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arXiv · 1505.07282

Network heterogeneity and node capacity lead to heterogeneous scaling of fluctuations in random walks on graphs

Kosmas Kosmidis·Moritz Beber·Marc-Thorsten Hütt
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Abstract

Random walks are one of the best investigated dynamical processes on graphs. A particularly fascinating phenomenon is the scaling relationship of fluctuations $σ$ with the average flux $\langle f \rangle $. Here we analyze how network topology and nodes with finite capacity lead to deviations from a simple scaling law $σ\sim \langle f \rangle ^α$. Sources of randomness are the random walk itself (internal noise) and the fluctuation of the number of walkers (external noise). We obtained exact results for the extreme case of a star network which are indicative of the behavior of large scale systems with a broad degree distribution.The latter are subsequently studied using Monte Carlo simulations. We find that the network heterogeneity amplifies the effects of external noise. By computing the `effective' scaling of each node we show that multiple scaling relationships can coexist in a graph with a heterogeneous degree distribution at an intermediate level of external noise. Finally, we analyze the effect of a finite capacity of nodes for random walkers and find that this also can lead to a heterogeneous scaling of fluctuations.

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Kosmas Kosmidis, Moritz Beber, Marc-Thorsten Hütt. 2015-05-27. Network heterogeneity and node capacity lead to heterogeneous scaling of fluctuations in random walks on graphs. https://doi.org/10.1142/s0219525915500071

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