New quantification of local transition heterogeneity of multiscale complex networks constructed from single-molecule...

A new measure is presented to quantify the local topographical feature, i.e., diversity in transitions from a state to the others, on complex networks. This measure is composed of two contributions: one is related to the number of outgoing links from a state (known as degree) and the other is related to heterogeneity in transition probabilities from a state to the others associated with the links. To illustrate the potential of the new measure, we apply it to the multiscale state space networks (SSNs) extracted directly from the single-molecule time series of protein fluctuation of the NADH:flavin oxidoreductase complex by using a recently developed technique [Li, C. B.; Yang, H.; Komatsuzaki, T. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 536]. We find that the multiscale SSN network structures dependent on the time scale of observation are not differentiated significantly in the topological feature of the SSNs where the connectivity pattern among the nodes is solely taken into account, but instead in the weighted properties of the network including the heterogeneous strengths of transitions and the resident probabilities of the nodes. The relationship of the transition heterogeneity with the anomalous diffusion observed in the single-molecule measurement is also discussed.