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Clustering in graphs and hypergraphs with categorical edge labels

Ilya Amburg, Nate Veldt, Austin R. Benson

arXiv:1910.09943v2cs.SIcs.LGphysics.soc-phstat.ML

TL;DR

Existing graph-clustering approaches for rich graph data do not naturally handle categorical edge labels, which can encode similarity in higher-order interactions. This paper develops an objective-based framework with polynomial-time optimization for two categories and approximation algorithms for more; LP rounding often exactly minimizes the objective in practice.

  • Problem

    Existing graph-clustering approaches for rich graph data do not naturally handle categorical edge labels, which encode discrete similarity and can occur in higher-order interactions.

    Source: §1 · p. 1

  • Method

    The framework clusters nodes by minimizing edges crossing cluster boundaries and intra-cluster category mistakes, and generalizes to hypergraphs.

    Source: §1 · p. 1, §2.1 · p. 2

  • Results

    For general problems, LP relaxations guarantee a 2-approximation or better; in practice, LP rounding often exactly minimizes the objective on large hypergraphs and outperforms baselines.

    Source: §1 · p. 2, §6 · pp. 10–11

  • Takeaways & Limitations

    The framework models higher-order interactions directly rather than approximating hyperedge cuts with weighted graph cuts.

    Source: §1 · p. 1

    In practice, maximum-flow formulations are often slower than the paper's LP relaxations.

    Source: §3.2 · p. 3

Abstract

from arXiv · show

Modern graph or network datasets often contain rich structure that goes beyond simple pairwise connections between nodes. This calls for complex representations that can capture, for instance, edges of different types as well as so-called "higher-order interactions" that involve more than two nodes at a time. However, we have fewer rigorous methods that can provide insight from such representations. Here, we develop a computational framework for the problem of clustering hypergraphs with categorical edge labels --- or different interaction types --- where clusters corresponds to groups of nodes that frequently participate in the same type of interaction. Our methodology is based on a combinatorial objective function that is related to correlation clustering on graphs but enables the design of much more efficient algorithms that also seamlessly generalize to hypergraphs. When there are only two label types, our objective can be optimized in polynomial time, using an algorithm based on minimum cuts. Minimizing our objective becomes NP-hard with more than two label types, but we develop fast approximation algorithms based on linear programming relaxations that have theoretical cluster quality guarantees. We demonstrate the efficacy of our algorithms and the scope of the model through problems in edge-label community detection, clustering with temporal data, and exploratory data analysis.

1 INTRODUCTION

The paper addresses clustering in networks with categorical edge labels and higher-order interactions, where existing approaches do not naturally handle this combination. It introduces an objective and algorithms whose tractability depends on the number of categories, then evaluates them across several applications.

1 INTRODUCTION

  • Existing rich-graph approaches do not naturally handle categorical edge labels, which can encode discrete similarity in both pairwise and higher-order interactions.Examples include connectivity patterns, publication venues, and shopping trips.Source: §1 · p. 1
  • The objective counts edges crossing cluster boundaries and intra-cluster label mistakes, and generalizes to labeled hyperedges.It seeks clusters that respect both edge-label coloring and network topology.Source: §1 · p. 1
  • Unlike related correlation-clustering objectives, this objective does not penalize placing unconnected nodes together, a difference linked to tractability, scalability, and hypergraph generalization.The objective is related to correlation clustering and chromatic correlation clustering.Source: §1 · p. 1
  • With two categories, the objective is polynomial-time solvable; with more than two, minimization is NP-hard and practical LP-based approximation algorithms are developed.The introduction describes approximation guarantees, but the supplied passages do not fully preserve the approximation-ratio expression.Source: §1 · pp. 1–2
  • Our LP-round algorithm often minimizes the NP-hard objective on hypergraphs with tens of thousands of edges in tens of seconds, and outperforms baselines on ground-truth tasks.The introduction also reports that the LP relaxation often rounds to a solution matching the lower bound and that the methods often run under 30 seconds on large hypergraphs.Source: §1 · p. 2
  • Applications include recovering ground-truth clusters, temporal community detection using binned timestamps, and exploratory analysis of cuisine-labeled ingredient hyperedges.For temporal clustering, the authors report good topological and temporal aggregation metrics.Source: §1 · p. 2

2 PRELIMINARIES AND RELATED WORK

The paper defines categorical edge clustering as assigning one category to each node cluster and minimizing edges that do not fit their label’s cluster. It distinguishes this objective from related clustering methods and positions it among approaches for graphs with categorical information.

