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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
Graph methods capture pairwise interactions, but simplicial complexes support higher-order structure and require topology-aware edge-space analysis. The paper introduces a normalized Hodge-Laplacian diffusion framework, then applies it to trajectory embeddings and edge-based PageRank, with explicit scope limitations for personalized edge influence.
Problem
Graph-based analysis does not account for simultaneous higher-order interactions, while line-graph edge walks are not directly tied to the simplicial complex’s Hodge-Laplacian topology.
Method
The paper introduces a normalized Hodge Laplacian related to random walks in the edge-space of simplicial complexes and applies it to embeddings and edge-based PageRank.
Results
The framework provides higher-order trajectory embeddings and a simplicial PageRank that analyzes edge importance and relations to global topology.
Takeaways & Limitations
The tools extend graph-based analysis to edge-signals while incorporating higher-order topology and complementing node-based methods.
Takeaways & Limitations
For non-indicator teleportation vectors, sign patterns make sums of absolute personalized PageRank vectors differ from the absolute PageRank of the combined vector, so the individual-edge influence interpretation is difficult.
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 extends graph-based diffusion analysis to simplicial complexes by linking a normalized Hodge Laplacian to edge-based random walks. It develops edge-space analytics and demonstrates them through trajectory embeddings and simplicial PageRank.
- Contributions: The paper introduces a normalized Hodge Laplacian related to random walks on 1-simplices, providing tools for signals in a simplicial complex’s edge-space.The framework aims to translate graph network-science methods to higher-order data while complementing node-based analysis.
- Applications: Two applications illustrate how the methodology incorporates higher-order topology into data analysis.The paper presents trajectory embeddings and simplicial PageRank as concrete applications of its random-walk model.
- Applications: The authors develop low-dimensional embeddings of edge-flows and trajectories as higher-order generalizations of diffusion maps and Laplacian eigenmaps.The embeddings represent edges and trajectories in Euclidean space for analyzing trajectory types and observed flows.
- Applications: They also derive a personalized PageRank variant for edges that measures their roles and topological importance within a simplicial complex.The analysis complements node-based graph signal processing and can indicate how strongly an edge participates in the cycle-space.
- Motivation: Graphs encode pairwise relationships, whereas simplicial complexes represent richer higher-order interactions with associated Hodge Laplacians.Simplicial complexes provide an algebraic framework for modeling relationships involving more than two nodes.
2. A short review of graphs, simplicial complexes, and Laplacians.
This section reviews graphs, simplicial complexes, oriented chains, edge-flows, and the Hodge 1-Laplacian. It establishes the vector-space and orientation conventions used for higher-order diffusion analysis.
- Simplicial complexes: A simplicial complex is a collection of simplices closed under taking subsets, with graphs as the special case containing only vertices and edges.This lets simplicial complexes encode higher-order relationships while retaining graph structure as their 1-skeleton.
- Simplicial complexes: Two k-simplices are upper adjacent when they share a co-face and lower adjacent when they share a common face; simplex degree counts co-faces.These relations describe how simplices connect through higher- and lower-dimensional structure.
- Orientations and chains: Orientations are fixed by vertex orderings for computation, and reversing a basis orientation changes its coefficient’s sign.For real-valued edge signals, signs record whether flow aligns or opposes the chosen reference orientation.
- Orientations and chains: The chain space represents k-chains as vectors of simplex coefficients, while the 1-chain space can be interpreted as edge-flows with signed values.Negative edge-flow values indicate direction opposite to the reference orientation.
- Hodge Laplacians: The Hodge 1-Laplacian is the paper’s primary operator, and the ordinary graph Laplacian appears as the degree-zero special case of the Hodge-Laplacian hierarchy.The hierarchy is constructed from boundary maps.
3. Diffusion processes on simplicial complexes.
The paper develops edge-based diffusion on simplicial complexes by lifting oriented edge-flows into a doubled state space and normalizing the Hodge 1-Laplacian. This construction preserves a connection between random walks, topology, and spectral structure.
- Diffusion processes on simplicial complexes: Edge diffusion is defined on 1-simplices while accounting for the simplicial complex’s higher-order topology.The approach is intended to extend graph-based diffusion tools to edge-space analysis.
