Avoiding a Spanning Cluster in Percolation Models

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.