2.1 Categorical edge clustering objective

  • A clustering partitions nodes into k category-specific groups and minimizes mistakes: edges crossing groups or placed in a group inconsistent with their label.The formulation extends to hypergraphs and can be modified for weighted edges, though the paper presents unweighted results for ease of notation.Source: §2.1 · p. 2

2.2 Relation to Correlation Clustering

  • Unlike chromatic correlation clustering and classical correlation clustering, this objective ignores non-edges and merges separate same-color clusters without worsening the objective.The authors interpret non-edges as missing information that may reflect unmeasured relationships.Source: §2.2 · p. 2
  • The objective is polynomial-time solvable for two categories and admits practical 2-approximation algorithms for more categories, unlike related methods described as harder or less practical.The proposed approximation approaches use LP relaxations small enough to solve quickly in practice.Source: §2.2 · p. 2

2.3 Additional related work

  • Existing categorical-feature clustering is not designed for graph data, while graph-attribute and multilayer methods do not connect edge types to cluster indicators.Source: §2.3 · p. 3

3 THE CASE OF TWO CATEGORIES

For two categories, the paper reduces categorical edge clustering on graphs and hypergraphs to minimum s-t cut. The hypergraph construction exactly represents hyperedge mistakes, while its more general cut formulation can be slower in practice than the LP relaxations.

3.1 An algorithm for graphs

  • For two-category graphs, the construction converts the labeled graph to a weighted unlabeled graph whose minimum s-t cut has the same value as the clustering objective.The source and sink correspond to the two category terminals, and a minimum cut directly gives a minimizer.Source: §3.1 · p. 3

3.2 An algorithm for hypergraphs

  • The hypergraph reduction is the first direct network-analysis application of a construction similar to one used for a Potts model in computer vision.Source: §3.2 · p. 3
  • A minimum s-t cut yields an optimal clustering for two-category hypergraphs, with each hyperedge represented by a subgraph in the reduction.The construction directly models hyperedge mistakes rather than approximating hyperedge cuts with weighted graph cuts.Source: §3.2 · p. 3
  • Both minimum-cut algorithms use O(T) vertices and edges, where T is the sum of hyperedge degrees, but maximum-flow formulations can be slower than the LP relaxations.The supplied passage gives a worst-case O(T^2) time bound.Source: §3.2 · p. 3
  • For unlabeled or ambiguously labeled edges, the objective can penalize only splitting their nodes across clusters, and the reductions connect those nodes to both terminals.Source: §3.2 · p. 4

4 MORE THAN TWO CATEGORIES

For more than two categories, minimizing the objective is NP-hard, so this section develops LP-rounding and multiway-cut approximations, plus a linear-penalty variant optimized by majority vote.

4.1 NP-hardness of Categorical Edge Clustering

  • Three-label Categorical Edge Clustering is NP-hard via a max-cut reduction: each gadget costs three mistakes exactly when its endpoints receive opposite red/blue clusters.The reduction equates a cut of at least K edges with a coloring having at most 4|E| − K mistakes.Source: §4.1 · p. 4

4.2 Algorithms based on LP relaxations

  • For more than two labels, LP rounding gives a 2-approximation, while randomized rounding achieves expected 2−1/k or 2−1/(r+1); with two labels, the LP is integral.The LP solver also finds solutions in seconds on hypergraphs with tens of thousands of hyperedges.Source: §4.2 · pp. 4–6

4.3 Algorithms based on multiway cut

  • Multiway-cut reductions translate a p-approximation into a p(r + 1)/2-approximation; a node-weighted reduction gives a tighter 2(1−1/k) LP bound, but its LPs are too large to be practical.The isolating-cuts approach can be expensive, while the LP relaxation performs better in solution quality and running time in the experiments.Source: §4.3 · p. 6

4.4 Approximation through a linear objective

  • Replacing the unit penalty per mistaken hyperedge with a per-node mismatch penalty makes majority vote optimal for the linear objective and an r-approximation for the original edge objective.The linear penalty for a hyperedge is at most r, compared with a penalty of 1 under the Categorical Edge Clustering objective.Source: §4.4 · pp. 6–7

5 EXPERIMENTS

The experiments evaluate the framework on real and synthetic graphs and hypergraphs, categorical community detection, temporal clustering, and a cooking hypergraph. Across these settings, LP performs strongly against baselines, while the cooking analysis explores trade-offs and ingredient clusters.

5.1 Analysis on Real Graphs and Hypergraphs

  • Experiments used LP-round and Cat-IsoCut against Majority Vote and chromatic-ball baselines on datasets spanning brain, drug, publication, cooking, and shopping interactions.For hypergraph comparisons, CB and LCB were adapted by reducing hypergraphs to edge-labeled graphs; LP was faster and more accurate than Cat-IsoCut across datasets.Source: §5.1 · p. 7
  • Across real graphs and hypergraphs, LP outperformed all other methods on every dataset, often finding exactly optimal solutions; Table 1 reports approximation guarantees, edge satisfaction, and running times.The LP was solvable within seconds or minutes on nearly all datasets; Walmart was an exception, with nearly 4 million variables and many more constraints.Source: §5.1 · p. 7, Table 1 · p. 8

