Agent-Based Triangle Counting: Unlocking Truss Decomposition, Triangle Centrality, and Local Clustering Coefficient

Abstract

Triangle counting in a graph is a fundamental problem with wide-ranging applications. It is crucial for understanding graph structure and serves as a basis for more advanced graph analytics. One key application is truss decomposition, a technique for identifying maximal, highly interconnected subgraphs, revealing structural cohesion and tight-knit communities in complex graphs. This facilitates analysis of relationships and information flow in fields such as social networks, biology, and recommendation systems. Using mobile agents or robots for tasks like truss decomposition and clustering coefficient computation is especially advantageous in decentralised environments with limited or unreliable communication. In such scenarios, agents can perform local computations without requiring an extensive communication infrastructure. This is valuable in contexts like disaster response, urban management, and military operations, where broadcast communication is impractical. In this paper, we address the triangle counting problem in an arbitrary anonymous graph using mobile agents. This method is extended as a subroutine to solve the truss decomposition problem and compute triangle centrality and the local clustering coefficient for each node. Our approach uses $n$ autonomous mobile agents, each starting at a different node of an $n$-node graph. These agents coordinate to collaboratively solve triangle enumeration, then truss decomposition, triangle centrality, and clustering coefficient. We assume a synchronous system where agents execute tasks concurrently, allowing time to be measured in rounds. The graph is anonymous (nodes have no IDs), but agents have distinct IDs and limited memory. Agents can perform local computations and communicate only when co-located. Our goal is to design algorithms that minimise both time and memory per agent, while enabling solutions to the above problems.