Exploring Graph and Digraph Persistence

1. Introduction

Computational topology and homological persistence provide swift algorithms to quantify geometrical and topological aspects of triangulable manifolds [1]. Let M be a triangulable manifold, given a continuous function $f:M\to \mathbb{R}$, persistent homology consists in studying the evolution of homological classes throughout the sublevel-set filtration induced by f on M. Indeed, in this context, the word persistence refers to the lifespan of homological classes along the filtration. In intuitive terms, this procedure allows us to see M using f as a lens capable of highlighting features of interest in M.

Many extensions and generalizations of topological persistence aim to augment the field of application of persistent homology. Such generalizations allow for studying objects different than triangulable manifolds, maintaining the valuable properties granted by the persistent homology framework, e.g., stability [2], universality [3], resistance to occlusions [4], computability [5].

In most cases, generalizations of persistent homology can be attained in two ways. On the one hand, it is possible to define explicit mappings from the object of interest to a triangulable manifold preserving the properties of the original object, see, e.g., [6]. On the other hand, theoretical generalizations aim to extend the definition of persistence to objects other than triangulable manifolds and functors other than homology. These generalizations come in many flavors that we shall briefly recall in Section 2.1. Here, we shall consider the rank-based persistence framework and its developments and applications detailed in [7,8,9]. This framework revolves around an axiomatic approach, that yields usable definitions of persistence in categories such as graphs and quivers, without requiring the usage of auxiliary topological constructions.

In the current manuscript, following the approach described in [9,10], we provide strategies to compute rank-based persistence on weighted graphs (either directed or undirected). Weighted (di)graphs are widely used in many real-world scenarios ranging from analysis of interactions in social networks to context-based recommendation. Modern machine learning provides specific neural architectures to process directed graphs, e.g., [11,12,13]. As neural networks assure maximum flexibility by learning directly from training data, graph theory offers a plethora of invariants that—albeit static—can be leveraged as descriptors of directed, weighted networks [14,15,16]. We provide an algorithmic approach to encode such features as persistence diagrams carrying compact information that can be leveraged complementarily or even be integrated in modern neural analysis pipelines.

Organization of the paper

Section 2 briefly outlines various generalizations of persistence, and the current methods taking advantage of persistent homology on graphs and directed graphs (digraphs). Section 3 recalls the essential concepts for the generalized theory of persistence used in this manuscript. Namely, categorical and indexing-aware persistence functions. These functions are based on graph-theoretical features. In particular, we introduce the concept of simple features in Section 4, and single-vertex features in Section 5. This latter class of features yields plug-and-play methods for applications: we provide an implementation of single-vertex feature persistence in Section 6, and apply it in the context of trust networks. Section 7 concludes the paper.

2. State of the art

Oftentimes, modern datasets can be endowed with a network-like structure. At the same time, topological persistence, thanks to its aforementioned mathematically provable properties, proved to be a valuable tool in many applications. Thus, generalizations of topological persistence and its adaptations to objects such as graphs and directed graphs are lively research lines.

2.1. Generalizations of persistence

The keystone of persistence theory is classically the persistence module, i.e. a functor from the category $(\mathbb{R},\le )$ to the category of modules on a fixed ring (generally a field), that catches the essential features of a filtered topological space or simplicial complex. As hinted previously, the filtration is mostly constituted by the nested sublevel-sets of a real filtering function. The modules are the images of the linear maps induced in homology by the inclusions. This setting provides great modularity: different filtering functions may catch different features [17]. The most used filtering function is the ball radius in building Vietoris-Rips or Čech complexes out of sampled objects [18].

Extensions of the classical persistence theory offer much greater flexibility. The first generalization was the passage from a single-parameter to multi-parameter filtering functions [19,20] and to different ranges for the filtering function [21]. However, we believe that the most impactful generalizations are in categorical terms. In [22], persistence theory is extended to functors from $(\mathbb{R},\le )$ to any Abelian category, in particular for what concerns the interleaving distance. This distance is also investigated in [3] for multidimensional domains, and extended in [23] to functors from a poset to an arbitrary category. Posets as domains are studied in [24] and [25] for a generalized notion of persistence diagram. The extension on which the present paper is based is exposed in [7] and aims to widen the range of phenomena that can give rise to persistence diagrams. This is done via rank functions which generalize the dimension of homology modules in classical persistence.

