Scaling of average weighted shortest path and average trapping time on the weighted extended dendrimer networks

Abstract

In this paper, the weighted extended dendrimer networks are constructed depending on a weight factor r. We derive the exact analytic formulas of the average weighted shortest path (AWSP) and the average trapping time (ATT) of the weighted networks. The expression for AWSP shows that: for 0<r<1, it grows with the logarithm of the network size Ng; for r=1, it grows with increasing size Ng as (lnNg)2; for r>1, it grows as a power-law function of the network size Ng with the exponent, represented by θ(r)=log2r. Then, the trapping problem for weighted-dependent walks taking place on a weighted extended dendrimer network is studied and the trap is fixed at the central node. In the large network, the ATT grows with increasing size Ng as (lnNg)4 for 0<r≤0.5 while ATT grows with the network order Ng as lnNg⋅Ng1+log2r for r>0.5. The obtained result displays that the efficiency of the trapping process depends on the weight factor r: the smaller the value of r is, the more efficient the trapping process is.