Fractal Dimensions of Networks

Eric Rosenberg has taught undergraduate and graduate courses in modelling and optimization at Princeton University, New Jersey Institute of Technology, and Rutgers University. He recently retired from AT&T Labs in Middletown, New Jersey, and has joined the faculty of Georgian Court University in Lakewood, New Jersey.He received a B.A. in Mathematics from Oberlin College and a Ph.D. in Operations Research from Stanford University, and has authored or co-authored 19 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.Dr. Rosenberg is the author of A Primer of Multicast Routing, and A Survey of Fractal Dimensionsof Networks, both published by Springer.New research and applications relating to fractal dimensions of networks can be sent to Dr. Rosenberg at FractalDimensionsOfNetworks@gmail.com.

• 1. Introduction.- 2. Networks: Introductory Material.- 3. Fractals: Introductory Material.- 4. Topological and Box Counting Dimensions.- 5. Hausdorff, Similarity, and Packing Dimensions.- 6. Computing the Box Counting Dimension.- 7. Network Box Counting Dimension.- 8. Network Box Counting Heuristics.- 9. Correlation Dimension.- 10. Computing the Correlation Dimension.- 11. Network Correlation Dimension.-  12. Dimensions of Infinite Networks.- 13. Similarity Dimension of Infinite Networks.- 14. Information Dimension.- 15. Network Information Dimension.- 16. Generalized Dimensions and Multifractals.- 17. Multifractal Networks.- 18. Generalized Hausdorff Dimensions of Networks.- 19. Lacunarity.- 20. Other Dimensions.- 21. Coarse Graining and Renormalization.- 22. Other Network Dimensions.- 23. Supplemental Material.-