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Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, Ali Jadbabaie
TL;DR
Edge walks built from line graphs are not directly tied to the Hodge Laplacian, leaving their relation to simplicial-complex topology unresolved. The paper introduces a normalized Hodge Laplacian for edge-space diffusion, with applications to edge trajectories and edge importance that respect higher-order topology.
Problem
Edge walks based on line graphs are not directly related to the Hodge Laplacian, so their properties do not inform the topology of the simplicial complex.
Source: §3.1 · p. 9Method
The paper introduces a normalized Hodge Laplacian and uses it to define diffusion processes on the edge-space of simplicial complexes.
Source: §1 · p. 2, §7 · p. 28Results
Applications demonstrate edge-trajectory embeddings and an edge-space PageRank for assessing edges’ roles in the global topology of a simplicial complex.
Source: §1 · p. 2, §7 · p. 28Takeaways & Limitations
Edge-space analysis complements node-based analysis and can highlight how much an edge belongs to the cycle-space of the complex.
Source: §1 · p. 2For non-indicator teleportation vectors, sign-pattern differences make individual-edge influence difficult to interpret, so the analysis focuses on local personalized PageRank vectors.
Source: §6.2 · p. 25
Abstract
from arXiv · showhide
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.
1. Introduction.
The paper develops a normalized Hodge Laplacian linked to random walks on simplicial-complex edges, extending analysis from nodes to edge-space signals. It illustrates this approach with trajectory embeddings and an edge-based variant of personalized PageRank.
1. Introduction.
- The normalized Hodge Laplacian links random walks on simplicial-complex edges to topology-aware analysis of edge signals, complementing typical node-based graph analysis.The framework aims to extract the relative importance of edges and edge-signals with respect to a complex’s higher-order topology.Source: §1 · p. 2
- Trajectory and edge-flow embeddings provide low-dimensional vector representations for differentiating trajectory types, while edge-based PageRank analyzes edge roles in global topology and cycle-space.The paper presents these as two applications that extend node-based analysis to signals defined on edges.Source: §1 · p. 2
1.1. Additional Background and Motivation.
- Graphs encode pairwise relationships but miss simultaneous interactions among more than two nodes; simplicial complexes model such higher-order interactions and have an associated Hodge Laplacian.Simplicial complexes have algebraic structure that distinguishes them from generic hypergraphs.Source: §1.1 · p. 3
- Although simplicial complexes have been applied across many fields, their Hodge Laplacian spectra and relation to dynamics remain little known, and random walks and diffusion are scarcely explored.The cited work describes exploration mainly from a pure-mathematics perspective [108].Source: §1.1 · p. 3
2. A short review of graphs, simplicial complexes, and Laplacians.
The review builds from graph Laplacians to simplicial complexes and their chain operators, culminating in the Hodge 1-Laplacian and its topological interpretation. It also presents the Hodge decomposition of chain spaces.
2.1. Graphs and the graph Laplacian.
- An undirected graph is encoded by symmetric adjacency A and degree matrix D, with L0=D−A; its spectrum reveals topological properties.Source: §2.1 · p. 4
2.2. Simplicial Complexes.
- An SC is a set of simplices closed under subsets; graphs are its at-most-1-dimensional case, while higher simplices encode higher-order vertex relationships.Source: §2.2 · p. 4
2.3. Oriented sipmlicial complexes and function spaces on simplicial complexes.
- Orientations provide computational bookkeeping; with real coefficients, Ck represents chains as vectors on oriented simplices, and reversing a basis orientation reverses its coefficient's sign.The text notes that real-valued coefficients are essential for data signals on edges and the associated Hodge theory.Source: §2.3 · pp. 5–6
2.4. Boundary and co-boundary maps.
- Boundary maps send chains to oriented boundary components, and their kernels describe cycles; B1 is the node-to-edge incidence matrix, with higher maps encoding simplex incidence.Source: §2.4 · pp. 6–7
