Simplicial Paths Lifting (Graph to Combinatorial)

Simplicial Paths as Higher-Order Motifs

A strong inductive bias for deep learning models is processing signals while respecting the local structure of their underlying space. Many real-world systems operate on asymmetric relational structures, leading to directed graph representations. However, most graph and topological models forcibly symmetrize these relationships, thereby losing critical information. While some graph neural networks have recently started incorporating asymmetric pairwise interactions, extending the topological deep learning (TDL) framework to account for asymmetric higher-order relationships remains unexplored.

Recent studies have examined cascading dynamics on networks at the simplicial level [2]. In Topological Data Analysis (TDA), the use of topological tools to address questions in neuroscience has generated interest in constructing topological spaces from digraphs to better understand the phenomena they support [3].

For this reason, we suggest using maximal simplicial paths, deerived from directed graphs, as cell of a combinatorial complex. Therefore, we are proposing a lifting from directed graphs to combinatorial complexes.

Next, we provide an introduction to the fundamental concepts underlying our approach. For a more comprehensive exploration of these basics, we refer the reader to [1]. To the best of our knowledge, this is the first lifting taking into account an higher-order notion of directionality in defining cells, differently from, e.g., taking directly as cells the simplices of a directed flag complex (see below).

Complexes

Directed Graphs

A directed graph (digraph) is a pair $G = (V,E)$ of a finite set $V$ of vertices and $E \subseteq [V]^2/\Delta_V$ is a relation, where $\Delta_V = {(v,v)|v \in V}$. Note that the relation is not necessarily symmetric. Quotienting by $\Delta_V$ we avoid loops on the graph, i.e., no edges $(v,v)$.

Abstract Simplicial Complexes

An abstract simplicial complex is a pair $K = (V, \Sigma)$, where $V$ is a finite set of vertices, and $\Sigma$ is a collection of subsets of $\Sigma$ such that for all element $\sigma \in \Sigma$, $\tau \subseteq \sigma$ implies $\tau \in \Sigma$. An element $\sigma$ of $\Sigma$ is an abstract simplex of $\mathcal{K}$. It is a k-simplex if $|\sigma| = k+1$. If $\tau \subseteq \sigma \in \mathcal{K}$, then $\tau$ is a face of $\sigma$. If the dimension $\tau$ is $\dim(\tau) = \dim(\sigma) - 1$, then it is a facet of $\sigma$. The dimension $\dim(\mathcal{K})$ of $\mathcal{K}$ is the maximal dimension of a simplex in $\mathcal{K}$.

There is a standard way of building an abstract simplicial complex from a graph.

Flag Complex

Given a graph $G$, its associated flag complex is the abstract simplicial complex whose $k$-simplices are formed by the $(k+1)$-cliques of the graph.

The following are the natural generalization of flag complexes for digraphs.

Directed Flag Complex

An ordered $k$-clique of a directed graph $G$ is a totally ordered $k$-tuple $(v_1, \dots, v_n)$ of vertices of $G$ with the property that $(v_i, v_{j}) \in E$ for $i < j$. Given a digraph $G$, its directed flag complex is the abstract simplicial complex whose simplices are all the directed $(k+1)$-cliques.

Simplicial Paths

Edge paths on digraphs

A path on a digraph is a sequence $(v_0, v_1, \dots, v_n)$ such that any consecutive pair $(v_i, v_{i+1}) \in E$, moving from a source vertex to a sink vertex. Directed graphs support various directed edge paths.

Directed cliques have an inherent directionality, which we exploit to extend the notion to higher-dimensional simplicial paths formed by sequences of simplices in the directed flag complex.

We will impose the notion of direction via face maps.

Face maps

Face maps uniquely identify the faces of the simplex by omitting the $i$th-vertex. Let $\sigma$ be an $n$-simplex. We denote by $\hat{d}_i$ the face map

$$ \hat{d}_i(\sigma) = \begin{cases} (v_0, \ldots, \hat{v}i, \ldots, v_n) & \text{if } i < n, \ (v_0, \ldots, v{n-1}, \hat{v}_n) & \text{if } i \geq n. \end{cases} $$

Directed Q-Connectivity

For an ordered simplicial complex $K$, let $(\sigma, \tau)$ be an ordered pair of simplices $\sigma \in K_s$ and $\tau \in K_t$, where $s, t \geq q$. Let $(\hat{d}_i, \hat{d}_j)$ be an ordered pair of the $ith$ and $jth$ face maps. Then $(\sigma, \tau)$ is $q$-near along $(\hat{d}_i, \hat{d}_j)$ if either of the following conditions is true:

$\sigma \leftrightarrow \tau$,
$\hat{d}_i(\sigma) \leftrightarrow \alpha \leftrightarrow \hat{d}_j(\tau)$, for some $q$-simplex $\alpha \in K$.

By closing the directed q-nearness transitively, the ordered pair $(\sigma, \tau)$ of simplices of $K$ is $q$-connected along $(\hat{d}_i, \hat{d}_j)$ if there is a sequence of simplices in $K$,

$$\sigma = \alpha_0, \alpha_1, \alpha_2, \ldots, \alpha_n, \alpha_{n+1} = \tau,$$

such that any two consecutive ones are $q$-near along $(\hat{d}_i, \hat{d}_j)$. The sequence of simplices is called a $q$-connection along $(\hat{d}_i, \hat{d}_j)$ between $\sigma$ and $\tau$ or $(q, \hat{d}_i, \hat{d}_j)$-connection, when the choices of $q$ and directions $\hat{d}_i$ and $\hat{d}_j$ are made. From now on we refer $(q, \hat{d}_i, \hat{d}_j)$ as $(q, i, j)$.

Theorem The relation of being $(q,i,j)$-connected is a preorder on $\Sigma_{\geq q}$.

Directions and Simplicial Paths as Topological Information

Instead of focusing on the path structure of the digraph, we look at the path structure of the high-dimensional simplices by exploring the $q$-connectivity preorder.

Different choices of $q,i,j$ allow to enphasize different features of directionality. For instance, $(1,0,2)$-connected paths of 2-simplices exhibit directed flows aligned with the directionality of the total order of the adjacent simplices. On the other hand, the $(1,1,2)$ preorder reveals circular flows around a source vertex

The $(q, i, j)$-connections exhibit different homotopical information compared to the original complex arising from the structure of the digraph. The following two digraphs span a $2$-dimensional directed flag complex homotopic to the $2$-sphere, making them indistinguishable by homology. However, by examining the $(1,0,2)$ and $(1,1,2)$ preorders, we can homotopically distinguish these complexes. The $(1,1,2)$ preorder, in particular, allows us to identify circular flows in both the upper and lower hemispheres. Specifically, the first complex has a circular flow only in the upper hemisphere, whereas the second complex exhibits circular flows in both the upper and lower hemispheres.

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