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The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes

Nicolas Fraiman, Sayan Mukherjee, Gugan Thoppe

Abstract

In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $ζ(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.

The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes

Abstract

In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to . Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.

Paper Structure

This paper contains 11 sections, 12 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview of Key Contributions
  3. Related Work
  4. Proof Outline
  5. Main Result
  6. Proofs
  7. Extensions
  8. Generic distribution for $d$-face weights
  9. Noisy perturbations in $d$-face weights
  10. Weighted $Y(n, p)$ complex
  11. Summary and Future Directions

Key Result

Theorem 1

(Informal summary of our Main Result) As $n \to \infty,$ the empirical measure related to $\{n w_u(\sigma): \sigma \in M_n\}$ converges to a deterministic measure $\mu$ related to the asymptotic shadow density of the $d$-dimensional Linial-Meshulam complex $Y(n, c/n),$$c \geq 0;$ see d:limiting.bulk

Figures (4)

  • Figure 1: Normalized histogram (yellow) of $\{n w_u(\sigma): \sigma \in M_n\}$ and the density (red) of the shadow based measure $\mu$ given in \ref{['d:limiting.bulk.measure']}.
  • Figure 2: Plot of $\{n w_u(\sigma) - d \log(n) + \log(d!) : \sigma \in M_n\}.$
  • Figure 3: Normalized histograms of $\{n p w(\sigma): \sigma \in M_{n, p}\}$ and $\{n w(\sigma): \sigma \in M_{n, p}\},$ depicted in yellow and blue, respectively. In the last panel, the two histograms coincide since $p = 1.$
  • Figure 4: Comparison of the scaled weight (blue) of the actual $d$-MSA with our conjecture (red). The values of $c_1$ and $c_2$ are $n/\binom{n - 1}{d}$ and $\binom{n}{d + 1} \mu x,$ respectively. The black curve denotes the value of $\mu x.$ Recall that $\mu x$ is $\zeta(3) \approx 1.2$ for $d=1$ and $1.56$ for $d=2$.

Theorems & Definitions (30)

  • Theorem
  • Definition 1: Spanning Acycle
  • Definition 2: Minimum Spanning Acycle
  • Definition 3: Shadow
  • Definition 4: Empirical Measure
  • Theorem 1: Main Result
  • Corollary 2
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 20 more