Searching efficiency of multiple walkers on the weighted networks

The weighted networks are realistic forms of networks, where a weight is attached to each link. In order to better study searching efficiency of multiple walkers we extend the study on the binary networks to the weighted networks. In this paper, the main aim is to measure the searching efficiency of multiple walkers on the weighted networks. Firstly, we review the theoretical foundations related to our study. Secondly, we introduce the heterogeneous mean-field (HMF) theory and the annealed network approach on the weighted networks. Then, we deduce the analysis formula of mean first parallel passage time (MFPPT). Finally, we study the global mean first parallel passage time (GMFPPT) and compare it to the searching efficiency of a single walker previously studied. The obtained result shows that the GMFPPT follows a uniform power law with the number of walkers. The key of this paper is to apply the heterogeneous mean-field (HMF) theory and the annealed weighted network approach to replace the weighted uncorrelated networks with the weighted fully connected networks and then construct a general probability transfer matrix on the weighted fully connected networks.