Random walks on complex networks with inhomogeneous impact

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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.