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A novel approach to graph distinction through GENEOs and permutants

Giovanni Bocchi, Massimo Ferri, Patrizio Frosini

TL;DR

These experiments show that GENEOs offer a good compromise between efficiency and computational cost in comparing r-regular graphs, while their actions on data are easily interpretable, and supports the idea that GENEOs could be a general-purpose approach to discriminative problems in Machine Learning.

Abstract

The theory of Group Equivariant Non-Expansive Operators (GENEOs) was initially developed in Topological Data Analysis for the geometric approximation of data observers, including their invariances and symmetries. This paper departs from that line of research and explores the use of GENEOs for distinguishing $r$-regular graphs up to isomorphisms. In doing so, we aim to test the capabilities and flexibility of these operators. Our experiments show that GENEOs offer a good compromise between efficiency and computational cost in comparing $r$-regular graphs, while their actions on data are easily interpretable. This supports the idea that GENEOs could be a general-purpose approach to discriminative problems in Machine Learning when some structural information about data and observers is explicitly given.

A novel approach to graph distinction through GENEOs and permutants

TL;DR

These experiments show that GENEOs offer a good compromise between efficiency and computational cost in comparing r-regular graphs, while their actions on data are easily interpretable, and supports the idea that GENEOs could be a general-purpose approach to discriminative problems in Machine Learning.

Abstract

The theory of Group Equivariant Non-Expansive Operators (GENEOs) was initially developed in Topological Data Analysis for the geometric approximation of data observers, including their invariances and symmetries. This paper departs from that line of research and explores the use of GENEOs for distinguishing -regular graphs up to isomorphisms. In doing so, we aim to test the capabilities and flexibility of these operators. Our experiments show that GENEOs offer a good compromise between efficiency and computational cost in comparing -regular graphs, while their actions on data are easily interpretable. This supports the idea that GENEOs could be a general-purpose approach to discriminative problems in Machine Learning when some structural information about data and observers is explicitly given.
Paper Structure (13 sections, 6 theorems, 9 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 6 theorems, 9 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Discussion
  4. Methods
  5. Acknowledgements
  6. Competing interests
  7. Author contributions
  8. Data and Code Availability

Key Result

Proposition 1

If $H \subseteq X^Y$ is a non-empty permutant for $T \colon G \to K$ then the operator $F_H:\Phi \to \mathbb{R}^Y$ defined as is a GENEO from $(\Phi, G)$ to $(\Psi, K)$, provided that $F_H(\Phi) \subseteq \Psi$ and that $C \ge |H|$.

Figures (4)

  • Figure 1: Model architecture after the forward selection algorithm.
  • Figure 2: Considering $\Lambda$ as a 4-cycle graph, the set $H^{\varphi_1}_{\Lambda}$ is the empty SP since there is no strict embedding of $\Lambda$ to $\Gamma_1$. In fact, the red dashed edge joins a node pair whose $\varphi$-value is 1, while each pair of non-adjacent nodes in $\Lambda$ has vanishing $\psi_0$-value. On the other hand, there are 8 strict embeddings of $\Lambda$ into $\Gamma_2$ (i.e. as many as the self-isomorphisms of $\Lambda$), hence we obtain an SP of size $|H^{\varphi_2}_{\Lambda}| = 8$.
  • Figure 3: The representations of the $\Lambda_j$ initially considered for analysis can be categorized into a limited number of classes. There are cycles, represented as $\Lambda_j$ for $j$ ranging from 1 to 8; stars, ranging from 9 to 12; complete graphs, ranging from 13 to 18; paths, ranging from 19 to 25; rigid graphs with no non-trivial self-isomorphism at 26 and 27 and cycles augmented with stars ranging from 29 to 56. Each $\Lambda_j$ also reports the accuracy of the operator $F_{\{j\}}$ using both colors and a numeric value.
  • Figure 4: The average execution time in function of the graph Size $N$ for the considered values of degree $r$: The GENEO-$t$ model (with $t$ being 1, 2 or 3) represents the method that aggregates the first $t$ operators identified through the forward selection process. The dotted and dashed lines represent respectively a linear and a quadratic growth.

Theorems & Definitions (16)

  • Definition 1: Group Equivariant Non-Expansive Operator
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4: Subgraph Permutant
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 5
  • ...and 6 more