Persistence partitions of real and synthetic networks

Determining network structures that are neither local nor global is an area of research that has received considerable attention. The study of these intermediate structures has been primarily concerned with the detection of network communities but also includes the examination of network roles, core and peripheral structure, etc. In an increasingly relevant line of research, persistent homology has also been used to analyze the shape of a network in terms of the network’s cycle structure and its higher-dimensional analogues. In this work, we bring these two perspectives together by introducing a non-parametric partition of a network derived from its persistent homology. This partition, which we call the network’s persistence partition, is defined using the concept of a persistence surface, assigning to each node a measure of its individual persistence relative to its position in the network. We examine the extent to which this partition aligns with standard notions of network roles defined via combinatorial equivalence. We then compare how persistence partitions relate to communities and to the core–periphery structure of a network. Our analysis draws on both real and synthetic networks and demonstrates that persistent homology can reveal distinctive structural features that are not detected by conventional methods.Author summary: The position a node occupies in a network influences quantities such as the node’s importance, role(s), community affiliation(s), etc. within the network. As we show in this paper, this position is possible to characterize relative to the voids and gaps found in the network. Here the voids and gaps refer to places left empty by the network’s cycles and their higher-dimensional analogues, which are the objects of study of persistent homology. To characterize this homology we introduce the notion of a node’s persistence surface, which reflects the order in which each network cycle, etc. are encountered as we move away from the node. Comparing these surfaces naturally leads to a partition of the network: nodes whose persistence surfaces coincide belong to the same class. The resulting persistence-based partition is uniquely determined by the choice of distance function on the network. It provides a way to analyze how a network’s persistent homology induces a node partition, and to compare this with partitions obtained from other network-science techniques. This includes node partitions by community, role, core and peripheral structure, etc. Some initial results and discussion relative to real and synthetic networks are described in this paper. A node’s position within a network shapes many of its properties, including its importance, functional role, and potential community or core–periphery affiliations. In this paper, we show that this positional information can be characterized in terms of the network’s voids and gaps—regions left empty by cycles and their higher-dimensional analogues, which are precisely the structures studied in persistent homology. To capture this relationship, we introduce the notion of a node’s persistence surface, which records the order in which the network’s cycles and higher-dimensional features are encountered as one expands outward from the node. Comparing these surfaces naturally leads to a partition of the network: nodes whose persistence surfaces coincide (equivalently, nodes that “see” the same homological features in the same order) belong to the same class. The resulting persistence-based partition, which is uniquely determined by the chosen distance function on the network, provides a principled way to understand how persistent homology induces a node equivalence relation. This perspective enables direct comparison with partitions arising from other areas of network science, including those based on communities, roles, core–periphery structure, and more. In this paper, we present initial results and examples illustrating how persistence-based partitions behave on both synthetic and real networks.