Condensation of edges on tree graphs induced by movement of a random walker

Condensation of edges, that is, the emergence of a finite fraction of highly connected vertices, is an interesting phenomenon found in a class of complex networks. We show that the addition of edges, due to stimulation by the movements of a random walker, can induce the condensation of edges on a tree graph. In this model, in the initial tree graph, the probability of bifurcation is a parameter that controls various complex structures, such as transition to the condensed phase and the value of the power-law exponent that describes the degree distribution. We detected the transition to the condensed phase by monitoring the growth property of the size of highly connected vertices (a complete subgraph) in a form similar to V-t((alpha-theta)/alpha), where a is an exponent which describes the number of vertices that the walker has visited by time t, V-t as similar to t(alpha). In condition 0<(alpha- theta)/alpha < 1, the size of the highly connected subgraph grows with time and can coexist with the rest of the graph with a power law P(k) similar to k(-gamma) in the degree distribution. We derived the relations between these power-law exponents, alpha,gamma, and zeta, which describe the time-dependence of the vertex degree as k similar to t(zeta), and showed that during the condensed phase, the properties of the evolving graph can be explained by the local structures created by the movements of the walker.