The effect of weight heterogeneity on the shortest path of weighted Cayley network

In this paper, we introduce the weighted Cayley network parameterized by three weight parameters r,q,g. The average weighted shortest path (AWSP) is calculated and its expression is derived. The results show that the AWSP of the network forms 10 laws with the range of weights r, q and g, and two laws based on single weight factor are generalized. From these 10 laws, we can see more clearly: When the average value of the weight factors (0<r,q,g<1) of the three branches is fixed, the larger the variance is, the larger the value of AWSP is; When the weight factor of any of the three branches is equal to 12, its AWSP value growth law is the same as that of the weight factor of 0 to 1, there is no special change; When any weight factor of the three branches is equal to 1, the network mainly grows with the logarithm of the network size. Meanwhile, the growth of the network has no boundary, but AWSP maintains a bounded state, and the growth rate of AWSP will be faster when the number of branch weights equal to 1 is more.