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Persistence Homology of Networks: Methods and Applications
Mehmet Emin Aktas, Esra Akbas, Ahmed El Fatmaoui
TL;DR
The paper addresses the challenge of comparing networks topologically when classical measures may miss connected components and holes. It reviews persistent homology, network filtrations, algorithms, and applications, synthesizing recent approaches and their conceptual distinctions. Its conclusion identifies scalability as a limitation for future progress.
Problem
Network comparison requires topology-aware similarity or distance measures because classical measures may overlook connected components and holes.
Method
The paper provides a conceptual review of persistent homology on complex networks, including mathematical background, filtrations, algorithms, and applications.
Results
The review synthesizes recent persistent-homology approaches for network analysis and emphasizes their conceptual distinctions.
Takeaways & Limitations
Persistent homology summarizes network topological features across multiple scales for network analysis.
Takeaways & Limitations
Scalability may constrain future progress in persistent-homology studies of networks.
Abstract
from arXiv · showhide
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks. In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.
1 Introduction
Information networks require topology-aware similarity measures because classical graph measures often rely on local node or edge information and may miss connected components or holes. The paper reviews persistent homology methods, filtrations, algorithms, and applications for network mining.
- Classical graph-theoretic similarity measures are often local and overlook network topology, including connected components and holes.
- Persistent homology measures topological features that persist across multiple scales by tracking changes in simplicial complexes.
- Persistent homology provides multiscale graph summaries unlike traditional metrics that describe graphs from specific angles.
- The review compares network filtrations according to properties such as sensitivity to directed, undirected, weighted, and unweighted networks.
- The paper highlights algorithms and applications that use persistent homology to solve network mining problems and concludes with future-work directions.
2 Preliminaries
The preliminaries represent networks as graphs and extend them into simplicial complexes whose homology captures connected components, holes, and higher-dimensional voids. Persistent homology tracks these features as simplices are added across parameter scales.
- 2.3 Simplicial homology: Homology associates vector spaces with topological objects, whose dimensions encode connected components, holes, and voids.
- 2.4 Persistent homology: Persistent homology adds simplices using a distance function and parameter, then tracks homology changes to identify features persisting across scales.
- 2.1 Graphs: Graphs represent objects as vertices and relationships as edges, with directed, undirected, weighted, metric, and dynamic variants.
- 2.2 Simplicial complex: A simplicial complex is a collection of simplices containing every simplex face and allowing intersections only along common faces.
- 2.2 Simplicial complex: The clique complex converts each k-clique of an undirected graph into a (k −1)-simplex.
3 Filtrations
The paper reviews network filtrations and compares their sensitivity to graph types and construction choices. Examples include Vietoris-Rips variants and directed Dowker filtrations, which can encode relationships to hub nodes.
- 3 Filtrations: The review compares network filtrations by properties including sensitivity to directed or undirected and weighted or unweighted network types.
- Vietoris-Rips filtration: The Vietoris-Rips filtration forms clique complexes from subgraphs containing edges whose weights are at most a threshold.
- Inverse Vietoris-Rips filtration: The inverse Vietoris-Rips filtration adds larger-weight edges earlier by decreasing the threshold from maximum to minimum weight.
- Dowker filtrations: Dowker sink and source filtrations use directed edges organized around sink or source vertices at a threshold.
- Dowker filtrations: Dowker sink and source filtrations may suit networks where simplices should form around particular hub nodes, and both generate the same persistent diagram.
3.3 Clique complex filtration (CCL)
The clique complex filtration builds a nested sequence of complexes by increasing simplex dimension. Vertices enter first, followed by edges, triangles, and higher-dimensional simplices.
- The construction starts with a graph G having n vertices and its clique complex Cl(G).
- The filtration forms nested clique complexes from Cl_0(G) through Cl_n(G)=Cl(G).
- Each complex is constructed from the skeletons of the clique complex.A jth skeleton contains simplices of dimension at most j.
- Vertices are added at δ = 0, edges at δ = 1, triangles at δ = 2, and so on.
3.4 Vertex-based clique filtration (VBCL)
The vertex-based clique filtration uses vertex weights to threshold a graph and its clique complex, while a related construction tracks k-clique communities through a connectivity graph.
- Vertex weights, rather than edge weights, determine threshold values in the vertex-based clique filtration.