5.2 Categorical Edge Community Detection

  • LP generally led on synthetic graph and 3-uniform hypergraph community detection, but hypergraph accuracy became sensitive when noise rose above 0.6.For the synthetic hypergraph experiment with L = 15, accuracy was nearly perfect for w ≤ 0.6; increasing w to 0.65 disrupted ground-truth recovery despite an integral LP solution.Source: Figure 3 · p. 8, §5.2 · p. 9
  • LP-Round consistently outperformed other methods in clustering nodes in both an email graph and a product co-purchasing hypergraph; with L = 20 and K varied from 50 to 500, LP always had better label-assignment scores and often better ARI scores.On the co-purchased-products hypergraph, LP-Round detected true departments at a much higher rate than the other methods.Source: Figure 5 · p. 9, Figure 4 · p. 9, §5.2 · p. 9

5.3 Temporal Community Detection

  • On the CollegeMsg network, LP was competitive with Graclus on normalized cut while finding clusters with smaller temporal deviations in their interior-edge timestamps.The timestamps were binned into edge labels, so the clustering sought groups of users exchanging messages within a short period.Source: §5.3 · pp. 9–10, Figure 6 · p. 10

5.4 Analysis of the Cooking Hypergraph

  • In the cooking hypergraph, LP outperformed Majority Vote on edge satisfaction and unused ingredients, while increasing β lowered edge satisfaction and reduced ingredient waste.At β = 10, LP edge satisfaction exceeded 0.64; at β = 170, LP identified 10 Brazilian-cuisine ingredients associated with 23 additional recipes, 21 of them Caipirinha variants.Source: §5.4 · p. 10, Figure 7 · p. 10

6 DISCUSSION

The framework combines strong practical performance with approximation guarantees, and often exactly minimizes its objective on large hypergraphs. Energy-minimization alternatives, however, apply only in specified special cases.

6 DISCUSSION

  • For general problems, LP relaxations provide 2-approximation or better guarantees, and often produce integral solutions that exactly minimize the objective.The exact solutions are certified through integral LP solutions.Source: §6 · pp. 10–11
  • The objective is polynomial-time solvable with two categories, while LP-rounding often solves the NP-hard objective exactly in practice.The reported exact solutions include hypergraphs with tens of thousands of edges, solved in tens of seconds.Source: §6 · pp. 10–11
  • The approach outperforms baselines substantially on problems evaluated against ground-truth labels.The LP-round algorithm also often certifies exact objective minima on hypergraphs with tens of thousands of edges in tens of seconds.Source: §6 · p. 11
  • Energy-minimization methods apply to two-category graphs and rank-3 hypergraphs, but not to more than two categories or general hypergraphs.For general hypergraphs, the penalties are no longer a semi-metric, which is needed for approximation algorithms.Source: §6 · p. 11

A CONNECTION TO ENERGY MINIMIZATION

The paper connects special instances of its clustering objective to energy minimization in computer vision. This connection supplies additional polynomial-time exact algorithms for those instances.

A CONNECTION TO ENERGY MINIMIZATION

  • The objective fits energy minimization for two-category graphs and rank-3 hypergraphs.The connection is to the energy minimization framework in computer vision.Source: A CONNECTION TO ENERGY MINIMIZATION · p. 12
  • Energy minimization uses minimum s-t cuts for binary and ternary variables satisfying a regularity property.The paper establishes that its objective has this property in the stated special cases.Source: A CONNECTION TO ENERGY MINIMIZATION · p. 12
  • This connection provides additional tools for solving those special instances exactly in polynomial time.These tools complement the algorithms developed in the paper.Source: A CONNECTION TO ENERGY MINIMIZATION · p. 12

A.1 Graphs with two categories

For two-category graphs, the objective can be expressed as a regular energy function and reduced to a minimum s-t cut. The resulting cut gives an optimal clustering.

A.1 Graphs with two categories

  • The clustering assigns each node one of two colors, encoded as a vector whose entries indicate each node's color.The paper writes the coloring as C = (x1, ..., xn), with each xi taking color 1 or 2.Source: §A.1 · p. 12
  • For two-category graphs, the Categorical Edge Clustering objective induces a regular energy function.The stated regularity condition is that diagonal terms sum to 1 while off-diagonal terms sum to 2.Source: §A.1 · p. 12
  • The reduction constructs a directed graph whose cut represents each original edge's energy, so a minimum s-t cut optimizes the clustering objective.The construction adds source and sink nodes and directs edges according to the original edge color.Source: §A.1 · pp. 12–13

A.2 Rank-3 hypergraphs with two categories

The energy-minimization construction extends to rank-3 hypergraphs with two categories. Its minimum s-t cut yields an optimal solution to the original hypergraph objective.

A.2 Rank-3 hypergraphs with two categories

  • The energy-minimization framework also handles rank-3 hypergraphs with two categories.The objective is regular when all projections into two variables are regular.Source: §A.2 · p. 13
  • The objective for rank-3 hypergraphs with two categories induces a regular energy function.The paper states this as a theorem.Source: §A.2 · p. 13
  • A constructed graph encodes the hypergraph objective so its minimum s-t cut optimizes the original clustering problem.Rank-3 hyperedges use one of two directed tree structures, chosen according to hyperedge color.Source: §A.2 · p. 13
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