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

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 $\gamma$ of power-law degree distribution $P(k)\sim k^{-\gamma}$, 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 $\gamma$ 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 $N$, which is obviously independent of $\gamma$ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where $\gamma$ influences qualitatively the MFPT of trapping problem.

💡 Research Summary

The paper investigates the trapping problem on a class of deterministic scale‑free graphs known as Koch networks. In the trapping problem a single trap is placed on a hub node (the node with the highest degree) and a random walker starts from any other node; the quantity of interest is the mean first‑passage time (MFPT), i.e., the average number of steps required for the walker to reach the trap for the first time.

Koch networks are generated recursively: starting from a triangle, each edge is replaced by a motif that introduces m new nodes (m ≥ 1) and connects them to the two end vertices of the edge. After t iterations the network contains

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