- At threshold δ, the 1-skeleton retains vertices with ω(v) ≤ δ and edges whose endpoint maximum weight is at most δ.
- The inverse vertex-based clique filtration reverses the thresholding direction for vertex-weighted cliques.
- k-clique community construction: For fixed k, the method detects k-clique communities and creates a connectivity graph with one vertex per k-clique.
- k-clique community construction: Two k-cliques are adjacent when they intersect in a (k − 1)-clique, meaning they share k − 1 vertices.
- k-clique community construction: This filtration focuses on the evolution of k-clique communities within the original graph.
3.6 Weighted simplex filtration (WS)
Weighted simplex filtrations assign appearance thresholds directly to simplices, whereas vertex-function filtrations generate nested subgraphs from vertex values and then take their clique complexes.
- Weighted simplex construction: Weighted simplex filtrations assign each simplex a weight based on relationship functions in a dissimilarity network.
- Weighted simplex construction: At threshold δ, Kδ contains the simplices appearing before or at δ.
- Weighted simplex construction: A valid filtration requires every face and simplex intersection to appear no later than the simplex containing them.
- Vertex-function-based construction: In a vertex-function-based filtration, sublevel graphs retain vertices with f(v) ≤ δ and edges whose endpoints are both retained.
- Vertex-function-based construction: The sublevel vertex-function filtration takes the clique complex of each sublevel graph.
- Vertex-function-based construction: A corresponding superlevel construction retains vertices with f(v) ≥ δ while δ decreases from ∞.
3.8 Intrinsic Čech filtration (IC)
The intrinsic Čech filtration is defined for metric graphs by taking nerves of open metric-ball covers at varying radii. Related metric-graph filtrations use distance-based sublevel or superlevel sets.
- Intrinsic Čech construction: The intrinsic Čech filtration applies only to metric graphs with geometric realization |G|.
- Intrinsic Čech construction: For each point x and radius δ, it forms the open ball B(x,δ) and the cover Uδ of all such balls.
- Intrinsic Čech construction: The complex Cδ is the nerve of Uδ, with a simplex whenever the corresponding balls have a nonempty common intersection.
- Intrinsic Čech construction: The resulting intrinsic Čech filtration is given by inclusion maps across scales.
- Functional metric graph construction: A related functional metric graph filtration uses geodesic distance from a fixed point and superlevel sets of that distance function.
- Functional metric graph construction: Depending on the problem, either superlevel or sublevel sets may be used.
3.10 Power filtration (POW)
The power filtration builds clique complexes from graph powers, where vertices become adjacent when their original graph distance is within a scale parameter. For dynamic networks, the related temporal filtration applies clique complexes to networks accumulated over time.
- Graph powers use distances derived from walks and paths, with path length defined by the number of edges.
- The rth power G_r preserves the original vertices and connects u and v exactly when d(u,v) ≤ r.Graph distance is the minimum path length between two vertices, optionally incorporating edge weights.
- The power filtration is the clique complex of each graph power G_r over an appropriate distance range.
- For a growing network sequence, the temporal filtration takes the clique complex of each network accumulated through time.
3.12 Zigzag simplicial filtration (ZSF)
Zigzag simplicial filtrations extend standard graph filtrations to dynamic settings by permitting both simplex insertion and deletion. Zigzag persistent homology then records topological features across these changing complexes.
- Dynamic graph vertex and edge deletions remove simplices, while insertions add simplices to the associated simplicial complexes.
- Zigzag filtrations allow simplicial complexes to become smaller, unlike standard filtrations that only expand with the scale parameter.
- The discontinuity points of a zigzag simplicial filtration must be locally finite.Each discontinuity point has a neighborhood containing only finitely many other discontinuity points.
- A scale-parameter condition relates neighboring graphs through inclusions with a sufficiently small epsilon.
- Zigzag persistent homology produces persistence barcodes or diagrams, including zero-dimensional barcodes that track connected components.
3.13 Digraph filtration using Persistent Path Homology (PPH)
Persistent path homology constructs chain complexes from directed weighted graphs using allowed, regular paths and a directed edge-weight filtration. It offers direction-sensitive network topology with stability and a conditional agreement with Dowker filtrations.
- Persistent path homology is sensitive to edge directions and is applied after constructing a filtration of directed weighted graphs.
