We studied the diffusion process of random walkers in networks formed by their traces. This model considers the rise and fall of links determined by the frequency of transports of random walkers. In order to examine the relation between the formed network and the diffusion process, a situation in which multiple random walkers start from the same vertex is investigated. The difference in diffusion rate of random walkers according to the difference in dimension of the initial lattice is very important for determining the time evolution of the networks. For example, complete subgraphs can be formed on a one-dimensional lattice while a graph with a power-law vertex degree distribution is formed on a two-dimensional lattice. We derived some formulae for predicting network changes for the 1D case, such as the time evolution of the size of nearly complete subgraphs and conditions for their collapse. The networks formed on the 2D lattice are characterized by the existence of clusters of highly connected vertices and their life time. As the life time of such clusters tends to be small, the exponent of the power-law distribution changes from γ ≃ 1–2 to γ ≃ 3.

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The power-law exponent γ describing the form of the degree distribution of networks is related to the mean degree of the networks. This relation provides a useful method to estimate this exponent because mean degrees can easily be obtained by the number of vertices and edges in the networks. We improved the relation obtained only by ordinary integral approximation, and examined its availability for number of vertices, minimum degree considered, and range of γ, taking also into consideration the number of vertices with minimum degree. As a result, we were able to estimate slight deviations of γ from 3, which are usually observed in numerical simulations of growing networks with linear preferential attachment. Furthermore, using this method, we were also able to predict to what extent γ changed by joining pre-existing vertices in growing networks or by imposing restrictions to prevent the gaining of new edges for certain vertices. For cases where γ < 2, we estimated the power-law exponent of degree distributions of networks formed by traces of random walkers from the increased rate of vertices with created edges.