Generalized Dimensions

Abstract

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

A multifractal is a fractal that cannot be characterized by a single fractal dimension such as the box counting dimension. The infinite number of fractal dimensions needed in general to characterize a multifractal are known as generalized dimensions. Generalized dimensions of geometric multifractals were proposed independently in 1983 by Grassberger [20] and by Hentschel and Procaccia [25]. They have been intensely studied (e.g., [21, 40, 61]) and widely applied (e.g., [39, 59]). Given N points from a geometric multifractal, e.g., the strange attractor of a dynamical system [9, 41], the generalized dimension D _q defined in [20, 25] is computed from a set of box sizes. For box size s, we cover the N points with a grid of boxes of linear size s, compute the fraction of the N points in box of the grid, discard any box for which , and compute the partition function value

(9.1)

where ${\mathcal {B}}(s)$ is the set of non-empty grid boxes, of linear size s, used to cover the N points. For q ≥ 0 and q ≠ 1, the generalized dimension D _q defined in [20, 25] of the geometric multifractal is

$$\displaystyle \begin{aligned} D_q \equiv \frac{1}{q-1} \lim_{s \rightarrow 0} \frac{ \log Z_q \big({\mathcal{B}}(s) \big) } {\log s} \; . {} \end{aligned} $$

(9.2)

When q = 0, this computation yields the box counting dimension $d_{ \stackrel {}{B}}$, so $D_0 = d_{ \stackrel {}{B}}$. When q = 1, after applying L’Hôpital’s rule we obtain the information dimension [13], so . When q = 2, we obtain the correlation dimension [23], so .

Generalized dimensions of a complex network were studied in [15, 34, 48, 49, 58, 67, 68]. Several of these studies employ the sandbox method, which we discuss at the end of this chapter. The method of [67] for computing D _q for $\mathbb {G}$ is the following. For a range of s, compute a minimal s-covering ${\mathcal {B}}(s)$. For , define , where N _j (s) is the number of nodes in . For $q \in {\mathbb {R}}$, use (9.1) to compute $Z_q \big ( {\mathcal {B}}(s) \big )$. (In [67], which uses a randomized box counting heuristic, $Z_q \big ( {\mathcal {B}}(s) \big )$ is the average partition function value, averaged over 200 random orderings of the nodes.) Typically, D _q is computed only for a small set of q values, e.g., integer q in [0, 10] or integer q in [−10, 10]. Then $\mathbb {G}$ has the generalized dimension D _q (for q ≠ 1) if for some constant c and for some range of s we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \log Z_q \big({\mathcal{B}}(s) \big) \approx (q-1){D_q} \log (s/\varDelta) + c \, . {} \end{array} \end{aligned} $$

(9.3)

However, as shown in [48], this definition is ambiguous, since different minimal s-coverings can yield different values of D _q .

FormalPara Example 9.1

Consider again the chair network of Fig. 8.2 , which shows two minimal 3-coverings and a minimal 2-covering. Choosing q = 2, for the covering $\widetilde {{\mathcal {B}}}(3)$ from (9.1) we have $Z_2 \big ( \widetilde {{\mathcal {B}}}(3) \big ) = (\frac {3}{5})^2 + (\frac {2}{5})^2 = \frac {13}{25}$, while for $\widehat {{\mathcal {B}}}(3)$ we have $Z_2 \big ( \widehat {{\mathcal {B}}}(3) \big ) = (\frac {4}{5})^2 + (\frac {1}{5})^2 = \frac {17}{25}$. For ${{\mathcal {B}}}(2)$ we have $Z_2 \big ( {{\mathcal {B}}}(2) \big ) = 2(\frac {2}{5})^2 + (\frac {1}{5})^2 = \frac {9}{25}$. If we use $\widetilde {{\mathcal {B}}}(3)$ then from (9.3) and the range s ∈ [2, 3] we obtain

$$\displaystyle \begin{aligned} D_2 = \left( \log \frac{13}{25} - \log \frac{9}{25} \right)/ (\log 3 - \log 2) \approx 0.907 \, . \end{aligned}$$

If instead we use $\widehat {{\mathcal {B}}}(3)$ and the same range of s we obtain

$$\displaystyle \begin{aligned} D_2 = \left( \log \frac{17}{25} - \log \frac{9}{25} \right)/ (\log 3 - \log 2) \approx 1.569 \, . \end{aligned}$$