- The construction starts with formal vertex sequences, then restricts them to regular paths whose consecutive vertices are distinct and whose consecutive pairs are directed edges.
- Allowed paths do not themselves form a chain complex because their boundaries may be disallowed, so invariant subspaces are introduced to resolve this issue.
- The resulting invariant path spaces form a chain complex and yield path homology groups.
- In undirected graphs, PPH and Dowker filtrations agree in dimension 1 under a square-free local condition, and PPH satisfies a stability result.
3.14 Generalizations of Vietoris-Rips filtration (GVR)
Ordered-tuple complexes generalize simplicial-complex persistence while retaining tuple order, enabling directed and preorder variants of Vietoris–Rips filtrations. The framework includes four such generalizations, each with a stability theorem.
- Ordered-tuple complexes are closed under deleting entries, while tuples with different orderings remain distinct.
- The framework defines chain complexes, homology, and k-dimensional ordered-tuple persistence homology for ordered-tuple complexes.
- Four generalizations of Vietoris–Rips filtrations are defined, and each case has a stability theorem.
- Directed Vietoris-Rips filtration: The directed Vietoris–Rips filtration includes an ordered tuple when f(v_i,v_j) ≤ t for all i ≤ j.
- Associated filtration of directed graphs: The associated directed-graph filtration includes vertices and directed edges when the relevant self- and pairwise weights are at most t.
- Preorder filtration: The preorder filtration consists of ordered tuples compatible with the filtration of ordered posets.
4 Algorithms and Applications
The review organizes persistent-homology applications in network science into single-graph and multiple-graph analyses, covering algorithms, filtrations, and network-mining uses. These applications extract structural, temporal, comparative, and domain-specific information from network topology.
- The review categorizes persistent-homology applications as single-graph or multiple-graph analyses and explains the algorithms and applications within each category.
- Single Graph Analysis: Removing each node and recording the resulting Betti-number change uses the drop in complexity to identify potential drug targets.The node whose removal causes the largest Betti-number drop is considered the most promising target.
- Single Graph Analysis: Persistent entropy and approximated von Neumann entropy recognize immune activation, while connectivity entropy does not distinguish activation from immune memory.Betti-number analysis also identifies a one-dimensional hole present in both activation and memory states.
- Single Graph Analysis: Vertex-based and k-clique filtrations analyze clique connectivity across degrees and weight thresholds, supporting interactive tracking of clique-community evolution.The approach is demonstrated on co-occurrence, brain, and collaboration networks.
- Multiple Graphs Analysis: Persistent-homology distances and vineyards capture differences between networks, temporal topological changes, and geometric models, with several studies reporting advantages over simpler or standard alternatives.Examples include persistence distortion distances growing roughly proportionally to noise, and persistent vineyards outperforming PCA and standard graph-theoretical methods in brain-network analysis.
- Multiple Graphs Analysis: Persistent homology supports network comparison and classification by distinguishing collaboration patterns, anonymized-network stability, and engineering communities with different research interests.Reported applications include structural similarity, temporal patterns, and comparisons of original versus anonymized online-social-network graphs.
5 Conclusion
The paper reviews persistent homology methods, algorithms, and applications for extracting multiscale topological features from networks. It identifies scalability, unproven filtration stability, and limited support for evolving networks as important future challenges.
- Review scope: The review examines research using persistent homology on networks and organizes methods for extracting their topological features.It also reviews applications in network-mining problems.
- Review scope: The reviewed studies primarily aim to extract network topological features that persist across multiple scales.This multiscale perspective is the implicit goal identified across most studies.
- Limitations and future work: Most reviewed studies use small networks with fewer than 1000 vertices, while scaling persistent-homology approaches to larger networks remains necessary.Scalability is identified as a concern for future progress.
- Limitations and future work: Some proposed filtrations still lack proven stability, limiting the theoretical guarantees available for those approaches.The review explicitly identifies filtration stability as an unresolved issue.
- Limitations and future work: Many filtration methods target static networks, although real-world networks evolve through continuously added and deleted edges.Existing methods therefore cannot generally be applied directly to large-scale evolving networks.
- Limitations and future work: The review highlights dynamic filtration algorithms and network or subnetwork embeddings as promising directions for persistent-homology research.These directions address evolving-network structure and representation problems.