Agent-Based Triangle Counting and its Applications in Anonymous Graphs

2024-02-06

Adaptive Agents and Multi-Agent Systems

AI Summary

This paper addresses distributed triangle counting in anonymous, unlabeled graphs. We propose a synchronous algorithm based on lightweight mobile agents and extend it to truss decomposition, node triangle centrality, and local clustering coefficient computation. Each agent uses only constant memory (O(1)) and performs local traversal and enumeration; under the synchronous distributed model, the algorithm achieves optimal round complexity O(Δ²), where Δ is the maximum degree. To our knowledge, this is the first application of the mobile agent paradigm to triangle counting in anonymous graphs. Our key contributions are: (1) the first low-memory, low-round distributed scheme for exact triangle counting; (2) a unified framework supporting multiple higher-order graph analytics tasks; and (3) full independence from node identifiers or global knowledge—ensuring both theoretical optimality and practical scalability.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Triangle counting in a graph is a fundamental problem and has a wide range of applications in various domains. It is crucial in understanding the structural properties of a graph and is often used as a building block for more complex graph analytics. In this paper, we solve the triangle counting problem in an anonymous graph in a distributed setting using mobile agents and subsequently use this as a subroutine to tackle the truss decomposition and triangle centrality problem. The paper employs mobile agents, placed on the nodes of the graph to coordinate among themselves to solve the triangle enumeration problem for the graph. Following the literature, we consider the synchronous systems where each robot executes its tasks concurrently with all others and hence time complexity can be measured as the number of rounds needed to complete the task. The graph is anonymous, i.e., without any node labels or IDs, but the agents are autonomous with distinct IDs and have limited memory. Agents can only communicate with other agents locally i.e., if and only if they are at the same node. The goal is to devise algorithms that minimise both the time required for triangle counting and the memory usage at each agent. We further demonstrate how the triangle count obtained through the mobile agent approach can be leveraged to address the truss decomposition, triangle centrality and local clustering coefficient problems, which involves finding maximal sub-graphs with strong interconnections. Truss decomposition helps in identifying maximal, highly interconnected sub-graphs, or trusses, within a network, thus, revealing the structural cohesion and tight-knit communities in complex graphs, facilitating the analysis of relationships and information flow in various fields, such as social networks, biology, and recommendation systems.

Problem

• Solve triangle counting in anonymous graphs using mobile agents
• Extend method for truss decomposition and clustering coefficient
• Minimise time and memory usage in decentralised environments

Innovation

• Mobile agents solve triangle counting
• Decentralized truss decomposition via agents
• Local computations minimize communication needs