Spatially embedded random networks

Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples

Combinatorial and Additive Number Theory Problem Sessions: '09--'19

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201

On Minrank and Forbidden Subgraphs

The minrank over a field $\mathbb{F}$ of a graph $G$ on the vertex set $\{1,2,\ldots,n\}$ is the minimum possible rank of a matrix $M \in \mathbb{F}^{n \times n}$ such that $M_{i,i} \neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an integer $n$, a graph $H$, and a field $\mathbb{F}$, let $g(n,H,\mathbb{F})$ denote the maximum possible minrank over $\mathbb{F}$ of an $n$-vertex graph whose complement contains no copy of $H$. In this paper we study this quantity for various graphs $H$ and fields $\mathbb{F}$. For finite fields, we prove by a probabilistic argument a general lower bound on $g(n,H,\mathbb{F})$, which yields a nearly tight bound of $\Omega(\sqrt{n}/\log n)$ for the triangle $H=K_3$. For the real field, we prove by an explicit construction that for every non-bipartite graph $H$, $g(n,H,\mathbb{R}) \geq n^\delta$ for some $\delta = \delta(H)>0$. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.Comment: 15 page

The Economics of Small Worlds

We examine a simple economic model of network formation where agents benefit from indirect relationships. We show that small-world features—short path lengths between nodes together with highly clustered link structures—necessarily emerge for a wide set of parameters

A caching game with infinitely divisible hidden material

We consider a caching game in which a unit amount of infinitely divisible material is distributed among $n\geq 2$ locations. A Searcher chooses how to distribute his search effort $r$ about the locations so as to maximize the probability she will find a given minimum amount $\bar{m} =1-m\leq r$ of the material. If the search effort $y_{i}$ invested by the Searcher in a given location $i$ is at least as great as the amount of material $x_{i}$ located there she finds all of it, otherwise the amount she finds is only $y_{i}$. In other words she finds $\min \left\{ x_{i},y_{i}\right\}$ in location $i$. We seek the randomized distribution of search effort that maximizes the probability of success for the Searcher in the worst case, hence we model the problem as a zero-sum win-lose game between the Searcher and a malevolent Hider who wishes to keep more than $m$ of the material. We show that in the case $r=\bar{m}$ the game has a geometric interpretation that for $n=2$ corresponds to a problem posed by W. H. Ruckle in his monograph [Geometric Games and Their Applications, Pitman, Boston, 1983]. We give solutions for the geometric game when $n=3$ for certain values of $m$, and bounds on the value for other values of $m$. In the more general case $r\geq \bar{m}$ we show that for $n=2$ the game reduces to Ruckle's gam