Diffusive Capture Process on Complex Networks

TL;DR

This paper investigates how a diffusing predator-prey process behaves on complex networks, revealing how network structure influences survival times and capture dynamics, with implications for understanding diffusion and search processes.

Contribution

It provides a detailed analysis of the diffusive capture process on various complex networks, highlighting the role of degree distribution moments in dynamical properties and deriving analytical relations for capture events.

Findings

Survival time scales as network size N.

Survival probability varies with degree exponent γ.

Number of capture events relates to degree distribution moments.

Abstract

We study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of $N$. We find that the life time <T>$ of a lamb scales as <T>\sim N$ and the survival probability $S(N\to \infty,t)$ becomes finite on scale-free networks with degree exponent $\gamma>3$. However, $S(N,t)$ for $\gamma<3$ has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. It suggests that the second moment of degree distribution <k^2>$ is the relevant factor for the dynamical properties in diffusive capture process. We numerically find that the normalized number of capture events at a node with degree $k$, $n(k)$, decreases as $n(k)\sim k^{-\sigma}$. When $\gamma<3$, $n(k)$ still increases anomalously for $k\approx k_{max}$. We analytically show that $n(k)$ satisfies the relation $n(k)\sim k^2P(k)$ and the total number of capture events $N_{tot}$ is proportional to <k^2>$, which causes the $\gamma$ dependent behavior of $S(N,t)$ and <T>$.