Expanded Complex Networks and their Percolations

Details Expanded Complex Networks and their Percolations Expanded Complex Networks and their Percolations

2003-12-31

arxiv.org/abs/cond-mat/0312712v3

Phys. Rev. E 70, 056106 (2004)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 4 figures. Revised version

Scientific paper

Given a complex network, its \emph{L-}paths correspond to sequences of $L+1$ distinct nodes connected through $L$ distinct edges. The \emph{L-}conditional expansion of a complex network can be obtained by connecting all its pairs of nodes which are linked through at least one \emph{L-}path, and the respective conditional \emph{L-}expansion of the original network is defined as the intersection between the original network and its \emph{L-}expansion. Such expansions are verified to act as filters enhancing the network connectivity, consequently contributing to the identification of communities in small-world models. It is shown in this paper for L=2 and 3, in both analytical and experimental fashion, that an evolving complex network with fixed number of nodes undergoes successive phase transitions -- the so-called \emph{L-}percolations, giving rise to Eulerian giant clusters. It is also shown that the critical values of such percolations are a function of the network size, and that the networks percolates for L=3 before L=2.

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