PaperlayerView exact revision on arXiv

Source-linked AI summary

Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

Uthsav Chitra, Benjamin J Raphael

arXiv:1905.08287v1cs.LGcs.DMcs.SIstat.ML

TL;DR

Graphs capture pairwise links, but hypergraph spectral methods have not fully addressed higher-order relationships, and existing linear Laplacians use only pairwise relations. The paper uses random walks to develop spectral tools for edge-dependent vertex weights. It shows that edge-independent walks reduce to graph walks, while edge-dependent weights support higher-order modeling and improved performance in the reported applications.

  • Problem

    Graphs can lose group structure, and existing linear hypergraph Laplacians use only pairwise relationships [2]; spectral theory for edge-dependent vertex weights is also lacking.

    Source: §1 · pp. 1–2

  • Method

    The paper uses random walks to derive a hypergraph Laplacian and spectral results for weights that vary by vertex and incident hyperedge.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    Random walks with edge-independent vertex weights are equivalent to graph walks, while edge-dependent weights model higher-order information and improve performance in the reported experiments.

    Source: §1 · p. 2, §7 · p. 11

  • Takeaways & Limitations

    The resulting Laplacian would allow hypergraph convolutional networks that currently use edge-independent weights to be extended to edge-dependent weights.

    Source: §7 · p. 12

    The walk definition allows self-loops; non-lazy walks are harder to analyze, and the paper gives only a weaker version of Theorem 4 for them.

    Source: §3 · p. 4

Abstract

from arXiv · show

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex $v$ has a weight $\gamma_e(v)$ for each incident hyperedge $e$ that describes the contribution of $v$ to the hyperedge $e$. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.

1 Introduction

Graphs represent pairwise links but can lose group structure, while hypergraph spectral methods have not fully captured higher-order relationships. The paper develops random-walk spectral results for edge-dependent vertex weights, including when graph-based walks suffice and when they do not.

1 Introduction

  • A graph’s pairwise representation can lose group structure, whereas a single hyperedge can represent a whole community.The introduction also gives protein complexes as an example of interactions among groups of proteins.Source: §1 · p. 1
  • The paper develops a spectral theory for hypergraphs whose vertex weights can vary by incident hyperedge, representing each vertex’s contribution to that edge.Such weights have been used in image segmentation, e-commerce, text ranking, image search, and 3D object classification.Source: §1 · p. 2
  • Existing linear hypergraph Laplacians use only pairwise relationships, and the lack of spectral theory has led to theoretically unsound assumptions in prior applications.The paper identifies examples involving Laplacians without spectral guarantees and an incorrect stationary distribution.Source: §1 · p. 2
  • With edge-independent vertex weights, hypergraph random walks are equivalent to walks on a clique graph, explaining why existing Laplacians [34, 43] do not use higher-order relations.The paper presents this as a generalization of earlier results.Source: §1 · p. 2
  • For edge-dependent weights, the paper derives a random-walk Laplacian that uses higher-order relations, is positive semi-definite, and satisfies a Cheeger inequality.It also derives a stationary-distribution formula and a mixing-time bound.Source: §1 · p. 2
  • The paper demonstrates applications to ranking authors in a citation network and players in a video game.The introduction says these applications use hypergraphs with edge-dependent vertex weights.Source: §1 · pp. 2–3

2 Graphs, Hypergraphs, and Random Walks

This section introduces hypergraphs with edge-dependent vertex weights and describes how random walks select edges and vertices. It also defines edge-independent weights and the clique graph used to relate hypergraphs to graphs.

