Percolation theory applied to measures of fragmentation in social networks

Details Percolation theory applied to measures of fragmentation in social networks Percolation theory applied to measures of fragmentation in social networks

2006-10-23

arxiv.org/abs/cond-mat/0610625v1

Phys. Rev. E 75, 046107 (2007)

Physics

Condensed Matter

Disordered Systems and Neural Networks

submitted to PRE

Scientific paper

10.1103/PhysRevE.75.046107

We apply percolation theory to a recently proposed measure of fragmentation $F$ for social networks. The measure $F$ is defined as the ratio between the number of pairs of nodes that are not connected in the fragmented network after removing a fraction $q$ of nodes and the total number of pairs in the original fully connected network. We compare $F$ with the traditional measure used in percolation theory, $P_{\infty}$, the fraction of nodes in the largest cluster relative to the total number of nodes. Using both analytical and numerical methods from percolation, we study Erd\H{o}s-R\'{e}nyi (ER) and scale-free (SF) networks under various types of node removal strategies. The removal strategies are: random removal, high degree removal and high betweenness centrality removal. We find that for a network obtained after removal (all strategies) of a fraction $q$ of nodes above percolation threshold, $P_{\infty}\approx (1-F)^{1/2}$. For fixed $P_{\infty}$ and close to percolation threshold ($q=q_c$), we show that $1-F$ better reflects the actual fragmentation. Close to $q_c$, for a given $P_{\infty}$, $1-F$ has a broad distribution and it is thus possible to improve the fragmentation of the network. We also study and compare the fragmentation measure $F$ and the percolation measure $P_{\infty}$ for a real social network of workplaces linked by the households of the employees and find similar results.

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