Generalized Hausdorff dimensions of a complex network

Abstract

The box counting dimension dB of a complex network G, and the generalized dimensions {Dq,q∈R} of G, have been well studied. However, the Hausdorff dimension dH of a geometric object, which generalizes dB by not assuming equal-diameter boxes, has not previously been extended to G. Similarly, the generalized Hausdorff dimensions {DqH,q∈R} of a geometric object (defined by Grassberger in 1985), which extend the generalized dimensions Dq by not assuming equal-diameter boxes, have not previously been extended to G. In this paper we first develop a definition of dH for G and compare dH to dB on both constructed and real-world networks. Then we extend Grassberger’s work by defining the generalized Hausdorff dimensions DqH of G, and computing the DqH vs. q multifractal spectrum for several networks. Given a minimal covering B(s) of G for a range S of box sizes, computing dH utilizes the diameter of each box in B(s) for s∈S, and computing DqH utilizes the diameter and mass of each box in B(s). Also, computing dB and Dq (for a given q) typically utilizes linear regression; in contrast, computing dH and DqH (for a given q) requires minimizing a function of one variable. Computational results show that dH can sometimes be more useful than dB in quantifying changes in the topology of a network. However, dH is harder to compute than dB, and DqH is less well behaved than Dq. We conclude that dH and DqH should be added to the set of useful metrics for characterizing a complex network, but they cannot be expected to replace dB and Dq.