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# Emergent metastable behavior in a resource-constrained network of exclusion processes

## Details

**Authors** Bhatia et al.

**Year** 2026

**Publisher** Physical Review E

**Discipline** Network Science

[Read it at the publisher](https://doi.org/10.1103/5wyh-tdc3) 
10.1103/5wyh-tdc3

## In authors' words

### Abstract

Inspired by the intrinsic organization observed in various biological and physical processes, which manifests through interacting subsystems connected by networks, where particles often compete for limited resources, we study a four-lane network system with a branching-merging geometry under resource-constrained conditions. The particle inflow in the network is regulated by the total number of particles considered in the system, quantified by a filling factor, while conflict between the particle flow at the merging point of the network is captured through a friction parameter. Utilizing mean-field approximations, we obtained the analytical expressions for the stationary-state attributes of the resource-constrained network, such as lane densities, flux, stationary phases, and phase boundaries. The analysis of systems' stationary-state behavior is facilitated by the construction of phase diagrams in the parameter space of the entry-exit rate for two distinct friction regimes. All theoretically obtained findings are validated by the extensive stochastic Monte Carlo simulations based on the Gillespie algorithm under the random sequential update. For a low friction regime, the system can exhibit up to five possible stationary phases, and its phase diagram features a metastable region corresponding to the passage lanes. In this region, the possible stationary phase exhibited by these lanes sensitively depends on the lanes' initial configuration or density, which is corroborated by spatiotemporal plots. In contrast, for a higher friction regime, the system stops exhibiting metastable behavior, and now the phase diagram becomes richer, supporting up to 11 possible stationary phases. In both regimes, the topology of the phase diagrams exhibits non-monotonic behavior in terms of complexity as well as the number of stationary phases as the reservoir feeds more particles to the network system. Lastly, the influence of the boundary rates and the friction parameter is investigated on the position and height of the shock, along with the examination of the finite-size effects, providing an additional insight into the underlying phase transitions of the system.

### What they found (results)

Modelling a four-lane branching–merging network whose lanes compete for a shared finite particle pool, the authors show up to five stationary phases with a metastable region under low friction, and up to eleven phases with metastability lost under high friction, verified by mean-field theory and Monte Carlo simulation.

## Commentary

### In short

The network's behaviour belongs to the whole and to no lane in it, and it arises because the parts are coupled through a shared pool. The phases themselves are boundaries: the system is in one regime or another, and the count of regimes changes with friction.

**Patterns it shows** D, S, R

**Added** 2026-08-03

**How to cite this** Bhatia et al. (2026). Emergent metastable behavior in a resource-constrained network of exclusion processes. Physical Review E.