Thus the method of [67] can yield different values of D ₂ depending on the minimal covering selected. □

To devise a computationally efficient method for selecting a unique minimal covering, first consider the maximal entropy criterion described in Chap. 8 . It is well known that entropy is maximized when all the probabilities are equal. A partition function is minimized when the probabilities are equal. To formalize this idea, for integer J ≥ 2, let P ( q ) denote the continuous optimization problem: subject to and for each j. It is proved in [48] that for q > 1, the solution of P ( q ) is for each j. Applying this result to $\mathbb {G}$, minimizing $Z_q\big ({\mathcal {B}}(s)\big )$ over all minimal s-coverings of $\mathbb {G}$ yields a minimal s-covering for which all the probabilities are, to the extent possible, equalized. Since , equal box probabilities means that all boxes in the minimal s-covering have the same number of nodes. The following definition [48] of an (s, q) minimal covering, for use in computing D _q , is analogous to the definition in [47] of a maximal entropy minimal s-covering, for use in computing .

FormalPara Definition 9.1

For $q \in {\mathbb {R}}$, the covering ${\mathcal {B}}(s)$ of $\mathbb {G}$ is an (s, q) minimal covering if (i) ${\mathcal {B}}(s)$ is a minimal s-covering and (ii) for any other minimal s-covering $\widetilde { {\mathcal {B}}}(s)$ we have $Z_q \big ({\mathcal {B}}(s) \big ) \leq Z_q \big ( \widetilde { {\mathcal {B}}}(s) \big )$. □

It is easy to modify any box counting method (in a manner analogous to Procedure 8.1 ) to compute an (s, q) minimal covering for a given s and q. However, this approach to eliminating ambiguity in the computation of a minimal s-covering is not particularly attractive, since it requires computing an (s, q) minimal covering for each value of q for which we wish to compute D _q . A better approach to resolving this ambiguity is to compute a lexico minimal summary vector [48], which summarizes an s-covering ${\mathcal {B}}(s)$ by the point $x \in {\mathbb {R}}^J$, where J ≡ B(s), where $x_{ \stackrel {}{j}} = N_j(s)$ for 1 ≤ j ≤ J, and where . (We use lexico instead of the longer lexicographically.) The vector x does not specify all the information in ${\mathcal {B}}(s)$; in particular, ${\mathcal {B}}(s)$ specifies exactly which nodes belong to each box, while x specifies only the number of nodes in each box. The notation $x = \sum {\mathcal {B}}(s)$ signifies that x summarizes the s-covering ${\mathcal {B}}(s)$ and that . For example, if N = 37, s = 3, and B(3) = 5, we might have $x = \sum {\mathcal {B}}(3)$ for x = (18, 7, 5, 5, 2). However, we cannot have $x = \sum {\mathcal {B}}(3)$ for x = (7, 18, 5, 5, 2) since the components of x are not ordered correctly. If $x = \sum {\mathcal {B}}(s)$ then each $x_{ \stackrel {}{j}}$ is positive, since $x_{ \stackrel {}{j}}$ is the number of nodes in box . The vector $x = \sum {\mathcal {B}}(s)$ a called a summary of ${\mathcal {B}}(s)$. By “x is a summary” we mean x is a summary of ${\mathcal {B}}(s)$ for some ${\mathcal {B}}(s)$. For $x(s) = \sum {\mathcal {B}}(s)$ and $q \in {\mathbb {R}}$, define

(9.4)

Thus for $x(s) = \sum {\mathcal {B}}(s)$ we have $Z\big (x(s),q\big ) = Z_q \big ( {\mathcal {B}}(s) \big )$, where $Z_q \big ( {\mathcal {B}}(s) \big )$ is defined by (9.1).

