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 →∞.

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 →∞. A convenient way to study such networks is to study how the “mass” of ${\mathbb {G}}_t$ scales with $\mathit {diam}( {\mathbb {G}}_t)$, where the “mass” of ${\mathbb {G}}_t$, which we denote by N _t , is the number of nodes in ${\mathbb {G}}_t$. The fractal dimension used in [73] to characterize $\{ {\mathbb {G}}_t \}_{t=1}^{\infty }$ is

$$\displaystyle \begin{aligned} \begin{array}{rcl} {d_{ {M}}} \equiv \lim_{t \rightarrow \infty} \frac {\log N_t} {\log \varDelta_{\hspace{0.07em} {t}}} \, , {} \end{array} \end{aligned} $$

(6.1)

and d _M is called the mass dimension. An advantage of d _M over the correlation dimension is that it is sometimes much simpler to compute the network diameter than to compute C(n, s) for each n and s, as is required to compute C(s) using (5.3 ).

A procedure is presented in [73] that uses a probability p to construct a network that exhibits a transition from fractal to non-fractal behavior as p increases from 0 to 1. For p = 0, the network does not exhibit the small-world property and has d _M  = 2, while for p = 1 the network does exhibit the small-world property and d _M  = ∞. The construction, illustrated by Fig. 6.1, begins with ${\mathbb {G}}_0$, which is a single arc, and p ∈ [0, 1]. Let ${\mathbb {G}}_t$ be the network after t steps. The network ${\mathbb {G}}_{t+1}$ is derived from ${\mathbb {G}}_{t}$. For each arc in ${\mathbb {G}}_{t}$, with probability p we replace the arc with a path of three hops (introducing the two nodes c and d, as illustrated by the top branch of the figure), and with probability 1 − p we replace the arc with a path of four hops (introducing the three new nodes c, d, and e, as illustrated by the bottom branch of the figure). For p = 1, the first three generations of this construction yield the networks of Fig. 6.2. For p = 0, the first three generations of this construction yield the networks of Fig. 6.3. This construction builds upon the construction in [51] of (u, v) trees.

Fig. 6.1

Network that transitions from fractal to non-fractal behavior

Fig. 6.2

Three generations with p = 1

Fig. 6.3

Three generations with p = 0

Let N _t be the expected number of nodes in ${\mathbb {G}}_t$, let A _t be the expected number of arcs in ${\mathbb {G}}_t$, and let Δ _t be the expected diameter of ${\mathbb {G}}_t$. The quantities N _t , A _t , and Δ _t depend on p, but for notational simplicity we omit that dependence. Since each arc is replaced by three arcs with probability p, and by four arcs with probability 1 − p, for t ≥ 1 we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_{t} &\displaystyle =&\displaystyle 3p A_{t-1} + 4 (1-p) A_{t-1} = (4-p)A_{t-1} \\ &\displaystyle = &\displaystyle (4-p)^2 A_{t-2} = \ldots = (4-p)^t A_0 = (4-p)^t \, , {} \end{array} \end{aligned} $$

(6.2)

where the final equality follows as A ₀ = 1, since ${\mathbb {G}}_0$ consists of a single arc.

Let be the number of new nodes created in the generation of ${\mathbb {G}}_t$. Since each existing arc spawns two new nodes with probability p and spawns three new nodes with probability 1 − p, from (6.2) we have

(6.3)

Since ${\mathbb {G}}_0$ has two nodes, for t ≥ 1 we have

(6.4)

Now we compute the diameter Δ _t of ${\mathbb {G}}_t$. We begin with the case p = 1. For this case, distances between existing node pairs are not altered when new nodes are added. At each time step, the network diameter increases by 2. Since Δ ₀ = 1 then Δ _t  = 2t + 1. Since N _t  ∼ (4 − p)^t, then the network diameter grows as the logarithm of the number of nodes, so ${\mathbb {G}}_t$ exhibits the small-world property for p = 1. From (6.1) we have d _M  = ∞.

Now consider the case 0 ≤ p < 1. For this case, the distances between existing nodes are increased. Consider an arc in the network ${\mathbb {G}}_{t-1}$, and the endpoints i and j of this arc. With probability p, the distance between i and j in ${\mathbb {G}}_{t}$ is 1, and with probability 1 − p, the distance between i and j in ${\mathbb {G}}_{t}$ is 2. The expected distance between i and j in ${\mathbb {G}}_{t}$ is therefore p + 2(1 − p) = 2 − p. Since each ${\mathbb {G}}_{t}$ is a tree, for t ≥ 1 we have

$$\displaystyle \begin{aligned} \varDelta_{\hspace{0.07em} {t}} = p \varDelta_{t-1} + 2 (1-p) \varDelta_{t-1} + 2 = (2-p) \varDelta_{t-1} + 2 \end{aligned}$$

and Δ ₀ = 1. This yields [73]

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta_{\hspace{0.07em} {t}} = \left( 1 + \frac{2}{1-p} \right)(2-p)^t - \frac{2}{1-p} \, . {} \end{array} \end{aligned} $$

(6.5)

From (6.1), (6.4), and (6.5),

$$\displaystyle \begin{aligned} \begin{array}{rcl} {d_{ {M}}} = \lim_{t \rightarrow \infty} \frac {\log N_t} {\log \varDelta_{\hspace{0.07em} {t}}} = \lim_{t \rightarrow \infty} \frac {\log [(4-p)^t + 1] } {\log \left[ \left( 1 + \frac{2}{1-p} \right)(2-p)^t - \frac{2}{1-p} \right]} = \frac{ \log (4-p)} {\log (2-p) } \, , \quad {} \end{array} \end{aligned} $$

(6.6)

so d _M is finite, and ${\mathbb {G}}_t$ does not exhibit the small-world property. For p = 0 we have $d_{ {M}} = \log 4/\log 2 = 2$. Note that $\log (4-p)/\log (2-p) \rightarrow \infty$ as p → 1.

