Fractal dimension on networks

Self-similarity of complex networks

Many real networks have two fundamental properties, scale-free property and small-world property. If the degree distribution of the network follows a power-law, the network is scale-free; if any two arbitrary nodes in a network can be connected in a very small number of steps, the network is said to be small-world.

The small-world properties can be mathematically expressed by the slow increase of the average diameter of the network, with the total number of nodes $N$,

$\left\langle l\right\rangle\sim\ln{N}$

where $l$ is the shortest distance between two nodes.

Equivalently, we obtain:

$N\sim e^{\left\langle l\right\rangle/l_0}$

where $l_0$ is a characteristic length.

For a self-similar structure, a power-law relation is expected rather than the exponential relation above. From this fact, it would seem that the small-world networks are not self-similar under a length-scale transformation.

However, analysis of a variety of real complex networks shows they are self-similar on all length scales, a conclusion derived from measuring a power-law relation between the number of boxes needed to cover the network and the size of the box, so called fractal scaling.

The methods for calculation of the dimension

Generally we calculate the fractal dimension using either the Box Counting Method or the Cluster Growing Method.

Fig.(1) Box Counting Method.

Fig.(2) Cluster Growing Method.

The Box Counting Method

Let $N_B$ be the number of boxes of linear size $l_B$, needed to cover the given network. The fractal dimension $d_B$ is then given by

$N_B\sim l_B^{-d_B}$

This means that the average number of vertices $\left\langle M_B\left(l_B\right)\right\rangle$ within a box of size $l_B$

$\left\langle M_B\left(l_B\right)\right\rangle \sim l_B^{d_B}$

By measuring the distribution of $N$ for different box sizes or by measuring the distribution of $\left\langle M_B\left(l_B\right)\right\rangle$ for different box sizes, the fractal dimension $d_B$ can be obtained by a power law fit of the distribution.

The Cluster Growing Method

One seed node is chosen randomly. If the minimum distance $l$ is given, a cluster of nodes separated by at most $l$ from the seed node can be formed. The procedure is repeated by choosing many seeds until the clusters cover the whole network. Then the dimension $d_f$ can be calculated by

$\left\langle M_C\right\rangle \sim l^{d_f}$

where $\left\langle M_C\right\rangle$ is the average mass of the clusters, defined as the average number of nodes in a cluster.

These methods are difficult to apply to networks since networks are generally not embedded in another space. In order to measure the fractal dimension of networks we add the concept of renormalization.

Fractal scaling in scale-free networks

Box-counting and Renormalization

Fig.(3) a, Demonstration of the box-counting and renormalization method for different $l_B$ in a sample network. b, Three stages in the renormalization scheme applied to real network data (WWW).

To investigate self-similarity in networks, we use the box-counting method and renormalization. Fig.(3a) shows this procedure using a network composed of 8 nodes.

For each size l_B, boxes are chosen randomly (as in the cluster growing method) until the network is covered, A box consists of nodes separated by a distance l < l_B. Then each box is replaced by a node(renormalization). The renormalized nodes are connected if there is at least one link between the unrenormalized boxes. This procedure is repeated until the network collapses to one node. Each of these boxes has an effective mass (the number of nodes in it) which can be used as shown above to measure the fractal dimension of the network. In Fig.(3b), renormalization is applied to a WWW network through three steps for l_B = 3.

Fig.(5) shows the invariance of the degree distribution P(k) under the renormalization performed as a function of the box size on the World Wide Web. The networks are also invariant under multiple renormalizations applied for a fixed box size l_B. This invariance suggests that the networks are self-similar on multiple length scales.

File:Skeleton of a network.jpg

Fig.(4) Skeleton of a Network.

Skeleton and Fractal Scaling

The fractal properties of the network can be seen in its underlying tree structure. In this view, the network consists of the skeleton and the shortcuts. The skeleton is a special type of spanning tree, formed by the edges having the highest betweenness centralities, and the remaining edges in the network are shortcuts. If the original network is scale-free, then its skeleton also follows a power-law degree distribution, where the degree can be different from the degree of the original network. For the fractal networks following fractal scaling, each skeleton shows fractal scaling similar to that of the original network. The number of boxes to cover the skeleton is almost the same as the number needed to cover the network.

Real-world fractal networks

Fig.(5) Invariance of the degree distribution of the WWW under the renormalization for different box sizes

Fig.(6) Fractal scaling analysis of WWW network. Red-the original network, Blue-the skeleton, and Orange-a random spanning tree.

Since fractal networks and their skeletons follow the relation

$\left\langle M_B\left(l_B\right)\right\rangle\sim l_B^{d_B}$,

we can investigate whether a network is fractal and what is the fractal dimension of the network. For example, the WWW, metabolic network, protein interaction network(PIN) of H. sapiens, and PIN of S. cerevisiaeare considered as fractal networks. Furthermore, the fractal dimensions measured are $d_B = 4.1,\mbox{ } 3.4,\mbox{ } 2.0, \mbox{ and } 1.8$ for the networks respectively. On the other hand, the Internet, actor network, and artificial models (for instance, the BA model) do not show the fractal properties.

Other definitions for network dimensions

The best definition of dimension for a complex network or graph depends on the application. For example, metric dimension is defined in terms of the resolving set for a graph. Definitions based on the scaling property of the "mass" as defined above with distance, or based on the complex network zeta function have also been studied.

For networks embedded in real space, one can define a dimension that characterizes the number of nodes that can be reached with an average Euclidean distance.