What are the statistical features of a network/graph without community structure? I want to filter out networks/graphs which are just a single community rather than interconnected communities. Each community can either be a random graph or a star graph, but there should no community structure. The definition is loose considering that these data sets can have multiple communities with a varying number of edges connecting them together or nodes  acting as bridging islands. (these networks are connected)
I thought of looking for networks where the average path length for at least 10% of the nodes is greater than 2. 
I am using `MATLAB' and any reference to useful libraries would be appreciated. 
 A: There are lots of ways to approach this.  Since you didn't provide many details, I would recommend looking at lots of different notions of what a "community" is and then finding the one that you think suits your needs best.
Here are some ideas and references that might help:
Here's an overview of several methods:
http://en.wikipedia.org/wiki/Community_structure.
Do a spectral clustering on the graph laplacian.
Use the adjacency matrix as a similarity matrix (a counterpart to a distance matrix) and then do a standard clustering method on that similarity matrix.
If you decide to use R, then I'd recommend the package statnet.  I don't use Matlab for network analysis, so I don't have package recommendations for that.
A: The best and easiest to understand statistical measure for when there are communities present in a network is the Q statistic in the method of "Modularity and community structure in networks" Newman 2006. It looks at a community structure's connectivity and compares the probability for such a connection structure to have arisen randomly. Low values indicate a deviation from this random case.
In my datasets this Q value distinguishes quite well between the networks with and without community structure.
