Question

l have an adjacency matrix of 16 by 16.

 Adjacency=[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
    [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
    [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

From this adjacency matrix l applied scipy algorithm to determine the connected components as follow :

from scipy.sparse.csgraph import connected_components 
supernodes=connected_components(Adjacency)

which returns 4 components :

 (4, array([0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 0], dtype=int32))

Now the algorithm returns 4 components (4 new nodes or 4 supernodes 0,1,2,3) and its associated adjacency matrix is of dim=(4,4)

My question is as follow :

Given the intial adjacency matrix of 16 by 16 and the connected components, how can l compute efficiently the new adjacency matrix ?

In other way, we need to merge all the nodes that are affected to the same connected component.

EDIT 1 : Here a concrete example. Given the following adjacency matrix of 6 nodes, dim=-6,6) :

Adjacency_matrix=[[0,1,1,0,0,1],
[1,0,0,1,0,0],
[1,0,0,0,1,1],
[0,1,0,0,1,0],
[0,0,1,1,0,0],
[1,0,1,0,0,0]]

Given three supernodes as follow :

supernodes[0]=[0,2]# supernode 0 merges node 0 and 2
supernodes[1]=[1,4]#supernode 1 merges node 1 and 4
supernodes[2]=[3,5]#supernode 2 merges node 3 and 5

The supposed output : Adjacency matrix of 3 supernodes dim=(3,3)

reduced_adjacency_matrix=[[0,1,1],
[1,0,1],
[1,1,0]]

What does it mean ?

For instance, consider the first supernodes[0]=[0,2]. The idea is as follow :

A) if i and j are in the same supernode then adjacency[i,j]=0

B)if i and j are in the same supernode and i or j has connection with other nodes other than i and j set 1

Thank you for your help.

Convert connected components to adjacency matrix

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