# graph_tool.spectral - Spectral properties¶

## Summary¶

 adjacency Return the adjacency matrix of the graph. laplacian Return the Laplacian matrix of the graph. incidence Return the incidence matrix of the graph. transition Return the transition matrix of the graph. modularity_matrix Return the modularity matrix of the graph. hashimoto Return the Hashimoto (or non-backtracking) matrix of a graph.

## Contents¶

graph_tool.spectral.adjacency(g, weight=None, index=None)[source]

Return the adjacency matrix of the graph.

Parameters
gGraph

Graph to be used.

weightEdgePropertyMap (optional, default: True)

Edge property map with the edge weights.

indexVertexPropertyMap (optional, default: None)

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

Returns
Acsr_matrix

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.

References

wikipedia-adjacency

http://en.wikipedia.org/wiki/Adjacency_matrix

Examples

>>> g = gt.collection.data["polblogs"]
>>> A = gt.adjacency(g)
>>> ew, ev = scipy.linalg.eig(A.todense())

>>> 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")


Adjacency matrix spectrum for the political blog network.

graph_tool.spectral.laplacian(g, deg='total', normalized=False, weight=None, index=None)[source]

Return the Laplacian matrix of the graph.

Parameters
gGraph

Graph to be used.

degstr (optional, default: “total”)

Degree to be used, in case of a directed graph.

normalizedbool (optional, default: False)

Whether to compute the normalized Laplacian.

weightEdgePropertyMap (optional, default: True)

Edge property map with the edge weights.

indexVertexPropertyMap (optional, default: None)

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

Returns
lcsr_matrix

The (sparse) Laplacian matrix.

Notes

The weighted Laplacian matrix is defined as

$\begin{split}\ell_{ij} = \begin{cases} \Gamma(v_i) & \text{if } i = j \\ -w_{ij} & \text{if } i \neq j \text{ and } (j, i) \in E \\ 0 & \text{otherwise}. \end{cases}\end{split}$

Where $$\Gamma(v_i)=\sum_j A_{ij}w_{ij}$$ is sum of the weights of the edges incident on vertex $$v_i$$. The normalized version is

$\begin{split}\ell_{ij} = \begin{cases} 1 & \text{ if } i = j \text{ and } \Gamma(v_i) \neq 0 \\ -\frac{w_{ij}}{\sqrt{\Gamma(v_i)\Gamma(v_j)}} & \text{ if } i \neq j \text{ and } (j, i) \in E \\ 0 & \text{otherwise}. \end{cases}\end{split}$

In the case of unweighted edges, it is assumed $$w_{ij} = 1$$.

For directed graphs, it is assumed $$\Gamma(v_i)=\sum_j A_{ij}w_{ij} + \sum_j A_{ji}w_{ji}$$ if deg=="total", $$\Gamma(v_i)=\sum_j A_{ji}w_{ji}$$ if deg=="out" or $$\Gamma(v_i)=\sum_j A_{ij}w_{ij}$$ deg=="in".

Note

For directed graphs the definition above means that the entry $$\ell_{i,j}$$ 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.

References

wikipedia-laplacian

http://en.wikipedia.org/wiki/Laplacian_matrix

Examples

>>> g = gt.collection.data["polblogs"]
>>> L = gt.laplacian(g)
>>> ew, ev = scipy.linalg.eig(L.todense())

>>> 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("laplacian-spectrum.svg")


Laplacian matrix spectrum for the political blog network.

>>> L = gt.laplacian(g, normalized=True)
>>> ew, ev = scipy.linalg.eig(L.todense())

>>> 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("norm-laplacian-spectrum.svg")


Normalized Laplacian matrix spectrum for the political blog network.

graph_tool.spectral.incidence(g, vindex=None, eindex=None)[source]

Return the incidence matrix of the graph.

Parameters
gGraph

Graph to be used.

vindexVertexPropertyMap (optional, default: None)

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

eindexEdgePropertyMap (optional, default: None)

Edge property map specifying the column indexes. If not provided, the internal edge index is used.

Returns
acsr_matrix

The (sparse) incidence matrix.

Notes

For undirected graphs, the incidence matrix is defined as

$\begin{split}b_{i,j} = \begin{cases} 1 & \text{if vertex } v_i \text{and edge } e_j \text{ are incident}, \\ 0 & \text{otherwise} \end{cases}\end{split}$

For directed graphs, the definition is

$\begin{split}b_{i,j} = \begin{cases} 1 & \text{if edge } e_j \text{ enters vertex } v_i, \\ -1 & \text{if edge } e_j \text{ leaves vertex } v_i, \\ 0 & \text{otherwise} \end{cases}\end{split}$

References

wikipedia-incidence

http://en.wikipedia.org/wiki/Incidence_matrix

Examples

>>> g = gt.random_graph(100, lambda: (2,2))
>>> m = gt.incidence(g)
>>> print(m.todense())
[[-1. -1.  0. ...  0.  0.  0.]
[ 0.  0.  0. ...  0.  0.  0.]
[ 1.  0.  0. ...  0.  0.  0.]
...
[ 0.  0. -1. ...  0.  0.  0.]
[ 0.  0.  0. ...  0.  0.  0.]
[ 0.  0.  0. ...  0.  0.  0.]]

graph_tool.spectral.transition(g, weight=None, index=None)[source]

Return the transition matrix of the graph.

