A multiobjective box-covering algorithm for fractal modularity on complex networks

Abstract

The box-covering method is widely used on measuring the fractal property on complex networks. The problem of finding the minimum number of boxes to tile a network is known as a NP-hard problem. Many algorithms have been proposed to solve this problem. All the current box-covering algorithms regard the box number minimization as the only objective. However, the fractal modularity of the network partition divided by the box-covering method, has been proved to be strongly related to the information transportation in complex networks. Maximizing the fractal modularity is also important in the box-covering method, which can be divided into two objectives: maximization of ratio association and minimization of ratio cut. In this paper, to solve the dilemma of minimizing the box number and maximizing the fractal modularity at the same time, a multiobjective discrete particle swarm optimization box-covering (MOPSOBC) algorithm is proposed. The MOPSOBC algorithm applies the decomposition approach on the two objectives to approximate the Pareto front. The proposed MOPSOBC algorithm has been applied to six benchmark networks and compared with the state-of-the-art algorithms, including two classical box-covering algorithms, four single objective optimization algorithms and six multiobjective optimization algorithms. The experimental results show that the MOPSOBC algorithm can get similar box numbers with the current best algorithm, and it outperforms the state-of-the-art algorithms on the fractal modularity and normalized mutual information.