Random Walks Performed by Topologically-Specific Agents on Complex Networks

TL;DR

The paper addresses how the topology of moving agents affects random-walk dynamics on complex networks by introducing topologically-specific agents that are small graphs (triangle, square, slashed square) and by constructing associated networks where each node represents a potential agent position. The authors define movement rules, build associated networks, and compare random walks across Erdős-Rényi, Barabási-Albert, and geometric networks, alongside hub-removal experiments. Key findings show that agent topology shapes displacement patterns and coverage: square agents often yield faster node coverage in ER/BA networks, while triangle agents perform comparatively better in GEO networks, with associated networks revealing topology-driven differences; hub removal generally has limited impact. This framework enables more realistic modeling of agents with internal structure on networks and suggests avenues for extending to additional agent types, interactions, and adaptive behaviors, with potential applications in transport, search, and network design.

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.

Figures (15)

• Figure 1: Three examples of topologically-specific moving agents: (a) triangle, (b) square, and (c) slashed square, consisting of relatively small graphs, henceforth considered for performing random walks in complex networks. These agents are allowed only to occupy a subgraph of the network provided their topology is completely contained in that subgraph. An agent can then move to new adjacent subgraphs of the networks where it fits, therefore performing a respective random walk.
• Figure 2: Illustration of the method adopted in this work for choosing a possible next move for a topologically-specific agent. The current position of the agent ($n=3$ in the network is shown in blue in (a). Three lists of first-neighbors are obtained for each of the agent nodes, including these nodes (b). Three node labels are respectively chosen from the lists with uniform probability, defining a respective subgraph from the larger network. The resulting adjacency matrix is compared to the adjacency matrix of the agent (c). In case this subgraph matrix contains the agent, as verified in (b), the agent can proceed to this possible new position, which is not the case in situation (d).
• Figure 3: A triangular agent (in blue) contained in a network (a) and some of its possible displacements involving 1 (b), 2 (c), and 3 (d) nodes. The arrows (in magenta) indicate the motion of each of the agent nodes. The nodes corresponding to the initial position of the agent are shown with orange borders.
• Figure 4: Example of a simple network (a) and a respective associated network (b). The original network nodes corresponding to each associated network node are indicated by colors as indicated in the legend. Note that the associated network is a weighted graph in which the weights of the edges are shown proportional to the transition probability of the agent.
• Figure 5: Illustration of an ER network (with 100 nodes and $<k=12>$) (a) and the respective associated networks obtained for the triangle (b), square (c), and slashed square (d) agents.