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Node and Edge Nonlinear Eigenvector Centrality for Hypergraphs

Francesco Tudisco, Desmond J. Higham

arXiv:2101.06215v2cs.SIcs.NAmath.NAphysics.data-an

TL;DR

Higher-order interactions call for centrality measures that represent hyperedges as well as nodes. This paper formulates a general nonlinear eigenvector framework for both, with existing models as special cases. It establishes conditions for unique solutions and efficient computation by a nonlinear power method.

  • Problem

    Existing centrality work has focused more on nodes than edges, while higher-order interactions require representations that can account for hyperedges.

    Source: §2.1 · p. 2, §1 · p. 2

  • Method

    The paper defines incidence-matrix-based node and hyperedge centralities using four nonlinear functions, and computes solutions with a Nonlinear Power Method.

    Source: §1 · p. 2

  • Results

    For homogeneous functions, nonlinear Perron–Frobenius theory provides existence, uniqueness, and computability guarantees, and the proposed framework includes clique-expansion and uniform-hypergraph tensor centralities as special cases.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework offers a rigorous, low-cost spectral approach to quantifying the importance of both nodes and hyperedges in general hypergraphs.

    Source: §3 · p. 12

    One stated guarantee regime, P2, assumes ρ = 1, differentiable positive maps, and a connected bipartite graph.

    Source: §2.6 · p. 6

Abstract

from arXiv · show

Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

1 Introduction

Hypergraphs represent higher-order interactions, but the appropriate representation depends on the question being asked. The paper develops a general nonlinear eigenvector framework for scoring both nodes and hyperedges, encompassing existing approaches.

1 Introduction

  • For coauthorship, graphs record pairwise collaboration, simplicial complexes record co-occurring subsets, and hypergraphs record complete author lists [36].In the hypergraph representation, a proper subset is not itself a hyperedge unless it was another complete author list.Source: §1 · p. 1
  • Clique expansion combines neighboring-node importances additively, whereas tensor eigenvector centrality combines them multiplicatively; these differ for larger uniform hyperedges.The two approaches coincide on standard graphs, where each hyperedge has exactly two nodes.Source: §1 · p. 2
  • The proposed incidence-matrix model uses four nonlinear functions to define node and hyperedge centralities for arbitrary hyperedge sizes, and transfers to simplicial complexes.Clique-expansion and uniform-hypergraph tensor centralities arise as particular choices of the nonlinear functions.Source: §1 · p. 2
  • The paper formulates centrality as a constrained nonlinear eigenvalue problem, establishes existence, uniqueness, and convergence results, and proposes a Nonlinear Power Method for computation.For graphs, the method yields a HITS-type iteration that assigns centrality to both nodes and edges.Source: §1 · p. 2

2 Results and Discussion

This section motivates centrality measures for edges alongside the more established measures for nodes. It introduces mutually reinforcing node–edge scores as a route that extends naturally to hypergraphs.

2.1 Notation and Motivation

  • Graph eigenvector centrality assigns each node importance proportional to the summed importance of its neighbors, represented by the positive Perron eigenvector of the adjacency matrix [10, 18].For an irreducible adjacency matrix, the solution is unique; the power method computes it to arbitrary precision when the matrix is primitive.Source: §2.1 · p. 2
  • Compared with node-centrality models, fewer models and methods quantify edge centrality, despite applications including link detection, edge prediction, and matrix completion [35].Source: §2.1 · pp. 2–3
  • A mutually reinforcing model lets edges inherit importance from their nodes and nodes from their participating edges, extending naturally to hyperedges of arbitrary size.Alternatives described here include line-graph edge scores and assigning each edge the sum of its nodes’ centralities [12].Source: §2.1 · p. 3

2.2 Hypergraphs

The paper represents a hypergraph with nodes, hyperedges, and an incidence matrix, while allowing node and edge weights. Its clique expansion connects nodes that share a hyperedge.

