New Topological Tool Splits Networks Into Persistence-Based Partitions

Networks are everywhere. The neurons of the brain, the routers of the internet, the proteins inside a single cell, and the friendships of an entire society can all be described as collections of nodes joined by edges. For decades, scientists have mined these structures for hidden patterns, and two questions have dominated the field: which nodes form tightly knit communities, and which nodes play similar roles within the overall architecture. A new study published in PLOS Complex Systems argues that both questions miss something important, and it proposes a way of finding it. By borrowing tools from a branch of mathematics called persistent homology, a team led by Abigail Jenkins, Nick Callor, Zachary M. Boyd, Taylor Gledhill, Raelynn Wonnacott and Benjamin Z. Webb has introduced what they call the persistence partition, a new way of carving a network into groups that reflects neither pure community structure nor pure role structure, but the shape of the network itself.

The central insight of the paper is that some of the most interesting structures in a network are neither local nor global. Communities are local: they describe clusters of nodes that are densely connected with one another and sparsely connected to the rest. Roles are closer to global: two nodes occupy the same role when they relate to the rest of the network in combinatorially equivalent ways, regardless of whether they are neighbors. Between these two poles lies a middle ground of structure that standard community detection and role analysis were never designed to capture. The researchers set out to fill that gap, and they did so by turning to topology, the mathematics of shape, holes and connectivity.

Persistent homology has become one of the most influential exports of applied topology. In essence, it watches how the holes in a shape appear and disappear as the shape is thickened or thinned. Applied to a network, it tracks the cycle structure of the graph and its higher-dimensional analogues: the triangles, tetrahedra and beyond that form when groups of nodes become mutually connected. Each such feature persists over a range of scales, and the length of that range is its persistence. Features with long persistence are considered robust, genuine components of the network’s shape, while features that flicker in and out over a narrow range are treated as noise. This multiscale character is what makes the method so attractive for data that has no single natural scale.

The new work converts this topological machinery into a partition of the network’s nodes. The key construct is the persistence surface, a function that assigns to each node a measure of its individual persistence relative to its position in the network. In other words, rather than asking only which cycles exist in the graph as a whole, the method asks how much of the network’s persistent topological structure each node is responsible for. Nodes are then grouped according to these measures, producing the persistence partition. Because the construction is non-parametric, it does not require the analyst to choose a number of groups in advance or to tune a resolution parameter, a freedom that distinguishes it from many clustering techniques.

Once the partition is defined, the obvious question is what, exactly, it is measuring. The authors examine the extent to which persistence partitions align with standard notions of network roles defined via combinatorial equivalence, the strictest and most classical definition of structural similarity between nodes. If the persistence partition simply reproduced role classes, it would be a redundant, if computationally convenient, proxy. The analysis shows that it does not. The partition captures information that role equivalence classes do not, suggesting that the topological persistence of a node encodes aspects of its structural situation that combinatorial symmetry alone cannot see.

The second comparison is with communities, the workhorse concept of network science. Community detection algorithms, from modularity optimization to stochastic block models, are built to find dense clusters, and they have been spectacularly successful across applications. Yet the persistence partition is not a community detection method in disguise. The researchers compare how persistence partitions relate to communities and find that the two views of the network diverge in informative ways. A node’s contribution to the persistent cycle structure of a graph can cut across community boundaries, revealing groupings that no density-based criterion would ever produce. This is precisely the kind of intermediate structure the authors set out to isolate.

The third comparison concerns core–periphery structure, another staple of network analysis. Many real systems, from financial networks to social webs, are organized around a dense core of highly connected nodes surrounded by a sparser periphery. Core–periphery detection asks which nodes belong to which region. The persistence partition offers a different lens on the same question, and the study shows that the topological grouping relates to core–periphery structure without collapsing into it. Nodes that carry the network’s persistent cycles may or may not be the nodes that a conventional core–periphery analysis would label as core members, and the discrepancies are themselves a source of insight.

Crucially, the authors do not rest their case on a single example. Their analysis draws on both real and synthetic networks, allowing them to test the method on empirical data with known structure and on controlled models where the ground truth can be manipulated. Across this range, the conclusion holds: persistent homology reveals distinctive structural features that are not detected by conventional methods. The synthetic networks serve as a laboratory, showing how the persistence partition responds when communities, roles or core–periphery organization are deliberately built in or removed, while the real networks demonstrate that the signal survives contact with the messiness of empirical data.

The significance of the work lies in what it adds to the toolbox rather than in what it replaces. Community detection, role analysis and core–periphery methods each answer a well-posed question, and none of them is rendered obsolete. What the persistence partition provides is a complementary description, grounded in the multiscale topology of the graph, that can be consulted alongside the others. In applications where the shape of the flow matters, such as understanding how information, disease or failure propagates through a system, the cycles that persistent homology tracks may be directly relevant, since cycles provide alternative routes and redundancy that trees and clusters do not.

The paper also contributes to a broader movement in network science toward structure at intermediate scales. As datasets grow and systems become more complex, researchers have increasingly recognized that a single partition, whether by community, role or core status, cannot exhaust the organization of a real network. The persistence partition joins a growing family of multiscale, topologically informed descriptions that treat a network as a geometric object with shape, not merely as a list of pairwise connections. Whether the method becomes a standard instrument will depend on how it performs across further domains, but the study makes a clear case that the holes in a network, and the nodes that sustain them, deserve a place in the analysis. For a field built on counting edges, it turns out that counting what persists between them can reveal an entirely new layer of structure.

Subject of Research: Persistent homology-based partitioning of real and synthetic networks

Article Title: Persistence partitions of real and synthetic networks

Article References: Jenkins, A., Callor, N., Boyd, Z. M., Gledhill, T., Wonnacott, R., & Webb, B. Z. (2026). Persistence partitions of real and synthetic networks. PLOS Complex Systems, 3(7), e0000109. https://doi.org/10.1371/journal.pcsy.0000109

Image Credits: AI Generated

DOI: 10.1371/journal.pcsy.0000109

Keywords: network science, persistent homology, topological data analysis, persistence partition, community detection, network roles, core-periphery structure, complex systems, graph theory, synthetic networks, cycle structure, multiscale analysis