A Survey of Fractal Dimensions of Networks

Description

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. Front Matter ....Pages i-xi Introduction (Eric Rosenberg)....Pages 1-6 Covering a Complex Network (Eric Rosenberg)....Pages 7-11 Network Box Counting Heuristics (Eric Rosenberg)....Pages 13-27 Lower Bounds on Box Counting (Eric Rosenberg)....Pages 29-37 Correlation Dimension (Eric Rosenberg)....Pages 39-44 Mass Dimension for Infinite Networks (Eric Rosenberg)....Pages 45-50 Volume and Surface Dimensions for Infinite Networks (Eric Rosenberg)....Pages 51-54 Information Dimension (Eric Rosenberg)....Pages 55-59 Generalized Dimensions (Eric Rosenberg)....Pages 61-67 Non-monotonicity of Generalized Dimensions (Eric Rosenberg)....Pages 69-75 Zeta Dimension (Eric Rosenberg)....Pages 77-79 Back Matter ....Pages 81-84