6.1 Transfinite Fractal Dimension

A deterministic recursive construction can be used to create a self-similar network, called a (u, v)-flower, where u and v are positive integers [51]. By varying u and v, both fractal and non-fractal networks can be generated. The construction starts at time t = 1 with a cyclic graph (a ring), with w ≡ u + v arcs and w nodes. At time t + 1, replace each arc of the time t network by two parallel paths, one with u arcs, and one with v arcs. Without loss of generality, assume u ≤ v. Figure 6.4 illustrates three generations of a (1, 3)-flower. The t = 1 network has four arcs. To generate the t = 2 network, arc a is replaced by the path {b} with one arc, and also by the path {c, d, e} with three arcs; the other three arcs in Fig. 6.4a are similarly replaced. To generate the t = 3 network, arc d is replaced by the path {p} with one arc, and also by the path {q, r, s} with three arcs; the other fifteen arcs in Fig. 6.4b are similarly replaced. The self-similarity of the (u, v)-flowers follows from an equivalent method of construction: generate the time t + 1 network by making w copies of the time t network, and joining the copies at the hubs.

Fig. 6.4

Three generations of a (1, 3)-flower

Let ${\mathbb {G}}_t$ denote the (u, v)-flower at time t. The number of arcs in ${\mathbb {G}}_t$ is A _t  = w ^t = (u + v)^t. The number N _t of nodes in ${\mathbb {G}}_t$ satisfies the recursion N _t  = wN _t−1 − w; with the boundary condition N ₁ = w we obtain [51]

$$\displaystyle \begin{aligned} \begin{array}{rcl} N_t = \left( \frac{w-2}{w-1} \right) w^{\, t} + \left( \frac{w}{w-1} \right) \, . {} \end{array} \end{aligned} $$

(6.7)

Consider the case u = 1. Let Δ _t be the diameter of ${\mathbb {G}}_t$. It can be shown [51] that for (1, v)-flowers and odd v we have Δ _t  = (v − 1)t + (3 − v)∕2 while in general, for (1, v)-flowers and any v,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta_{\hspace{0.07em} {t}} \sim (v-1)t \, . {} \end{array} \end{aligned} $$

(6.8)

Since N _t  ∼ w ^t then $\varDelta _{\hspace {0.07em} {t}} \sim \log N_t$, so (1, v)-flowers enjoy the small-world property. By (6.1), (6.7), and (6.8), for (1, v)-flowers we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {d_{ {M}}} = \lim_{t \rightarrow \infty} \frac{\log N_t}{\log \varDelta_{\hspace{0.07em} {t}}} = \lim_{t \rightarrow \infty} \frac{\log w^{\, t}}{\log t} = \infty \, , {} \end{array} \end{aligned} $$

(6.9)

so (1, v)-flowers have an infinite mass dimension.

We want to define a new type of fractal dimension that is finite for (1, v)-flowers and for other networks whose mass dimension is infinite. For (1, v)-flowers, from (6.7) we have

$$\displaystyle \begin{aligned} N_t \sim w^{\, t} = (1+v)^t \end{aligned}$$

as t →∞, so $\log N_t \sim t \log (1+v)$. From (6.8) we have Δ _t  ∼ (v − 1)t as t →∞. Since both $\log N_t$ and Δ _t behave like a linear function of t as t →∞, but with different slopes, let be the ratio of the slopes, so

(6.10)

From (6.10), (6.8), and (6.7), as t →∞ we have

(6.11)

from which we obtain

(6.12)

Define α _t  ≡ Δ _t+1 − Δ _t . From (6.12),

(6.13)

Writing N _t  = N(Δ _t ) for some function N(⋅), we have

$$\displaystyle \begin{aligned} N_{t+1} = N(\varDelta_{t+1}) = N(\varDelta_{\hspace{0.07em} {t}} + \alpha_{ \hspace{0.05em} {t}}) \, . \end{aligned}$$

From this and (6.13) we have

(6.14)

which says that, for t ≫ 1, when the diameter increases by α _t , the number of nodes increases by a factor which is exponential in . As observed in [51], in (6.14) there is some arbitrariness in the selection of e as the base of the exponential term , since from (6.10) the numerical value of depends on the logarithm base. If (6.14) holds as t →∞ for a sequence of self similar graphs $\{ {\mathbb {G}}_t \}$ then is called the transfinite fractal dimension, since this dimension “usefully distinguishes between different graphs of infinite dimensionality” [51]. Self-similar networks such as (1, v)-flowers whose mass dimension d _M is infinite, but whose transfinite fractal dimension is finite, are called transfinite fractal networks, or simply transfractals. Thus (1, v)-flowers are transfractals with transfinite fractal dimension .

Finally, consider (u, v)-flowers with u > 1. It can be shown [51] that Δ _t  ∼ u ^t. Using (6.7) we have

$$\displaystyle \begin{aligned} \lim_{t \rightarrow \infty} \frac {\log N_t } {\log \varDelta_{\hspace{0.07em} {t}}} = \lim_{t \rightarrow \infty} \frac {\log w^{\, t} } {\log u^t} = \frac {\log (u+v)} {\log u} \, , \end{aligned}$$

so

$$\displaystyle \begin{aligned} {d_{ {M}}} = \frac{\log (u+v)}{\log u} \, . \end{aligned}$$

Since d _M is finite, these networks are fractals, not transfractals, and these networks do not enjoy the small-world property.