2.1 Equivalence of persistence surfaces

The notion of a network’s persistence curve was recently introduced to study the differences among complex networks [23]. Here, we build on this concept, introducing the idea of a persistence surface (see Definition 3). The goal is to use these surfaces to determine a nonparametric partition of the network nodes that respects the network’s homotopic structure.

2.2 Persistence curves

Definition 1. (Persistence Curves) Let G=(V,E) be the graph of a network with nonempty vertex and edge sets. For each , consider the set of all persistence intervals, . Note that when , the interval is half-open: . Without loss of generality, we order so that for , and if then j < k. The dimension-D persistence curve of is the linear interpolation in of the set of points .

Persistence curves allow us to visualize how many intervals of a given persistence there are in the network. This is demonstrated in the following example.

Example 2. (Persistence Curves of the Gundangborn Genealogical Network) The Gundangborn network shown in Fig 2 (left) is a genealogical network consisting of thirty-two individuals where edges indicate either a parent-child or union relationship [27] (see S1 Text and S1 Data). The dimension-1 persistence intervals of the network are . The corresponding cycles are shown in Fig 2 (left) in red and blue, respectively. The intervals are both shown in blue in Fig 2 (right), with the corresponding birth and death persistence curves in purple and cyan, respectively. The death persistence curve is constructed by outlining (interpolating) the upper endpoints of the persistence intervals, and the birth persistence curve outlines the lower endpoints.

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Fig 2. Left: Genealogical network for the Gundangborn of Arnhem Land, Australia consisting of thirty-two individuals (see).

The relations are of two types: either parent-child relations or union relationships but are not distinguished here. The cycles generating the [1,2] and [1,3] persistence intervals are shown in red and blue, respectively. Right: The dimension-1 persistence intervals and the birth and death persistence curves for the network are shown in blue, purple, and cyan, respectively.

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2.3 Persistence surfaces

A network’s persistence curve gives us a global picture of its structure, describing how the network’s cycles are related (or their equivalent higher-dimensional analogs). Starting from a given vertex it is possible to use this notion to describe the homological structure of the network as one moves farther from v. The result is the local persistence structure of the network as viewed from the point of view of a specific vertex. To define this structure, we will use the concept of an ego-graph, which allows us to view the network centered at a specific vertex restricted to a specific radius.

Definition 2. (Ego-Graph) Let G=(V,E) be a network, and let . The ego-graph of G centered at v is the induced subgraph of G with vertex set , i.e., all vertices reachable by a path of length at most r from vertex v, and all edges that exist between these vertices in the original network G.

In general, for a network G=(V,E) and a vertex we can always form the sequence of persistence curves for the ego-graphs where is the diameter of the network. If , i.e., G is disconnected, we can simply restrict our attention to the graph’s connected components where the diameter is finite. This sequence of networks can be used to create a persistence surface of the vertex v in the following way.

Definition 3. (Persistence Surface) Let G=(V,E) be a network of diameter and let . The persistence surface of v is the linear interpolation of all persistence curves of the ego-graphs for in this order.

Example 3. (Ego-Graph of the Gundangborn Network) Consider the Gundangborn network G=(V,E) in Fig 2 (left) with the vertex . In Fig 3 the ego-graphs are shown from top to bottom on the left-hand side of the left column, where the vertex v is labeled. On the right side of the column are the persistence curves corresponding to each ego-graph, respectively, from top to bottom.

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Fig 3. Left: The ego-graphs for the vertex v in the Gundangborn network (see Fig 2, top) are shown left for together with the corresponding persistence curve, shown right.

Following the arrows, the persistence surface is constructed by stacking the persistence curves (see Example 3 for details). The purple and cyan surfaces are the persistence surface created using birth and death times, respectively.

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To create the persistence surface we arrange the persistence curves along the y-axis, which indicates the ego-graph radius. The other axes indicate the persistence interval index (x-axis) and the birth or death time (z-axis). This results in a piecewise linear surface. Note that for a fixed radius , the cross-section y = r produces the persistence curve of the ego-graph at radius r.