- Beyond graphs: keeping track of orientations: Line-graph walks provide edge states but lose the direct relationship between the walk and the simplicial complex’s Hodge Laplacian.The line-graph Laplacian is not directly related to the Hodge Laplacian, so its walk does not encode the complex’s topology in the desired way.
- Beyond graphs: keeping track of orientations: The Hodge 1-Laplacian’s orientation-dependent positive and negative entries prevent it from serving directly as a Markov transition matrix.Unlike vertices, edges carry nontrivial orientations, creating the normalization challenge.
- Lifting of edge-flows and matrix operators: Each edge-flow is lifted into a space containing both orientations, doubling the state space to |D1| = 2|C1| before applying and projecting the operator.The lifted representation separates flow magnitude from direction and treats both edge orientations as states.
- The normalized Hodge 1-Laplacian and edge-space random walks: The normalized Hodge 1-Laplacian has a stochastic lifting, yielding a random walk on the lifted edge space with upper and lower adjacent transitions.The construction uses a standard-form normalization and supports a normalized Hodge decomposition.
- Spectral properties: The alternating subspace is invariant under the lifted transition matrix, so the normalized Hodge Laplacian’s spectrum is contained in the lifted walk’s spectrum.Eigenvectors transfer from the normalized Hodge operator to the lifted transition matrix through the lifting map.
- Spectral properties: The normalized Hodge 1-Laplacian has bounded spectral support, paralleling the normalized graph Laplacian.Near-zero eigenvectors of related operators correspond to nearly harmonic functions in the normalized Hodge decomposition.
4. Constructing simplicial complexes and computation.
The paper describes several ways to construct simplicial complexes from data and emphasizes sparse, iterative computation for normalized Hodge-Laplacian analyses. Matrix-vector products can be computed in time linear in the complex size.
- Constructing simplicial complexes: Simplicial complexes may be induced from collections of sets, point clouds, or graph structure, depending on the data representation.Examples include author sets, Stack Overflow tags, and clique complexes derived from graphs.
- Computational aspects: Constructing a complex from point clouds or graph triangles can be computationally demanding, although real-world graph structure often permits faster practical algorithms.The stated worst-case triangle-enumeration running time is O(n3/2), where n1 is the number of edges.
- Computational aspects: For complexes with maximum simplex size three, normalized Hodge-Laplacian systems are typically sparse and can often be handled with iterative methods.Approximate solutions are described as sufficient for typical data applications.
- Computational aspects: Applying the normalized Hodge Laplacian to a vector costs O(n0 + n1 + n2) = O(|X|), linear in the size of the complex.The cost follows from sparse incidence matrices and diagonal scaling operations.
- Hodge decomposition: Computing the normalized Hodge decomposition reduces to solving least-squares problems because the decomposition is orthogonal.The decomposition is applied to edge-flow vectors in R^n1.
2 B2w −c∥2. (4.2)
The section addresses computational aspects of least-squares problems used in the Hodge decomposition, including sparsity, nontrivial kernels, and residual-based computation.
- When ker(L1) is non-trivial, the least-squares problems may be non-unique, but the residual vectors suffice to compute the decomposition.
- The decomposition is obtained by solving least-squares problems associated with the gradient and curl components.
2 B2w∗, h = c −g −r. (4.3)
The normalized Hodge decomposition separates an edge flow into weighted gradient, curl, and harmonic components with distinct topological interpretations.
- The gradient component is a weighted cut-space flow with zero signed circulation along every cyclic path.
- The curl component consists of weighted local circulations around filled triangles, analogous to curl in vector calculus.
- The harmonic component represents global circulation that is neither a gradient nor a combination of local curl flows.
- Harmonic edge-flow projections identify global cycles directly related to the simplicial complex’s homology.
5. Application I: edge flow and trajectory embeddings.
The paper develops harmonic spectral embeddings for edges and trajectories using the normalized Hodge 1-Laplacian, then applies them to synthetic and ocean-drifter data.
- 5.1. Synthetic data example.: A simplicial complex with two holes has two zero eigenvalues, whose harmonic functions encircle the respective holes.
