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Approximating Simplet Frequency Distribution for Simplicial Complexes

Hamid Beigy, Mohammad Mahini, Salman Qadami, Morteza Saghafian

TL;DR

This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector and presents a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm.

Abstract

Simplets, constituting elementary units within simplicial complexes (SCs), serve as foundational elements for the structural analysis of SCs. Previous efforts have focused on the exact count or approximation of simplet count rather than their frequencies, with the latter being more practical in large-scale SCs. This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector. In addition, we present a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm. We also present a simple algorithm for this purpose and justify the theoretical bounds with experiments on some random simplicial complexes.

Approximating Simplet Frequency Distribution for Simplicial Complexes

TL;DR

This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector and presents a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm.

Abstract

Simplets, constituting elementary units within simplicial complexes (SCs), serve as foundational elements for the structural analysis of SCs. Previous efforts have focused on the exact count or approximation of simplet count rather than their frequencies, with the latter being more practical in large-scale SCs. This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector. In addition, we present a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm. We also present a simple algorithm for this purpose and justify the theoretical bounds with experiments on some random simplicial complexes.
Paper Structure (7 sections, 6 theorems, 3 equations, 2 figures)

This paper contains 7 sections, 6 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximating the SFD Vector
  4. Sample Complexity of Approximating the SFD Vector
  5. Simplet Uniform Sampling Algorithm
  6. The SFD Vector Approximation Algorithm
  7. Conclusion

Key Result

theorem 1

Let $\mathcal{R}= \{ \mathcal{S}_i~|~1 \leq i \leq N_m \}$ be a family of all simplet sets where $N_m$ is the number of simplet types with at most $m$ vertices, and $D = \mathcal{S}_\mathcal{K}^m$. Then, we have $VC(D, \mathcal{R}) = 1$.

Figures (2)

  • Figure 1: The set of all $18$ simplet types with at least two and at most four vertices.
  • Figure 2: The SFD vector and the number of at most 4-vertices simplets for two sample SCs. The simplet types in the table refer to the types in Figure \ref{['fig:simplets']}.

Theorems & Definitions (9)

  • theorem 1: VC Dimension of Simplets
  • proof
  • theorem 2
  • Proposition 3
  • lemma 1
  • proof
  • lemma 2
  • proof
  • corollary 1: Time Complexity of $(\epsilon,\delta)$-approximation of SFD vector