In an effort to improve the appearance of the surface, we choose a viewing window of where I is the number of persistence intervals in , R is the maximum distance from v to any other vertex, and T is the maximum finite death time of any persistence interval in . The process of first creating the ego graphs then persistence curves, and then the persistence surface is indicated in Fig 3. The result is the persistence surface shown at the bottom right. Note that the purple and cyan colors represent the persistence curves and surfaces created when using the birth and death times of the persistence intervals, respectively.

To create a vertex partition of the graph G=(V,E) we require the following characterization of a persistence surface in terms of what we refer to as ledges.

Definition 4. (Persistence Surface Ledge) Let be the set of ego-graphs for a vertex in a network given by G = (V,E), where is the diameter of G. Let be such that the multiset of persistence intervals present in is not equivalent to the multiset of persistence intervals present in . In other words, the ego-graph contains at least one additional persistence interval more than the ego-graph . If this is the case a ledge is formed at radius i + 1. This ledge is visually represented by a ridge or shelf in the persistence surface at r = i + 1.

The sequence of ledges is the list of radii at which ledges form as r increases from 0 to d. The sequence of ledge heights is the death times of the longest persistence interval corresponding to the ledge for .

In dimensions one and two, a vertex’s persistence surface measures a vertex’s relation to the cycles and closed singular surfaces in the network, respectively, as we move further and further from the vertex. This allows us to incrementally observe the persistent homology of the network as we increase our distance from a given vertex. Specifically, each ledge at radius of the surface corresponds to a different cycle or set of cycles with death-time . Since the surface is ordered in such a way that the radius increases at each step, the ledges occur in order (moving along the x-axis) of the cycles that become trivial as the radius of the ego-graph increases.

We note that just as persistence intervals belong to a specific dimension, so do persistence curves and persistence surfaces. To simplify illustrating our method, we consider only persistence surfaces for dimensions one and two in this paper. The dimension one persistence surfaces of the Gundangborn network are considered in the following example.

Example 4. (Persistence Surfaces of the Gundangborn Genealogical Network) The Gundangborn network G=(V,E) has |V| = 34 individuals each with their own persistence surface . The surfaces of the vertices can be grouped according to type (see Fig 4):

1. i. the surface has two ledges that both remain in line with the top of the surface;
2. ii. the surface has one ledge; and
3. iii. the surface has two ledges, the first of which peaks above the top of the surface before dropping back down.

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Fig 4. Shown left, center, and right are two distinct persistent surfaces for each of the surface types (i)-(iii) of the Gundangborn network considered in Example 2.

Each surface type corresponds to two different vertices in Fig 2 (top) colored red, green, and yellow; respectively.

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These surfaces suggest that each individual in the network plays one of three roles relative to the network’s persistent homology within the network corresponding to (i)-(iii). If we color these individuals in (i)-(iii) red, green, and yellow, respectively, the result of this partition is shown in Fig 2 (top). The red vertices see the five-cycle “first,” represented by the first ledge, and then “later” see the larger seven-cycle represented by the second ledge. Each of the green vertices see both cycles at the same time-step, corresponding to the single ledge of their surfaces. The yellow vertices see the seven-cycle first, corresponding to the first large ledge, and then later see the smaller five-cycle corresponding to the second ledge. Thus, the types of persistence surfaces reflect the order in which each vertex first “sees” the cycles in a given network in terms of distance.

The vertex partition described in Example 4 is relative to the cycle structure or higher-dimensional analogues of the network. In Section 3, we consider how well the elements of this partition align with the standard notions of communities and roles in real and synthetic networks. In the following, we formalize the method used in Example 4 to partition the vertex set of the Gundangborn network.

2.4 Persistence surface partitioning algorithm

To formalize the grouping of surfaces described in Example 4 we define an equivalence relation on the set of persistence surfaces of a network.

Definition 5. (Persistence Partition) Suppose G=(V,E) has vertices . The persistence surfaces and are topologically equivalent if they have the same sequence of ledge heights . In this case, we write . This is an equivalence relation that induces a partition of V, where each is a collection of vertices with topologically equivalent surfaces. For fixed dimension we refer to this collection as the persistence partition of G, where each is a persistence cell of G.