2.5. Hodge Laplacians.
- The graph Laplacian is the k=0 case, while the paper focuses on L1; ker(Lk) has dimension equal to the kth Betti number, counting k-dimensional holes.The kth Betti number counts the k-dimensional holes of the simplicial complex.Source: §2.5 · p. 8
2.6. The Hodge decomposition.
- The Hodge decomposition splits Ck orthogonally into boundary, harmonic, and co-boundary components, with its C1 form noted as useful for data insights.Nonzero elements of ker(Lk) represent non-trivial kth homology classes.Source: §2.6 · p. 8
3. Diffusion processes on simplicial complexes.
This section develops diffusion on simplicial complexes through lifted oriented edge-flows and a normalized Hodge 1-Laplacian related to random walks on lifted edges. It connects eigenvectors of edge-coupling operators to the normalized Hodge decomposition and describes computation using sparse matrix-vector products.
3. Diffusion processes on simplicial complexes.
- A line-graph walk on edges does not retain the connection to the simplicial complex’s Hodge topology, while the raw Hodge 1-Laplacian’s orientation-dependent signed entries do not directly define Markov transitions.The section focuses on diffusion between 1-simplices, or edges.Source: §3 · p. 8, §3.1 · p. 9
3.1. Beyond graphs: keeping track of orientations.
- Lifting separates edge-flow magnitude, which can be related to probability, from its orientation sign, while representing both orientations of each edge in the larger state space.Each original edge contributes two states in the lifted space.Source: §3.1 · p. 9, §3.2 · p. 9
3.2. Lifting of edge-flows and matrix operators.
- The matrix-lifting construction represents an operator’s action as lifting, applying a linear transformation in the larger space, and projecting back to the original space.The lifted negative Hodge Laplacian is a nonnegative symmetric adjacency matrix for a graph on the two orientations of each edge.Source: §3.2 · pp. 10–11
3.3. The normalized Hodge 1-Laplacian and edge-space random walks.
- The normalized Hodge 1-Laplacian has a stochastic lifting that combines upper- and lower-adjacent edge transitions, defining an edge-based random walk in the lifted space.The chosen standard-form normalization admits a normalized Hodge decomposition; other lifted operators and transition weights could yield different normalizations.Source: §3.3 · pp. 12–13
- The spectrum of the normalized Hodge operator is contained in that of its lifted stochastic matrix and has bounded support.The alternating-function subspace is invariant under the lifted transition matrix, and eigenvectors of the normalized operator induce lifted eigenvectors with the same eigenvalue.Source: §3.3 · p. 14
3.4. Normalized Hodge decompositions.
- The normalized Hodge decomposition corresponds to the standard Hodge decomposition, with harmonic functions associated with the zero eigenvalue.The normalized and standard decompositions have isomorphic respective subspaces, and the symmetrized normalized operator also has a decomposition under the standard inner product.Source: §3.4 · p. 14
- Eigenvectors of G1 and G2 with eigenvalue λ correspond to eigenvectors of the normalized Hodge 1-Laplacian with eigenvalue λ.G1 captures lower-adjacent edge couplings, while G2 captures upper-adjacent edge couplings.Source: §3.4 · p. 15
- Near-zero eigenvalues of G1 or G2 correspond to near-zero eigenvalues of the normalized Hodge 1-Laplacian, representing nearly harmonic functions.Source: §3.4 · p. 15
- After a simplicial complex is constructed, computations reduce to sparse matrix-vector products with sparsity linear in the number of complex elements.The computational cost of higher-order interactions exceeds graph-based methods but is often not prohibitive; the same sparse-product approach extends to higher-order Hodge Laplacians, although complex size can grow.Source: §3.4 · p. 15
4. Constructing simplicial complexes and computation.
Simplicial complexes can be constructed from several kinds of data, and their normalized Hodge computations exploit sparse structure. The section also explains how edge-flow decompositions are computed and interpreted.
