Difference between modularity and clustering in graphs

Question

The definition of modularity given on wikipedia is as follows.

http://en.wikipedia.org/wiki/Modularity_(networks)

Modularity is one measure of the structure of networks or graphs. It was designed to measure the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules.

And clustering coefficient.

http://en.wikipedia.org/wiki/Clustering_coefficient

In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971;

I am facing a doubt here. For a given graph, if the value of modularity is high, shouldn't the value of clustering coefficient be high as well? Is it possible that modularity is extremely high and clustering coefficient is very low?

Also, I have certain graphs for which this is happening. Why is it? Modularity is a measure of nodes forming clusters. And clustering coefficient is similar as well. Then, why the difference?

Thanks!

Answer 1

Both concepts are heuristics to measure clustering. One looks at edge densities in given clusters compared to edge densities between clusters, the other looks at density of triangles compared to induced density of 2-paths. Very different concepts, and both may work very well in many applications.

But it is very easy to construct graphs with very high modularity and very low clustering coefficient: Just take a number of complete balanced bipartite graphs with no edges between each other, and make each their own cluster.

No triangles, so clustering coefficient 0. Very dense (1/2) in each cluster, so very high modularity (1/2 or so if I read the definition correctly).

Upshot: if you want to measure sexual clustering in a strictly heterosexual network, better not use the clustering coefficient...