graph_tool.spectral.adjacency#

graph_tool.spectral.adjacency(g, weight=None, vindex=None, operator=False, csr=True)[source]#

Return the adjacency matrix of the graph.

Parameters:
gGraph

Graph to be used.

weightEdgePropertyMap (optional, default: None)

Edge property map with the edge weights.

vindexVertexPropertyMap (optional, default: None)

Vertex property map specifying the row/column indices. If not provided, the internal vertex index is used.

operatorbool (optional, default: False)

If True, a scipy.sparse.linalg.LinearOperator subclass is returned, instead of a sparse matrix.

csrbool (optional, default: True)

If True, and operator is False, a scipy.sparse.csr_matrix sparse matrix is returned, otherwise a scipy.sparse.coo_matrix is returned instead.

Returns:
Acsr_matrix or AdjacencyOperator

The (sparse) adjacency matrix.

Notes

For undirected graphs, the adjacency matrix is defined as

\[\begin{split}A_{ij} = \begin{cases} 1 & \text{if } (j, i) \in E, \\ 2 & \text{if } i = j \text{ and } (i, i) \in E, \\ 0 & \text{otherwise}, \end{cases}\end{split}\]

where \(E\) is the edge set.

For directed graphs, we have instead simply

\[\begin{split}A_{ij} = \begin{cases} 1 & \text{if } (j, i) \in E, \\ 0 & \text{otherwise}. \end{cases}\end{split}\]

In the case of weighted edges, the entry values are multiplied by the weight of the respective edge.

In the case of networks with parallel edges, the entries in the matrix become simply the edge multiplicities (or twice them for the diagonal, for undirected graphs).

Note

For directed graphs the definition above means that the entry \(A_{ij}\) corresponds to the directed edge \(j\to i\). Although this is a typical definition in network and graph theory literature, many also use the transpose of this matrix.

Note

For many linear algebra computations it is more efficient to pass operator=True. This makes this function return a scipy.sparse.linalg.LinearOperator subclass, which implements matrix-vector and matrix-matrix multiplication, and is sufficient for the sparse linear algebra operations available in the scipy module scipy.sparse.linalg. This avoids copying the whole graph as a sparse matrix, and performs the multiplication operations in parallel (if enabled during compilation).

References

Examples

>>> g = gt.collection.data["polblogs"]
>>> A = gt.adjacency(g, operator=True)
>>> N = g.num_vertices()
>>> ew1 = scipy.sparse.linalg.eigs(A, k=N//2, which="LR", return_eigenvectors=False)
>>> ew2 = scipy.sparse.linalg.eigs(A, k=N-N//2, which="SR", return_eigenvectors=False)
>>> ew = np.concatenate((ew1, ew2))

>>> figure(figsize=(8, 2))
<...>
>>> scatter(real(ew), imag(ew), c=sqrt(abs(ew)), linewidths=0, alpha=0.6)
<...>
>>> xlabel(r"$\operatorname{Re}(\lambda)$")
Text(...)
>>> ylabel(r"$\operatorname{Im}(\lambda)$")
Text(...)
>>> tight_layout()
>>> savefig("adjacency-spectrum.svg")
../_images/adjacency-spectrum.svg

Adjacency matrix spectrum for the political blogs network.#