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Higher-Order Networks Representation and Learning: A Survey

Hao Tian, Reza Zafarani

TL;DR

This survey addresses the need to move beyond dyadic network representations by examining three main higher-order formalisms—motifs, simplicial complexes, and hypergraphs—along with their foundations, algorithms, and applications. It highlights motif-based methods for frequency analysis, clustering, representation learning, and link prediction, and then details simplicial complexes and their homology/cohomology, Hodge Laplacians, and random complex models for topological data analysis. It also covers hypergraphs, including tensor representations, cuts, random walks, downgrading to dyadic graphs, and modern learning approaches such as hypergraph neural networks, complemented by datasets and tools. Collectively, the paper provides a structured, cross-domain view of higher-order network representations and learning techniques, guiding researchers toward data sources, scalable methods, and practical ML applications across science and engineering.

Abstract

Network data has become widespread, larger, and more complex over the years. Traditional network data is dyadic, capturing the relations among pairs of entities. With the need to model interactions among more than two entities, significant research has focused on higher-order networks and ways to represent, analyze, and learn from them. There are two main directions to studying higher-order networks. One direction has focused on capturing higher-order patterns in traditional (dyadic) graphs by changing the basic unit of study from nodes to small frequently observed subgraphs, called motifs. As most existing network data comes in the form of pairwise dyadic relationships, studying higher-order structures within such graphs may uncover new insights. The second direction aims to directly model higher-order interactions using new and more complex representations such as simplicial complexes or hypergraphs. Some of these models have long been proposed, but improvements in computational power and the advent of new computational techniques have increased their popularity. Our goal in this paper is to provide a succinct yet comprehensive summary of the advanced higher-order network analysis techniques. We provide a systematic review of its foundations and algorithms, along with use cases and applications of higher-order networks in various scientific domains.

Higher-Order Networks Representation and Learning: A Survey

TL;DR

This survey addresses the need to move beyond dyadic network representations by examining three main higher-order formalisms—motifs, simplicial complexes, and hypergraphs—along with their foundations, algorithms, and applications. It highlights motif-based methods for frequency analysis, clustering, representation learning, and link prediction, and then details simplicial complexes and their homology/cohomology, Hodge Laplacians, and random complex models for topological data analysis. It also covers hypergraphs, including tensor representations, cuts, random walks, downgrading to dyadic graphs, and modern learning approaches such as hypergraph neural networks, complemented by datasets and tools. Collectively, the paper provides a structured, cross-domain view of higher-order network representations and learning techniques, guiding researchers toward data sources, scalable methods, and practical ML applications across science and engineering.

Abstract

Network data has become widespread, larger, and more complex over the years. Traditional network data is dyadic, capturing the relations among pairs of entities. With the need to model interactions among more than two entities, significant research has focused on higher-order networks and ways to represent, analyze, and learn from them. There are two main directions to studying higher-order networks. One direction has focused on capturing higher-order patterns in traditional (dyadic) graphs by changing the basic unit of study from nodes to small frequently observed subgraphs, called motifs. As most existing network data comes in the form of pairwise dyadic relationships, studying higher-order structures within such graphs may uncover new insights. The second direction aims to directly model higher-order interactions using new and more complex representations such as simplicial complexes or hypergraphs. Some of these models have long been proposed, but improvements in computational power and the advent of new computational techniques have increased their popularity. Our goal in this paper is to provide a succinct yet comprehensive summary of the advanced higher-order network analysis techniques. We provide a systematic review of its foundations and algorithms, along with use cases and applications of higher-order networks in various scientific domains.
Paper Structure (47 sections, 20 equations, 1 figure, 4 tables)

This paper contains 47 sections, 20 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary of Graphs
  3. Network Motifs
  4. Foundation and Algorithms
  5. Motif Frequency
  6. Motif Matrix and Motif Cuts
  7. Motif Clustering Coefficient
  8. Use Cases and Applications
  9. Capturing Functionalities in Networks
  10. Network Classification
  11. Network Models
  12. Clustering
  13. Representation Learning
  14. Link Prediction
  15. Simplicial Complexes
  16. ...and 32 more sections

Figures (1)

  • Figure 1: Comparison among Higher-Order Network Representations. (a). A simple graph (left) along with its motif graph (right) of 3-Cliques (triangles). With the same set of nodes, motif graphs transform edges into "membership in given motifs." From the example, the given motif is a triangle, so there are two triangles detected and only one edge $(v_1,v_3)$ shared by both triangles. The edge $(v_4,v_4)$ that does not belong to any motif will be ignored. (b). A Simplicial Complex, including a 0-simplex (single node), a 2-simplex (triangle), and a 3-simplex (tetrahedron). Note that any face (sub-simplex) of an existing simplex is also included in the simplicial complex, for example, $(v_5,v_6,v_7)$. (c). A Hypergraph. Unlike simplicial complexes, any subedge of hyperedges does not have to appear in the edge set, i.e., $(v_1,v_2,v_3)$ and $(v_2,v_3)$ are two different hyperedges.

Theorems & Definitions (7)

  • Definition 3.1: Network motif
  • Definition 4.1: Simplicial Complex
  • Definition 4.2: Boundary Operator
  • Definition 4.3: Hodge Laplacian
  • Definition 4.4: Degree of a Simplex
  • Definition 4.5: Random 2-Complex
  • Definition 5.1: $s$-walk