Let $x \in {\mathbb {R}}^K$ for some positive integer K. Let $right(x) \in {\mathbb {R}}^{K-1}$ be the point obtained by deleting the first component of x. For example, if x = (18, 7, 5, 5, 2) then right(x) = (7, 5, 5, 2). Similarly, we define right ²(x) ≡ right(right(x)), so right ²(7, 7, 5, 2) = (5, 2). Let $u \in {\mathbb {R}}$ and $v \in {\mathbb {R}}$ be numbers. We say that u ≽ v (in words, u is lexico greater than or equal to v) if ordinary inequality holds, that is, u ≽ v if u ≥ v. Thus 6 ≽ 3 and 3 ≽ 3. Now let $x \in {\mathbb {R}}^K$ and $y \in {\mathbb {R}}^K$. We define lexico inequality recursively: we say that y ≽ x if either (i) or (ii) and right(y) ≽ right(x). For example, for x = (9, 6, 5, 5, 2), y = (9, 6, 4, 6, 2), and z = (8, 7, 5, 5, 2), we have x ≽ y and x ≽ z and y ≽ z.

FormalPara Definition 9.2

Let $x = \sum {\mathcal {B}}(s)$. Then x is lexico minimal if (i) ${\mathcal {B}}(s)$ is a minimal s-covering and (ii) if $\widetilde { {\mathcal {B}}}(s)$ is a minimal s-covering distinct from ${\mathcal {B}}(s)$ and $y = \sum \widetilde { {\mathcal {B}}}(s)$ then y ≽ x. □

The following two theorems are proved in [48].

FormalPara Theorem 9.1

For each s there is a unique lexico minimal summary.

FormalPara Theorem 9.2

Let $x = \sum {\mathcal {B}}(s)$ . If x is lexico minimal then ${\mathcal {B}}(s)$ is (s, q) minimal for all sufficiently large q.

Analogous to Procedure 8.1 , Procedure 9.1 below shows how, for a given s, the lexico minimal x(s) can be computed by a simple modification of whatever box counting method is used to compute a minimal s-covering.

FormalPara Procedure 9.1

Let ${\mathcal {B}}_{\min }(s)$ be the best s-covering obtained over all executions of whatever box counting method is utilized. Suppose we have executed box counting some number of times, and stored ${\mathcal {B}}_{\min }(s)$ and $x_{\min }(s) = \sum {\mathcal {B}}_{\min }(s)$, so $x_{\min }(s)$ is the current best estimate of a lexico minimal summary vector. Now suppose we execute box counting again, and generate a new s-covering ${\mathcal {B}}(s)$ using B(s) boxes. Let $x = \sum {\mathcal {B}}(s)$. If ${B}(s) < B_{\min }(s)$, or if ${B}(s) = B_{\min }(s)$ and $x_{\min }(s) \succeq x$, then set ${\mathcal {B}}_{\min }(s) = {{\mathcal {B}}}(s)$ and $x_{\min }(s) = x$. □

Procedure 9.1 shows that the only additional steps, beyond the box counting method itself, needed to compute x(s) are lexicographic comparisons, and no evaluations of the partition function $Z_q \big ({\mathcal {B}}(s) \big )$ are required. By Theorems 9.1 and 9.2, the summary vector x(s) is unique and also “optimal” (i.e., (s, q) minimal) for all sufficiently large q. Thus an attractive way to resolve ambiguity in the choice of minimal s-coverings is to compute x(s) for a range of s and use the x(s) vectors to compute D _q , using Definition 9.3 below.

FormalPara Definition 9.3

For q ≠ 1, the complex network $\mathbb {G}$ has the generalized dimension D _q if for some constant c and for some range of s we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \log Z\big(x(s),q \big) \approx (q-1) {D_q} \hspace{0.02em} \log (s/\varDelta) + c \, , {} \end{array} \end{aligned} $$

(9.5)

where $x(s) = \sum {\mathcal {B}}(s)$ is lexico minimal. □

FormalPara Example 9.2 (Continued)

Consider again the chair network of Fig. 8.2 . Choose q = 2. For s = 2 we have $x(2) = \sum {{\mathcal {B}}}(2) = (2, 2, 1)$ and $Z \big ( x(2), 2 \big ) = \frac {9}{25}$. For s = 3 we have $\widetilde {x}(3) = \sum \widetilde {{\mathcal {B}}}(3) = (3,2)$ and $Z \big ( \widetilde {x}(3), 2 \big ) = \frac {13}{25}$. Over the range s ∈ [2, 3], from Definition 9.3 we have $D_2 = \log (13/9)/\log (3/2) \approx 0.907$. For this network, not only is the value of D _q dependent on the minimal s-covering selected, but even the overall shape of the D _q vs. q curve depends on the minimal s-covering selection. For x(2) = (2, 2, 1) we have

$$\displaystyle \begin{aligned} Z\big(x(2), q \big) = 2 \left( \frac{2}{5} \right)^q + \left (\frac{1}{5} \right)^q \, . \end{aligned}$$

