Expanded Complex Networks and their Percolations

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Given a complex network, its L-paths correspond to sequences of L+1 distinct nodes connected through L distinct edges. The L-conditional expansion of a complex network can be obtained by connecting all its pairs of nodes which are linked through at least one L-path, and the respective conditional L-expansion of the original network is defined as the intersection between the original network and its 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 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.