2 Graphs, Hypergraphs, and Random Walks

  • In the standard hypergraph walk, a walker chooses an incident edge proportional to its weight, then chooses a vertex in that edge uniformly and moves there.The walk is described step-by-step in the cited passages.Source: §2 · p. 3
  • The edge-dependent model generalizes the standard walk by replacing uniform vertex selection with a fixed probability distribution over the selected edge’s vertices.This motivates the definition of edge-dependent vertex weights.Source: §2 · p. 3
  • An edge-dependent hypergraph assigns each hyperedge a weight and each incident vertex a weight for that particular hyperedge.The vertex weight represents that vertex’s contribution to the hyperedge, and the walker selects a vertex with probability proportional to that weight.Source: §2 · p. 3
  • Weights are edge-independent when each vertex has the same contribution to every incident edge; weights are trivial when all such contributions equal 1.Source: §2 · p. 3
  • The vertex-weight matrix records γe(v), while the hyperedge-weight matrix records ω(e) for incident vertex-edge pairs; diagonal degree matrices record vertex and hyperedge degrees.Source: §2 · p. 3
  • The clique graph connects every pair of vertices sharing a hyperedge, and the paper assumes hypergraphs are connected through this graph.Source: §2 · p. 3

3 Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

The section defines a random walk using edge and vertex weights, then establishes when that walk can be represented on a clique graph. It also gives limits on such representations and a partial spectral comparison for edge-dependent weights.

3 Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

  • A walker chooses an incident edge proportional to ω(e), then chooses a vertex in it proportional to its edge-dependent weight and moves there.The transition matrix is written in matrix form, and the definition permits self-loops, making the walk lazy.Source: §3 · p. 4
  • The walk definition allows self-loops; non-lazy analysis is significantly harder, and the Supplement proves only a weaker version of Theorem 4 for non-lazy walks.Source: §3 · p. 4
  • For edge-independent vertex weights, some edge weights on the clique graph make the hypergraph walk equivalent to a graph walk.This generalizes prior results; the proof relies on time-reversibility, which also yields a formula for the clique-graph weights.Source: §3 · p. 5
  • An edge-dependent hypergraph can have a walk that no choice of clique-graph edge weights can represent; simulations suggest such walks are common, but that observation is not formalized.Source: §3 · p. 5
  • Edge-dependent hypergraphs cannot represent every Markov chain on the same vertex set, and their walks form a hierarchy beyond the reversible-versus-irreversible distinction.In general, a walk using a hyperedge of cardinality k cannot be reduced to one using only hyperedges of cardinality at most k−1.Source: §3 · p. 5
  • For a specific clique-graph weighting, the second-smallest Laplacian eigenvalues of the hypergraph and graph are close.The authors describe this as a partial answer to whether the two walks can be close.Source: §3 · pp. 5–6

4 Stationary Distribution and Mixing Time

The section derives stationary distributions and a mixing-time bound for random walks on hypergraphs with edge-dependent vertex weights. It also identifies how the formulas specialize to edge-independent and trivial weights, and how the bound depends on reachability-related parameters.

4.1 Stationary Distribution

  • The stationary-distribution formula uses an edge-dependent proportionality constant ρ_e and multiplies each incident-edge term by γ_e(v).The factors ρ_e can be computed in O(|E|^3 + |E|^2 · |V|) time, and serve as the appropriate scaling despite arbitrary rescaling of vertex weights.Source: §4.1 · p. 6
  • With edge-independent weights, the formula recovers Zhou et al.’s stationary distribution; with trivial weights, π_v = d(v)/Σ_v d(v).The edge-independent case has ρ_e = (Σ_v d(v))^-1.Source: §4.1 · p. 6

4.2 Mixing Time

  • The mixing-time bound for a hypergraph random walk with edge-dependent vertex weights has the same dependence on 1/Φ², log(1/ϵ), and log(1/√d_min) as the graph bound.Source: §4.2 · pp. 6–7
  • Small β_1 and β_2 correspond to vertices that are hard to reach, and their presence increases the mixing time.Source: §4.2 · p. 7

5 Hypergraph Laplacian

The section defines a random-walk-based hypergraph Laplacian and shows how its ability to represent higher-order relations depends on edge-dependent vertex weights. It also gives a clique-graph approximation result for the Laplacian’s second-smallest eigenvalue.