For $\widetilde {x}(3) = (3, 2)$ we have

$$\displaystyle \begin{aligned} Z\big( \widetilde{x}(3), q \big) = \left( \frac{3}{5} \right)^q + \left( \frac{2}{5} \right)^q \, . \end{aligned}$$

Over the range s ∈ [2, 3], from (9.5) we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widetilde{D}_q \equiv \left( \frac{1}{q-1} \right) \left( \frac { \log \left( \frac{3^q + 2^q}{5^q} \right) - \log \left( \frac{(2)(2^q)+1}{5^q} \right) } { \log (3/\varDelta) - \log (2/\varDelta) } \right) = \frac { \log \left( \frac{3^q + 2^q}{(2)(2^q)+1} \right) } {\log(3/2) (q-1) } \, . \end{array} \end{aligned} $$

(9.6)

If for s = 3 we instead choose the covering $\widehat {{\mathcal {B}}}(3)$ then for $\widehat {x}(3) = (4, 1)$ we have

$$\displaystyle \begin{aligned} Z\big( \widehat{x}(3), q \big) = \left( \frac{4}{5} \right)^q + \left( \frac{1}{5} \right)^q \, .\end{aligned} $$

Again over the range s ∈ [2, 3], but now using $\widehat {x}(3)$ instead of $\widetilde {x}(3)$, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{D}_q \equiv \left( \frac{1}{q-1} \right) \left( \frac { \log \left( \frac{4^q + 1^q}{5^q} \right) - \log \left( \frac{(2)(2^q)+1}{5^q} \right) } { \log (3/\varDelta) - \log (2/\varDelta) } \right) = \frac { \log \left( \frac{4^q + 1}{(2)(2^q)+1} \right) } {\log(3/2) (q-1) } \, .\vspace{-3pt} \end{array} \end{aligned} $$

(9.7)

Figure 9.1 plots $\widetilde {D}_q$ vs. q, and $\widehat {D}_q$ vs. q over the range 0 ≤ q ≤ 15. Neither curve is monotone non-increasing: the $\widetilde {D}_q$ curve (corresponding to the lexico minimal summary vector $\widetilde {x}(3) = (3,2)$) is unimodal, with a local minimum at q ≈ 4.1, and the $\widehat {D}_q$ curve is monotone increasing. □

Fig. 9.1

Two plots of the generalized dimensions for the chair network

The fact that neither curve in Fig. 9.1 is monotone non-increasing is remarkable, since it is well known that for a geometric multifractal, the D _q vs. q curve is monotone non-increasing [20]. The shape of the D _q vs. q curve will be explored further in Chap. 10 . We next show that the x(s) summary vectors can be used to compute D _∞ ≡lim_q→∞ D _q . Let $x(s) = \sum {\mathcal {B}}(s)$ be lexico minimal, and let be the first element of x(s). It is proved in [48] that

(9.8)

We can use (9.8) to compute D _∞ without having to compute any partition function values. It is well known [41] that, for geometric multifractals, D _∞ corresponds to the densest part of the fractal. Similarly, (9.8) shows that, for a complex network, D _∞ is the factor that relates the box size s to , the number of nodes in the box in the lexico minimal s-covering for which is maximal.

