Branching Ratios of Input Trees for Directed Multigraphs

TL;DR

This paper introduces the concept of the branching ratio of input trees in directed multigraphs, linking it to the largest eigenvalue of the subgraph's adjacency matrix, with implications for understanding graph growth.

Contribution

It defines the branching ratio for input trees in directed multigraphs, proves its existence, and relates it to the Perron-Frobenius eigenvalue, providing foundational properties and asymptotic insights.

Findings

Branching ratio exists for every node.

Branching ratio equals the largest eigenvalue of the subgraph's adjacency matrix.

Provides asymptotic growth analysis of input trees.

Abstract

We define the branching ratio of the input tree of a node in a finite directed multigraph, prove that it exists for every node, and show that it is equal to the largest eigenvalue of the adjacency matrix of the induced subgraph determined by all upstream nodes. This real eigenvalue exists by the Perron-Frobenius Theorem for non-negative matrices. We motivate our analysis with simple examples, obtain information about the asymptotics for the limit growth of the input tree, and establish other basic properties of the branching ratio.