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I am aware of the "scale free", "existence of community structure" and"small world" phenomena that commonly occur in real world networks, but what are some other properties associated with these kinds of networks, and what metrics can be used to evaluate them?

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    $\begingroup$ Node degree distribution, node/edge centrality, node/edge betweenness centrality, are some of them. Also, scale-free node degree follows asymptotically a power law distribution $P(k) \sim k^{-\gamma}$ . $\endgroup$ – Digio Dec 31 '17 at 13:28
  • $\begingroup$ I like to think about microscale phenomena, on the level of diads and triads in a network. At the diad level you can think about mutuality of ties: if we condition on A->B does the edge B->A exist with higher or lower probability than if we don't condition? At the triad level, common patterns to look for are transitivity and cyclicality. To measure transitivity you would look at the probability of A->C with and without conditioning on A->B and B->C. Similarly, to measure cyclicality you could look at the probability of C->A with and without conditioning on A->B and B->C. $\endgroup$ – Moormanly Dec 31 '17 at 15:28
  • $\begingroup$ @Digio could you talk more about "node/edge centrality, node/edge betweenness centrality"? How are they different from non-complex networks. Also, I am aware about the power law distribution. That is what I had implied by saying I knew about the "scale free" phenomena. $\endgroup$ – Melsauce Dec 31 '17 at 22:19
  • $\begingroup$ @Moormanly Thanks, that was helpful. That said, it was a mistake on my part that I forgot to mention I was referring to undirected networks, for which these would not hold. Could you point me towards properties of undirected complex networks that you're aware of? $\endgroup$ – Melsauce Dec 31 '17 at 22:28
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One place to look for interesting statistics of networks might be the R package ergm which is used to fit exponential-family random graph models to graphs. In particular, there is a list of optional terms to include in the model starting on page 50 of the linked docs.

Additionally, degree distributions (Pareto-Zipf, Yule, Waring, Poisson, etc) and centrality measures (degree, betweenness, eigenvector, etc) may be of interest.

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