Higher Order Laplacians: Extensions & Applications

• Higher Order Laplacians are operator families that extend classical graph Laplacians by incorporating polyadic interactions and acting on higher-dimensional simplices or nodes.
• They include constructions such as generalized node Laplacians, discrete Hodge Laplacians, and cross-order Laplacians, each tailored to different network dynamics and topological structures.
• These frameworks enable advanced analysis in synchronization, diffusion, and machine learning, offering improved modeling for complex systems and data-driven applications.

Searching arXiv for key higher-order Laplacian references.

Higher-order Laplacians are operator families that extend the graph Laplacian beyond dyadic networks, either by incorporating polyadic interactions among nodes or by acting directly on signals defined on higher-dimensional simplices. In current usage, the term covers at least three distinct constructions. First, for node-based dynamics on hypergraphs or simplicial complexes, higher-order interactions can be compressed into generalized node Laplacians such as $L^{(d)}_{ij}=dK_i^{(d)}\delta_{ij}-A^{(d)}_{ij}$, and in special synchronization settings into a multiorder Laplacian that plays the role of the ordinary graph Laplacian for linear stability (Battiston et al., 6 Oct 2025). Second, for signals on edges, triangles, and higher simplices, the central object is the discrete Hodge Laplacian $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$, built from boundary and coboundary operators and tied to Hodge decomposition, harmonic modes, and higher-order diffusion (Battiston et al., 6 Oct 2025, Baccini et al., 2022, Reitz et al., 2020). Third, in hypergraph-based diffusion and renormalization, one also encounters Laplacians defined on simplices or hyperedges through cross-order adjacency graphs, including cross-order Laplacians $\mathbf{L}^{\times}_{(k,m)}$ that govern diffusion on $k$-simplices mediated by $m$-simplices (Nurisso et al., 2024). The subject therefore does not admit a single canonical definition; rather, it comprises several operator frameworks adapted to different notions of higher-order structure, dynamics, and data representation.

1. Historical and conceptual landscape

The ordinary graph Laplacian

$L=D-A$

acts on functions on vertices and encodes only pairwise adjacency. Higher-order Laplacians arise when this dyadic description is inadequate: simplicial complexes retain faces and incidence across dimensions, hypergraphs encode general $N$-ary interactions, and higher-order fractional or polyharmonic operators generalize the Laplacian in the PDE sense rather than in the combinatorial sense (Aktas et al., 2021, Ros-Oton et al., 2014, Gesztesy et al., 2014).

A central distinction separates two broad operator traditions. In one tradition, the state remains node-based, but the coupling reflects polyadic interactions. The resulting operators are generalized node Laplacians, multiorder Laplacians, or cross-order Laplacians on adjacency graphs of simplices (Battiston et al., 6 Oct 2025, Nurisso et al., 2024). In the other tradition, the state lives on $k$-simplices themselves, and the operator is built from boundary calculus. This yields the combinatorial or weighted Hodge Laplacian, together with its down and up components, and related operators such as the topological Dirac operator (Battiston et al., 6 Oct 2025, Baccini et al., 2022, Reitz et al., 2020).

The distinction is substantive. A generalized node Laplacian acts on node states but incorporates higher-order interactions indirectly through co-membership statistics. A Hodge Laplacian acts on $k$-cochains and encodes adjacency through shared $(k-1)$-faces and $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$0-cofaces. This difference determines what phenomena the operator can resolve. Node Laplacians are natural in synchronization and master-stability settings, whereas Hodge Laplacians support curl-free, divergence-free, and harmonic decompositions and connect directly to topology (Battiston et al., 6 Oct 2025).

The term “higher-order Laplacian” is also used in other, non-combinatorial senses. In analysis, higher-order fractional Laplacians $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$1 with $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$2 interpolate polyharmonic and fractional regimes and exhibit oscillatory effects absent for $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$3 (Ros-Oton et al., 2014, Abatangelo et al., 2016, Abatangelo et al., 2022, Covi et al., 2020, Cao et al., 2019). In spectral graph theory, “higher accuracy” graph Laplacians $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$4 are finite-difference-inspired operators built from open-path matrices rather than simplices or hyperedges (Yoon, 6 Apr 2025). These usages are mathematically legitimate but conceptually separate from higher-order network Laplacians in the simplicial and hypergraph sense.

