Random walks on complex networks with inhomogeneous impact

TL;DR

This paper introduces a random walk model on complex networks with inhomogeneous impact, revealing how impact variability affects activity fluctuations and the universality of scaling exponents.

Contribution

It presents a new model incorporating impact variability in random walks on complex networks, explaining non-universal fluctuation scaling exponents.

Findings

Impact variability leads to non-universal alpha values.

Universal alpha=1 occurs under strong external drive.

Analytical and numerical results confirm the model's predictions.

Abstract

In many complex systems, for the activity f(i) of the constituents or nodes i, a power-law relationship was discovered between the standard deviation sigma(i) and the average strength of the activity: sigma(i) ~ <f(i)>^alpha; universal values alpha = 1/2 or 1 were found, however, with exceptions. With the help of an impact variable we introduce a random walk model where the activity is the product of the number of visitors at a node and their impact. If the impact depends strongly on the node connectivity and the properties of the carrying network are broadly distributed (like in a scale free network) we find both analytically and numerically non-universal alpha values. The exponent always crosses over to the universal value of 1 if the external drive dominates.