Affiliations: Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK. E-mails: {ra.gkal, djh}@maths.strath.ac.uk
Note: [] Corresponding author: Gabriela Kalna, Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK.
Abstract: The clustering coefficient has been used successfully to summarise important features of unweighted, undirected networks across a wide range of applications in complexity science. Recently, a number of authors have extended this concept to the case of networks with non-negatively weighted edges. After reviewing various alternatives, we focus on a definition due to Zhang and Horvath that can be traced back to earlier work of Grindrod. We give a natural and transparent derivation of this clustering coefficient and then analyse its properties. One attraction of this version is that it deals directly with weighted edges and avoids the need to discretise, that is, to round weights up to 1 or down to 0. This has the advantages of (a) retaining all edge weight information, and (b) eliminating the requirement for an arbitrary cutoff level. Further, the extended definition is much less likely to break down due to a ‘divide-by-zero’. Using our new derivation and focusing on some special cases allows us to gain insights into the typical behaviour of this measure. We then illustrate the idea by computing the generalised clustering coefficients, along with the corresponding weighted degrees, for pairwise correlation gene expression data arising from microarray experiments. We find that the weighted clustering and degree distributions reveal global topological differences between normal and tumour networks.
Keywords: Bioinformatics, computational graph theory, microarray data, network topology, range dependent random graph, small world network, Grindrod–Zhang–Horvath clustering coefficient
