Double Percolation Phase Transition in Clustered Complex Networks

Abstract

The internal organization of complex networks often has striking consequences on either their response to external perturbations or on their dynamical properties. In addition to small-world and scale-free properties, clustering is the most common topological characteristic observed in many real networked systems. In this paper, we report an extensive numerical study on the effects of clustering on the structural properties of complex networks. Strong clustering in heterogeneous networks induces the emergence of a core-periphery organization that has a critical effect on the percolation properties of the networks. We observe a novel double phase transition with an intermediate phase in which only the core of the network is percolated and a final phase in which the periphery percolates regardless of the core. This result implies breaking of the same symmetry at two different values of the control parameter, in stark contrast to the modern theory of continuous phase transitions. Inspired by this core-periphery organization, we introduce a simple model that allows us to analytically prove that such an anomalous phase transition is, in fact, possible.

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Plain Language Summary

The internal organization of complex networks often has striking consequences for their response to external perturbations or their dynamical properties. Percolation theory has played a prominent role in understanding the anomalous behaviors observed in networks after the failure of their constituent parts. However, the role of clustering in percolation properties of networks is still unclear, despite being one of the most common topological patterns of real complex networks. We perform a thorough study of the effect of clustering on the structural properties of networks and the consequences on their percolation properties.

We find that strong clustering in heterogeneous networks induces a core-periphery organization that has a critical effect on the percolation properties of networks. We observe a novel “double percolation” phase transition with an intermediate state in which only the core is percolated and a final phase in which the periphery percolates, regardless of the core; this break between core and periphery in terms of percolation can arise because of clustering. Inspired by this core-periphery organization, we introduce a simple model that allows us to prove analytically that such an anomalous phase transition is possible. This fact is contrary to the common belief that it is not possible to have two or more consecutive continuous phase transitions associated with the same symmetry breaking.

Our work motivates the search for multiple phase transitions in phenomena beyond percolation and also provides a new definition of the core and periphery of a network. We expect that our findings may have applications in studying the spread of disease epidemics.