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Avoiding a Spanning Cluster in Percolation Models

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Science  08 Mar 2013:
Vol. 339, Issue 6124, pp. 1185-1187
DOI: 10.1126/science.1230813

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Limits in Percolation Models

Slight changes in the number of connections within a network that form at random (for example, connections in social networks) can lead to a huge increase in connectivity, a phenomenon termed "explosive percolation." These percolation transitions are often studied with Erdös and Rényi models, in which edges connecting pairs of vertices in a network are added randomly or according to a rule. Whether these transitions are continuous in nature has been the subject of several recent studies. Cho et al. (p. 1185; see the Perspective by Ziff) examined the effect of avoiding bridge bonds that create a spanning cluster (one that completes the percolation path) on the continuity of transitions for a d-dimensional lattice (up to six dimensions). Analytical arguments and numerical studies reveal a critical value for the number of bonds m below which the percolation transition is continuous and above which it is discontinuous. The critical value depends on d and on the fractal dimension of the bridge bonds of the clusters.

Abstract

When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as can happen when an epidemic spreads. Recently, an explosive percolation (EP) model was introduced to understand such phenomena. The order of the EP transition has not been clarified in a unified framework covering low-dimensional systems and the mean-field limit. We introduce a stochastic model in which a rule for dynamics is designed to avoid the formation of a spanning cluster through competitive selection in Euclidean space. We use heuristic arguments to show that in the thermodynamic limit and depending on a control parameter, the EP transition can be either continuous or discontinuous if d < dc and is always continuous if ddc, where d is the spatial dimension and dc is the upper critical dimension.

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