2.2. Persistent homology for graphs and digraphs

Data frequently come in the form of graphs or digraphs, and researchers were often attracted by the possibility of applying persistence to the analysis of networks. The classical strategy consists in building a filtered simplicial complex from a weighted graph, and then compute persistent homology. This can be done in several ways [6]. Of course, the most direct way is to consider the graph itself as a 1-dimensional simplicial complex (or CW-complex if loops and multiple edges are allowed). This is done, e.g., in [26], where a connection between Laplacian, persistence, and network circuit graphs is drawn. [27] does persistence on discrete curvature for the evaluation of graph generative models. The clique complex is also a very natural way of transforming a graph into a possibly multidimensional complex. This latter construction is leveraged in [28] for the study of deep neural networks, in [29] for the analysis of the dynamics of scientific knowledge, and also in [30] and [31] (among many others) for the representation and study of the human connectome. Persistent homology of clique-communities complexes is applied to the study of general types of complex networks in [32,33]. Other constructions of simplicial complexes—e.g. independent sets, neighborhoods, dominating sets—are also discussed and utilized in applications but, to the best of our knowledge, not in connection with persistence.

The construction of simplicial complexes from digraphs poses similar but not coinciding problems than its undirected counterpart. In [34] directed cliques are used to investigate the structural and functional topology of the brain. A differently flavored homology is built in [35,36], where the role of simplices is played by directed paths. Directed paths are also at the base of an application of Hochschild homology [37, Ch.1]—through the use of connectivity digraphs and within the framework of [7]—for a new persistence homology for digraphs in [38].

3. Graph-theoretical persistence

To the best of our knowledge, current applications of persistence to graphs and digraphs leverage homology of some kind. Here, we provide a non-necessarily-homological approach to the study of graphs and digraphs. We recall the basic notions of the extension of topological persistence to graph theory introduced in [7,8]. For an encompassing view on classical topological persistence, we refer the reader to [39,40].

3.1. Categorical persistence functions

Definition 1. 

[7, Def. 3.2] Let $\mathbf{C}$ be a category. We say that a lower-bounded function $p:Morph\left(\mathbf{C}\right)\to \mathbb{Z}$ is a categorical persistence function if, for all $u_1\to u_2\to v_1\to v_2$, the following inequalities hold:

• $p(u1\to v1)\le p(u2\to v1)$ and $p(u2\to v2)\le p(u2\to v1)$
• $p(u2\to v1)-p(u1\to v1)\ge p(u2\to v2)-p(u1\to v2)$

With $\mathbf{C}=(\mathbb{R},\le )$, the graphs of such functions have the appearance of superimposed triangles typical of Persistent Betti Number functions [39, Sect. 2], so they can be condensed into persistence diagrams [7, Def. 3.16].

In the remainder, we shall write “(di)graph” for “graph (respectively digraph)”, extending this notation to cases such as sub(di)graph et similia. Symmetrically and when no confusion arises, we shall denote with $\mathbf{Gr}$ the category $\mathbf{Graph}$ or the category $\mathbf{Digraph}$, with monomorphisms as morphisms. We shall use the noun edge for edges of a graph and arcs of a digraph.

Filtered (di)graphs$(G,f)$ are pairs consisting of a (di)graph $G=(V,E)$ and a filtering function $f:V\cup E\to \mathbb{R}$ on the edges and extended to each vertex as the minimum of the values of its incident edges. For any $u\in \mathbb{R}$, the sublevel (di)graph$G_{f,u}$ (briefly $G_u$ if no confusion occurs) is the sub(di)graph induced by the edges e such that $f\left(e\right)\le u$.

Definition 2. 