5 Hypergraph Laplacian

  • With edge-independent weights, the walk is determined by clique-graph pairwise weights and the Laplacian equals that of an undirected graph.Source: §5 · pp. 7–8
  • With edge-dependent vertex weights, transition probabilities generally cannot be reduced to one pairwise clique-graph weight, so the Laplacian encodes higher-order relations.Source: §5 · p. 8
  • The random-walk-based Laplacian is positive semidefinite and supports a Cheeger inequality for hypergraphs.It is defined using the walk’s transition matrix and stationary distribution.Source: §5 · pp. 7–8

5.1 Approximating the Hypergraph Laplacian with a Graph Laplacian

  • The clique-graph approximation places the Laplacians’ second-smallest eigenvalues within a vertex-weight-dependent factor c(H), but the proposed edge weights’ tightness is unknown.Source: §5.1 · p. 8
  • Zhang et al. obtain state-of-the-art object-classification results with a variant of the graph Laplacian, and the approximation theorem offers theoretical evidence for this outcome.Source: §5.1 · p. 8

6 Experiments

The experiments illustrate how edge-dependent vertex weights alter author rankings and compare a hypergraph random-walk ranker with established methods on Halo 2 match outcomes. The Halo 2 ranker performs significantly better than MC3 and clique-graph walks, and comparably to TrueSkill.

6.1 Citation Network

  • The citation network treats authors as vertices and papers as hyperedges, assigning weight 2 to first and last authors and 1 to middle authors.For alphabetically ordered papers, all author weights are 1; hyperedge weights equal citation count plus 1.Source: §6.1 · p. 9
  • The two citation-network rankings had Kendall τ correlation 0.77; first- or last-author positions on highly cited papers typically raised ranks under edge-dependent weights, while middle-author positions lowered them.The example is intended to illustrate alternative rankings, not establish that this simple weighting scheme is always best in practice.Source: §6.1 · p. 9

6.2 Rank Aggregation

  • The Halo 2 evaluation builds player rankings from free-for-all matches and tests them on 1-v-1 outcomes, comparing an edge-dependent hypergraph walk with clique-graph walks, MC3, and TrueSkill.The dataset contains 31,028 free-for-all matches and 5,093 1-v-1 matches among 5,507 players.Source: §6.2 · p. 10
  • The rank-aggregation method uses match scores to set positive vertex weights and variance-inspired hyperedge weights; further optimization of these simple choices may improve performance.MC3 uses partial rankings but not score information, while TrueSkill models each player's skill with a normal distribution.Source: §6.2 · pp. 10–11
  • On 8.9% of 1-v-1 matches, the hypergraph method predicted the outcome correctly while TrueSkill did not; overall, it significantly outperformed MC3 and clique-graph walks and was comparable to TrueSkill.The authors could not identify a specific pattern in the matches where the hypergraph method was correct and TrueSkill was not.Source: §6.2 · p. 11, Table 2 · p. 11

7 Conclusion

The paper develops a random-walk spectral theory for hypergraphs with edge-dependent vertex weights, while showing that edge-independent walks can reduce to graph walks. The concluding and supplementary passages state scope for future work and conditions for related equivalence results.

7 Conclusion

  • The paper develops a random-walk spectral theory for edge-dependent vertex weights, showing theoretically and experimentally how they model higher-order information and improve hypergraph-algorithm performance.It also establishes that edge-independent random walks on hypergraphs are equivalent to random walks on graphs.Source: §7 · p. 11
  • Further work includes evaluating more applications, sharpening graph-Laplacian approximation bounds, and relating the linear hypergraph Laplacian to nonlinear Laplacians.Source: §7 · p. 11
  • The paper's hypergraph Laplacian could extend hypergraph convolutional networks to edge-dependent vertex weights.The cited approaches implement graph convolution through a graph Laplacian or use hypergraphs with edge-independent vertex weights.Source: §7 · p. 12

A Incorrect Stationary Distribution in Earlier Work

B Proof of Teorem 4

  • For edge-independent vertex weights, a random walk on a hypergraph is equivalent to one on its clique graph with suitable edge weights.The proof passage characterizes graph equivalence for finite irreducible reversible Markov chains.Source: B Proof of Teorem 4 · p. 16

C Non-Lazy Random Walks on Hypergraphs

D Relationships between Random Walks on Hypergraphs and Markov Chains on Vertex Set

Random walks with edge-independent vertex weights are equivalent to graph walks, while edge-dependent weights permit walks that cannot be represented at lower hyperedge cardinality. The section also shows that not every lazy Markov chain on a vertex set comes from such a hypergraph walk.

