Distributed Butterfly Analysis using Mobile Agents

TL;DR

This paper introduces distributed, agent-based algorithms for butterfly counting in bipartite graphs, enabling efficient detection of dense substructures without prior graph knowledge, with extensions to general graphs.

Contribution

The paper presents novel distributed algorithms for butterfly counting in bipartite and general graphs, including leader election and spanning tree construction with optimal complexity.

Findings

Efficient butterfly counting in bipartite graphs within O(Δ) rounds.

Distributed algorithms operate with minimal memory, only O(log λ) bits per agent.

Techniques extend naturally to general graphs with similar complexity bounds.

Abstract

Butterflies, or 4-cycles in bipartite graphs, are crucial for identifying cohesive structures and dense subgraphs. While agent-based data mining is gaining prominence, its application to bipartite networks remains relatively unexplored. We propose distributed, agent-based algorithms for \emph{Butterfly Counting} in a bipartite graph $G((A,B),E)$. Agents first determine their respective partitions and collaboratively construct a spanning tree, electing a leader within $O(n \log \lambda)$ rounds using only $O(\log \lambda)$ bits per agent. A novel meeting mechanism between adjacent agents improves efficiency and eliminates the need for prior knowledge of the graph, requiring only the highest agent ID $\lambda$ among the $n$ agents. Notably, our techniques naturally extend to general graphs, where leader election and spanning tree construction maintain the same round and memory complexities. Building on these foundations, agents count butterflies per node in $O(\Delta)$ rounds and compute the total butterfly count of $G$ in $O(\Delta+\min\{|A|,|B|\})$ rounds.