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 $\mathrm{\Omega }\in {\mathbb{R}}^{C\times C}$, the coaffiliation $H\in {\mathbb{R}}^{I\times O}$ is computed as

$H=C^{(i)}\mathrm{\Omega }{C}^{(o)⊺}$

Dimension:

C_i : $(\ast ,I,C)$

* 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 : $(\ast ,I,C)$

O denotes number of vertices in the sink set. If the same set of vertices emits and receives edges, then $I=O$.

L : $(\ast ,C,C)$

As above.

Output : $(\ast ,I,O)$

As above.

Parameters:

CTensor

Community affiliation of vertices in the source set. Each slice is a matrix $C^{(i)}\in {\mathbb{R}}^{ItimesC}$ that encodes the uncertainty in each vertex’s community assignment. $C_{jk}^{(i)}$ 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_o is set equal to C_i.

LTensor or None (default None)

The inter-community coupling matrix $\mathrm{\Omega }$, 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.