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Possible simple extensions?

Posted by MHumphries on 01 Jul 2008 at 15:25 GMT

An intriguing paper - I particularly commend the effort to make the network analysis useful, by testing the use of the normalised clustering coefficient γ (= C/C_random) as a biomarker. I have a quick suggestion that the authors can apply to their existing data-sets and may throw up something interesting....

Though not stated explicitly, the authors have (I believe) used the Watts-Strogatz form of the clustering coefficient, namely

C_ws = 1/n * sum(C_i),

which is the average taken over all n individual node’s clustering coefficients, each given by

C_i = E / [0.5*k(k-1)]

where E is the number of edges that actually exist between the node’s immediate neighbours, and the denominator is the maximum number of possible edges (where k is the node’s degree).

The problem with this form is that it can obscure some global statistical properties of the networks, and may indeed have partially obscured the very property (abnormal functional organisation of the brain) the authors were looking for. I suggest they may want to also test the alternative form of clustering coefficient in wide use:

C_tri = [3 x number of triangles in network] / [number of paths of length 2]

which gives a global measure of “clustering”, but is also implicitly a measure of the fraction of completed 3-node motifs (triangles) in an undirected network. A simple way to compute this is given by Keeling (1999, Proc Roy Soc B. 266, 859-867)

C_tri = trace(G^3) / [||G^2|| - trace(G^2)]

where G is the network’s matrix (i.e. the adjacency matrix) and ||.|| means a sum over all elements in the matrix.

Certainly the values given by C_ws and C_tri are often in good agreement, but can be an order of magnitude different for some networks (I – shameless plug alert - give some examples of this in Humphries, M. D. & Gurney, K. Network 'small-world-ness': A quantitative method for determining canonical network equivalence PloS One, 2008, 3, e0002051). You may also want to look at using small-world-ness based on C_tri as well, of course...

Why use C_tri? Because, as it measures the fraction of 3-node motifs, it is often a good indicator of the behaviour of the network as a dynamic system: for example, see
Prill et al (2005, PLoS Biol, 3, e343) for the relationship between motif composition and system stability; and the brilliant paper by Zhigulin (2004, Phys Rev Lett, 92, 238701) on the necessary minimal motifs for periodic and chaotic/aperiodic dynamics to appear in the whole network. This is essentially just a quick way of beginning a motif-based analysis of the data!