2.2 Hypergraphs

  • A hypergraph incidence matrix B has rows for nodes, columns for hyperedges, and B_i,e = 1 when node i belongs to hyperedge e.Source: §2.2 · p. 3
  • Node and hyperedge weight functions can encode external values, such as item prices and the profit from a purchased list, through diagonal matrices N and W.Source: §2.2 · p. 3
  • When every hyperedge has two nodes, BWB^T = A + D; more generally, BWB^T = A_H + D_H for the hypergraph’s clique-expansion graph.The clique-expansion graph connects nodes that occur together in a hyperedge, with weighted cliques representing those hyperedges.Source: §2.2 · p. 3

2.3 Node and edge nonlinear hypergraph eigenvector centrality

The model couples nonnegative node and hyperedge scores through functions of the importance of their incident counterparts. Four nonlinear functions allow flexible score-combination choices, with existence, uniqueness, and computability guarantees under stated conditions.

2.3 Node and edge nonlinear hypergraph eigenvector centrality

  • Each hyperedge score depends on the importance of its nodes, and each node score on the importance of its incident hyperedges; weighted ℓp norms generalize the corresponding sums.Source: §2.3 · pp. 3–4
  • Four functions f, g, ϕ, and ψ define the constrained nonlinear equations, allowing different ways to combine node and hyperedge importances.Source: §2.3 · p. 4
  • With all four functions set to identity, the equations give a HITS-like model in which nodes and hyperedges inherit importance from one another.Source: §2.3 · p. 4
  • For homogeneous functions, nonlinear Perron–Frobenius theory provides existence, uniqueness, and computability guarantees for the model’s nonlinear singular vectors.Source: §2.3 · p. 4
  • The framework includes previously proposed eigenvector centrality models as special cases while allowing additional flexibility.Source: §2.3 · p. 4

2.4 The linear case: eigenvector centrality for graph and line graph

In the linear case, the incidence-matrix model connects node and edge centralities to eigenvector centrality on a graph and its line graph. The same connection extends to hypergraphs through clique-expansion and line-graph structures.

2.4 The linear case: eigenvector centrality for graph and line graph

  • For graphs, the incidence-matrix products yield diagonally perturbed adjacency matrices for the graph and its line graph, whose Perron eigenvectors are the node and edge scores.The identities use BWB⊤ and B⊤NB; for an unweighted graph, B⊤B = A(e) + Δ, with Δ = 2I.Source: §2.4 · pp. 4–5
  • For general hypergraphs, the linear node-edge model is tightly connected to eigenvector centrality on the clique-expansion graph and its line graph.The line-graph adjacency and degree matrices are denoted H and Δ_H.Source: §2.4 · p. 5

2.5 Tensor-based eigenvector centrality for uniform hypergraphs and its extension

Tensor-based eigenvector centralities for uniform hypergraphs are a special case of the nonlinear singular-vector framework. The paper uses that connection to extend tensor-like centrality to non-uniform hypergraphs.

2.5 Tensor-based eigenvector centrality for uniform hypergraphs and its extension

  • Tensor-based eigenvector centrality for uniform hypergraphs is a special case of the nonlinear singular-vector framework, which extends it to non-uniform hypergraphs.The extension uses logarithmic- and exponential-based nonlinear functions.Source: §2.5 · p. 5
  • Tensor eigenvector centrality uses a k-index adjacency tensor whose entries encode hyperedge weights for k-uniform hypergraphs.When k = 2, this tensor coincides with the graph adjacency matrix.Source: §2.5 · p. 5
  • For a k-uniform hypergraph, the choices f(x) = x, g(x) = x^(1/(p+1)), ψ(x) = e^x, and ϕ(x) = ln(x) recover a tensor eigenvector centrality solution.The theorem assumes unit node weights and a positive solution of (2).Source: §2.5 · p. 6

2.6 Existence, uniqueness and computation of nonlinear hypergraph centralities

The paper applies nonlinear Perron–Frobenius theory to establish existence, uniqueness, and computation guarantees for nonlinear hypergraph centralities. Its Nonlinear Power Method converges under stated conditions, with weaker requirements than the linear power method.

2.6 Existence, uniqueness and computation of nonlinear hypergraph centralities

  • Under either P1 or P2, Theorem 2.2 gives unique positive solutions x and y, up to scaling, and unique positive λ and μ.P2 requires ρ = 1, differentiable positive maps, and a connected bipartite graph.Source: §2.6 · p. 6
  • The positive solutions defining the hypergraph centralities are called nonlinear Perron singular vectors.The existence, uniqueness, positivity, maximality, and computation analysis follows from nonlinear Perron–Frobenius theory for multihomogeneous mappings.Source: §2.6 · p. 6
  • The nonlinear power method converges to the unique positive solutions under P1 or P2, without requiring aperiodicity; depending on the functions' homogeneity degree, global convergence may hold even for disconnected graphs.Under P1, convergence is linear.Source: §2.6 · p. 7

2.7 Experiments

The experiments compare three nonlinear centrality models on sunflower and real-world hypergraphs. They show that model choice can change node and edge rankings, and that edge weight does not necessarily track spectral importance.

