# Novel Algorithms for Sampling Abstract Simplicial Complexes

## TL;DR

This paper introduces dual algorithms for sampling abstract simplicial complexes, including a generative sampler and a local ergodic random walk, with formulas for exact probabilities and empirical validation.

## Contribution

It presents novel sampling algorithms for abstract simplicial complexes, including a heuristic-based generative sampler and a local ergodic random walk with known transition probabilities.

## Key findings

- The generative sampler balances combinatorial multiplicities effectively.
- The local ergodic random walk has well-characterized autocorrelation.
- Numerical tests demonstrate the efficacy of the proposed methods.

## Abstract

We provide dual algorithms for sampling the space of abstract simplicial complexes on a fixed number of vertices. We develop a generative and descriptive sampler designed with heuristics to help balance the combinatorial multiplicities of the states and more widely sample across the space of nonisomorphic complexes. We provide a formula for the exact probabilities with which this algorithm will produce a requested labeled state, and compare with an existing benchmark. We also design a highly conductive local ergodic random walk with known transition probabilities. We characterize the autocorrelation of the walk, and numerically test it against our sampler to illustrate its efficacy.