arXiv subjects

Stephen J. Young

Publications and source records attributed to Stephen J. Young.

26 records · Page 2Linked to original sources

arXiv2018

Model of Cognitive Dynamics Predicts Performance on Standardized Tests

In the modern knowledge economy, success demands sustained focus and high cognitive performance. Research suggests that human cognition is linked to a finite resource, and upon its depletion, cognitive functions such as self-control and decision-making may decline. While fatigue, among other factors, affects human activity, how cognitive performance evolves during extended periods of focus remains poorly understood. By analyzing performance of a large cohort answering practice standardized test questions online, we show that accuracy and learning decline as the test session progresses and recover following prolonged breaks. To explain these findings, we hypothesize that answering questions consumes some finite cognitive resources on which performance depends, but these resources recover during breaks between test questions. We propose a dynamic mechanism of the consumption and recovery of these resources and show that it explains empirical findings and predicts performance better than alternative hypotheses. While further controlled experiments are needed to identify the physiological origin of these phenomena, our work highlights the potential of empirical analysis of large-scale human behavior data to explore cognitive behavior.

cs.CY↗

arXiv2018

On the Structure of Isometrically Embeddable Metric Spaces

Since its popularization in the 1970s the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space.

math.CO↗

arXiv2017

Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

math.AC↗

arXiv2015

Posets with cover graph of pathwidth two have bounded dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant $d$ such that if $P$ is a poset with cover graph of $P$ of pathwidth at most $2$, then $\dim(P)\leq d$. We answer this question in the affirmative by showing that $d=17$ is sufficient. We also show that if $P$ is a poset containing the standard example $S_5$ as a subposet, then the cover graph of $P$ has treewidth at least $3$.

math.CO↗

arXiv2015

Connectivity and Giant Component of Stochastic Kronecker Graphs

Stochastic Kronecker graphs are a model for complex networks where each edge is present independently according the Kronecker (tensor) product of a fixed matrix k-by-k matrix P with entries in [0,1]. We develop a novel correspondence between the adjacencies in a general stochastic Kronecker graph and the action of a fixed Markov chain. Using this correspondence we are able to generalize the arguments of Horn and Radcliffe on the emergence of the giant component from the case where k = 2 to arbitrary k. We are also able to use this correspondence to completely analyze the connectivity of a general stochastic Kronecker graph.

math.CO↗

arXiv2014

Dimensionality of social networks using motifs and eigenvalues

We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an $m$-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when $m$ scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.

cs.SI↗

arXiv2013

Towards a weighted version of the Hajnal-Szemerédi Theorem

For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that guarantees the existence of a K_r-factor such that every factor has total edge weight at least tr(r-1)/2? We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.

math.CO↗

arXiv2009

On the Stanley Depth of Squarefree Veronese Ideals

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $\sdepth(M)$, and conjectured that $\depth(M) \le \sdepth(M)$ for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J \subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1\le d\le n < 5d+4$, then $\sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d$, and if $d\geq 1$ and $n\ge 5d+4$, then $d+3\le \sdepth(I_{n,d}) \le \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d$.

math.AC↗