Modularity and anti-modularity in networks with arbitrary degree distribution

Background: Much work in systems biology, but also in the analysis of social network and communication and transport infrastructure, involves an in-depth analysis of local and global properties of those networks, and how these properties relate to the function of the network within the integrated system. Most often, systematic controls for such networks are difficult to obtain, because the features of the network under study are thought to be germane to that function. In most such cases, a surrogate network that carries any or all of the features under consideration, while created artificially and in the absence of any selective pressure relating to the function of the network being studied, would be of considerable interest. Results: Here, we present an algorithmic model for growing networks with a broad range of biologically and technologically relevant degree distributions using only a small set of parameters. Specifying network connectivity via an assortativity matrix allows us to grow networks with arbitrary degree distributions and arbitrary modularity. We show that the degree distribution is controlled mainly by the ratio of node to edge addition probabilities, and the probability for node duplication. We compare topological and functional modularity measures, study their dependence on the number and strength of modules, and introduce the concept of anti-modularity: a property of networks in which nodes from one functional group preferentially do not attach to other nodes of that group. We also investigate global properties of networks as a function of the network's growth parameters, such as smallest path length, correlation coefficient, small-world-ness, and the nature of the percolation phase transition. We search the space of networks for those that are most like some well-known biological examples, and analyze the biological significance of the parameters that gave rise to them. Conclusions: Growing networks with specified characters (degree distribution and modularity) provides the opportunity to create surrogates for biological and technological networks, and to test hypotheses about the processes that gave rise to them. We find that many celebrated network properties may be a consequence of the way in which these networks grew, rather than a necessary consequence of how they work or function.