different levels of abstraction to model dynamics on temporal networks. In this illustration, a walker steps from node 1 to node 2 and then jumps to one of the latter’s three neighbors. When studying dynamics on temporal networks, researchers tend to either (a) perform numerical simulations on empirical networks using the observed times at which edges are active between nodes or (b) develop mathematically-tractable Markovian models that neglect temporal patterns (typically by considering only aggregations of the interactions between nodes). In this chapter, we aim at finding (c) an intermediate level of modelling, in which we replace the sequence of activation times by a stochastic model that preserves a system’s inter-event distribution. We thereby balance the amount of data included in the description with the description’s simplicity
and biological systems. Because of the ubiquity of network representations, the study of networks has emerged as one of the fundamental building blocks in the study of complex systems [6, 13, 43]. Unfortunately, most empirical studies of networks thus far have been based on observations of snapshots of systems. Similarly, most theoretical efforts have focused on static network properties (
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