How the power-law form of the degree distribution of networks can survive in the equilibrium state of number of nodes is an open question, since the deletion of nodes seriously affects the degree distribution. In this paper, we introduce a network evolution rule based on a random walk to describe the replacement of nodes in networks. Numerical results show that model assumptions support the power-law form of degree distribution even when the replacement of vertices achieves an equilibrium state of the number of vertices, and the distribution is described by the distribution of the maximum degree that each node experiences in its life. We showed additionally that the reduction of the addition probability of nodes causes an edge condensation to hub nodes. A novel feature of the model is that the visit frequency and recurrence property of the walker to a vertex determines the degree and lifetime of the vertex in an equilibrium state of the number of vertices.
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