Robustness of weighted networks with the harmonic closeness against cascading failures

In order to overcome the limitation of existing weighting methods and mitigate the effect of cascading failures, the harmonic closeness is adopted to define the node weight whose strength is controlled by a weight parameter θ, so that the initial load can be obtained by the node weight. We find that regardless of the average degree in Barabási–Albert (BA networks), Newman–Watts (NW networks), and Erdos–Renyi networks (ER networks), the critical threshold Tc achieves the minimum value under optimal θ. In these artificial networks except for NW and ER networks with big tolerance parameter T, the bigger the value of θ, the smaller the value of the normalized avalanche size CFN. Through the comparison of different methods, a key finding is that the value of Tc obtained by the method proposed in this paper is significantly smaller than the ones by methods concerning the degree and the betweenness in artificial and real networks. In the range of big T, our method results in the smallest value of CFN in the networks mentioned above compared with previous methods. These results may be helpful for optimizing the distribution of the initial loads in real-life systems, and extending the research on cascading failures in the light of the harmonic closeness.