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Chris Wells

Publications and source records attributed to Chris Wells.

10 recordsLinked to original sources
arXiv2026

A finite victory over de Bruijn-Erd\H{o}s in interval discrepancy

We study a finite form of the classical interval discrepancy problem. Starting from the unit interval, one repeatedly splits an existing interval into two until $n$ intervals have been produced. The discrepancy of such a process is the maximum, over all intermediate stages, of the ratio between the longest interval and the shortest interval. A theorem of de Bruijn and Erd\H{o}s from 1949 shows that this ratio must approach $2$ as $n\to\infty$, and they give a sharp construction achieving this bound. For fixed $n$, their construction gives the upper bound $\text{disc}(n)\leq 2-\frac{3}{2n}+O(1/n^2)$. In this paper, we improve the first-order term of this bound. Specifically, we construct a strategy, called \emph{lex-merge}, with $\text{disc}(n)\leq 2-\frac{4\ln 2}{n}+O(1/n^2)$. We prove also the lower bound $\text{disc}(n)\geq 2-\frac{6\ln 2}{n}-O(1/n^2)$, showing that the first-order term in this improvement over the de Bruijn--Erd\H{o}s construction has the correct order of magnitude. We conjecture that the lex-merge strategy is optimal for every $n$.

math.CO↗
arXiv2026

A discrete view of Gromov's filling area conjecture

A compact metric surface $M$ isometrically fills a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for every $x,y\in C=\partial M$; that is, $M$ does not introduce any ``shortcuts'' between points on its boundary. Gromov's filling area conjecture in differential geometry from 1983 asserts that among all compact, orientable Riemannian surfaces which isometrically fill the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov demonstrated that this is indeed the case if $M$ is homeomorphic to the disk. While Gromov's conjecture has since been verified in some other cases, the full conjecture remains unresolved. In this paper, we consider a discrete analogue of Gromov's problem, which is likely natural to those who study graph embeddings on arbitrary surfaces. Using standard graph-theoretic tools, such as Menger's theorem, we obtain reasonable asymptotic bounds on this discrete variant. We then demonstrate how these discrete bounds can be translated to the continuous setting, showing that any isometric filling of the Riemannian circle of length $2\pi$ has surface area at least $1.36\pi$ (the hemisphere has surface area $2\pi$). This appears to be the first quantitative lower-bound on Gromov's problem that applies to arbitrary isometric fillings.

math.DG↗
arXiv2024

Facets in the Vietoris--Rips complexes of hypercubes

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.

math.AT↗
arXiv2024

Enabling Contextual Soft Moderation on Social Media through Contrastive Textual Deviation

Automated soft moderation systems are unable to ascertain if a post supports or refutes a false claim, resulting in a large number of contextual false positives. This limits their effectiveness, for example undermining trust in health experts by adding warnings to their posts or resorting to vague warnings instead of granular fact-checks, which result in desensitizing users. In this paper, we propose to incorporate stance detection into existing automated soft-moderation pipelines, with the goal of ruling out contextual false positives and providing more precise recommendations for social media content that should receive warnings. We develop a textual deviation task called Contrastive Textual Deviation (CTD) and show that it outperforms existing stance detection approaches when applied to soft moderation.We then integrate CTD into the stateof-the-art system for automated soft moderation Lambretta, showing that our approach can reduce contextual false positives from 20% to 2.1%, providing another important building block towards deploying reliable automated soft moderation tools on social media.

cs.CL↗
arXiv2024

Frogs, hats and common subsequences

Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from the same alphabet as $W$. We are interested in the average length of the longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be $\gamma(W)\cdot n+o(n)$ for some constant $\gamma(W)$. Bukh and Cox recently developed an interacting particle system, dubbed the frog dynamics, which can be used to compute the constant $\gamma(W)$ for any fixed word $W$. They successfully analyzed the simplest case of the frog dynamics to find an explicit formula for the constants $\gamma(12\cdots k)$. We continue this study by using the frog dynamics to find an explicit formula for the constants $\gamma(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation of the PushTASEP on the ring where some clocks are identical. Interestingly, exclusion processes with correlated clocks of this type appear to have not been analyzed before. Our analysis leads to a seemingly new combinatorial object which could be of independent interest: frogs with hats!

math.CO↗
arXiv2024

Phase transitions in isoperimetric problems on the integers

Barber and Erde asked the following question: if $B$ generates $\mathbb Z^d$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\operatorname{Cay}(\mathbb Z^d,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of $d=1$. The key is to show that the structure of the cylinder $\mathbb Z\times(\mathbb Z/k\mathbb Z)$ can be mimicked in certain Cayley graphs on $\mathbb Z$, leading to a phase transition. We do, however, show that Barber--Erde's question for Cayley graphs on $\mathbb Z$ has a positive answer if one is allowed to ignore finitely many sets.

math.CO↗
arXiv2023

The maximum number of odd cycles in a planar graph

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?

math.CO↗
arXiv2018

Discovering Political Topics in Facebook Discussion threads with Graph Contextualization

We propose a graph contextualization method, pairGraphText, to study political engagement on Facebook during the 2012 French presidential election. It is a spectral algorithm that contextualizes graph data with text data for online discussion thread. In particular, we examine the Facebook posts of the eight leading candidates and the comments beneath these posts. We find evidence of both (i) candidate-centered structure, where citizens primarily comment on the wall of one candidate and (ii) issue-centered structure (i.e. on political topics), where citizens' attention and expression is primarily directed towards a specific set of issues (e.g. economics, immigration, etc). To identify issue-centered structure, we develop pairGraphText, to analyze a network with high-dimensional features on the interactions (i.e. text). This technique scales to hundreds of thousands of nodes and thousands of unique words. In the Facebook data, spectral clustering without the contextualizing text information finds a mixture of (i) candidate and (ii) issue clusters. The contextualized information with text data helps to separate these two structures. We conclude by showing that the novel methodology is consistent under a statistical model.

stat.AP↗
arXiv2013

My Brain is Full: When More Memory Helps

We consider the problem of finding good finite-horizon policies for POMDPs under the expected reward metric. The policies considered are {em free finite-memory policies with limited memory}; a policy is a mapping from the space of observation-memory pairs to the space of action-memeory pairs (the policy updates the memory as it goes), and the number of possible memory states is a parameter of the input to the policy-finding algorithms. The algorithms considered here are preliminary implementations of three search heuristics: local search, simulated annealing, and genetic algorithms. We compare their outcomes to each other and to the optimal policies for each instance. We compare run times of each policy and of a dynamic programming algorithm for POMDPs developed by Hansen that iteratively improves a finite-state controller --- the previous state of the art for finite memory policies. The value of the best policy can only improve as the amount of memory increases, up to the amount needed for an optimal finite-memory policy. Our most surprising finding is that more memory helps in another way: given more memory than is needed for an optimal policy, the algorithms are more likely to converge to optimal-valued policies.

cs.AI↗