Degree correlations of percolating clusters in random networks

Most networks consist of several components such that only nodes in each component are connected by a path. Then the degree correlation of each connected component is different from that of the whole network. Recent works have formalized the joint probability of degrees in the giant component(GC) by the generating function method, investigating the average degree of nearest neighbor nodes of degree k nodes [Tishby et al. PRE 2018] and the assortativityr defined by Pearson's coefficient for nearest degrees [Bialas and Oles PRE 2008]. As shown in previous works , the GCs extracted from the configuration model and from Erdos-Renyi random graphscan have the negative degree-degree correlation (disassortativity) even though the whole network does not possess any degree correlation. In this study, we focus on GCs formed by the site percolationprocesses in networks. Analyzing the GC formed by site percolationon uncorrelatednetworks with arbitrary degree distributionp(k), we discuss the generality of the disassortativity of GCs [Mizutaka and Hasegawa PRE 2018]. Deriving the general expression for the assortativityr of the GC, we prove that r<0 above the percolation thresholdif the third moment of p(k) is finite. The result is persistent at percolation thresholdif the fourth moment of p(k) is finite. In addition, it has been shown that the average degree of nearest neighbor nodes of degree k nodes at the percolation thresholdis inversely proportional to k on the GC for arbitrary p(k) with a finite third moment. These results mean that the GC formed by the site percolationprocess on uncorrelatedrandom networks possesses disassortativity above and at the percolation threshold. This supports the previous report [Yook et al. PRE 2005] about the relation between fractality and disassortativity of real-world networks in that the GC is fractal at the percolation threshold.