We refer to the process of finding a network’s persistence birth surfaces and using these to determine the network’s persistence partition as the Persistence Surface Algorithm (PSA). We use the program Ripser (from the Python package Ripser) [28] to calculate the persistence intervals of a network given by G=(V,E). Ripser has a computational and spatial complexity of O((n + m)³) where n = |V| and m = |E|. Here, the number n + m is the number of network simplices.

Using the Persistence Surface Algorithm requires that we calculate the persistent homology for each ego-graph in the network, resulting in the Ripser algorithm being run at most d times, where d is the diameter of the network. Because this is repeated for each node, finding the network’s persistence surfaces has a computational and spatial complexity O(dn(n + m)³). While this may be prohibitively expensive for large networks, we note that a network’s persistence partition can be approximated by limiting the size of a vertex’s ego-graph to some maximal radius . This yields what may be viewed as a local persistence partition, whose computation has temporal and spatial complexity where denotes the maximum number of vertices in an ego–graph over all , and denotes the maximum number of edges among these ego-graphs.

A natural question is whether persistence partitions capture traditional community structure or, alternatively, particular role structures or core–periphery organization within a network. This is investigated in the following section.

3.1 Persistence equivalence and role detection

Classical approaches to role detection in social networks are grounded in positional analysis, which seeks to group vertices (actors) that share similar patterns of relationships with other vertices (actors). These methods partition network vertices into roles based on various equivalence relations [9,10,13].

More recently, new techniques have been developed that assign roles to vertices using stochastic equivalence and network embeddings. Stochastic equivalence partitions vertices according to a probability distribution: two vertices share the same role if exchanging them yields a graph with the same distribution. Network embedding methods, in contrast, learn low-dimensional representations of networks to uncover role structures, employing tools such as low-rank matrix factorization, random-walk–based algorithms, and deep learning methods [11].

Among these approaches, the methods most closely aligned with the concept of a persistence partition are those that define explicit equivalence relations. In each, the vertices of a network G=(V,E) are partitioned into the sets where the vertices in are assumed to play the same role for each . The principal forms of this type of equivalence are structural equivalence, automorphic equivalence, and regular equivalence, with structural equivalence being the most restrictive (see [9], for instance).

Definition 6. (Structural Equivalence) For the graph G = (V,E) let be the set of neighbors of . Two vertices are structurally equivalent if .

Since structural equivalence is an equivalence relation, the vertex set V of the graph G=(V,E) is partitioned into equivalence classes . We refer to this collection as the structural partition of G, where each is a structural cell of G.

The second type of equivalence considered in the literature is automorphic equivalence which is related to graph symmetries, which are known to be ubiquitous in real-world networks [29]. Symmetries of a graph G=(V,E) can be understood in terms of automorphisms that preserve adjacencies. More intuitively, an automorphism describes how parts of a graph can be interchanged without affecting the structure of the graph. These parts (i.e., subgraphs) are symmetric and together with the automorphism constitute a graph symmetry or vertices that are equivalent with respect to symmetry.

Definition 7. (Automorphism Equivalence) An automorphism of a graph G = (V,E) is a permutation of V such that the adjacency matrix A = A(G) satisfies

for each pair of vertices i and j. The orbit of a vertex is the vertex set

where ℓ is the smallest positive integer such that is the identity. Vertices in the same orbit are automorphically equivalent with respect to .

If G=(V,E) has an automorphism , the orbits of partition the vertices of G into the equivalence class of automorphic vertices . This is referred to as the automorphic partition induced by , where each is an automorphic cell of G. As a graph may have many graph automorphisms it may have many different automorphic partitions. Our interest in both structural and automorphic equivalence is whether such vertices are persistence equivalent.

Definition 8. (Persistence Equivalence) For the graph G = (V,E) let be its persistence partition. Vertices in the same persistence cell are said to be persistence equivalent for .

