Fractality of Massive Graphs: Scalable Analysis with Sketch-Based Box-Covering Algorithm

Abstract

Analysis and modeling of networked objects are fundamental pieces of modern data mining. Most real-world networks, from biological to social ones, are known to have common structural properties. These properties allow us to model the growth processes of networks and to develop useful algorithms. One remarkable example is the fractality of networks, which suggests the self-similar organization of global network structure. To determine the fractality of a network, we need to solve the so-called box-covering problem, where preceding algorithms are not feasible for large-scale networks. The lack of an efficient algorithm prevents us from investigating the fractal nature of large-scale networks. To overcome this issue, we propose a new box-covering algorithm based on recently emerging sketching techniques. We theoretically show that it works in near-linear time with a guarantee of solution accuracy. In experiments, we have confirmed that the algorithm enables us to study the fractality of million-scale networks for the first time. We have observed that its outputs are sufficiently accurate and that its time and space requirements are orders of magnitude smaller than those of previous algorithms.