Shortest Paths in a Weighted Simplicial Complex

TL;DR

The paper addresses shortest-path computation in weighted simplicial complexes by introducing weighted $d$-complexes and $d$-paths, extending graph-theoretic methods to higher-dimensional adjacency. It develops generalized BFS/DFS for discovery, a Dijkstra-like algorithm for non-negative weights with $O((n+m)\log n)$ time, and a Bellman--Ford-type method for negative weights with negative-cycle detection, along with rigorous correctness and complexity analyses. A real-world multi-agent example demonstrates the modeling power of the framework. Overall, the work provides a foundational bridge between graph theory and higher-dimensional combinatorial structures, with potential applications in distributed computing, sensor networks, and simplicial neural networks.

Abstract

Simplicial complexes are extensively studied in the field of algebraic topology. They have gained attention in recent time due to their applications in fields like theoretical distributed computing and simplicial neural networks. Graphs are mono-dimensional simplicial complex. Graph theory has application in topics like theoretical computer science, operations research, bioinformatics and social sciences. This makes it natural to try to adapt graph-theoretic results for simplicial complexes, which can model more intricate and detailed structures appearing in real-world systems. Though seemingly obvious, we did not find any previous work that looked into this prospect of simplicial complexes. In this article, we define the concept of weighted simplicial complex and $d$-path in a simplicial complex. Both these concepts have the potential to have numerous real-life applications. We start by adapting the Depth-First Search and Breadth-First Search algorithms for our setup. Next, we provide two novel algorithms to find the shortest paths in a weighted simplicial complex. The core principles of our algorithms align with those of Dijkstra$^\prime$s algorithm and Bellman-Ford algorithm for graphs. Hence, this work lays a building block for the sake of integrating graph-theoretic concepts with abstract simplicial complexes.

Figures (5)

• Figure 1: Example of a finite abstract simplicial complex
• Figure 2: Weighted 2-Complex and Neighbour
• Figure 3: Degree and Set of Neighbours
• Figure 4: Example of a 2-Path in a Complex. Here $P$= $(\{1,2\}$,$\{2,6\}$,$\{5,6\}$,$\{5,10\}$,$\{10,11\}$,$\{11,13\})$ is a 2-path from $\{1,2\}$ to $\{11,13\}$.
• Figure 5: Diagram explaining the execution of the algorithm using an example of finding $d_{w}(\{1,3\},\{4,7\})$ in the complex

Theorems & Definitions (16)

• Definition 1: Finite Abstract Simplicial Complex
• Definition 2: Weighted $d$-complex
• Definition 3: $d$-path and $d$-cycle
• Lemma 1: Correctness of BFS in $d$-Complexes
• proof
• Lemma 2: Complexity of BFS
• proof
• Lemma 3: DFS Visits and Forest Structure
• proof
• Lemma 4: Complexity of DFS