[8, Def. 5] The natural pseudodistance of filtered (di)graphs $(G,f)$ and $(G^{\prime },f^{\prime })$ is

$$\delta \left((G,f),(G^{\prime },f^{\prime })\right)=\left\{\begin{array}{cc}+\infty \hfill & \mathrm{if}\mathcal{H}=\varnothing \hfill \\ \underset{\varphi \in \mathcal{H}}{inf}\underset{e\in E}{sup}|f\left(e\right)-f^{\prime }\left(\varphi \left(e\right)\right)|\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where $\mathcal{H}$ is the set of (di)graph isomorphisms between G and $G^{\prime }$.

Let p be a categorical persistence function on $\mathbf{Gr}$, $(G,f)$, $(G^{\prime },f^{\prime })$ any filtered (di)graphs, $\mathcal{D}$, ${\mathcal{D}}^{\prime }$ the respective persistence diagrams, and d be the bottleneck distance [39, Sect. 6][7, Def. 3.24].

Definition 3. 

[8, Sect. 2] The categorical persistence function p is said to be stable if the inequality

$$d(\mathcal{D},{\mathcal{D}}^{\prime })\le \delta ((G,f),(G^{\prime },f^{\prime }))$$

holds. Moreover, the bottleneck distance is said to be universal with respect to p, if it yields the best possible lower bound for the natural pseudodistance, among the possible distances between $\mathcal{D}$ and ${\mathcal{D}}^{\prime }$, for any $(G,f)$, $(G^{\prime },f^{\prime })$.

3.2. Indexing-aware persistence functions

The difference between any of the categorical persistence functions introduced in [8] and the functions of the present subsection (presented originally in [9]) is that the former comes from a functor defined on $\mathbf{Gr}$, while the latter strictly depends on the filtration, thus descending from a functor defined on $(\mathbb{R},\le )$.

Definition 4. 

[9, Def. 5] Let p be a map assigning to each filtered (di)graph $(G,f)$ a categorical persistence function $p_{(G,f)}$ on $(\mathbb{R},\le )$, such that $p_{(G,f)}=p_{(G^{\prime },f^{\prime })}$ whenever an isomorphism between $(G,f)$ and $(G^{\prime },f^{\prime })$ compatible with the functions $f,f^{\prime }$ exists.

All the resulting categorical persistence functions $p_{(G,f)}$ are called indexing-aware persistence functions (ip-functions for brevity). The map p itself is called an ip-function generator.

Definition 5. 

[9, Def. 6, Prop. 1] Let p be an ip-function generator on $\mathbf{Gr}$. The map p itself and the resulting ip-functions are said to be balanced if the following condition is satisfied. Let $G=(V,E)$ be any (di)graph, f and g two filtering functions on G, and $p_{G,f)}$ and $p_{(G,g)}$ their ip-functions. If a positive real number h exists, such that ${sup}_{e\in E}|f\left(e\right)-g\left(e\right)|\le h$, then for all $(u,v)\in {\Delta }^{+}$ the inequality $p_{(G,f)}(u-h,v+h)\le p_{(G,g)}(u,v)$ holds.

Proposition 6. [9, Thm. 1] Balanced ip-functions are stable.

We now recall our main ip-function generators: steady and ranging sets.

Definition 7. 

[9, Def. 8] Given a (di)graph $G=(V,E)$, any function $\mathcal{F}:2^{V\cup E}\to \{true,false\}$ is called a feature. Conventionally, for $X\subset V\cup E$ we set $\mathcal{F}\left(X\right)=false$ in a sub(di)graph $G^{\prime }=(V^{\prime },E^{\prime })$ if $X⊈V^{\prime }\cup E^{\prime }$.

Remark 8. 

The second part of Def. 7 was added for use in some of the following proofs.

Let $\mathcal{F}$ be a feature.

Definition 9. 

[9, Def. 8] We call $\mathcal{F}$-set any $X\subset V\cup E$ such that $\mathcal{F}\left(X\right)=true$. In a weighted (di)graph G, we shall say that $X\subset V\cup E$ is an $\mathcal{F}$-set at level$w\in \mathbb{R}$ if it is an $\mathcal{F}$-set of $G_w$.

Definition 10. 

[9, Def. 9] We define the maximal feature $m\mathcal{F}$ associated with $\mathcal{F}$ as follows: for any $X\subseteq (V\cup E)$, $m\mathcal{F}\left(X\right)=true$ if and only if $\mathcal{F}\left(X\right)=true$ and there is no $Y\subseteq (V\cup E)$ such that $X\subset Y$ and $\mathcal{F}\left(Y\right)=true$.

Definition 11. 

[9, Def. 10] A set $X\subseteq V\cup E$ is a steady$\mathcal{F}$-set (s$\mathcal{F}$-set for brevity) at $(u,v)$ if it is an $\mathcal{F}$-set at all levels w with $u\le w\le v$.

We call X a ranging$\mathcal{F}$-set (r$\mathcal{F}$-set) at $(u,v)$ if there exist levels $w\le u$ and $w^{\prime }\ge v$ at which it is an $\mathcal{F}$-set.

Let $S_{(G,f)}^{\mathcal{F}}(u,v)$ be the set of s$\mathcal{F}$-sets at $(u,v)$ and let $R_{(G,f)}^{\mathcal{F}}(u,v)$ be the set of r$\mathcal{F}$-sets at $(u,v)$.

Definition 12. 

[9, Def. 11] For any filtered (di)graph $(G,f)$ we define ${\sigma }_{(G,f)}^{\mathcal{F}}$ (resp. ${\varrho }_{(G,f)}^{\mathcal{F}}$) as the function which assigns to $(u,v)\in {\mathbb{R}}^2,u<v$, the number $|S_{(G,f)}^{\mathcal{F}}(u,v)|$ (resp. $|R_{(G,f)}^{\mathcal{F}}(u,v)|$).

We denote by ${\sigma }^{\mathcal{F}}$ and ${\varrho }^{\mathcal{F}}$ the maps assigning ${\sigma }_{(G,f)}^{\mathcal{F}}$ and ${\varrho }_{(G,f)}^{\mathcal{F}}$ respectively to $(G,f)$.

Proposition 13. [9, Prop. 2] The maps ${\sigma }^{\mathcal{F}}$ and ${\varrho }^{\mathcal{F}}$ are ip-function generators.

The next definition will be generalized in Section 4.

Definition 14. 

[9, Def. 13] We say that a feature $\mathcal{F}$ is monotone if

• for any (di)graphs $G^{\prime }=(V^{\prime },E^{\prime })\subset G^{\prime \prime }=(V^{\prime \prime },E^{\prime \prime })$, and any $X\subseteq (V^{\prime }\cup E^{\prime })$, $\mathcal{F}\left(X\right)=true$ in $G^{\prime \prime }$ implies $\mathcal{F}\left(X\right)=true$ in $G^{\prime }$
• in any (di)graph $\overline{G}=(\overline{V},\overline{E})$, for any $Y\subset X\subseteq \overline{V}\cup \overline{E}$, $\mathcal{F}\left(X\right)=true$ implies $\mathcal{F}\left(Y\right)=true$.

Proposition 15. [9, Prop. 3] If $\mathcal{F}$ is monotone, then ${\sigma }^{\mathcal{F}}={\varrho }^{\mathcal{F}}$

Proposition 16. [9, Prop. 4] If $\mathcal{F}$ is monotone, then the ip-function generators ${\sigma }^{\mathcal{F}}={\varrho }^{\mathcal{F}}$ are balanced.

4. Simple features

We now define a class of features extending the class of monotone features (Def. 14).

Definition 17. 

Let $G=(V,E)$ be a (di)graph. A feature $\mathcal{F}$ is said to be simple if, for $X\subseteq (V\cup E)$ and for sub(di)graphs $H_1\subseteq H_2\subset H_3\subseteq G$ the following condition holds:

if $\mathcal{F}\left(X\right)=true$ in $H_3$ and $\mathcal{F}\left(X\right)=false$ in $H_2$, then $\mathcal{F}\left(X\right)=false$ in $H_1$

For the remainder of this section, let $(G,f)$ be a weighted (di)graph, $G=(V,E)$, and $\mathcal{F}$ a simple feature in G.

Lemma 18. 

Let $X\subseteq (V\cup E)$. Then, either there is no value u for which $\mathcal{F}\left(X\right)=true$ in $G_u$, or $\mathcal{F}\left(X\right)=true$ in $G_u$ for all $u\in [u_1,v_1)$, where $u_1$ is the lowest value u such that in the sub(di)graph $G_u=(V_u,E_u)$ one has $\mathcal{F}\left(X\right)=true$ in $G_u$, and $v_1$ is either the lowest value $v>u_1$ for which $\mathcal{F}\left(X\right)=false$ in $G_v$ or $+\infty$.

Proof. 

Assume that there is at least a value $\overline{u}$ such that $\mathcal{F}\left(X\right)=true$ in $G_{\overline{u}}$. There surely are values w such that $\mathcal{F}\left(X\right)=false$ in $G_w$: at least the values beneath the minimum attained by f; let $\overline{w}<\overline{u}$ be such that $\mathcal{F}\left(X\right)=false$ in $G_{\overline{w}}$; for any $w^{\prime }<\overline{w}$ set $H_1=G_{w^{\prime }}$, $H_2=G_{\overline{w}}$, $H_3=G_{\overline{u}}$. Then, by Def. 17, $\mathcal{F}\left(X\right)=false$ in $G_{w^{\prime }}$. So between two values for which $\mathcal{F}\left(X\right)=true$ there cannot be one such that $\mathcal{F}\left(X\right)=false$. □

Definition 19. 

The interval $[u_1,v_1)$ of Lemma 19, i.e. the widest interval for which $\mathcal{F}\left(X\right)=true$ in $(G,f)$, is called the $\mathcal{F}$-interval of X in $(G,f)$.

Proposition 20. 

${\sigma }^{\mathcal{F}}={\varrho }^{\mathcal{F}}$

Proof. 

By Lemma 19, since ${\sigma }^{\mathcal{F}}$ and ${\varrho }^{\mathcal{F}}$ would differ only if, for at least one set X, there were values $w_1<w_2<w_3$ for which $\mathcal{F}\left(X\right)=true$ in $G_{w_1}$ and in $G_{w_3}$ but $\mathcal{F}\left(X\right)=false$ in $G_{w_2}$. □

It is easy to prove that the following features are simple.

Example 21. 

For a graph G, the (rather trivial) feature ${\mathcal{D}eg}^d$ for which ${\mathcal{D}eg}^d\left(X\right)=true$ if and only if X is a singleton containing a vertex of a fixed degree d is simple. Also the feature ${\mathcal{D}eg}^{\ge d}$ for which ${\mathcal{D}eg}^{\ge d}\left(X\right)=true$ if and only if X is a singleton containing a vertex whose degree is $\ge d$ is simple, with half-lines as ${\mathcal{D}eg}^{\ge d}$-intervals. Analogous features are defined for a digraph in terms of indegree and outdegree.

Example 22. 

For a (di)graph G, $\mathcal{C}onn\left(X\right)=true$ if and only if X is a set of vertices inducing a connected subgraph (resp. a strongly connected subdigraph) of G. $\mathcal{C}onn$ is simple and its $\mathcal{C}onn$-intervals are half-lines.

Example 23. 

For a (di)graph G, $\mathcal{C}omp\left(X\right)=true$ if and only if X is a set of vertices inducing a connected component (resp. a strong component) of G. $\mathcal{C}omp$ is simple; for the $\mathcal{C}omp$-interval $[u_1,v_1)$ of a set X, the value $v_1$ either is $+\infty$ or the value at which some other vertex joins the component (resp. strong component) induced by X.

Proposition 24. 

For a (di)graph G, any monotone feature $\mathcal{F}$ is simple.

Proof. 

Let $G=(V,E)$ be a (di)graph and $\mathcal{F}$ be a monotone feature. For $H_1=(V_1,E_1)\subseteq H_2=(V_2,E_2)\subset H_3\subseteq G$ and $X\subseteq (V\cup E)$ assume that $\mathcal{F}\left(X\right)=true$ in $H_3$ and $\mathcal{F}\left(X\right)=false$ in $H_2$. By condition 1 of Def. 14, $X⊈V_2\cup E_2$ (recall the convention of Def. 7), so also $X⊈V_1\cup E_1$ and $\mathcal{F}\left(X\right)=false$ in $H_1$. □

Lemma 25. 

