Signed Magnetic Laplacian

• Signed Magnetic Laplacian is defined as a Laplacian-type operator on graphs where edge transport carries a sign or unit-modulus phase, unifying classical Laplacian forms.
• It incorporates gauge invariance and switching phenomena, enabling accurate spectral partitioning and analysis on signed, weighted, and directed networks.
• Applications span spectral graph theory, fractals, higher-dimensional Hodge theory, and non-Hermitian operators, providing robust tools for modern graph learning and analysis.

A signed magnetic Laplacian is a Laplacian-type operator in which transport across an edge carries either a sign or a more general unit-modulus phase. On a signed graph $\Sigma=(\Gamma,\sigma)$ with $\sigma:E\to\{+1,-1\}$, the basic operator is the signed Laplacian

$L(\Sigma)=D(\Sigma)-A(\Sigma),$

where $A(\Sigma)$ is the signed adjacency matrix and $D(\Sigma)$ is the usual degree diagonal; in this form it is the real $\pm1$ special case of a magnetic Laplacian $L^\theta=D-A^\theta$ with edge phases $e^{i\theta_{ij}}$ (Reff, 2011). More broadly, the literature uses the expression for a family of Hermitian or sectorial operators on signed, directed, weighted, fractal, or simplicial structures, all built to encode sign, orientation, flux, or holonomy in the Laplacian itself (He et al., 2022).

1. Algebraic definitions and competing conventions

For a simple signed graph $\Sigma=(\Gamma,\sigma)$, with vertex set $V=\{v_1,\dots,v_n\}$ and $\sigma:E\to\{+1,-1\}$0, the signed adjacency matrix is

$\sigma:E\to\{+1,-1\}$1

and the signed Laplacian is

$\sigma:E\to\{+1,-1\}$2

with $\sigma:E\to\{+1,-1\}$3 formed from the unsigned vertex degrees. This operator is real symmetric and positive semidefinite, and it unifies the ordinary Laplacian and the signless Laplacian by the all-positive and all-negative signings, respectively (Reff, 2011).

A second convention, common in spectral partitioning on signed weighted graphs, defines the standard Laplacian as $\sigma:E\to\{+1,-1\}$4 with signed weights $\sigma:E\to\{+1,-1\}$5, and the signed Laplacian as

$\sigma:E\to\{+1,-1\}$6

Its quadratic form is

$\sigma:E\to\{+1,-1\}$7

where $\sigma:E\to\{+1,-1\}$8. In this convention, $\sigma:E\to\{+1,-1\}$9 is positive semidefinite and is explicitly the real-phase form of a magnetic Laplacian, while $L(\Sigma)=D(\Sigma)-A(\Sigma),$0 may have negative eigenvalues when negative weights are present (Knyazev, 2017). The coexistence of these conventions means that “signed Laplacian” is not uniform terminology; in practice, “signed magnetic Laplacian” often refers to any Laplacian whose edge interaction is twisted by a sign or phase.

A normalized version also appears prominently. For a weighted signed graph $L(\Sigma)=D(\Sigma)-A(\Sigma),$1, the normalized signed Laplacian is

$L(\Sigma)=D(\Sigma)-A(\Sigma),$2

or equivalently $L(\Sigma)=D(\Sigma)-A(\Sigma),$3. Its spectrum lies in $L(\Sigma)=D(\Sigma)-A(\Sigma),$4, and it is the normalized $L(\Sigma)=D(\Sigma)-A(\Sigma),$5-magnetic Laplacian obtained by restricting magnetic phases to $L(\Sigma)=D(\Sigma)-A(\Sigma),$6 and $L(\Sigma)=D(\Sigma)-A(\Sigma),$7 (Atay et al., 2014).

2. Switching, balance, and gauge structure

The central structural feature is switching, the signed-graph analogue of a magnetic gauge transformation. Given $L(\Sigma)=D(\Sigma)-A(\Sigma),$8, the switched signature is

$L(\Sigma)=D(\Sigma)-A(\Sigma),$9

If $A(\Sigma)$0, then

$A(\Sigma)$1

so the spectrum is invariant under switching (Reff, 2011). This is the discrete $A(\Sigma)$2 form of gauge invariance.

A signed graph is balanced when every cycle has positive sign. Balanced components are precisely those that are switching-equivalent to all-positive signings, and the Laplacian kernel detects them: $A(\Sigma)$3 where $A(\Sigma)$4 is the number of balanced connected components. Thus the multiplicity of the zero eigenvalue is the number of balanced components, not the number of connected components as in the unsigned case (Reff, 2011).

The same mechanism appears in the normalized setting. A signing is called coherent when every cycle has positive sign; for connected graphs, coherence is equivalent to the existence of a non-trivial kernel for the twisted Laplacian and to conjugacy with the usual unsigned Laplacian by a vertex sign flip (Gournay, 2014). A broader symmetry mechanism for normalized signed Laplacians includes ordinary bipartiteness of unsigned graphs as a special case, so spectral symmetry about $A(\Sigma)$5 is no longer tied only to unsigned bipartite structure (Atay et al., 2014).

