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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Consider the network \({\mathbb {G}} = ( {\mathbb {N}}, {\mathbb {A}})\) where \(\mathbb {N}\) is the set of nodes connected by the set \(\mathbb {A}\) of arcs. Let \(N \equiv \vert {\mathbb {N}} \vert \) be the number of nodes, and let \(A \equiv \vert {\mathbb {A}} \vert \) be the number of arcs. (We use “≡” to denote a definition.)
Eric Rosenberg

Chapter 2. Covering a Complex Network

Abstract
The previous chapter showed how the box counting and Hausdorff dimensions of a geometric object Ω are computed from a covering of Ω. With this background, we can now consider what it means to cover a complex network \(\mathbb {G}\), and how a fractal dimension can be computed from a covering of \(\mathbb {G}\). We require some definitions.
Eric Rosenberg

Chapter 3. Network Box Counting Heuristics

Abstract
In this chapter we examine several methods for computing a minimal set of diameter-based boxes, or a minimal set of radius-based boxes, to cover \(\mathbb {G}\).
Eric Rosenberg

Chapter 4. Lower Bounds on Box Counting

Abstract
Consider a box counting heuristic using radius-based boxes, e.g., Maximum Excluded Mass Burning. There is no guarantee that the computed https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-90047-6_4/461732_1_En_4_IEq1_HTML.gif is minimal or even near minimal.
Eric Rosenberg

Chapter 5. Correlation Dimension

Abstract
Extending the definition to a complex network, we say that \(\mathbb {G}\) has correlation dimension \(d_{ \stackrel {}{C}}\) if the fraction C(s) of nodes at a distance less than s from a random node follows the scaling law.
Eric Rosenberg

Chapter 6. Mass Dimension for Infinite Networks

Abstract
In this chapter we consider a sequence \(\{ {\mathbb {G}}_t \}_{t=1}^{\infty }\) of complex networks such that \(\varDelta _{\hspace {0.07em} {t}} \equiv \mathit {diam}( {\mathbb {G}}_t ) \rightarrow \infty \) as t →.
Eric Rosenberg

Chapter 7. Volume and Surface Dimensions for Infinite Networks

Abstract
In this chapter we study two very early (1988) definitions of the dimension of an infinite network.
Eric Rosenberg

Chapter 8. Information Dimension

Abstract
The information dimension d I of a network extends the concept of the information dimension of a probability distribution.
Eric Rosenberg

Chapter 9. Generalized Dimensions

Abstract
A multifractal is a fractal that cannot be characterized by a single fractal dimension such as the box counting dimension.
Eric Rosenberg

Chapter 10. Non-monotonicity of Generalized Dimensions

Abstract
In Chap. 9 , we showed that the value of D q for a given q depends in general on which minimal s-covering is selected, and we showed that this ambiguity can be eliminated by using the unique lexico minimal summary vectors x(s).
Eric Rosenberg

Chapter 11. Zeta Dimension

Abstract
In this final chapter we consider the use of the zeta function
$$\displaystyle {\zeta }(\alpha ) = \sum _{i=1}^{\infty } i^{-\alpha } $$
to define the dimension of a network.
Eric Rosenberg

Backmatter

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