A Survey of Fractal Dimensions of Networks

Many different fractal dimensions have been proposed for networks. In A Survey of Fractal Dimensions of Networks the theory and computation of the most important of these dimensions are reviewed, including the box counting dimension, the correlation dimension, the mass dimension, the transfinite fractal dimension, the information dimension, the generalized dimensions (which provide a way to describe multifractals), and the sandbox method (for approximating the generalized dimensions). The book describes the use of diameter-based and radius-based boxes, and presents several heuristic methods for box counting, including greedy coloring, random sequential node burning, and a method for computing a lower bound. We also discuss very recent results on resolving ambiguity in the calculation of the information dimension and the generalized dimensions, and on the non-monotonicity of the generalized dimensions. 

Anyone interested in the theory and application of networks will want to read this Brief. This includes anyone studying, e.g., social networks, telecommunications networks, transportation networks, ecological networks, food chain networks, network models of the brain, or financial networks.

Eric Rosenberg received a B.A. in Mathematics from Oberlin College and a Ph.D. in Operations Research from Stanford University. He works at AT&T Labs in Middletown, New Jersey (email: ericr@att.com). Dr. Rosenberg has taught undergraduate and graduate courses in optimization at Princeton University and New Jersey Institute of Technology. He has authored or co-authored 17 patents and has published in the areas of convex analysis and nonlinearly constrained optimization, computer aided design of integrated circuits and printed wire boards, telecommunications network design and routing, and fractal dimensions of networks. He is the author of A Primer of Multicast Routing (Springer Briefs in Computer Science, 2012).