Related papers: Drawing with Distance

A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+1$)-coloring for graphs with maximum average…

We consider a special case of the generalized P\'{o}lya's urn model introduced by Benaim et al (2013). Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete…

Generating realistic artificial preference distributions is an important part of any simulation analysis of electoral systems. While this has been discussed in some detail in the context of a single electoral district, many electoral…

There is a class of models for pol/mil/econ bargaining and conflict that is loosely based on the Median Voter Theorem which has been used with great success for about 30 years. However, there are fundamental mathematical limitations to…

For the plain Polya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls assuming that the initial number of black balls is large. Depending on the initial number of white balls,…

The most studied and accepted pseudometric for probabilistic processes is one based on the Kantorovich distance between distributions. It comes with many theoretical and motivating results, in particular it is the fixpoint of a given…

We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich…

The map of elections framework is a methodology for visualizing and analyzing election datasets. So far, the framework was restricted to elections that have equal numbers of candidates, equal numbers of voters, and where all the (ordinal)…

The goal of this thesis is to study the use of the Kantorovich-Rubinstein distance as to build a descriptor of sample complexity in classification problems. The idea is to use the fact that the Kantorovich-Rubinstein distance is a metric in…

We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a…

Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…

A P\'olya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots,d\}$ for $d\in \mathbb{N}$. At each time step, a random ball is chosen uniformly in…

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections…

For the most general Polya urn schemes, we establish the almost sure convergence of its composition. The only requirement is that there are always enough balls of both colors, so that the extractions can be indefinitely pursued according to…

Wasserstein 1 optimal transport maps provide a natural correspondence between points from two probability distributions, $\mu$ and $\nu$, which is useful in many applications. Available algorithms for computing these maps do not appear to…

We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a distance in chaotic unidimensional maps. Based on that…

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Under fairly mild assumptions on the…

In this article, we define the transport dimension of probability measures on $\mathbb{R}^m$ using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski…

Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$…

The generalized P\`olya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with diagonal replacement matrix, which arise in…