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Branching Ratios of Input Trees for Directed Multigraphs

Paolo Boldi, Ian Stewart

TL;DR

The paper studies the growth of input trees in finite directed multigraphs by defining the branching ratio δ(i) as δ(i) = lim_{ℓ→∞} (a_i(ℓ))^{1/ℓ}. It proves δ(i) exists for every node and equals the Perron eigenvalue ρ(i) of the upstream subnetwork 𝒰(i), tying a combinatorial growth rate to spectral data. It then develops both strong results for strongly connected networks (where all δ(i) coincide and equal ρ(i)) and a general framework for arbitrary networks, showing δ(i) = ρ(i) and that the asymptotics of a_i(ℓ) are governed by the dominant upstream SCCs, with precise periodic or polynomial refinements depending on the network's period structure. The work employs Perron–Frobenius theory, the upstream principle, and block-structured decompositions to derive detailed asymptotics: in irreducible cases, a_i(ℓ) ∼ C_r ρ^ℓ with r = ℓ mod h, while in general a_i(ℓ) ∼ R_s(ℓ) ρ^ℓ for s = ℓ mod g, with g the lcm of relevant periods. These results have implications for synchronization phenomena in biology and for message-passing growth in distributed systems, connecting local input-tree growth to global spectral properties.

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.

Branching Ratios of Input Trees for Directed Multigraphs

TL;DR

The paper studies the growth of input trees in finite directed multigraphs by defining the branching ratio δ(i) as δ(i) = lim_{ℓ→∞} (a_i(ℓ))^{1/ℓ}. It proves δ(i) exists for every node and equals the Perron eigenvalue ρ(i) of the upstream subnetwork 𝒰(i), tying a combinatorial growth rate to spectral data. It then develops both strong results for strongly connected networks (where all δ(i) coincide and equal ρ(i)) and a general framework for arbitrary networks, showing δ(i) = ρ(i) and that the asymptotics of a_i(ℓ) are governed by the dominant upstream SCCs, with precise periodic or polynomial refinements depending on the network's period structure. The work employs Perron–Frobenius theory, the upstream principle, and block-structured decompositions to derive detailed asymptotics: in irreducible cases, a_i(ℓ) ∼ C_r ρ^ℓ with r = ℓ mod h, while in general a_i(ℓ) ∼ R_s(ℓ) ρ^ℓ for s = ℓ mod g, with g the lcm of relevant periods. These results have implications for synchronization phenomena in biology and for message-passing growth in distributed systems, connecting local input-tree growth to global spectral properties.

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.
Paper Structure (26 sections, 20 theorems, 124 equations, 10 figures)

This paper contains 26 sections, 20 theorems, 124 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Biological Background
  4. Distributed Systems
  5. Summary of Paper and Main Results
  6. Graph-Theoretic Preliminaries
  7. Adjacency Matrix
  8. Walks
  9. Input Trees
  10. Strongly Connected Components
  11. Upstream Subnetwork
  12. Number of Walks of Given Length
  13. Reducible and Irreducible Matrices
  14. Definition of Branching Ratio
  15. Bounds
  16. ...and 11 more sections

Key Result

Proposition 2.8

$$ (1) If $(x_0,e_1, x_1, e_2,x_2, \ldots, e_\ell, x_\ell)$ is a walk from $j=x_\ell$ to $i=x_0$ then all $x_t$ and $e_t$ are included in ${\mathcal{U}}(i)$. (2) ${\mathcal{U}}(i)$ consists precisely of the vertices and edges of all walks that terminate at vertex $i$. (3) If $b_\ell(i)$ is the numbe (4) ${\mathcal{U}}(i)$ is the subnetwork induced by the set of all the vertices of all SCCs of ${\m

Figures (10)

  • Figure 1: Fibonacci circuit.
  • Figure 2: Input trees for the Fibonacci circuit, up to level $\ell= 4$.
  • Figure 3: Left: Fibonacci circuit feeding forward to a single node with two self-loops. Right: Single node with two self-loops feeding forward to a Fibonacci circuit.
  • Figure 4: Left: A simple $n$-cycle.
  • Figure 5: Left: A directed graph with six nodes. Right: Input tree of node 1.
  • ...and 5 more figures

Theorems & Definitions (58)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Proposition 2.8
  • Corollary 2.9
  • ...and 48 more