Volume 118, Number 6, June 2017
|Number of page(s)||7|
|Section||Interdisciplinary Physics and Related Areas of Science and Technology|
|Published online||28 August 2017|
Heterogeneous micro-structure of percolation in sparse networks
1 Department of Mathematics, King's College London - Strand, London WC2R 2LS, UK
2 Centre for Networks and Collective Behaviour, Department of Mathematical Sciences, University of Bath Bath, BA2 7AY, UK
Received: 23 June 2017
Accepted: 28 July 2017
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erdős-Rényi random graphs. Interpreted in terms of the application of percolation to real-world processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow.
PACS: 89.75.Hc – Networks and genealogical trees / 64.60.ah – Percolation / 05.70.Fh – Phase transitions: general studies
© EPLA, 2017
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