graph_tool.spectral.transition

graph_tool.spectral.transition(g, weight=None, vindex=None, operator=False, csr=True)

Return the transition 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:

Tcsr_matrix or TransitionOperator

The (sparse) transition matrix.

Notes

The transition matrix is defined as

\[T_{ij} = \frac{A_{ij}}{k_j}\]

where \(k_i = \sum_j A_{ji}\), and \(A_{ij}\) is the adjacency matrix.

In the case of weighted edges, the values of the adjacency matrix are multiplied by the edge weights.

Note

For directed graphs the definition above means that the entry \(T_{ij}\) 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

[wikipedia-transition]

https://en.wikipedia.org/wiki/Stochastic_matrix

Examples

>>> g = gt.collection.data["polblogs"]
>>> T = gt.transition(g, operator=True)
>>> N = g.num_vertices()
>>> ew1 = scipy.sparse.linalg.eigs(T, k=N//2, which="LR", return_eigenvectors=False)
>>> ew2 = scipy.sparse.linalg.eigs(T, 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("transition-spectrum.svg")

Transition matrix spectrum for the political blogs network.