To conclude this chapter, we consider the sandbox method for approximating D _q . The sandbox method, originally designed to compute D _q for geometric multifractals obtained by simulating diffusion-limited aggregation on a lattice [64, 65, 71], overcomes a well-recognized [1] limitation of using box counting to compute generalized dimensions: spurious results can be obtained for q ≪ 0. This will happen if some box probability is close to zero, for then when q ≪ 0 the term will dominate the partition sum . The sandbox method has also been shown to be more accurate than box counting for geometric fractals with known theoretical dimensions [62]. To describe the sandbox method, note that for a geometric multifractal for which D _q exists, by (9.1) and (9.2) we have, as s → 0,

The sandbox method approximates as follows [62]. Let ${\widetilde {\mathbb {N}}}$ be a randomly chosen subset of the N points and define ${\widetilde {N}} \equiv \vert {\widetilde {\mathbb {N}}} \vert$. With M(n, r) defined by (7.1 ) and (7.2 ), define

$$\displaystyle \begin{aligned} \textit{avg}\big( p^{q-1}(r) \big) \equiv \frac{1}{{\widetilde{N}} } \sum_{n \in {\widetilde{\mathbb{N}}}} \left( \frac{M(n,r)}{N} \right)^{q-1} \, , {} \end{aligned} $$

(9.9)

where the notation avg(p ^q−1(r)) is chosen to make it clear that this average uses equal weights of $1/{\widetilde {N}}$. Let L be the linear size of the lattice. The essence of the sandbox method is the approximation, for r ≪ L,

(9.10)

Note that is a sum over the set of non-empty grid boxes, and the weight applied to is . In contrast, avg(p ^q−1(r)) is a sum over a randomly selected set of sandpiles, and the weight applied to (M(n, r)∕N)^q−1 is $1/{\widetilde {N}}$. Since the $\widetilde {N}$ sandpile centers are chosen from the N points using a uniform distribution, the sandpiles may overlap. Because the sandpiles may overlap, and the sandpiles do not necessarily cover all the N points, in general $\sum _{n \in {\widetilde {\mathbb {N}}}} M(n,r) \neq N$, and we cannot regard the values $\{ M(n,r)/N \}_{n \in {\widetilde {\mathbb {N}}}}$ as a probability distribution. Let β be the spacing between adjacent lattice positions (e.g., between adjacent horizontal and vertical positions for a lattice in ${\mathbb {R}}^2$).

FormalPara Definition 9.4

For q ≠ 1, the sandbox dimension function [62] of order q is the function of r defined for β ≤ r ≪ L by

$$\displaystyle \begin{aligned} {D_{\stackrel{}{q}}^{\, {\textit{sandbox}}}}(r/L) \equiv \frac{1}{q-1} \frac{ \log \textit{avg}\big( p^{q-1}(r) \big) } {\log (r/L) } \; . {} \end{aligned} $$

(9.11)

□

For a given q ≠ 1 and lattice size L, the sandbox dimension function does not define a single sandbox dimension, but rather a range of sandbox dimensions, depending on r. It is not meaningful to define $\lim _{r \rightarrow 0} {D_{\stackrel {}{q}}^{\, {\textit {sandbox}}}}(r/L)$, since r cannot be smaller than the spacing β between lattice points. In practice, for a given q and L, a single value $D_{\stackrel {}{q}}^{\, {\textit {sandbox}}}$ of the sandbox dimension of order q is typically obtained by computing $D_{\stackrel {}{q}}^{\, {\textit {sandbox}}}(r/L)$ for a range of r values, and finding the slope of the $\log \textit {avg}\big ( p^{q-1}(r) \big )$ vs. $\log (r/L)$ curve. The estimate of $D_{\stackrel {}{q}}^{\, {\textit {sandbox}}}$ is 1∕(q − 1) times this slope.

The sandbox method was applied to complex networks in [34]. The box centers are randomly selected nodes. There is no firm rule in [34] on the number of random centers to pick: they use $\widetilde {N} \equiv \vert {\widetilde {\mathbb {N}}} \vert = 1000$ random nodes, but suggest that $\widetilde {N}$ can depend on N. For a given q ≠ 1, they compute avg(p ^q−1(r)) for a range of r values. Adapting (9.10) to a complex network $\mathbb {G}$, for r ≪ Δ we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \log {\textit{avg}\big( p^{q-1}(r) \big)} \sim (q-1) {D_{\stackrel{}{q}}^{\, {\textit{sandbox}}}} \log(r/\varDelta) \, . {} \end{array} \end{aligned} $$

(9.12)

In [34], linear regression is applied to (9.12) to compute $D_{\stackrel {}{q}}^{\, {\textit {sandbox}}}$.

The sandbox method was applied to undirected weighted networks in [58]. The calculation of the sandbox radii in [58] is similar to the selection of box sizes discussed in Sect. 3.2 .