the random walks performed in the expansion phase. After some integer $\beta\ge2$ steps of random walk, gene pairs with strong direct connections and/or strong indirect connections through other genes tend to see more probability flow exchanges, suggesting higher probabilities of belonging to the same gene modules. The expansion operation for the $\beta$-step random walk corresponds to the matrix power operation [$${\mathbf{C}}\rightarrow{\mathbf{C}}^\beta.$$]{}

The MCL algorithm performs the above two operations iteratively until convergence. Non-zero entries in the convergent matrix ${\mathbf{C}}$ connect gene pairs belonging to the same cluster, whereas all inter-cluster edges attain the value zero, so that cluster structure can be obtained directly from this matrix [@Van-Dongen:2001; @Enright:2002].

#### Weighted Gene Co-expression Network Analysis

With higher than average correlation or edge densities within clusters, genes from the same cluster typically share more neighbouring (i.e. correlated) genes. The weighted number of shared neighbouring genes hence can be another measure of gene function similarity. This information is captured in the so-called topological overlap matrix ${\mathbf{\Omega}}$, first defined in [@ravasz2002] for binary networks as [$$\omega_{ij}=\frac{A_{ij}+\sum_u
    A_{iu}A_{uj}}{\mathrm{min}(k_i,k_j)+1-A_{ij}},$$]{} where $A$ is the (binary) adjacency matrix of the network and $k_i=\sum_uA_{iu}$ is the connectivity of vertex $X_i$. The $\sum_uA_{iu}A_{uj}$ term represents vertex similarity through neighbouring genes, 