Impact of degree heterogeneity on the behavior of trapping in Koch networks

Zhang, Zhongzhi; Gao, Shuyang; Xie, Wenlei

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent [Formula: see text] of power-law degree distribution [Formula: see text] , which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent [Formula: see text] in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number [Formula: see text] , which is obviously independent of [Formula: see text] and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where [Formula: see text] influences qualitatively the MFPT of trapping problem.