Effect of Closed Paths in Complex networks on Six Degrees of Separation and Disorder

Abstract

Milgram Condition proposed by Aoyama et al. plays an important role on the analysis of "six degrees of separation". We have shown that the relations between Milgram condition and the generalized clustering coefficient, which was introduced as an index for measuring the number of closed paths by us, are absolutely different in scale free networks (Barabasi and Albert) and small world networks (Watts and Strogatz, Watts). This fact implies that the effect of closed paths on information propagation is different in both networks. In this article, we first investigate the difference and pursuit what is a crucial mathematical quantity for information propagation. As a result we find that a sort of "disorder" plays more important role for information propagation than partially closed paths included in a network. Next we inquired into it in more detail by introducing two types of intermediate networks. Then we find that the average of the local clustering coefficient and the generalized clustering coefficients $C_{(q)}$ have some different functions and important meanings, respectively. We also find that $C_{(q)}$ is close to the propagation of information on networks. Lastly, we show that realizability of six degrees of separation in networks can be understood in a unified way by disorder.