Generating random graphs in biased Maker-Breaker games

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=\Theta\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all spanning trees having maximum degree $\Delta=O(1)$ with a bare path of linear length (a bare path in a tree $T$ is a path with all interior vertices of degree exactly two in $T$)

Efficient winning strategies in random-turn Maker-Breaker games

We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A $p$-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is $p$). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chv\'atal and Erd\H os in 1987), the Hamilton cycle game and the $k$-vertex-connectivity game (both played on the edge set of $K_n$). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of $p$ for which he typically wins the game, assuming optimal strategies of both players.Comment: 20 page

Finding paths in sparse random graphs requires many queries

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in $G\sim \mathcal G(n,p)$ when $p=\frac{1+\varepsilon}{n}$ for some fixed constant $\varepsilon>0$. This random graph is known to have typically linearly long paths. To have $\ell$ edges with high probability in $G\sim \mathcal G(n,p)$ one clearly needs to query at least $\Omega\left(\frac{\ell}{p}\right)$ pairs of vertices. Can we find a path of length $\ell$ economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length $\ell=\Omega\left(\frac{\log\left(\frac{1}{\varepsilon}\right)}{\varepsilon}\right)$ with at least constant probability in $G\sim \mathcal G(n,p)$ with $p=\frac{1+\varepsilon}{n}$ must query at least $\Omega\left(\frac{\ell}{p\varepsilon \log\left(\frac{1}{\varepsilon}\right)}\right)$ pairs of vertices. This is tight up to the $\log\left(\frac{1}{\varepsilon}\right)$ factor.Comment: 14 page

A leap of faith: *Scale, critical realism and *emergence in the geography of religion

This dissertation explores the role of scale in human geography through a study involving a critical realist investigation of the geography of religious adherence. Using the contributions of a critical realist framework of stratification, emergence, and pluralistic methodologies, religious adherence is studied at the scales of the individual adherent, the church, and within local associations of churches. Analysis was performed through a study of two denominational congregations and an independent congregation in Harrison County, West Virginia and used a combination of surveys and in-depth interviews with religious adherents, pastors and local denominational leaders. The conceptual framework of this dissertation stands in contrast to traditional studies of the geography of religious adherence which rely on the quantification of denominationally collected attendance statistics aggregated to the scale of county boundaries and displayed as choropleth maps. Importantly, the traditional approach lacks the capacity to jump scale and is only valuable for making general assumptions at regional or national scales. Furthermore, these studies are embedded with the scaled problems associated with ecological fallacy and the Modifiable Areal Unit Problem.;This study demonstrates that the geography of religious adherence in Harrison County is emergent and irreducible. Emergent congregational and denominational powers and properties are facilitated through scaled structures and hierarchies, with mechanisms rooted in, but not reducible to, the scale of the adherent. Because questions pertaining to adherents, churches and church hierarchies are unique to the powers and mechanisms functioning at each stratum, methodological pluralism is required to understand a robust geography of religion. In contrast to traditional GOR studies, a critical realist approach has the capacity to reveal the scaled linkages and complex processes that operate between adherents, congregations and denominations. By incorporating ecclesiastical emergence into GOR, religionists gain a valuable tool to examine the substantial ways in which religion impacts social, economic and environmental life. This study also makes contributions to the broader debate about scale in human geography by suggesting that a framework of emergence provides a valuable contribution and addition to acknowledging and understanding the complex dimensions of scale

IPC: A Benchmark Data Set for Learning with Graph-Structured Data

Benchmark data sets are an indispensable ingredient of the evaluation of graph-based machine learning methods. We release a new data set, compiled from International Planning Competitions (IPC), for benchmarking graph classification, regression, and related tasks. Apart from the graph construction (based on AI planning problems) that is interesting in its own right, the data set possesses distinctly different characteristics from popularly used benchmarks. The data set, named IPC, consists of two self-contained versions, grounded and lifted, both including graphs of large and skewedly distributed sizes, posing substantial challenges for the computation of graph models such as graph kernels and graph neural networks. The graphs in this data set are directed and the lifted version is acyclic, offering the opportunity of benchmarking specialized models for directed (acyclic) structures. Moreover, the graph generator and the labeling are computer programmed; thus, the data set may be extended easily if a larger scale is desired. The data set is accessible from \url{https://github.com/IBM/IPC-graph-data}.Comment: ICML 2019 Workshop on Learning and Reasoning with Graph-Structured Data. The data set is accessible from https://github.com/IBM/IPC-graph-dat