4.1. Constructing simplicial complexes from data.
- Simplicial complexes are built top-down from sets, bottom-up from point clouds, or intermediately as graph clique complexes determined by their 1-skeleton.Point-cloud constructions include Vietoris–Rips and Čech complexes; clique complexes use graph triangles as 2-simplices. Triangle enumeration has worst-case O(n_1^{3/2}) time, though practical algorithms are typically faster on real-world graphs.Source: §4.1 · pp. 15–16
4.2. Solving L1 systems and the cost of matrix-vector multiplication.
- A normalized Hodge-Laplacian matrix-vector product costs O(n_0+n_1+n_2)=O(|X|), linear in the size of the complex.The cost follows from the sparsity of the incidence matrices.Source: §4.2 · p. 16
4.3. Computation and interpretation of the Hodge Decomposition for edge flows.
- The normalized edge-flow Hodge decomposition reduces to least-squares residuals, enabling sparse iterative solvers such as LSQR or LSMR without recovering the minimizers.This residual-based calculation avoids needing to recover p or w even when ker(L1) is non-trivial.Source: §4.3 · pp. 16–17
- The decomposition separates weighted gradients with zero cyclic flow, local curl circulations around filled triangles, and harmonic global cycles tied directly to simplicial-complex homology.Source: §4.3 · p. 17
5. Application I: edge flow and trajectory embeddings.
The section extends Laplacian-based spectral embedding from graph nodes to edges and flows using the Hodge 1-Laplacian, and applies harmonic embeddings to ocean drifter trajectories around Madagascar.
5.1. Synthetic data example.
- On the synthetic complex, two holes correspond to two zero eigenvalues whose harmonic functions encircle the holes.The extracted flow features derive from the complex’s encoded topology.Source: §5.1 · pp. 18–19
- Projecting each oriented edge onto the harmonic subspace gives coordinates that indicate its alignment with global circular flows, with signs depending on reference orientation.The embedding is the projection e ↦ H^⊤e; positive and negative coordinates indicate aligned and oppositely oriented flow, respectively.Source: §5.1 · p. 18
- Trajectory embeddings group paths with similar harmonic projections, distinguishing their topological properties in the synthetic example.Trajectories around the lower-left obstacle share similar embeddings, as do two other groups of three trajectories.Source: §5.1 · pp. 19–20
5.2. Analysis of ocean drifter data.
- The approach depends on the chosen simplicial-complex model, so its interpretation is context-dependent and is not limited to planar complexes.Edges and faces can be modeled from observed flow patterns; whether that complex provides a useful interpretation depends on the application.Source: §5.2 · p. 20
- The two-dimensional harmonic space captures Madagascar’s island-scale flow split and a localized southwest circulation; most trajectories project only weakly onto the latter.The first harmonic function separates northbound and southbound currents east of Madagascar, while the second reflects a loopy current near southwest Madagascar.Source: Figure 5.3 · p. 21, §5.2 · p. 21
6. Application II: PageRank on simplicial complexes.
The section defines edge-space PageRank from the normalized Hodge 1-Laplacian, using a lifted random walk and projection to measure edge importance. It applies harmonic PageRank to political book co-purchases, finding topological information complementary to graph-based PageRank.
6.1. Background: (personalized) PageRank on graphs.
- Graph PageRank defines π through transition probability α and teleportation probability 1 − α; personalized PageRank restarts at a selected node, with π indicating its influence on other nodes.The section extends this random-walk-based centrality idea to edges in simplicial complexes.Source: §6.1 · p. 22, §6 · p. 22
6.2. PageRank vectors on simplicial complexes.
- The method solves PageRank in the lifted walk space, projects it onto oriented edges, and computes the projected vector from L1 without constructing the lifted transition matrix.The projected vector is expressible using the shifted inverse of L1; generalized PageRank uses (κI + L1)^−1 as an edge-signal filter.Source: §6.2 · pp. 22, 24
- Projected PageRank can have negative entries because projection takes a difference between orientation-specific probabilities; for an indicator input, orientation changes can make the generalized vector nonnegative.For an indicator on edge [i, j], absolute entry values can represent that edge’s influence on other edges; this interpretation is harder for non-indicator inputs, so the analysis focuses on edge-localized personalization.Source: §6.2 · pp. 24–25
- In the synthetic complex, a clique edge had a localized, small-magnitude vector, a bridge edge had larger localized components, and a cycle edge had strong influence around the cycle.Hodge decomposition linked these patterns to structure: the bridge’s PageRank lay in the weighted cut-space, while the cycle edge had a high harmonic projection.Source: Figure 6.1 · p. 24, §6.2 · pp. 25–26
- Edge importance can be aggregated by the 2-norm of each personalized PageRank vector or its Hodge-subspace projections, including the harmonic component.Harmonic components are associated with cycles not induced by 2-simplices, and edges with substantial projection into L1’s null space can have strong harmonic PageRank.Source: §6.2 · p. 26, Figure 6.2 · p. 26