A partition P is a refinement of the partition Q if every cell (element) of P is a subset of some cell (element) of Q. Two partitions are incomparable if neither is a refinement of the other. It is known that structural equivalence is stricter than automorphic equivalence meaning that if is an automorphic partition of G then it is a refinement of its structural partition [13]. Similarly, automorphic equivalence is stricter than persistence equivalence.

Theorem 1. (Persistence and Automorphic Equivalence) Suppose is an automorphic partition of the graph G. Then is a refinement of the persistence partition of G in any dimension .

Proof. Let the graph where the edge set . Since then or G is isomorphic to meaning G is the graph except the vertices of have been relabeled. Thus, if then both have isomorphic ego-graphs for all . Because the ego-graphs determine the persistence curves in any dimension which in turn determine the persistence surface, then both and have the same persistence surface and are in the same partition element of the network’s persistence partition.

Given that the relation if is an equivalence relation on the vertices of V, then all vertices in the same orbit of are in the same persistence cell. □

As automorphisms preserve adjacencies between vertices of a graph, Theorem 1 indicates that if two vertices are symmetric, then they are structurally indistinguishable via some automorphism. Therefore, each ego-graph of each vertex is structurally identical, so both vertices have the same persistence surface and are therefore in the same persistence cell.

With this result in mind, we consider two generalizations of symmetries: equitable partitions and latent symmetries. Equitable partitions have a history of being used in network analysis to partition networks into distinct roles (see, for instance, [30]). Latent symmetries, however, represent a new class of symmetries that have been recently developed [31] but have yet to be applied to role analysis.

While classically associated with the spectral properties of a graph [32], equitable partitions have more recently been connected with dynamical phenomena such as synchronization and network growth [33–36]. The definition is as follows:

Definition 9 (Equitable Partition). An equitable partition of a graph G=(V,E) with adjacency matrix is a partition of V with the property that for all , the sum

is constant for any .

For a simple graph G=(V,E), an equitable partition can be thought of as dividing the vertices of the graph into the partition where every vertex in has the same number of neighbors in for all . A graph can have many different equitable partitions. To define the notion of equitable equivalence we use the a graph’s coarsest equitable partition, which has the useful property that it is unique for any graph [37].

Definition 10. (Equitable Equivalence) An equitable partition of a graph G is the graph’s coarsest equitable partition if, for any other equitable partition on G, the cell for some . Vertices in the same cell of the coarsest equitable partition are said to be equitably equivalent.

We refer to the elements of a graph’s coarsest equitable partition as its equitable cells. As shown in [32], every automorphism induces an equitable partition. This implies that automorphic equivalence is stricter than equitable equivalence.

Theorem 2. (Equitable and Automorphic Equivalence) Suppose is an automorphic partition of the graph G. Then is a refinement of the coarsest equitable partition of G.

Proof. The automorphic partition of G has the property that the set is an equitable partition of G. Since any equitable partition is a refinement of the graph’s coarsest equitable partition by Definition 10 the result follows. □

A second generalization of the notion of a graph symmetry is the notion of a latent symmetry. Latent symmetries are defined as automorphisms of a network with graph G=(V,E) after it has been transformed via an isospectral reduction. Isospectral reductions were first developed in [38] and have been used more recently to improve eigenvalue estimates, determine the dynamic stability of dynamical systems, perform link prediction on real-world networks, characterize cospectral vertices, etc. (see [39–42]). An isospectral reduction of a graph (matrix) is a transformation that reduces the size of a graph while preserving its spectral properties, e.g., eigenvalues and eigenvectors (see [43] for a detailed description and further applications).

An isospectral reduction of a graph (matrix) is defined as follows:

Definition 11. (Isospectral Reduction) The isospectral reduction of a matrix over the proper subset is the matrix

where is the complement of S in N. The isospectral reduction of a graph G=(V,E) with adjacency matrix A over the set is defined as the graph with adjacency matrix .

In Definition 11 the set is the set of rational functions in the variable . The parameter is used because the eigenvalues of the matrix and the smaller reduced matrix are typically the same. In order not to violate the Fundamental Theorem of Algebra, the entries of cannot be scalars but instead are objects that can carry more information.