Let $f,g:E\to \mathbb{R}$ be filtering functions on a (di)graph G, and assume that there exists a positive real number h such that ${sup}_{e\in E}\left|f\left(e\right)-g\left(e\right)\right|\le h$. Then, for real numbers $\overline{u},\overline{v}$ such that $\overline{u}<\overline{v}-h$, the sublevel (di)graph $G_{f,\overline{u}}$ is a sub(di)graph of $G_{g,\overline{v}}$ and $G_{g,\overline{u}}$ is a sub(di)graph of $G_{f,\overline{v}}$.

Proof. 

For any $e\in E$, if $f\left(e\right)\le \overline{u}$ then $g\left(e\right)\le \overline{u}+h<\overline{v}$; analogously, if $g\left(e\right)\le \overline{u}$ then $f\left(e\right)\le \overline{u}+h<\overline{v}$. □

The following Proposition generalizes Prop. 16 (i.e. [9, Prop. 4]) with essentially the same proof.

Proposition 26. 

The ip-function generators ${\sigma }^{\mathcal{F}}={\varrho }^{\mathcal{F}}$ are balanced, hence stable.

Proof. 

With the same hypotheses as in Lemma 25, let $X\subseteq (V\cup E)$ be such that $\mathcal{F}\left(X\right)=true$ in $G_{f,w}$ at all levels $w\in [u-h,v+h]$, with $u<v$. For any $\overline{w}\in [u,v]$ we now show that $\mathcal{F}\left(X\right)=true$ in $G_{g,\overline{w}}$. In fact, by Prop. 25$G_{f,u-h}$ is a sub(di)graph of $G_{g,\overline{w}}$, which is a sub(di)graph of $G_{f,v+h}$. Since $\mathcal{F}$ is simple and $\mathcal{F}\left(X\right)=true$ both in $G_{f,u-h}$ and in $G_{f,v+h}$, we necessarily have that $\mathcal{F}\left(X\right)=true$ in $G_{g,\overline{w}}$. Therefore there is an injective map (actually an inclusion map) from $S_{(G,f)}^{\mathcal{F}}(u-h,v+h)$ to $S_{(G,g)}^{\mathcal{F}}(u,v)$, therefore ${\sigma }_{(G,f)}^{\mathcal{F}}(u-h,v+h)\le {\sigma }_{(G,g)}^{\mathcal{F}}(u,v)$. Stability comes from Prop. 6. □

Remark 27. 

The maximal version of a simple feature (Def. 10) is generally not simple: an example is the feature $\mathcal{I}$ identifying independent sets of vertices, which is monotone (hence simple) while the ip-functions corresponding to its maximal version $m\mathcal{I}$ are not balanced ([9, Sect. 2.4, Appendix, Fig. 14, Fig. 15]), so $m\mathcal{I}$ is not simple. Still, some are.

Proposition 28. 

For a (di)graph G, the feature $m\mathcal{M}$[9, Sect. 2.4], i.e. such that $m\mathcal{M}\left(X\right)=true$ if and only if X is a maximal matching, is simple.

Proof. 

A matching of a graph is a matching also of any supergraph. If, for $H_1=(V_1,E_1)\subseteq H_2=(V_2,E_2)\subset H_3\subseteq G$, we have $m\mathcal{M}\left(X\right)=true$ in $H_3$ and $m\mathcal{M}\left(X\right)=false$ in $H_2$, this means that X is a maximal matching in $H_3$ but is not in $H_2$. There are only two ways of not being a maximal matching in $H_2$: either $X⊈V_2\cup E_2$ or $H_2$ contains a matching $X^{\prime }\supset X$. The latter is impossible because in that case, X would not be maximal in $H_3$ either. So $X⊈V_1\cup E_1\subseteq V_2\cup E_2$ and $m\mathcal{M}\left(X\right)=false$ in $H_1$. □

Two variations on the notion of matching for digraphs are the following.