This suggests a general principle: the spectrally relevant data are not individual edge signs but gauge-invariant cycle data. In the signed case those data reduce to $A(\Sigma)$6-valued cycle products; in magnetic language they are fluxes or holonomies.

3. Spectral theory on signed graphs

For the unsigned-degree convention $A(\Sigma)$7, several spectral invariants depend explicitly on sign. If $A(\Sigma)$8 denotes the number of negative edges incident to $A(\Sigma)$9, then for a connected signed graph of order $D(\Sigma)$0,

$D(\Sigma)$1

and further lower bounds involve signed triangle counts $D(\Sigma)$2 and the number of balanced components through $D(\Sigma)$3 (Reff, 2011). These estimates are sign-sensitive in a way that classical unsigned bounds are not.

A key extremal statement is that for a connected underlying graph $D(\Sigma)$4,

$D(\Sigma)$5

with equality if and only if $D(\Sigma)$6 is switching-equivalent to the all-negative signing. Since $D(\Sigma)$7, the signless Laplacian is the spectral-radius maximizer over the switching class (Reff, 2011).

The smallest eigenvalue is also controlled by sign geometry. For the twisted signed Laplacian, the signed isoperimetric quantity $D(\Sigma)$8, built from boundary size and the minimal number of edge deletions required to make induced subgraphs coherently signed, satisfies a Cheeger-type estimate

$D(\Sigma)$9

where $\pm1$0 is the maximal degree (Gournay, 2014). This is the signed analogue of an isoperimetric bound for the first eigenvalue.

Interlacing under graph operations remains available but acquires sign dependence. For Laplacians of signed graphs, deleting a vertex yields

$\pm1$1

where $\pm1$2 are eigenvalues of the original graph and $\pm1$3 those of the vertex-deleted graph; analogous bounds are obtained for signed cycles, net-Laplacians, and normalized net-Laplacians, with different inequalities for deleting positive and negative edges (Guragain et al., 2023).

Normalized spectra exhibit further structure. The normalized signed Laplacian can have symmetric spectra in situations more general than unsigned bipartiteness, and that spectral symmetry is linked to damped two-periodic solutions of the discrete-time heat equation on the graph (Atay et al., 2014).

4. Directed, signed, and weighted magnetic Laplacians

On directed signed graphs, the main difficulty is to encode sign and direction while retaining a Hermitian operator suitable for spectral methods. One construction, introduced for MSGNN, starts from a signed directed adjacency matrix $\pm1$4, forms the symmetrized signed adjacency

$\pm1$5

the absolute degree

$\pm1$6

and a phase matrix

$\pm1$7

The Hermitian signed-magnetic adjacency is

$\pm1$8

and the magnetic signed Laplacians are

$\pm1$9

These operators are Hermitian and positive semidefinite, the normalized spectrum lies in $L^\theta=D-A^\theta$0, and the construction reduces to the standard, signed, and magnetic Laplacians in the appropriate undirected or unsigned limits (He et al., 2022).

A different, parameter-free construction is the Sign-Magnetic Laplacian of SigMaNet. For directed graphs with arbitrary real weights, it uses

$L^\theta=D-A^\theta$1

and a complex matrix

$L^\theta=D-A^\theta$2

leading to

$L^\theta=D-A^\theta$3

These operators are Hermitian, positive semidefinite, and positively homogeneous in the edge weights; on unweighted directed graphs, $L^\theta=D-A^\theta$4 coincides with the magnetic Laplacian at charge $L^\theta=D-A^\theta$5 (Fiorini et al., 2022).

In SDGCL, a signed directed graph $L^\theta=D-A^\theta$6 is encoded by a symmetrized adjacency $L^\theta=D-A^\theta$7 and a complex phase matrix $L^\theta=D-A^\theta$8 whose angles distinguish the nine possible sign-direction types between ordered node pairs. The Hermitian adjacency

$L^\theta=D-A^\theta$9

induces unnormalized and normalized magnetic Laplacians

$e^{i\theta_{ij}}$0

These are Hermitian and positive semidefinite, and the phase parameter $e^{i\theta_{ij}}$1 is used as a Laplacian-level augmentation in contrastive learning (Ko et al., 2023).

These directed constructions make the term “signed magnetic Laplacian” genuinely plural. They agree on the use of complex Hermitian edge transport to encode direction, but differ in how they normalize degree, whether they retain signed magnitudes in the symmetric part, and whether a tunable magnetic charge $e^{i\theta_{ij}}$2 is present.

5. Fractals, continuous magnetic fields, and higher-dimensional Hodge theory

The graph-based formalism extends beyond finite graphs. On the Sierpinski gasket, a magnetic Laplacian is defined through the Dirichlet-form derivation $e^{i\theta_{ij}}$3 and a real 1-form $e^{i\theta_{ij}}$4 by

$e^{i\theta_{ij}}$5

At graph approximation level $e^{i\theta_{ij}}$6, the discrete magnetic energy is

$e^{i\theta_{ij}}$7

with corresponding magnetic graph Laplacian

$e^{i\theta_{ij}}$8

If $e^{i\theta_{ij}}$9, then $\Sigma=(\Gamma,\sigma)$0, and one literally recovers a signed Laplacian. For locally exact forms with a local Coulomb gauge, the renormalized graph magnetic energies converge to the fractal magnetic form, and the magnetic Laplacian has the same spectral asymptotics as the ordinary Laplacian (Hyde et al., 2016).