Definition 12. (Latent Equivalence) Suppose is an isospectral reduction of the graph G=(V,E). If there exists an automorphism , then the vertices of that are in the same orbit of are referred to as latently equivalent.

The notion of latent equivalence is based on the idea of a latent symmetry described in [31]. It can be shown that latent equivalence defines an equivalence relation. Moreover, consistent with previous results, automorphic equivalence is stricter than latent equivalence, at least for the undirected graphs considered here.

Theorem 3. (Latent and Automorphic Equivalence) Latent equivalence defines an equivalence relation on the vertices of a graph. Furthermore, given an automorphic partition of the graph G, this partition is a refinement of the latent partition of G.

Proof. Corollary 3.4 of [42] states that two vertices in an undirected graph are latently symmetric if and only if they are cospectral. As cospectrality is an equivalence relation the result follows. □

Example 5. (Automorphic, Persistent, Equitable, and Latent Partitions) Consider the graph G=(V,E) in Fig 5 shown in each of the four panels. The graph has the automorphic partition , the persistence partition , the coarsest equitable partition , and the latent partition . Vertices in the same cell of a given partition share the same color. The automophic partition, shown top center, is a refinement of each of the other partitions as guaranteed by Theorems 1–3. As the graph G is a regular graph its coarsest equitable partition is a single set E₁ = V.

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Fig 5. Different partitions of the graph G=(V,E) are shown where vertices in the same cell share the same color for a given partition.

Top Center: An automorphic partition is shown. Bottom Left: The persistence partition is shown. Bottom Center: The coarsest equitable partition consisting of a single cell is shown. Bottom Right: The latent partition is shown. The automorphic partition is a refinement of each of the persistence, equitable, and latent partition, respectively (cf. Theorem 1).

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In Fig 5, the latent partition of the graph constitutes a refinement of its persistence partition, which itself refines the coarsest equitable partition of the graph. This ordering need not hold universally. As illustrated in Fig 6, the persistence and latent partitions may be incomparable. It remains an open problem to determine whether any nontrivial structural relations exist among a graph’s persistence, coarsest equitable, and latent partitions, such as those established for automorphic and related partitions in Theorems 1–3.

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Fig 6. Left: The persistent partition of the graph G=(V,E) is shown where vertices in the same persistent cell have the same color.

Right: The latent partition is shown where vertices in the same persistence cell have the same color. Although the two partitions have the same number of cells, the two partitions are incomparable.

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Since structural equivalence is stricter than automorphic equivalence combining this with Theorems 1–3 yields the following corollary.

Corollary 1. (Role Equivalence and Refinements) For a graph G=(V,E) the automorphic, persistence, equitable, and latent partitions of G are all refinements of the structural partition of G.

For example, the structural partitions of the graphs in Figs 5 and 6 are trivial: each cell contains exactly one vertex. Consequently, these structural partitions refine every other partition on their respective graphs. In contrast, in Fig 8 (the Zachary Karate Club network), the structural partition contains two nontrivial cells, corresponding to the orange and green vertex sets, respectively.

Finally, it is worth noting that regular equivalence is a relaxation of structural equivalence that defines roles by how nodes are similarly connected to equivalent types of other nodes, rather than requiring them to be identically connected [10]. However, we do not consider regular equivalence here, as determining a graph’s regular partition can require knowledge of the roles of each vertex in the graph (see [12], for instance). Because this goes beyond the information provided by the graph’s topology alone, it falls outside the scope of our analysis.

3.2 Persistence partitions of some synthetic networks

Partitions of real-world networks are often created to detect the community, role, or the core and peripheral structure of a network. Here our goal is to understand whether the PSA finds the partition structure either found or embedded in two network models. The two models we consider are (i) the degree corrected stochastic block model (DCSBM) and (ii) random geometric graphs (RGG).