2. Generalized node Laplacians for polyadic interactions

In higher-order network dynamics, hypergraphs and simplicial complexes provide the basic structural objects. Hypergraphs encode arbitrary polyadic interactions, while simplicial complexes impose downward closure, meaning that whenever a higher-order simplex is present, all lower-dimensional faces are also present (Battiston et al., 6 Oct 2025). For node dynamics with nonpairwise coupling, one may write interactions through an adjacency matrix $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$5 for pairwise terms and adjacency tensors $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$6 for higher-order terms. A representative example is the generalized Kuramoto model

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$7

where the tensors encode different triadic interaction types (Battiston et al., 6 Oct 2025).

The key reduction consists in constructing order-specific adjacency-like matrices $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$8, where $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$9 counts how many $\mathbf{L}^{\times}_{(k,m)}$0-body interactions contain both $\mathbf{L}^{\times}_{(k,m)}$1 and $\mathbf{L}^{\times}_{(k,m)}$2, and generalized degrees $\mathbf{L}^{\times}_{(k,m)}$3, which count how many $\mathbf{L}^{\times}_{(k,m)}$4-body hyperedges involve node $\mathbf{L}^{\times}_{(k,m)}$5 (Battiston et al., 6 Oct 2025). This yields the generalized order-$\mathbf{L}^{\times}_{(k,m)}$6 Laplacian

$\mathbf{L}^{\times}_{(k,m)}$7

For instance,

$\mathbf{L}^{\times}_{(k,m)}$8

In the identical-frequency Kuramoto setting, the graph Laplacian then generalizes to the multiorder Laplacian

$\mathbf{L}^{\times}_{(k,m)}$9

This operator is positive semidefinite and has zero row and column sums, so it retains the structural properties expected of a Laplacian (Battiston et al., 6 Oct 2025).

The role of these operators is primarily dynamical. In the classical pairwise case, linear stability of synchrony is controlled by the graph Laplacian. In the higher-order Kuramoto case, the multiorder Laplacian “extracts all the information relevant for synchronization stability,” packages it into a single matrix, and determines whether full synchrony is stable (Battiston et al., 6 Oct 2025). For more general node dynamics,

$k$0

one again obtains order-specific generalized Laplacians, but they cannot generally be merged into one universal multiorder Laplacian because different coupling functions must be handled separately in the variational equations (Battiston et al., 6 Oct 2025). The resulting obstruction is noncommutativity: several generalized Laplacians may appear simultaneously and fail to commute, preventing straightforward simultaneous diagonalization and limiting direct master-stability generalizations.

A related but distinct node-centric construction appears in cross-order diffusion. For a higher-order network $k$1, the cross-order adjacency number between $k$2-simplices $k$3 through order $k$4 is

$k$5

with adjacency matrix $k$6 and degree vector given by row sums. The associated cross-order Laplacian is

$k$7

a symmetric positive semidefinite operator whose zero-eigenvalue multiplicity equals the number of connected components of the adjacency graph $k$8 (Nurisso et al., 2024). This framework includes ordinary graph diffusion as $k$9, but also edge-through-triangle, triangle-through-edge, and higher variants.

3. Hodge Laplacians on simplicial complexes

When the state variable is defined on edges, triangles, or higher simplices, the natural language is algebraic topology. Let $m$0 denote the free abelian group of $m$1-chains, and let the boundary operator $m$2 act by

$m$3

with $m$4 (Baccini et al., 2022). In matrix form this becomes the incidence matrix $m$5, and dually one has coboundary operators represented by $m$6, where $m$7 can be introduced for normalization (Baccini et al., 2022).

In this setting, the $m$8-th weighted Hodge Laplacian is

$m$9

with

$L=D-A$0

where the adjoint is defined with respect to weighted inner products on cochains (Baccini et al., 2022). In the unweighted combinatorial setting, this corresponds to the familiar formula

$L=D-A$1

and

$L=D-A$2

(Reitz et al., 2020). For dynamics on $L=D-A$3-simplices, the review on collective dynamics writes the discrete Hodge Laplacian as

$L=D-A$4

with the linearized topological Kuramoto model

$L=D-A$5

(Battiston et al., 6 Oct 2025).

