Related papers: Impartial redistricting: a Markov chain approach t…
We consider the computational task of sampling a bit string $x$ from a distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the…
Using the recently introduced declination function, we estimate the net number of seats won in the US House of Representatives due to asymmetries in vote distributions. Such asymmetries can arise from combinations of partisan gerrymandering…
We compare and contrast fourteen measures that have been proposed for the purpose of quantifying partisan gerrymandering. We consider measures that, rather than examining the shapes of districts, utilize only the partisan vote distribution…
Ensembles of random legislative districts are a valuable tool for assessing whether a proposed district plan is an outlier or gerrymander. Expert witnesses have presented these in litigation using various methods, and unsurprisingly, they…
Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution.…
The analysis of parametrised systems is a growing field in verification, but the analysis of parametrised probabilistic systems is still in its infancy. This is partly because it is much harder: while there are beautiful cut-off results for…
This note outlines three intellectually distinct but not mutually exclusive strategies for measuring partisan gerrymandering: partisan symmetry, efficiency gap, and algorithmic sampling.
The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich…
We give a closed form of the discrete-time evolution of a recombination transformation in population genetics. This decomposition allows to define a Markov chain in a natural way. We describe the geometric decay rate to the limit…
Many studies involving large Markov chains require determining a smaller representative (aggregated) chains. Each {\em superstate} in the representative chain represents a {\em group of related} states in the original Markov chain.…
The Political Districting Problem is mapped to a $q$-state Potts model in which the constraints can be written as interactions between sites or external fields acting on the system. Districting into $q$ voter districts is equivalent to…
Long-run average optimization problems for Markov decision processes (MDPs) require constructing policies with optimal steady-state behavior, i.e., optimal limit frequency of visits to the states. However, such policies may suffer from…
We propose a deterministic denoising algorithm for discrete-state diffusion models. The key idea is to derandomize the generative reverse Markov chain by introducing a variant of the herding algorithm, which induces deterministic state…
It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly…
The parameters of a discrete stationary Markov model are transition probabilities between states. Traditionally, data consist in sequences of observed states for a given number of individuals over the whole observation period. In such a…
One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent…
This paper considers a Markov-modulated duplication-deletion random graph where at each time instant, one node can either join or leave the network; the probabilities of joining or leaving evolve according to the realization of a finite…
Redistricting practitioners must balance many competing constraints and criteria when drawing district boundaries. To aid in this process, researchers have developed many methods for optimizing districting plans according to one or more…
We consider the problem of controlling a fully specified Markov decision process (MDP), also known as the planning problem, when the state space is very large and calculating the optimal policy is intractable. Instead, we pursue the more…
In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior…