Network decomposition into Modules

It is widely expected that neural systems have a hierarchical, modular structure [7, 12]. Network methods offer one route to decomposing a network into its component modules. Various methods are available to calculate such decompositions [19]: spectral methods [6, 19, 20], such as clustering (eg. via k-means) of the eigenvectors of the graph Laplacian [8,9], or of the modularity matrix [21]; and by the Newman-Girvan agglomerative or divisive methods [22, 23], in which links are successively added or pruned to form sub-networks. One variant is the so-called fast Newman agglomerative algorithm [24, 25], with stability optimisation [26]. All these methods seek to maximise some modularity metric, such as Newman’s Q [20–23], which measures the fraction of within-module links compared to that expected for a random network with the same degree distribution. Thus the random graph, with Q = 0, is taken as a null model that should not have any clustering tendency. Higher Q (eg. Q > 0.4) may indicate good partitions, while small or near zero Q values indicate poor decompositions or a weak tendency to form modules in the network.

Given that a primary purpose of neurons is to transmit signals over the network of connections and to process information, it seems likely that measures related to information or signal flows on the network may lead to insights regarding possible neuron circuits. The InfoMap [27–30] algorithm performs a modular decomposition of the network by mapping the probability flows that the network structure induces, specifically by launching a large number of random walks onto the network of nodes and their weighted links. A monte carlo like sampling of these walks then estimates the fraction of signal flows through each node and link. Modules emerge as regions of the network in which the random walker has proportionally larger residence times. The random walk trajectories are described using signal coding theory [27, 30], specifically a Huffman code. This measures the information cost of describing the random walk trajectory on any modular partition of the network. A search for the most efficient, or minimum code length, yields the modular partition of the network. Such a process can serve as a proxy for information flows on the network which, for a nervous system, is likely to more appropriately characterise network functions over the available structure. Tests of the InfoMap method showed that it produced reliable decompositions of a variety of real-world networks [27]. Thus we adopted the InfoMap method herein. It is available as a set of C⁺⁺ codes [30] that can be compiled locally on a desktop computer.