Definition 13. (Degree Corrected Stochastic Block Model) A degree corrected stochastic block model is a model that creates an undirected graph G=(V,E) with planted clusters. The vertex set V is partitioned into communities where a symmetric matrix determines the probability that vertices and are connected. These probabilities are modified by the parameters so that there is a directed edge from vertex u to vertex v with probability

where is choosen such that . The resulting directed graph is undirected to create the simple graph G=(V,E).

Example 6. (DCSMB and persistence partitions) Consider the stochastic block model network with two communities C₁ and C₂ where . The probability of edges between vertices of the same community is , and the probability of edges between communities is . We choose each uniformly at random for each k = 1,2,…,100. A realization of this DCSBM is shown in Fig 7 (left). In contrast, choosing each results in the standard SMB model. A realization of this SBM is shown in Fig 7 (center) using , and .

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Fig 7. Left: A degree corrected stochastic block model (DCSMB) network on two embedded communities C₁ and C₂ of size fifty each is shown.

The network has fourteen distinct cells, indicated by node color, which are not always contained in either C₁ and C₂. Center: A stochastic block model network (SBM) also on two embedded communities C₁ and C₂ of size fifty each is shown (see Example 6 for details). The network has sixteen distinct cells, indicated by node color, which similarly are not always contained in either C₁ and C₂. Right: The four cells in the network’s persistence partition of a random geometric graph are shown where cells are indicated by color. The larger three cells are non-localized, spreading across the entire network (see Example 7).

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Running PSA in dimension-2 on this network results in the colored persistence partition indicated in Fig 7 (left, center). What can be seen here, and what happens in general, is that cells of persistence partitions need not belong to either C₁ or C₂ but may have vertices in both.

To understand the extent to which these persistence cells are aligned with DCSBM and SMB communities, i.e., contained within C₁ or C₂, we compare the adjusted mutual information (AMI) score of the persistence cells against the DCSBM and SMB communities to the AMI scores of one thousand random permutations of the vertex labels against the DCSBM and SMB communities, respectively.

We find that the persistence cells have an average AMI score of 0.032 for the degree-corrected stochastic block model (DCSBM), indicating that the relationship between persistence cells and the embedded communities in this model is negligible. In contrast, for the standard stochastic block model (SBM), we observe an AMI score of 0.37, reflecting a substantially higher relationship. However, even in this case, the correspondence is only moderate and does not constitute a meaningful match between the persistence cells and the embedded communities.

Next, we consider random geometric graphs, which are defined as follows.

Definition 14. (Random Geometric Graph) A random geometric graph (RGG) model is a model that creates an undirected graph G = (V, E) where the vertices are randomly sampled from the uniform distribution spanning an underlying metric space. Vertices are connected by an edge if and only if the distance between them is less than a parameter r > 0.

Example 7. (RGG and persistence partitions) In Fig 7, an RGG is shown together with its persistence cells for vertices a placed uniformly in a subset of . In this specific realization there are four persistence cells shown as red, purple, yellow, and two blue vertices are spread throughout the network cutting across the embedded communities, i.e., local subgraphs with high modularity. This is in contrast to what is typically seen in the SBM model we considered in Example 6. While the precise reason for this is unknown, this indicates that persistence cells can be quite different from standard communities, and it is an open question as to how the number, size, and distribution of persistence cells depend on the parameters |V| = n and r > 0 in an RGG model.

3.3 Persistence partitions of two real-world networks

Zachary’s karate club is a social network that models the social interactions of members of a karate club. This network is of particular interest as the club eventually split, with roughly half of the students following the club instructor and the other half following the club president [44] (see S2 Text and S2 Data). This split is recovered almost exactly by maximizing the modularity over all two-community partitions (see Fig 8, top left). We refer to these as the instructor community and president community, which are shown in red and blue, respectively.

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Fig 8. Zachary’s karate club is shown both left and right, top and bottom with the club’s instructor and president indicated by and , respectively.

Top Left: Maximizing modularity over all two-community partitions results in those shown in red and blue. Top Right: The twelve persistence cells of Zachary’s karate club are shown, indicated by color. Bottom Left: The automorphic, coarsest equitable, and latent partitions of the network are shown; in this case, all three coincide. Colors denote the cells of the partition, with the exception that each blue vertex forms its own singleton cell. Bottom Right: The core-periphery structure of the network is shown created using isospectral reductions.

