coaffiliation: Community coaffiliation#
- hypercoil.functional.graph.coaffiliation(C: Tensor, C_o: Tensor | None = None, L: Tensor | None = None, exclude_diag: bool = True, normalise_coaffiliation: bool = False) Tensor[source]#
Coaffiliation of vertices under a community structure.
Given community affiliation matrices \(C^{(i)} \in \mathbb{R}^{I \times C}\) for source nodes and \(C^{(o)} \in \mathbb{R}^{O \times C}\) for sink nodes, and given a matrix of inter-community coupling coefficients \(\Omega \in \mathbb{R}^{C \times C}\), the coaffiliation \(H \in \mathbb{R}^{I \times O}\) is computed as
\(H = C^{(i)} \Omega C^{(o)\intercal}\)
- Dimension:
- C_i : \((*, I, C)\)
* denotes any number of preceding dimensions, I denotes number of vertices in the source set, and C denotes the total number of communities in the proposed partition.
- C_o : \((*, I, C)\)
O denotes number of vertices in the sink set. If the same set of vertices emits and receives edges, then \(I = O\).
- L : \((*, C, C)\)
As above.
- Output : \((*, I, O)\)
As above.
- Parameters:
- CTensor
Community affiliation of vertices in the source set. Each slice is a matrix \(C^{(i)} \in \mathbb{R}^{I \ times C}\) that encodes the uncertainty in each vertex’s community assignment. \(C^{(i)}_{jk}\) denotes the probability that vertex j is assigned to community k. If this is binary-valued, then it reflects a deterministic assignment.
- C_oTensor or None (default None)
Community affiliation of vertices in the sink set. If None, then it is assumed that the source and sink sets are the same, and
C_ois set equal toC_i.- LTensor or None (default None)
The inter-community coupling matrix \(\Omega\), mapping the probability of affiliation between communities. Each entry \(L_{ij}\) encodes the probability of a vertex in community i connecting with a vertex in community j. If None, then a strictly assortative structure is assumed (equivalent to L equals identity), under which nodes in the same community preferentially coaffiliate while nodes in different communities remain disaffiliated.
- exclude_diagbool (default True)
Indicates that self-links are not factored into the coaffiliation.
- normalise_coaffiliationbool (default False)
Normalise all community assignment weights to max out at 1.
- Returns:
- CTensor
Coaffiliation matrix for each input community structure.