These operators support the Hodge decomposition

$L=D-A$6

with harmonic $L=D-A$7-cochains in the kernel, and the nonzero spectrum partitioned into up and down sectors (Baccini et al., 2022). A key structural property in the weighted setting is

$L=D-A$8

which implies that every nonzero eigenvector belongs either to the up part or the down part (Baccini et al., 2022). In dynamical terms, the review on higher-order networks states that divergence-free and curl-free components evolve independently via $L=D-A$9 and $N$0, while harmonic modes remain stationary (Battiston et al., 6 Oct 2025).

The Hodge framework also admits a Dirac formulation. The topological Dirac operator $N$1, built from boundary operators, is self-adjoint and satisfies

$N$2

in the $N$3 weighted simplicial setting (Baccini et al., 2022), and more generally

$N$4

in the review formulation (Battiston et al., 6 Oct 2025). This identifies $N$5 as a square root of the block-diagonal Hodge Laplacian and provides a dimension-coupling differential operator complementary to the dimensionwise Laplacians.

An important extension concerns normalization and weights. Weighted simplicial complexes are introduced to represent higher-order data without the information loss of unweighted closure: bare affinity weights $N$6 are lifted recursively to positive topological weights $N$7, which define diagonal metric matrices $N$8 and hence weighted adjoints and weighted Hodge Laplacians (Baccini et al., 2022). With the choice

$N$9

the resulting normalized weighted Hodge Laplacians have spectra in

$k$0

which enables cross-dimensional spectral comparison through Gibbs densities, higher-order spectral entropy, and relative entropy (Baccini et al., 2022).

4. Diffusion, renormalization, and cross-dimensional operators

A standard dynamical interpretation of higher-order Laplacians is diffusion on simplices. For a signal $k$1 on $k$2-simplices, the weighted Hodge diffusion equation is

$k$3

and in the eigenbasis one has

$k$4

(Baccini et al., 2022). This identifies the higher-order spectrum as the organizer of diffusion time scales, return probabilities, and entropy-like observables. In the renormalization-group approach to simplicial complexes, the low-eigenvalue density of the normalized up-Laplacian

$k$5

defines a higher-order spectral dimension $k$6 through

$k$7

with explicit formulas for deterministic Apollonian and pseudo-fractal simplicial complexes (Reitz et al., 2020). In those models, the spectral dimension depends on simplex order $k$8, so one complex may possess a family of spectral dimensions rather than a single one.

Cross-order diffusion generalizes this picture by fixing a diffusion order $k$9 and an interaction order $k$0, then evolving

$k$1

on $k$2-simplices through $k$3-simplex-mediated adjacency (Nurisso et al., 2024). The corresponding heat kernel

$k$4

and normalized density matrix

$k$5

support entropy and susceptibility diagnostics for scale invariance (Nurisso et al., 2024). The renormalization procedure groups $k$6-simplices using diffusion at time $k$7, then induces a coarse-graining of vertices and simplices in the original higher-order network. This is not a matrix renormalization law for the Laplacian itself, but a diffusion-informed combinatorial coarse-graining.

A more radical hypergraph diffusion formalism is proposed in “Hypergraph Laplacians in Diffusion Framework” (Aktas et al., 2021). There, for a fixed simplex dimension $k$8, the operator

$k$9

aggregates interactions between $(k-1)$0-simplices through simplices of all other dimensions, with

$(k-1)$1

A second operator,

$(k-1)$2

is a block matrix coupling simplices of all dimensions through all dimensions. These constructions are explicitly described as diffusion-motivated higher-order coupling operators rather than classical Hodge Laplacians (Aktas et al., 2021). The paper does not establish a full spectral theory for them, but it broadens the operator landscape by allowing diffusion through arbitrary simplex orders and cross-dimensional couplings.

5. Learning and data-analytic formulations

Higher-order Laplacians have also entered machine learning as operators for higher-order message passing and spectral filtering. In “Higher-order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes,” simplicial complexes are reduced to a flower-petals model in which vertices constitute a core and each simplex order $(k-1)$3 forms a separate petal (Huang et al., 2023). The higher-order incidence matrix

$(k-1)$4

induces a two-step random walk from vertices to $(k-1)$5-simplices and back, yielding the normalized adjacency

$(k-1)$6

and the FP Laplacian

$(k-1)$7

(Huang et al., 2023). These operators act on vertex features but encode co-membership in higher-order simplices. They are symmetric positive semidefinite, and $(k-1)$8 is always an eigenvalue with eigenvector $(k-1)$9 (Huang et al., 2023).

