Random Walks Performed by Topologically-Specific Agents on Complex Networks

Abstract

Random walks by single-node agents have been systematically conducted on various types of complex networks in order to investigate how their topologies can affect the dynamics of the agents. However, by fitting any network node, these agents do not engage in topological interactions with the network. In the present work, we describe random walks on complex networks performed by agents that are actually small graphs. These agents can only occupy admissible portions of the network onto which they fit topologically, hence their name being taken as topologically-specific agents. These agents are also allowed to move to adjacent subgraphs in the network, which have each node adjacent to a distinct original respective node of the agent. Given a network and a specific agent, it is possible to obtain a respective associated network, in which each node corresponds to a possible instance of the agent and the edges indicate adjacent positions. Associated networks are obtained and studied respectively to three types of topologically-specific agents (triangle, square, and slashed square) considering three types of complex networks (geometrical, Erdős-Rényi, and Barabási-Albert). Uniform random walks are also performed on these structures, as well as networks respectively obtained by removing the five nodes with the highest degree, and studied in terms of the number of covered nodes along the walks. Several results are reported and discussed, including the fact that substantially distinct associated networks can be obtained for each of the three considered agents and for varying average node degrees. Respectively to the coverage of the networks by uniform random walks, the square agent led to the most effective coverage of the nodes, followed by the triangle and slashed square agents. In addition, the geometric network turned out to be less effectively covered.