- 5.1. Synthetic data example.: Each oriented edge maps to its coordinates in the harmonic subspace through projection onto the harmonic eigenvectors.
- 5.1. Synthetic data example.: The edge embedding’s coordinate signs indicate alignment with clockwise or counter-clockwise harmonic flows around the holes.
- 5.1. Synthetic data example.: Trajectory embeddings differentiate topological properties by grouping trajectories with similar projections onto harmonic flows.
- 5.1. Synthetic data example.: The trajectory framework supports clustering, similarity scoring across different lengths, and filtering noisy trajectories.
- 5.2. Analysis of ocean drifter data.: The method depends on constructing a simplicial complex whose faces and weights provide a context-dependent interpretation of the trajectory data.
- 5.2. Analysis of ocean drifter data.: For Madagascar drifters, the first harmonic separates currents flowing north and south of the island, while the second captures a localized southwest circulation.
6. Application II: PageRank on simplicial complexes.
The paper extends PageRank from graph nodes to simplicial-complex edges using the normalized Hodge 1-Laplacian, yielding topology-sensitive measures of edge importance. Applications show localized, cycle-related, and co-purchasing patterns that are largely distinct from graph-based PageRank.
- PageRank on simplicial complexes: Projected PageRank can be computed directly with L1, without explicitly constructing the lifted random-walk matrix.This follows from the relationship between the lifted walk and the projection onto alternating edge functions.
- PageRank on simplicial complexes: The edge-space PageRank solves a shifted linear system involving the normalized Hodge 1-Laplacian and a projected teleportation vector.The formulation uses (βI + L1)π1 = (β −2)x, with x obtained by projecting a probability vector into the edge space.
- Synthetic data example: In the synthetic example, PageRank distinguishes a low-influence edge in a large clique, a high-magnitude bridge, and cycle edges whose influence spreads around the cycle.The differing patterns arise from the complex's topology and the harmonic, gradient, and curl components of the Hodge decomposition.
- Political book co-purchasing data: For book co-purchasing data, harmonic PageRank highlights edges connecting around holes, including 18 co-purchases with harmonic PageRank greater than 0.4.The complex is built by filling each triangle in the co-purchasing graph as a 2-simplex; harmonic values use β = 2.5.
7. Discussion.
The discussion frames normalized Hodge-Laplacian diffusion as a principled extension of graph-based analysis to simplicial complexes and identifies open questions about its broader theoretical scope.
- Discussion: Normalized Hodge-Laplacian diffusion extends graph-based methods to simplicial complexes while respecting higher-order topology.The framework focuses on diffusion in edge-space, including edge-trajectory embeddings and simplicial extensions of PageRank for edge importance.
- Future research: The paper focuses on diffusion-based embeddings and centrality measures, while other random-walk learning tasks remain future directions.The authors specifically suggest translating additional random-walk tools from graphs to simplicial complexes.
- Open theory: A clean generalization of graph connectivity and Cheeger-like concepts to simplicial complexes remains open.Some higher-order Cheeger-like inequalities can be impossible because simplicial-complex homology may contain torsion absent from graphs.
- Open theory: Incorporating near-harmonic components into simplicial-complex data analysis remains a challenge.The authors describe their framework as a starting point for addressing this issue.
| {z } Plower
The appendix establishes that the constructed edge-walk matrices are stochastic transition matrices, including lower- and upper-adjacency components and their convex combinations.
- Plower: The resulting half-weighted sum 1/2(Plower + Pupper) is a stochastic matrix.The proof combines the stochasticity of the lower and upper components.
- Plower: The forward and backward walk matrices are transition matrices for weighted graph walks, so any convex combination is also valid.The matrices correspond to weighted adjacency matrices for forward and backward walks.
- Plower: The lower-adjacency component is column stochastic by construction.This is stated directly in the stochasticity argument for the lifted normalized Hodge 1-Laplacian.
- Plower: The upper-adjacency component is shown to be column stochastic by separately treating edges with zero and positive upper degree.For positive degree, normalization accounts for the 3·deg([i, j]) nonzero entries contributed by co-faces.