D Relationships between Random Walks on Hypergraphs and Markov Chains on Vertex Set

E Proof of Teorem 6

The proof establishes positive edge constants used in the stationary-distribution result for hypergraphs with edge-dependent vertex weights. It extends the construction from equal edge weights to rational and then real weights.

E Proof of Teorem 6

  • The proof extends existence of the required constants from equal edge weights to rational weights by edge replication, then to real weights by continuity.Source: E Proof of Teorem 6 · pp. 23–24
  • Theorem 6 gives positive constants ρe for the stationary distribution of a random walk on a hypergraph with edge-independent vertex weights.Source: E Proof of Teorem 6 · p. 25
  • Computing the constants ρe takes time O(|E|^3 + |E|^2 · |V|).Source: E Proof of Teorem 6 · p. 25

F Proof of Teorem 8

The proof derives a mixing-time bound for random walks on hypergraphs with edge-dependent vertex weights. It uses a general bound for lazy Markov chains and Cheeger-inequality arguments.

F Proof of Teorem 8

  • The proof invokes a bound for finite irreducible Markov chains whose self-transition probabilities are at least δ, using the smallest stationary probability πmin.Source: F Proof of Teorem 8 · p. 26
  • The derivation uses a Cheeger inequality and relations involving the chain's Dirichlet form to obtain the bound.Source: F Proof of Teorem 8 · p. 26
  • Theorem 8 provides a mixing-time bound for a random walk on a hypergraph with edge-dependent vertex weights.Source: F Proof of Teorem 8 · pp. 26–27

G Proof of Teorem 11

Theorem 11 compares a hypergraph random walk with a walk on its weighted clique graph. Its proof relates their transition and stationary quantities to obtain a spectral bound.

G Proof of Teorem 11

  • Theorem 11 relates the second-smallest Laplacian eigenvalues of a hypergraph and its clique graph when the vertex weights are normalized so that ρe = 1.Source: G Proof of Teorem 11 · p. 27
  • The proof bounds transition and stationary-probability ratios using vertex weights, then applies Chung's theorem to obtain the comparison.Source: G Proof of Teorem 11 · pp. 27–28
  • The clique graph's edge weights are chosen so its random walk and the hypergraph walk have the same stationary distribution.Source: G Proof of Teorem 11 · p. 28

H Rank Aggregation Experiments with Synthetic Data

The synthetic experiments test random-walk rank aggregation on partial rankings generated from skill-based scores. The weighted hypergraph algorithm outperforms both baselines, with its largest gains when each partial ranking contains less information.

H Rank Aggregation Experiments with Synthetic Data

  • The procedure selects a subset in which each player is included with probability p, then independently draws each selected player’s score from N(0.2 · i, σ) and scales it by a uniform factor c ∈ [1/3, 3].Source: H Rank Aggregation Experiments with… · p. 29
  • The method treats players as vertices and partial rankings as hyperedges, weighting vertices by exponentiated scores and edges by score standard deviation plus 1.The baselines are MC3 and a rank aggregation algorithm using the clique graph.Source: H Rank Aggregation Experiments with… · p. 29
  • With n = 100 and σ = 1, the experiments test p = 0.03, 0.05, and 0.07, using enough hyperedges to include every player at least once.The evaluation compares estimated and true rankings using weighted Kendall τ correlation.Source: H Rank Aggregation Experiments with… · p. 29
  • The weighted hypergraph algorithm outperforms MC3 and the clique-graph algorithm in every tested case, with the largest gains at small p.The clique-graph algorithm performs much worse than both, which the authors say suggests it is not a good approximation of the hypergraph.Source: H Rank Aggregation Experiments with… · p. 29, Figure 5 · p. 29
Loading 1905.08287v1…