Percolation behaviors of finite components on complex networks

Percolation behavior is of wide applicability and provides insight into functional structure of complex networks. Different from the percolation behaviors of giant component (GC), the finite components make one more clearly explore network percolation behaviors and critical phenomena from a microscopic perspective, especially for large-scale network systems. Here we focus on the percolation behaviors of small component pi(s) with the size s = 1, 2, 3, ... under different failure scenarios such as random attack, localized attack, target attack and intentional attack with limited knowledge. We find theoretically and via simulation that finite components all show the peak shape which is different from GC for random networks including random regular network, Erdos-Renyi networks and scale-free networks. In particular, we find a new general scaling relationship between 1/s 1/p(max), p(max) represents the value of p (non-failure fraction of initial nodes) corresponding to the peak point of pi(s) in the network. This finding also provides a potential approach for determining the critical threshold and fill the gap between finite components and GC on the percolation process.

Automated summary

Percolation behavior and critical phenomena are important characteristics of complex networks. Studying the percolation behavior of finite components provides insights into the overall network from a microscopic perspective and offers a new approach to determine the critical threshold.