Non-monotonicity of the generalized dimensions of a complex network

Computing the generalized dimensions Dq of a complex network requires covering the network by a minimal number of “boxes” of size s, for a range of s. We show that, unlike the case for a geometric multifractal, for a complex network the shape of the Dq vs. q curve can be monotone increasing, or monotone decreasing, or even have both a local maximum and a local minimum, depending on the range of box sizes used to compute Dq. We provide insight into this behavior by deriving a simple closed-form expression for the derivative of Dq at q=0. The estimate depends on the ratio of the geometric mean of the box masses (where the mass of a box is the number of nodes it contains) to the arithmetic mean of the box masses.