Bipartivity index
The
bipartivity index
quantifies the extent to which individual nodes contribute to the global bipartivity of a network [2]. Intuitively, in a perfectly bipartite network, there are no closed walks of odd length.
Mathematically, the network bipartivity \(β(G)\) is defined as the proportion of even-length closed walks to the total number of closed walks:
\[
β(G) = \frac{\langle SC \rangle_{\mathrm{even}}}{\langle SC \rangle}
= \frac{\langle SC \rangle_{\mathrm{even}}}{\langle SC \rangle_{\mathrm{even}} + \langle SC \rangle_{\mathrm{odd}}}
= \frac{\sum_{k=1}^N \cosh(λ_k)}{\sum_{k=1}^N e^{λ_k}} \in \left( \frac{1}{2}, 1 \right],
\]
where \(\langle SC \rangle\) denotes the subgraph centralization, which can be decomposed into contributions from even- and odd-length closed walks, and \(λ_k\) is the \(k\)-th eigenvalue of the adjacency matrix \(A\). Note that \(β(G) = 1\) if and only if the network \(G\) is bipartite.
The contribution of an individual node \(i\) to the network bipartivity, denoted \(c_β(i)\), can be computed using the node-level subgraph centrality:
\[
c_β(i) = \frac{\sum_{k=1}^N v_k(i)^2 \cosh(λ_k)}{\sum_{k=1}^N v_k(i)^2 e^{λ_k}},
\]
where \(v_k(i)\) is the \(i\)-th component of the eigenvector \(v_k\) corresponding to eigenvalue \(λ_k\) of \(A\). Thus, bipartivity index captures the extent to which node \(i\) participates in even-length closed walks relative to all closed walks passing through it.