OFFPRINT Fractal boundaries of complex networks

We introduce the concept of the boundary of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundary nodes seen from a given node of complex networks. We find that for both Erdős-Rényi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes B follows a power law probability density function which scales as B. The clusters formed by the boundary nodes seen from a given node are fractals with a fractal dimension df ≈ 2. We present analytical and numerical evidences supporting these results for a broad class of networks. Copyright c © EPLA, 2008 Many complex networks are “small world” due to the very small average distance d between two randomly chosen nodes. Often d∼ lnN , where N is the number of nodes [1–6]. Thus, starting from a randomly chosen node following the shortest path, one can reach any other node in a very small number of steps. This phenomenon is called “six degrees of separation” in social networks [4]. That is, for most pairs of randomly chosen people, the shortest “distance” between them is not more than six. Many random network models, such as Erdős-Rényi network (ER) [1], Watts-Strogatz network (WS) [5] and scale-free network (SF) [3,6–8], as well as many real networks, have been shown to possess this small-world property. Much attention has been devoted to the structural properties of networks within the average distance d from a given node. However, almost no attention has been given to nodes which are at distances greater than d from a given node. We define these nodes as the boundaries of the network and study the ensemble of boundaries formed by all possible starting nodes. An interesting question is: how many “friends of friends of friends etc. . . . ” has one at a distance greater than the average distance d? What is their probability distribution and what is the structure of the boundaries? The boundaries have an important (a)E-mail: [email protected] role in several scenarios, such as in the spread of viruses or information in a human social network. If the virus (information) spreads from one node to all its nearest neighbors, and from them to all next nearest neighbors and further on until d, how many nodes do not get the virus (information), and what is their distribution with respect to the origin of the infection? In this letter, we find theoretically and numerically that the nodes at the boundaries, which are of order N , exhibit similar fractal features for many types of networks, including ER and SF models as well as several real networks. Song et al. [9] found that some networks have fractal properties while others do not. Properties of fractal networks were also studied [10,11]. Here we show that almost all model and real networks including non-fractal networks have fractal features at their boundaries. Figure 1 demonstrates our approach and analysis. For each “root” node, we call the nodes at distance l from it “nodes in shell l”. We choose a random root node and count the number of nodes Bl at shell l. We see that B1 = 10, B2 = 11, B3 = 13, etc. . . . We estimate the average distance (diameter) d≈ 2.9 by averaging the distances between all pairs of nodes. After removing nodes with l < d≈ 2.9, the network is fragmented into 12 clusters, with sizes s3={1, 1, 2, 5, 1, 3, 1, 1, 8, 1, 2, 3}. Note that the