6.4. Analysis of political book co-purchasing data.
- Across β ∈ (2.05, 2.67), harmonic PageRank rankings had mean Spearman correlation ρ = 0.78, indicating ranking consistency across the tested parameter range.This β range corresponds to teleportation parameter α ∈ (0.75, 0.975).Source: §6.4 · p. 27
- The harmonic scores identified 18 co-purchases above 0.4; 15 linked books in the same category, and 9 linked conservative books, suggesting opinion gaps within political clusters.The analysis treated co-purchased book cliques as 2-simplices; harmonic PageRank highlights edges connecting around holes, or sets of books never bought together.Source: §6.4 · p. 27
- Harmonic PageRank was essentially uncorrelated with edge scores from node-PageRank sums, node-PageRank differences, and line-graph PageRank (ρ = 0.04, −0.08, and 0.08, respectively).The comparison used graph-based PageRank with teleportation parameter α = 0.85.Source: §6.4 · pp. 27–28, Figure 6.4 · p. 28
7. Discussion.
Traditional graph-based models can fall short for increasingly complex datasets, motivating richer frameworks such as simplicial complexes that capture polyadic interactions. This work introduces a normalized Hodge Laplacian for principled diffusion on simplicial complexes, focusing on edge-space embeddings and PageRank extensions.
Discussion
- The normalized Hodge Laplacian enables diffusion on simplicial complexes that respects their topology, with applications to edge-trajectory embeddings and edge-importance measures.The paper notes that topological tools have mostly been used in static contexts, while this framework brings them to dynamical processes.Source: §7 · p. 28
- Future work includes comparing random-walk variants with graph-based counterparts and studying how higher-order topology affects convergence, mixing, and other walk properties.The authors also propose extending random-walk tools beyond diffusion-based embeddings and centrality to other learning tasks.Source: §7 · p. 28
- For graphs, personalized PageRank finds low-conductance sets via a local Cheeger inequality, but a clean generalization of these concepts to simplicial complexes remains open.This connection motivates the paper’s personalized polyadic walks, while graph random walks are also linked to connectivity and near 0th-order homology through spectral and isoperimetric inequalities.Source: §7 · pp. 28–29
- Hodge-Laplacian spectra connect to expansion in simplicial complexes [97], but torsion can make some higher-order Cheeger-like inequalities impossible.Properly incorporating near-harmonic components into data analysis remains a challenge, and the framework is presented as a starting point.Source: §7 · p. 29
| {z } Plower
The proof establishes stochastic transitions for weighted lower-adjacency walks and verifies that the upper-adjacency component is column stochastic, yielding a stochastic lifting of the normalized Hodge 1-Laplacian.
| {z } Plower
- Forward and backward lower-adjacency walks define transition matrices on weighted adjacency matrices, so any convex combination of them is also a valid transition matrix.The lower adjacency matrices correspond to walks along edges.Source: | {z } Plower · pp. 35, 37
- Plower is column stochastic by construction, and proving Pupper column stochastic completes the stochastic lifting of the normalized Hodge 1-Laplacian.Source: | {z } Plower · p. 38
- For Pupper, zero-degree edge columns are stochastic after scaling the selected identity column by 1/2, while positive-degree columns sum to 3 deg([i, j]) before D4^-1 scaling makes them stochastic.Each co-face contributes three nonzero entries for edges with a different orientation relative to that co-face.Source: | {z } Plower · p. 38