Parameters
gGraph

Graph to be used.

weightEdgePropertyMap (optional, default: True)

Edge property map with the edge weights.

indexVertexPropertyMap (optional, default: None)

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

Returns
Tcsr_matrix

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.

References

wikipedia-transition

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

Examples

>>> g = gt.collection.data["polblogs"]
>>> T = gt.transition(g)
>>> ew, ev = scipy.linalg.eig(T.todense())

>>> 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 blog network.

graph_tool.spectral.modularity_matrix(g, weight=None, index=None)[source]

Return the modularity matrix of the graph.

Parameters
gGraph

Graph to be used.

weightEdgePropertyMap (optional, default: True)

Edge property map with the edge weights.

indexVertexPropertyMap (optional, default: None)

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

Returns
BLinearOperator

The (sparse) modularity matrix, represented as a LinearOperator.

Notes

The modularity matrix is defined as

$B_{ij} = A_{ij} - \frac{k^+_i k^-_j}{2E}$

where $$k^+_i = \sum_j A_{ji}$$, $$k^-_i = \sum_j A_{ij}$$, $$2E=\sum_{ij}A_{ij}$$ 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.

References

newman-modularity

M. E. J. Newman, M. Girvan, “Finding and evaluating community structure in networks”, Phys. Rev. E 69, 026113 (2004). DOI: 10.1103/PhysRevE.69.026113 [sci-hub, @tor]

Examples

>>> g = gt.collection.data["polblogs"]
>>> B = gt.modularity_matrix(g)
>>> B = B * np.identity(B.shape[0])  # transform to a dense matrix
>>> ew, ev = scipy.linalg.eig(B)

>>> 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("modularity-spectrum.svg")


Modularity matrix spectrum for the political blog network.

graph_tool.spectral.hashimoto(g, index=None, compact=False)[source]

Return the Hashimoto (or non-backtracking) matrix of a graph.

Parameters
gGraph

Graph to be used.

indexVertexPropertyMap (optional, default: None)

Edge property map specifying the row/column indexes. If not provided, the internal edge index is used.

compactboolean (optional, default: False)

If True, a compact $$2|V|\times 2|V|$$ version of the matrix is returned.

Returns
Hcsr_matrix

The (sparse) Hashimoto matrix.

Notes

The Hashimoto (a.k.a. non-backtracking) matrix [hashimoto] is defined as

$\begin{split}h_{k\to l,i\to j} = \begin{cases} 1 & \text{if } (k,l) \in E, (i,j) \in E, l=i, k\ne j,\\ 0 & \text{otherwise}, \end{cases}\end{split}$

where $$E$$ is the edge set. It is therefore a $$2|E|\times 2|E|$$ asymmetric square matrix (or $$|E|\times |E|$$ for directed graphs), indexed over edge directions.

If the option compact=True is passed, the matrix returned has shape $$2|V|\times 2|V|$$, where $$|V|$$ is the number of vertices, and is given by

$\begin{split}\boldsymbol h = \left(\begin{array}{c|c} \boldsymbol A & -\boldsymbol 1 \\ \hline \boldsymbol D-\boldsymbol 1 & \boldsymbol 0 \end{array}\right)\end{split}$

where $$\boldsymbol A$$ is the adjacency matrix, and $$\boldsymbol D$$ is the diagonal matrix with the node degrees [krzakala_spectral].

References

hashimoto

Hashimoto, Ki-ichiro. “Zeta functions of finite graphs and representations of p-adic groups.” Automorphic forms and geometry of arithmetic varieties. 1989. 211-280. DOI: 10.1016/B978-0-12-330580-0.50015-X [sci-hub, @tor]

krzakala_spectral

Florent Krzakala, Cristopher Moore, Elchanan Mossel, Joe Neeman, Allan Sly, Lenka Zdeborová, and Pan Zhang, “Spectral redemption in clustering sparse networks”, PNAS 110 (52) 20935-20940, 2013. DOI: 10.1073/pnas.1312486110 [sci-hub, @tor], arXiv: 1306.5550

Examples

>>> g = gt.collection.data["football"]
>>> H = gt.hashimoto(g)
>>> ew, ev = scipy.linalg.eig(H.todense())
>>> figure(figsize=(8, 4))
<...>
>>> 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("hashimoto-spectrum.svg")


Hashimoto matrix spectrum for the network of American football teams.