Comparing the Switch and Curveball Markov Chains for Sampling Binary Matrices with Fixed Marginals

by   Corrie Jacobien Carstens, et al.

The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better convergence rate for reaching the stationary (uniform) distribution in certain cases. Some experimental evidence for this has been given in the literature. In this note we give a spectral gap comparison between two switch-based chains and the Curveball chain. In particular, this comparison allows us to conclude that the Curveball Markov chain is rapidly mixing whenever one of the two switch chains is rapidly mixing. Our analysis directly extends to the case of sampling binary matrices with forbidden entries (under the assumption of irreducibility). This in particular captures the case of sampling simple directed graphs with given degrees. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tetali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest. We also obtain an improved bound on the smallest eigenvalue for the switch Markov chain studied by Greenhill for uniformly sampling simple directed regular graphs.


page 1

page 2

page 3

page 4


Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences

We show that the switch Markov chain for sampling simple undirected, as ...

On the Switch Markov Chain for Strongly Stable Degree Sequences

We show that the switch Markov chain for sampling undirected and biparti...

Mixing time for uniform sampling of binary matrices with fixed row and column sums using the trade algorithm

The trade algorithm, which includes the curveball and fastball implement...

Switch-based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

We consider the irreducibility of switch-based Markov chains for the app...

Concentration Bounds for Co-occurrence Matrices of Markov Chains

Co-occurrence statistics for sequential data are common and important da...

Learning from non-irreducible Markov chains

Most of the existing literature on supervised learning problems focuses ...

A Fast Scheme for the Uniform Sampling of Binary Matrices with Fixed Margins

Uniform sampling of binary matrix with fixed margins is an important and...

Please sign up or login with your details

Forgot password? Click here to reset