Strong clustering and degree distributions

Question

I didn't find an anywhere good explanation of what "typical properties of complex networks such as heterogeneous degree distributions and strong clustering" is?

In "Hyperbolic Geometry of Complex Networks" (2010) there is a correspondence between:

• The hierarchical organization and heterogeneous degree distribution.
• The metric structure and strong clustering.

But does strong clustering mean like some special metric structure?

Thanks in advance for your reply!

Answer 1

I do not think there is any consensus on a precise definition for heterogeneous degrees and strong clustering, but there is a consensus on the following weakly defined criteria:

• heterogeneous degree distribution means that degrees span several orders of magnitude, with a very high variance that makes the average degree of little interest; this distribution if sometimes well (but often poorly) fitted by a power law with an exponent typically around $2$.
• strong clustering is generally measured as the average node clustering coefficient, which is the fraction of a node neighbours linked by an edge, aka the density of the sub-graph induced by their neighbours; if this average is orders of magnitude larger than the one in a comparable random graph, clustering is considered high, or strong; considered comparable graphs are generally the ones with the same number of vertices and edges, obtained with the ER model, or (more rarely) the ones with the same degree sequence, obtained with the CM model.

Hierarchical structure and presence of an underlying metric are interpretations for these empirical observations: various models that rely on such underlying principle lead to graphs with these features, which makes them good candidates for being their causes. Other interpretations do exist, though, like preferential attachment or node similarities, for instance.