Trapping in complex networks

TL;DR

This paper studies how random walkers are trapped in complex networks, revealing how trap placement and network type influence particle survival over time through theoretical and simulation analysis.

Contribution

It provides a detailed analysis of trapping dynamics in ER and SF networks, highlighting the effects of trap location, network size, and degree distribution on particle density evolution.

Findings

In ER networks, particle density decays exponentially at short times.

In SF networks, trap location drastically affects particle survival.

Particle density decay in SF networks depends on degree distribution exponent ter

Abstract

We investigate the trapping problem in Erdos-Renyi (ER) and Scale-Free (SF) networks. We calculate the evolution of the particle density $\rho(t)$ of random walkers in the presence of one or multiple traps with concentration $c$. We show using theory and simulations that in ER networks, while for short times $\rho(t) \propto \exp(-Act)$, for longer times $\rho(t)$ exhibits a more complex behavior, with explicit dependence on both the number of traps and the size of the network. In SF networks we reveal the significant impact of the trap's location: $\rho(t)$ is drastically different when a trap is placed on a random node compared to the case of the trap being on the node with the maximum connectivity. For the latter case we find $\rho(t)\propto\exp\left[-At/N^\frac{\gamma-2}{\gamma-1}\av{k}\right]$ for all $\gamma>2$, where $\gamma$ is the exponent of the degree distribution $P(k)\propto k^{-\gamma}$.