Multiplex network and clustering algorithm

The multiplex network represents a multilayer network where the same nodes exist in each layer; the network encodes both the PPI relationships and phenotype relationships between the genes in the network. The gene-PPI-phenotype multiplex network was created by stacking the PPI network and gene-phenotype network layers, generated as detailed in the previous sections, such that each node in one layer is connected to itself in the other layer.

The Louvain algorithm is a modularity maximization approach that is commonly used to detect modules in a network and has been shown to perform well on biological networks68. For the individual PPI and phenotype layers, the Louvain algorithm was used to maximize the modularity, H, defined by:

Here, ec is the total number of edges in community c, m is the total number of edges in the network, and Kc is the sum of the degree of the nodes in community c. To maximize modularity, which the Louvain algorithm is useful for, then is to maximize the difference between the actual number of edges and the expected number of edges in a community. In the equation, γ, is the resolution parameter which controls the size of the communities. The Louvain algorithm was applied 1000 times with different random seeds at a range of resolutions and the partition with the globally optimal modularity was chosen (Fig. S3). The Louvain algorithm can be easily extended to be applicable to a multiplex network. In this case the overall modularity, which the algorithm will try to maximize, is the sum of the modularity of each layer weighted by some constant:

We set both layers to have equal weights in order to have equal contribution from the PPI and phenotype layers. The louvain-igraph Python package was used to run the Louvain algorithm (https://louvain-igraph.readthedocs.io/).

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