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Computing eulerian magnitude homology

Giuliamaria Menara, Luca Manzoni

TL;DR

This work addresses the problem of computing the ranks of the first-diagonal Eulerian magnitude homology groups $EMH_{k,k}(G)$ for graphs. It establishes that the problem is $ #W[1]$-complete by reducing to subgraph isomorphism via a family of critical subgraphs, and then presents the First Diagonal Algorithm (FDA), a BFS-based method that builds the Eulerian magnitude chains $EMC_{k,k}(G)$ and $EMC_{k-1,k}(G)$ to obtain $EMH_{k,k}(G)$ through kernel computation. The authors show the FDA runs in $O(n (N_v)^L)$ operations, where $L$ is the graph diameter and $N_v$ the maximum degree, offering practical efficiency for graphs with small diameter despite worst-case exponential behavior. This work connects magnitude-homology computations to subgraph-structure enumeration, and paves the way for future enhancements inspired by persistent homology to broaden the applicability of magnitude-based topology in network analysis.

Abstract

In this paper tackle the problem of computing the ranks of certain eulerian magnitude homology groups of a graph G. First, we analyze the computational cost of our problem and prove that it is #W[1]-complete. Then we develop the first diagonal algorithm, a breadth-first-search-based algorithm parameterized by the diameter of the graph to calculate the ranks of the homology groups of interest. To do this, we leverage the close relationship between the combinatorics of the homology boundary map and the substructures appearing in the graph. We then discuss the feasibility of the presented algorithm and consider future perspectives.

Computing eulerian magnitude homology

TL;DR

This work addresses the problem of computing the ranks of the first-diagonal Eulerian magnitude homology groups for graphs. It establishes that the problem is -complete by reducing to subgraph isomorphism via a family of critical subgraphs, and then presents the First Diagonal Algorithm (FDA), a BFS-based method that builds the Eulerian magnitude chains and to obtain through kernel computation. The authors show the FDA runs in operations, where is the graph diameter and the maximum degree, offering practical efficiency for graphs with small diameter despite worst-case exponential behavior. This work connects magnitude-homology computations to subgraph-structure enumeration, and paves the way for future enhancements inspired by persistent homology to broaden the applicability of magnitude-based topology in network analysis.

Abstract

In this paper tackle the problem of computing the ranks of certain eulerian magnitude homology groups of a graph G. First, we analyze the computational cost of our problem and prove that it is #W[1]-complete. Then we develop the first diagonal algorithm, a breadth-first-search-based algorithm parameterized by the diameter of the graph to calculate the ranks of the homology groups of interest. To do this, we leverage the close relationship between the combinatorics of the homology boundary map and the substructures appearing in the graph. We then discuss the feasibility of the presented algorithm and consider future perspectives.

Paper Structure

This paper contains 12 sections, 5 theorems, 10 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. List of abbreviations
  4. Background
  5. Graph terminology and notation
  6. Eulerian magnitude homology
  7. Eulerian magnitude homology computational cost
  8. W[1]-completeness
  9. First Diagonal Algorithm
  10. Complexity of eulerian magnitude chain computation
  11. Discussion
  12. Conclusions and perspectives

Key Result

Lemma 2.1

Let $G$ be a graph, and $k > \ell$ non-negative integers. Then $EMC_{k,\ell}(G) \cong 0.$

Figures (6)

  • Figure 1: Graph $G$
  • Figure 2: Subgraph $H(\bar{x})$ induced by $[\bar{x}]=[0,1,2,3,4,5] \in EMH_{5,5}(G)$. The edges in the path $(0,1,2,3,4,5)$ are represented in black. Since the removal of each vertex causes the length of the induced path to decrease, it means all grey edges $\{x_{i-1},x_{i+1}\}$ are contained in the induced graph. The dashed edges do not play a role in the homology computation.
  • Figure 3: Subgraph $H$ induced by the $(4,4)$-$EMH$ cycle $[0,1,2,3,4]-[0,1,2',3,4]$. In this case, the edge $(1,3)$ cannot be present in order to have $\partial_{4,4}(0,1,2,3,4) \neq 0$ and $\partial_{4,4}(0,1,2',3,4) \neq 0$, but $\partial_{4,4}((0,1,2,3,4)-(0,1,2',3,4)) = 0$. The black edges are in the support of the two paths, the grey edges needed to be added for the differential to vanish, and the dashed edges do not play a role in the homology computation.
  • Figure 4: Subgraph $H$ induced by the $(4,4)$-$EMH$ cycle $\bar{x}_1 - \bar{x}_2 + \bar{x}_3 - \bar{x}_4 + \bar{x}_5 - \bar{x}_6 + \bar{x}_7 - \bar{x}8$, where $\bar{x}_1=(0,1,2,3,4)$, $\bar{x}_2=(0,1',2,3,4)$, $\bar{x}_3=(0,1',2',3,4)$, $\bar{x}_4=(0,1',2',3',4)$, $\bar{x}_5=(0,1',2,3',4)$, $\bar{x}_6=(0,1,2,3',4)$, $\bar{x}_7=(0,1,2',3',4)$, $\bar{x}_8=(0,1,2',3,4)$. In this case, the edges $(0,2)$, $(1,3)$ and $(2,4)$ cannot be present in order to have $\partial_{4,4}\bar{x}_i \neq 0$ but $\partial_{4,4}(\sum_i (-1)^i \bar{x_i}) = 0$.
  • Figure 5: Socio-centric social network with individuals A, B, C, D, E, F, G, H. The diameter of this graph is $L=3$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1: c.f. hepworth2015categorifying
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Lemma 3.1: giusti2024eulerian
  • proof
  • ...and 9 more