Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7374595 | Physica A: Statistical Mechanics and its Applications | 2018 | 29 Pages |
Abstract
The box counting dimension dB of a complex network G, and the generalized dimensions {Dq,qâR} 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,qâR} 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 sâS, 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Eric Rosenberg,