2.7 Experiments

  • The experiments compare linear, log-exp, and max models, with log-exp extending tensor eigenvector centrality to non-uniform hypergraphs.The linear model corresponds to eigenvector centrality on clique-expanded graphs; max approximates a softmax with α = 10.Source: §2.7 · p. 7

2.7.1 Illustrative example: hypergraph sunflower

  • On the uniform sunflower, the core receives greater node centrality than other nodes when the homogeneity degree β is positive, while the edge scores are constant across petals.As β approaches 0, node centralities tend to coincide; Figure 1a illustrates the models on eight three-node petals.Source: §2.7.1 · pp. 8–9, Figure 1 · p. 8
  • In the non-uniform sunflower, all models rank the core highest, but linear favors nodes in larger petals, log-exp favors smaller petals, and max gives petal nodes equal scores.Edge centrality is proportional to edge size for linear, inversely proportional for log-exp, and constant across edges for max.Source: Figure 1 · p. 8, §2.7.1 · p. 9

2.7.2 Real-world hypergraph data

  • The real-world tests use Math Stack Exchange co-tags and Walmart trips, represented as weighted hypergraphs.The first uses tags and question tag-sets; the second uses products and co-purchased product sets.Source: §2.7.2 · p. 9
  • Across the two real-world hypergraphs, the models show no apparent linear correlation in their node and edge centrality assignments.Figure 2 compares pairwise model scores; Figure 3 compares top-k node rankings for k from 1 to 100.Source: §2.7.2 · pp. 9–10, Figure 2 · p. 10, Figure 3 · p. 11
  • Larger edge weights do not correspond to greater spectral importance, especially under the linear and max models.The paper illustrates this with edge-weight versus centrality scatter plots for Math Stack Exchange and Walmart trips.Source: §2.7.2 · p. 11, Figure 4 · p. 12

3 Conclusion

The conclusion argues that hypergraphs restore interaction detail that pairwise network views omit. It reports that the proposed node-edge spectral centralities have a theoretical basis and can be computed by a globally convergent iteration built from matrix-vector products.

3 Conclusion

  • The proposed constrained eigenvalue problems have a sound theory, and an efficient globally convergent iteration computes the measures using matrix-vector products.The conclusion points to Theorems 2.2 and 2.3 and Algorithm 1.Source: §3 · p. 12
  • The method extends spectral centrality measurement to general hypergraphs to quantify the importance of both nodes and hyperedges.The conclusion characterizes the extension as rigorous and of very little cost.Source: §3 · p. 12
  • Pairwise network views trade fine detail for simplicity, while higher-order interactions can reintroduce some of that detail.The conclusion frames understanding computational expense and ease of assimilating results as costs and benefits to consider.Source: §3 · p. 12

4 Methods

The methods section relates the proposed nonlinear eigenproblem to tensor eigenvectors in a special case and uses multihomogeneous Perron–Frobenius theory to establish solution properties. It also outlines the proof strategy for convergence of the nonlinear power method.

4 Methods

  • For uniform hypergraphs, the proof reduces the proposed node equation to the adjacency-tensor eigenvalue problem under the specified function choices.The derivation uses the incidence and weight matrices and the product of the other nodes' scores within each hyperedge.Source: §4 · pp. 12–13
  • The existence and uniqueness argument treats the associated mapping as order-preserving and multihomogeneous, then invokes multihomogeneous Perron–Frobenius results.The proof discusses positivity under assumptions P1 and P2.Source: §4 · p. 13
  • The convergence proof transforms the eigenvalue mapping into an order-preserving multihomogeneous mapping with a primitive Jacobian, then applies a Perron–Frobenius result.It relates the transformed mapping's eigenvectors to those of the original mapping.Source: §4 · p. 13

Funding

The work acknowledges EPSRC Programme Grant EP/P020720/1 support.

Funding

  • DJH was supported by EPSRC Programme Grant EP/P020720/1.Source: Funding · p. 14
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