In the continuum, a sign-changing magnetic field can create an “attractive magnetic edge.” For the semiclassical operator

$\Sigma=(\Gamma,\sigma)$1

on $\Sigma=(\Gamma,\sigma)$2, with magnetic field

$\Sigma=(\Gamma,\sigma)$3

discontinuous across the smooth closed curve $\Sigma=(\Gamma,\sigma)$4, low-energy eigenfunctions are exponentially localized near $\Sigma=(\Gamma,\sigma)$5, and the low-energy spectrum is described by an effective one-dimensional edge operator (2207.13391). This is the continuous analogue of a signed magnetic interface: the sign change in the field, rather than a boundary, generates localized spectral states.

Allowing the magnetic potential itself to be complex yields a non-selfadjoint extension. For

$\Sigma=(\Gamma,\sigma)$6

if $\Sigma=(\Gamma,\sigma)$7 is relatively form-bounded with respect to the real magnetic part, the operator is defined as an m-sectorial operator via a closed sectorial form. In two dimensions, sufficient growth conditions on $\Sigma=(\Gamma,\sigma)$8, $\Sigma=(\Gamma,\sigma)$9, or $V=\{v_1,\dots,v_n\}$0 guarantee compact resolvent, and for non-critical complex magnetic fields one can construct semiclassical pseudomodes by a WKB method; such pseudomodes do not exist when the field is real-valued (Krejcirik et al., 2024). This is an extension from self-adjoint signed or magnetic Laplacians to genuinely non-Hermitian magnetic operators.

A higher-dimensional discrete generalization appears for Hodge Laplacians on weighted simplicial complexes. The up-, down-, and Hodge Laplacians can be rewritten as signed Schrödinger operators on the graph of simplices, with edge signs $V=\{v_1,\dots,v_n\}$1 determined by combinatorial incidence and a potential $V=\{v_1,\dots,v_n\}$2. In the Hodge case, this potential is Forman curvature, yielding a discrete Weitzenböck-type decomposition in signed magnetic form (Bartmann et al., 11 Aug 2025).

6. Applications, cospectrality, and methodological debates

Signed magnetic Laplacians serve both as structural objects and as algorithmic primitives. In unsigned graph theory they interpolate between the ordinary Laplacian and the signless Laplacian through all-positive and all-negative signings (Reff, 2011). In the theory of 2-lifts and higher-degree combinatorial Laplacians, the twisted Laplacian provides a signed analogue of the Cheeger framework and an entry point for higher-dimensional Hodge operators (Gournay, 2014). In graph learning, MSGNN, SigMaNet, and SDGCL use Hermitian signed magnetic Laplacians as the spectral backbone of GNNs or contrastive encoders (He et al., 2022).

Not every Laplacian-like operator on a signed graph is gauge invariant. For the signed neighbourhood corona, switching preserves adjacency and Laplacian spectra but does not, in general, preserve net-Laplacian spectra, so the net-Laplacian is not a gauge-invariant magnetic operator in the same sense as $V=\{v_1,\dots,v_n\}$3 (Shamsher et al., 2023). This distinction matters when one wants spectral information to depend only on cycle holonomy rather than on a chosen representative of a switching class.

Cospectrality also limits spectral identifiability. Non-isomorphic signed graphs can be Laplacian-cospectral, including families produced by partial transpose, and signed neighbourhood corona constructions preserve adjacency, Laplacian, and net-Laplacian cospectrality under suitable hypotheses (Shamsher et al., 2022). A plausible implication is that inverse problems for signed magnetic Laplacians are at least as non-unique as in ordinary spectral graph theory.

A recurrent methodological debate concerns which operator is most appropriate for partitioning signed graphs. One line of work argues that for spectral partitioning the standard Laplacian $V=\{v_1,\dots,v_n\}$4 may be preferable to the positive-semidefinite signed Laplacian $V=\{v_1,\dots,v_n\}$5: examples show that the leading nontrivial eigenvectors of $V=\{v_1,\dots,v_n\}$6 can yield meaningless partitions, whereas negative eigenvalues of $V=\{v_1,\dots,v_n\}$7 can sharpen spectral gaps and make the Fiedler vector easier to compute (Knyazev, 2017). This does not negate the operator-theoretic value of the signed or magnetic Laplacian; it identifies a task-dependent distinction between “repulsion as negative stiffness” and “repulsion as phase consistency.”

Taken together, these results support an umbrella view. A signed magnetic Laplacian is not a single matrix but a class of gauge-theoretic Laplacians whose edge transport ranges from $V=\{v_1,\dots,v_n\}$8 signs to general unit phases, whose diagonal may encode ordinary degree, absolute degree, or signed degree, and whose analytic realizations extend from finite signed graphs to fractals, simplicial complexes, and non-selfadjoint magnetic Schrödinger operators.