This construction is distinct from Hodge theory. FP Laplacians do not act on cochains and do not rely on boundary/coboundary operators. Instead, they define one node-space Laplacian per simplex order and support per-order spectral filtering,

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$00

with learned filter magnitudes interpreted as higher-order interaction strengths (Huang et al., 2023). This suggests that, in data-analytic settings, higher-order Laplacians can function as order-specific smoothness operators even when the task remains node-centric.

For manifold-valued data on hypergraphs, a different extension replaces Euclidean differences by Riemannian logarithmic maps. In “$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$01-Laplacians for Manifold-valued Hypergraphs,” one defines a Fréchet hyperedge gradient

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$02

using Fréchet means on the input and output sets of an oriented hyperedge, and a pairwise gradient

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$03

using all cross-pairs (Stokke et al., 14 Jul 2025). These induce isotropic and anisotropic hypergraph $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$04-Laplacians, for example the Fréchet anisotropic operator

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$05

which generalize graph Laplacians to higher-arity relations and nonlinear target manifolds (Stokke et al., 14 Jul 2025). This is a higher-order Laplacian in the hypergraph sense, not in the Hodge sense.

6. Analytical higher-order Laplacians and broader extensions

Outside network theory, higher-order Laplacians also denote higher powers or fractional powers of the Euclidean Laplacian. For the higher-order fractional Laplacian $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$06 with $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$07, the operator is defined through the Fourier multiplier

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$08

(Ros-Oton et al., 2014, Abatangelo et al., 2016, Covi et al., 2020). This analytical regime displays oscillatory phenomena absent for $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$09: failure of general maximum principles for $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$10 with $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$11 odd, failure of polarization and Pólya–Szegő inequalities, and even sign-changing first eigenfunctions on suitable domains (Abatangelo et al., 2016, Abatangelo et al., 2022). At the same time, one retains strong structural results, including Pohozaev identities

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$12

for smooth bounded domains (Ros-Oton et al., 2014), unique continuation for noninteger orders (Covi et al., 2020), and Liouville and classification results for mixed operators

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$13

(Cao et al., 2019).

The higher-order local operator $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$14 also supports a spectral theory of self-adjoint extensions. For arbitrary finite-volume open sets $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$15, the Krein–von Neumann extension $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$16 of the minimally defined polyharmonic operator has positive-eigenvalue counting function bounded by

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$17

(Gesztesy et al., 2014). This is a different branch of higher-order Laplacian theory, but it underscores that the phrase encompasses both combinatorial and analytical generalizations.

A final caveat concerns terminology. “Graph Laplacians with Higher Accuracy” introduces graph matrices $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$18 motivated by $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$19-th order finite-difference approximations of the one-dimensional Laplacian, with

$L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$20

where $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$21 counts open paths of length $L_{[n]}=L_{[n]}^{down}+L_{[n]}^{up}$22 (Yoon, 6 Apr 2025). These operators are higher-order only in approximation accuracy, not in simplicial, hypergraph, or PDE order. A similar terminological ambiguity appears in discrete geometry, where “higher order approximations” of tetrahedral Laplacians refer to curvature-sensitive corrections rather than polyharmonic operators (Liao, 21 Jan 2025). Such uses are best distinguished from the mainstream higher-order network and analytical meanings.

In current mathematical practice, then, “higher-order Laplacians” denotes a family of related but non-equivalent operator frameworks. Generalized node Laplacians compress polyadic interactions into node-space stability operators (Battiston et al., 6 Oct 2025). Hodge Laplacians act on cochains and expose topological mode structure (Baccini et al., 2022, Reitz et al., 2020). Cross-order and hypergraph Laplacians organize diffusion on simplices or hyperedges through arbitrary interaction orders (Nurisso et al., 2024, Aktas et al., 2021, Stokke et al., 14 Jul 2025). Analytical higher-order fractional and polyharmonic Laplacians generalize the Euclidean operator in spectral, variational, and boundary-value theory (Ros-Oton et al., 2014, Gesztesy et al., 2014, Abatangelo et al., 2022). Their shared theme is the extension of Laplacian structure beyond pairwise, first-order, node-centric settings; their differences lie in the state space, the incidence structure they encode, and the phenomena—synchronization, diffusion, topology, learning, or elliptic analysis—they are designed to capture.