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The Effects of Structural Heterogeneity on the Scaling of AMFPT. For the BA scale-free network with size N, the AMFPT <<[T.sub.ij]>> of PRW is much greater than that of SRW; see Figure 5(a).
As the actual average searching path length, the AMFPT <<[T.sub.ij]>> of the random walk on the network can be regarded as one generalization of the average shortest path length, which, to some extent, characterizes network searchability [46].
The numerical result presented in Figure 5(a) shows that the AMFPT <<[T.sub.ij]>> satisfies <<[T.sub.ij]>> ~ N for either PRW or SRW on the BA network.
Accordingly, we obtained the closed-form formulas of AMFPT between all node pairs and observed that the average over MFPTs from an arbitrary node to all other target nodes is equal to the AMFPT [see (30) and (45)].
Through the comparison of PRW and SRW in networks, we revealed the CPD-based assortativity of network structure and found that the structural heterogeneity/homogeneity has a considerable impact on the scaling of MFPT and AMFPT. If we consider various random walks as search strategies applied to target problems, the MPFT between source and target characterizes search efficiency.
For random walks on the BA scale-free network, the AMFPT <<[T.sub.ij]>> [marked as x] is equal to the average over MFPTs from one randomly chosen node i to all other N - 1 nodes [marked as [degrees]].
Caption: FIGURE 5: (a) AMFPT <<[T.sub.ij]>> versus network size N for PRW and SRW on the BA scale-free network.
Accordingly, based on the two formulas for MFPT, we get the analytical formulas of the average over MFPTs (AMFPTs) between all node pairs.