# Graph-Facilitated Resonant Mode Counting in Stochastic Interaction Networks

## TL;DR

This paper introduces a graph-theoretic and real root counting approach to efficiently identify stochastic oscillations and resonant modes in complex networks, simplifying analysis beyond small systems.

## Contribution

It presents a novel method combining graph theory and root counting algorithms to determine stochastic resonance properties in large networks.

## Key findings

- Resonant modes depend on the squared Jacobian matrix $J^2$.
- Graph-theoretic tools simplify analysis of stochastic behavior in large networks.
- Chemical reaction networks with multiple resonant modes can be identified easily.

## Abstract

Oscillations in a stochastic dynamical system, whose deterministic counterpart has a stable steady state, are a widely reported phenomenon. Traditional methods of finding parameter regimes for stochastically-driven resonances are, however, cumbersome for any but the smallest networks. In this letter we show by example of the Brusselator how to use real root counting algorithms and graph theoretic tools to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix $J^2$ , unlike deterministic oscillations which are determined by $J$. By using graph theoretic tools, analysis of stochastic behaviour for larger networks is simplified and chemical reaction networks with multiple resonant modes can be identified easily.