magnetic_laplacian_matrix

magnetic_laplacian_matrix(G, *, nodelist=None, normalized=False, q=0.25, weight='weight')

Returns the magnetic Laplacian matrix of DiGraph G.

The magnetic Laplacian matrix (also called the q-magnetic Laplacian) is a Hermitian matrix for directed graphs that encodes edge directionality via complex phases [1].

Given a weighted directed graph, \(G = (V, E, W)\), with \(W\) the weighted adjacency matrix, the symmetrized weighted adjacency matrix is defined as \(W' = 0.5 (W + W^{T})\). A skew-symmetric term \(\delta\) is introduced to encode directionality, where

..math::

delta_{jk} = begin{cases}

+1 & text{if} j to k text{ is an edge and } k to j text{ is not}, \ -1 & text{if} k to j text{ is an edge and } j to k text{ is not}, \ 0 & text{if both or neither edge is present.} end{cases}

Then, the magnetic Laplacian matrix is defined as:

\[L^{(q)} := D - H^{(q)}\]

where \(H^{(q)}\) is the Hermitian adjacency matrix with entries \(H^{(q)}_{jk} = W'_{jk} e^{2\pi i q \delta_{jk}}\), and \(D\) is the degree matrix associated with the symmetrized weight adjacency matrix \(W'\).

If normalized is True, compute the normalized version using the Moore-Penrose inverse \(D^{+}}\) of the degree matrix \(D\). The normalized formula is then:

\[L^{(q)}_{norm} = (D^{+})^{0.5} L^{(q)} (D^{+})^{0.5}\]

Parameters:

GDiGraph

A directed graph

nodelistlist, optional (default=list(G))

Node ordering for row/columns.

normalizedbool, optional (default=False)

Bool that encodes if return the magnetic Laplacian or the normalized magnetic Laplacian. If True returns the normalized version.

qfloat, optional (default=0.25)

The phase of the magnetic potential is the charge parameter 0 <= q <= 0.5. At q=0 returns the standard Laplacian.

weightstring or None, optional (default=’weight’)

Edge attribute key for weights. If None, all edges have weight 1.

Returns:

LSciPy sparse array (complex dtype)

The magnetic Laplacian matrix of G if not normalized and the normalized version if normalized

Raises:

ValueError

If q is not between 0 and 0.5

NetworkXNotImplemented

If G is undirected or a multigraph

References

[1]

Fanuel, M., Alaíz, C. M., Fernández, Á., & Suykens, J. A. (2018). Magnetic eigenmaps for the visualization of directed graphs. Applied and Computational Harmonic Analysis, 44(1), 189–199. <https://doi.org/10.1016/j.acha.2017.01.004>