Module identification methods

Complex systems are thought to have a hierarchical modular organization, meaning modules have in themselves a modular structure and can therefore be decomposed or partitioned in subsequent sub-modules. In particular, studies have revealed that brain functional networks have a hierarchical modular organization [Felleman and Van Essen, 1991, Meunier et al., 2009, Modha and Singh, 2010, Ferrarini et al., 2011]. This finding is based on the idea that cortical connectivity has two main roles in neural processing: there are strong intracortical connections between neurones with small physical separation, on the order of a few hundred micrometers, and there is another set of connections with larger physical separation linking different cortical areas [Nicoll and Blakemore, 1993, Varela et al., 2001, Sporns and Zwi, 2004]. Such

modular structure of segregated interconnected groups of nodes has been observed in networks of functional connectivity both for non-human [Hilgetag et al., 2000, Modha and Singh, 2010] and human data [Meunier et al., 2009, He et al., 2009, Ferrarini et al., 2011]. Hierarchical structure can give information on the developmental stage of the brain, for example, in their 2009 fMRI paper Supekar et al. found that children have significantly less hierarchical structure than young adults [Supekar et al., 2009]. Brain networks of patients with schizophrenia are also believed to show a loss in hierarchical organization [Bullmore and Sporns, 2009].

Each algorithm to identify the modular structure of the network will produce a particular partition of the network. A quantitative measure Q called modularity was first introduced by Newman and Girvan to asses the strength of the community structure in a given network [Girvan and Newman, 2001]. Modularity is defined originally for an unweighted, undirected network as

Q= 1 2m X B∈D X i,j∈B aij − k_ik_j 2m

where a_ij are the entries of the adjacency matrix, k_i is the degree of node i, m is the total number of edges, and B is an index that runs over all the modules found by a given partition D [Girvan and Newman, 2001, Meunier et al., 2009]. There are many methods to find the modular structure of networks, several of which are based on optimizing Q. We present the most relevant for our research along with their ad- vantages and limitations.

The Louvain method is a two phase method that leads to a hierarchical decom- position of the network [Blondel et al., 2008]. The first phase consists of a greedy optimization in which communities of nodes are found by locally optimizing mod- ularity. The second phase consists of building a new network whose nodes are the previously found communities. The output of the algorithm is a series of partitions, a partition for each step. The Louvain method has been proven to be exception- ally efficient in terms of computation time, outperforming other partition methods.

Such low computation times allow the method to analyze networks of considerable size [Blondel et al., 2008]. However, since the method is based on greedy optimiza- tion it cannot ensure that a global maximum of modularity has been obtained. It has also been shown that although it identifies the highest levels of modularity with great precision, the resolution of the method needs to be changed in order to expose further partitions [Meunier et al., 2009].

The Qcut graph partitioning algorithm was developed by Ruan and Zhang to find communities of highly interconnected nodes [Ruan and Zhang, 2008]. This is another method based on optimizing the network’s modularity. It applies a standard spectral clustering algorithm recursively until a maximum value of Q is found. It has been demonstrated that the overall network clustering accuracy of the algorithm

Qcut is better than that of other methods based on hierarchical clustering [Ruan and Zhang, 2008], however it can neglect possible sub-modular structures. Its accuracy on functional node roles results has not been proven and cannot be ensured so far [Joyce et al., 2010].

A third method is based on the weighted gene coexpression network analysis (WGCNA) approach developed by Zhang and Horvath [Zhang and Horvath, 2005], and it is called weighted voxel coactivation network analysis (WVCNA) [Mumford et al., 2010]. This method relies on the measure of topological overlap [Ravasz et al., 2002] which considers not only the adjacency between two voxels, but also the overlap in the sets of neighbours of the two voxels. The topological overlap is defined by Ravasz et al. as

W_ij = lij +aij

min{k_i, k_j}+ 1−a_ij

where a_ij ∈ [0,1] are the entries of the adjacency matrix A, l_ij = P

uaiuauj, and

k_i = P

uaiu is the node’s degree. If i and j are connected to each other and all of

the neighbours of i are also neighbours of j, then W_ij = 1. If on the contrary, i and

j are not connected and do not have any neighbours in common then W_ij = 0. We note that each term of l_ij represents the strength of the connection between nodes i

and j through another node,u. The total value of lij reflects the degree to which the

neighbourhoods of i and j overlap.

Average linkage hierarchical clustering with the topological overlap dissimilar- ity measure is used to cluster similar nodes in a dendogram whose branches deter- mine the clusters or modules. The clusters in the resulting dendrogram are then identified using the dynamic branch cutting algorithm of Langfelder et al. which identifies modules based not only by absolute height in the dendogram, but also on their shape [Langfelder et al., 2008]. One of the strengths of WGCNA is that it less susceptible to noise than methods based on pairwise correlations since the topologi- cal overlap considers a much larger array of correlations. Several advantages of this method were analyzed by Mumfordet al., the most important being that modularity- based algorithms favour larger modules whereas WVCNA is capable of finding both large and small modules. It was also shown that WGCNA module identification is more reliable than ICA based methods [Mumford et al., 2010].