An a posteriori measure of network modularity

Measuring the modularity of networks, and how it deviates from random expectations, important to understand their structure and emerging properties. Several measures exist to assess modularity, which when applied to the same network, can return both different modularity values (i.e. different estimates of how modular the network is) and different module compositions (i.e. different groups of species forming said modules). More importantly, as each optimization method uses a different optimization criterion, there is a need to have an a posteriori measure serving as an equivalent of a goodness-of-fit. In this article, I propose such a measure of modularity, which is simply defined as the ratio of interactions established between members of the same modules vs. members of different modules. I apply this measure to a large dataset of 290 ecological networks representing host-parasite (bipartite) and predator-prey (unipartite) interactions, to show how the results are easy to interpret and present especially to a broad audience not familiar with modularity analyses, but still can reveal new features about modularity and the ways to measure it.