Example 29. 

For a digraph $G=(V,E)$, $\mathcal{PK}\left(X\right)=true$ if and only if X is a set of arcs, any two of which have neither the same head nor the same tail. Such a set X is said to be path-like.

Example 30. 

For a digraph $G=(V,E)$, $\mathcal{PS}\left(X\right)=true$ if and only if X is a set of arcs, such that for any two of them the head of one is not the tail of the other. Such a set X is said to be path-less.

Proposition 31. 

The features $\mathcal{PK}$ and $\mathcal{PS}$ are monotone, hence simple.

Proof. 

Straightforward. □

Proposition 32. 

The features $m\mathcal{PK}$ and $m\mathcal{PS}$ are simple.

Proof. 

Same as for Prop. 28. □

Figure 1 shows the functions ${\sigma }^{m\mathcal{PK}}={\varrho }^{m\mathcal{PK}}$ (middle) and ${\sigma }^{m\mathcal{PS}}={\varrho }^{m\mathcal{PS}}$ (right) for the same toy example (left) used in [9].

5. Single-vertex features

Some features are of particular interest in applications: those that may be true on singletons consisting of vertices. They can be categorized as point-wise if they depend on the single vertex, local if they depend on the k-neighborhood of the vertex for some k, or global if they depend on the whole (di)graph.

Example 33. 

See Example 21 for examples of point-wise features.

Example 34. 

Another point-wise feature for a digraph G is $\mathcal{S}ou$, for which $\mathcal{S}ou\left(X\right)=true$ if and only if X consists of a single vertex (that we call a source) whose outdegree is larger than its indegree. An analogous feature identifies a sink, where the indegree exceeds the outdegree. Variations on these features can be built by specifying that the outdegree (respectively indegree) exceeds a fixed fraction of the total degree of the vertex. The usual example with its ip-functions ${\sigma }^{\mathcal{S}ou}$ (middle) and ${\varrho }^{\mathcal{S}ou}$ is shown in Figure 2. ${\sigma }^{\mathcal{S}ou}$ and ${\varrho }^{\mathcal{S}ou}$ are not balanced: see the Appendix.

Example 35. 

A local feature is the one identifying a hub (see [9, Sect. 2.5, 4]) in a (di)graph, i.e. a vertex whose degree (resp. outdegree) exceeds the one of each of its neighbors. The corresponding ip-functions $\sigma$ and $\varrho$ are unbalanced [9, Appendix].

Example 36. 

In [10, Sect. 4.1.4] a local feature is defined for the graph $G=(V,E)$ representing a gray-tone image, where each vertex is a pixel and is adjacent to each of its eight neighboring pixels. The filtering function $f:E\to \mathbb{R}$ maps each edge to the minimum of the intensities of the incident pixels. The feature ${\mathcal{G}}_{m,n}^{k,t}$ (short: $\mathcal{G}$) maps each vertex v to $true$ if and only if more than m and less than n pixels at distance $\le k$ have intensity $<t$. The function ${\sigma }^{\mathcal{G}}$ is proved to be balanced [41, Thm. 4.3.1].

Example 37. 

A global feature in a digraph G is $\mathcal{R}oot$ for which $\mathcal{R}oot\left(X\right)=true$ if and only if X consists of a vertex (that we call a root) from which there is a directed path to each vertex of the connected component containing it. Figure 3 shows the same weighted digraph of the previous examples, with its ip-functions ${\sigma }^{\mathcal{R}oot}$ (middle) and ${\varrho }^{\mathcal{R}oot}$. ${\sigma }^{\mathcal{R}oot}$ and ${\varrho }^{\mathcal{R}oot}$ are not balanced: see the Appendix.

6. Computational Experiments

Large, directed, weighted graphs are largely used in representing context-based rating, from recommendation engines to trust networks.

We implemented the single-vertex features described in Section 5 as part of the Generalized Persistence Analysis Python package available at https://github.com/LimenResearch/gpa.

