# An almost-solvable model of complex network dynamics

## TL;DR

This paper introduces RandLOE, a novel model for large multi-agent network dynamics combining deterministic laws and randomness, which allows for analytic study and stable simulation beyond traditional stochastic differential equations.

## Contribution

The paper presents RandLOE, a new operator-based model for complex networks that extends the logistic equation framework and enables analysis of diverse dynamical behaviors.

## Key findings

- RandLOE can simulate large networks efficiently.
- The model exhibits cycles of expansion and contraction.
- It provides a new analytical approach outside stochastic differential equations.

## Abstract

We discuss a specific model, which we refer to as RandLOE, of a large multi-agent network whose dynamic is prescribed via a combination of deterministic local laws and random exogenous factors. The RandLOE approach lies outside the framework of Stochastic Differential Equations, but lends itself to analytic examination as well as to stable simulation even for relatively large networks. RandLOE is based on the logistic operator equation (LOE), which is a multidimensional dynamical system extending the classical logistic equation via an operator-algebraic interaction term. The network is defined by interpreting the LOE variable as an adjacency matrix of a complete graph. Depending on the choice of parameters, it can display a number of essentially distinct dynamical characteristics: e.g. cycles of expansion and contraction.