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The persistence partition of the unweighted Zachary’s karate club also recovers this split to some degree (see Fig 8, right). In particular, all of the network’s twelve persistence cells except one belong to either the instructor’s community or the president’s community. We refer to the exception as the mixed community, which is shown in purple. The president and instructor each form their own persistence cell, and the other cells form the peripheral structure of the network on the side of the instructor and/or president. In this way of thinking, each individual performs a specific type of role within the network relative to the instructor and president. For instance, the individuals in the dark green community interact only with the president and the second most well-connected individual in the president’s community by degree.

To provide a point of comparison, we analyze the network’s core–periphery structure (see, e.g., [16,45,46]). Detection is performed using the method proposed in [18], which is based on isospectral reduction (see Definition 11). Applying this method using the criterion that selects all vertices except those with the smallest degree centrality yields the core–periphery structure shown in Fig 8 (bottom right). This procedure identifies seven hierarchical levels. This includes the core followed by successive peripheral layers, corresponding to the vertices colored red, yellow, purple, orange, green, indigo, and dark blue. The innermost core (red vertices) contains both the club’s instructor and president, along with several members occupying paths between them. As one moves outward, the peripheral layers expand from these central vertices. This pattern differs substantially from that of the network’s persistence partition resulting in two partitions are incomparable. The key difference is that the core–periphery structure is less refined, merging the orange and green vertices of the automorphic/equitable/latent partition, whereas these are separated in the persistence partition.

Currently, it is unknown whether persistence partitions correspond more to specific roles in real-world networks, to standard notions of communities, or are better suited for describing core and peripheral structures (cf. the New York taxicab network, SMB and RGG models, and the Zachary karate club, respectively).

As a second example, the New York taxicabs network G=(V,E) consists of |V| = 265 taxi zones or neighborhoods in New York City where there is an edge between neighborhoods i and j if there was a taxicab that picked up a passenger in neighborhood i and dropped them off in neighborhood j [47] (see S3 Text and S3 Data). Here we consider the edges to be undirected and unweighted, although, as mentioned in the introduction, both directionality and edge weight could be incorporated into our persistence surface algorithm. For simplicity, we only consider the undirected and unweighted version of this network, ignoring the directionality and frequency of travel.

The persistence partition of the New York taxicab network is shown in Fig 9, where color is used to indicate each of the cells. The network itself is not shown, but rather the neighborhood map of New York City. This is to emphasize the fact that persistence cells are often not spatially connected but are spread throughout the city’s five boroughs. That is, there are neighborhoods that are distant from one another, but nevertheless see the same types of traffic dynamics. We note that this interpretation suggests that persistence partitions represent specific transportation roles, as opposed to the more classical notion of communities, such as the city’s boroughs.

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Fig 9. The persistence partition of the New York taxicab network is shown where color is used to indicate each of the cell.

The network consists of vertices representing city neighborhoods and edges indicating whether taxi cabs move passengers from one neighborhood to another. The network itself is not shown but rather the neighborhood map of New York City is shown to scale. The majority of the ten persistence cells are spread throughout each of the five New York City boroughs. The base layer of the map can be found at https://d37ci6vzurychx.cloudfront.net/misc/taxi_zones.zip. The dataset is publicly available under the New York City Open Data Terms of Use: https://opendata.cityofnewyork.us/overview/#termsofuse.

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Similarly to Zachary’s karate club, the taxicab’s persistence cells do not appear to represent standard communities, rather specific types of communities that are more aligned with roles or functions, each determined by structure rather than modularity. In the context of the New York City taxicab network, this distinction may be useful in the process of designing and building city roads and other traffic-related infrastructure, as topologically similar regions may experience similar traffic patterns. If a particular infrastructure works well in neighborhood i, we might predict that the same infrastructure may be useful in region j if neighborhoods i and j are in the same persistence cell.