6.1. Algorithm

We consider a directed, weighted graph $G=(V,E,\Omega )$ and apply a transformation function on the weights defined on its edges. Transformed weights induce a sublevel-set filtration. See Figure 4. For each sublevel set, we compute a single-vertex feature (sink in Figure 4). Finally, we compute the persistence in terms of the considered single vertex feature, realized by each vertex throughout the filtration. Thus, the persistence associated to each vertex is the largest interval where the feature holds, as detailed in [9].

The software interface we adopted for this algorithm is provided at https://github.com/LimenResearch/gpa/tree/master/gpa/examples/bitcoin/bitcoin.py.The user instantiate a directed weighted graphs, specify a transformation function to be applied to the weights to obtain the filtration, and finally chooses the single-vertex feature among the implemented ones, namely sources and sinks. These features are implemented by adding two simple methods to the WeightedGraph class for the computation of the feature itself, and to apply the computation to each step of the filtration.

6.2. Datasets

The OTC and Bitcoin Alpha datasets consist of user-to-user trust networks of Bitcoin traders on the OTC and Alpha platform, respectively [42]. Users can express a vote of trust on other users by assigning them a integer rating varying from $-10$ (fraudulent user) to 10 (completely trustworthy), excluding the value 0. We utilized the data and ground truth provided in https://github.com/horizonly/Rev2-model/tree/master. There, fraudulent and trustworthy users are identified by considering the platform founders votes.

6.3. Results

We computed the steady persistence of the source single-vertex feature considering the sublevel set filtration induced on the Bitcoin Alpha network $A=(V_A,E_A,{\Omega }_A)$ by the function

$$f\left(\omega \right)=max\left({\Omega }_A\right)-(\omega +|min\left({\Omega }_A\right)\left|\right),$$

where ${\Omega }_A$ is the set of all weights defined on the network’s edges.

For each sublevel of the filtration and each of its nodes, we say that a vertex v is a sink if the sum of the weights of its in-edges is larger the sum of the weights defined on its out-edges. In symbols:

$$Si\left(v\right)=\left(\sum _{e\in in\_edges\left(v\right)}{\omega }_e>\sum _{e\in out\_edges\left(v\right)}{\omega }_e\right).$$

This computation gives rise to the persistence diagram in Figure 5 (left panel). The persistence associated with the networks’ nodes allows us to draw the histogram of the values associated with trustworthy and fraudulent users, respectively, See Figure 5 (right panel). The histogram reveals how nodes representing fraudulent users attain high persistence levels (with mean $>12$ and most of them being represented by cornerline in the persistence diagram, i.e., points with infinite persistence, as showcased in Figure 5), while trustworthy users’ persistence is typically finite with mean $\approx 6$ and median 4.

7. Conclusions

The availability of massive data represented as graphs expedited the development of computational methods specialized in handling data endowed with the complex structures arising from pairwise interactions.

We provide a computational recipe to generate compact representations of large networks according to graph-theoretical features or custom properties defined on the networks’ vertices. To do this, we leverage a generalization of topological persistence adapted to work on weighted (di)graphs. In this framework, we widen the notion of monotone feature on (di)graphs—i.e., feature respecting inclusion—and thus, well suited to generate filtrations and persistence diagrams. Simple features are a special case of features, whose value can change from true to false only once in a filtration of a (di)graph. We prove that in a (di)graph any monotone feature is simple and that simple features give rise to stable indexing-aware persistence functions. We then provide examples of simple features. Turning our attention to application, single-vertex features allow the user to compute the persistence of a weighted (di)graph for any feature, focused on singletons. There, we provide several examples, characterizing single-vertex features as point-wise, local, or global descriptors of a network.

Computational experiments—supporting our belief that single-vertex feature can be easily integrated in applications—show how the sink feature characterizes the users of trust networks. When considering the persistence of sink vertices, fraudulent users are, in the majority of cases, represented by half-lines with infinite persistence, while trusted users are associated with finite persistence. To replicate our results, we provide a Python implementation of the algorithm for the computation of single-vertex feature persistence integrating it in the package originally supporting [8,9].

We believe that straightforward implementation to compute local and global descriptors of networks arising from easily engineerable features can complement and help gaining control on current approaches based on artificial neural architectures and learned features.