Diffusive Capture Process on Complex Networks

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>$ofalambscalesas<T>\sim N$ 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>$istherelevantfactorforthedynamicalpropertiesindiffusivecaptureprocess.Wenumericallyfindthatthenormalizednumberofcaptureeventsatanodewithdegree$k$,$n(k)$,decreasesas$n(k)\sim k^{-σ}$.When$γ<3$,$n(k)$stillincreasesanomalouslyfor$k\approx k_{max}$.Weanalyticallyshowthat$n(k)$satisfiestherelation$n(k)\sim k^2P(k)$andthetotalnumberofcaptureevents$N_{tot}$isproportionalto<k^2>$, which causes the $\gamma$ dependent behavior of $S(N,t)$ and <T>$.

9 pages, 6 figures

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