Boundary Perturbation Analysis of Complex Networks

In order to clarify the relation between network structure and dynamical phenomena in networks, we investigate the first passage and first return times of a random walker and the response to a boundary perturbation in complex networks. We introduce an efficient calculation method for the first passage time distribution (FPTD) and first return time distribution (FRTD) by introducing a perfectly absorbing node. Using this method, we calculate the FPTD and FRTD in small-world networks and study the relation between these distributions and the characteristics of network structures. We find that the characteristics of the FPTD and FRTD depend on the number of shortcuts in the network, and that the mean first passage time is not determined by the shortest path length alone. We also derive the relation between the FPTD and the complex admittance in boundary perturbation experiments, and show that the Cole--Cole diagram represents characteristic behavior depending on complex networks. The characteristics of the network structure can be understood by performing boundary perturbation experiments.