Network Science: Connectivity Algorithm

1. Reachability Matrix

1.2 Powers of tadjacency matrix

Powers of the graph adjacency matrix: Computing $#\text{walks} between two nodes

• \(A_{uv} = 1\) if \(u \in \mathcal{N}(v)\)

• Let \(P_{uv}^{(K)} = \# \text{walks}\) of length \(K\) between \(u\) and \(v\).
  \(P_{uv}^{(K)}\) 表示点 \(u\) 到点 \(v\) 经过 \(K\) 步达到的路径（方案）个数，参考 Floyd 算法。

\[P_{uv}^{(2)} = A_{uv}^{(2)} = \sum_{j} A_{uj} A_{jk} \qquad \text{where } A_{uj} A_{jk} = 1 \text{ represents } u \to j \to k \text{ is a feasible path} \]

• \(\mathbf{P}^{(K)} = \mathbf{A}^k\)

Compute \(P_{uv}^{(2)}\):

• Step 1 : Compute \(\# \text{walk}\) of length 1 between each of \(u\)'s neighbor and \(v\)

• Step 2 : Sum up these \(\# \text{walks}\) across \(u\)'s neighbors

  • \(A^{(P)}_{uv} = \sum \limits_i {A}_{ui} \cdot P_{iv}^{(1)} = \sum \limits_i {A}_{ui} \cdot {A}_{iv} = {A}^2_{u v}\)

Compute \(\#\text{walk}\) between two nodes: use adjacency matrix powers

• \(A_{uv}\) specifies \(\#\text{walk}\) of length \(1\) (direct neighborhood) between \(u\) and \(v\).

• \(A_{uv}^{2}\) specifies \(\#\text{walk}\) of length \(2\) (neighbor of neighbor) between \(u\) and \(v\)

• \(A_{uv}^{l}\) specifies \(\#\text{walk}\) of length \(l\)

1.2 Reachability Matrix

Theorem (On Boolean Adjacency Matrices and Reachability): If \( \mathbf{A}\) is the Boolean adjacency matrix for a directed graph \(G\) with \(n\) nodes and no parallel arcs, then \(\mathbf{A}^{(m)}[i,j]=1\) if and only if there is a path of length \(m\) from node \(n_i\) to node \(n_j\)

We can define a reachability matrix \(\mathbf{R}\) by:

\[\mathbf{R} = \mathbf{A} \or \mathbf{A}^{(2)} \or \mathbf{A}^{(3)} \or \cdots \or \mathbf{A}^{(N)} \]

where Then \(n_j\) is reachable from \(n_i\) is and only if entry \(i,j\) in \(\mathbf{R}\) is positive.

2. Components

References:

Binary Relation, Graph Algorithms, site