Andrew Beveridge

Random Minimum Length Spanning Trees in Regular Graphs

r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that .

Network of Thrones

T he international hit HBO series Game of Thrones, adapted from George R. R. Martin’s epic fantasy novel series A Song of Ice and Fire, features interweaving plotlines and scores of characters. With so many people to keep track of in this sprawling saga, it can be a challenge to fully understand the dynamics between them. To demystify this saga, we turn to network science, a new and evolving branch of applied graph theory that brings together traditions from many disciplines, including sociology, economics, physics, computer science, and mathematics. It has been applied broadly across the sciences, the social sciences, the humanities, and in industrial settings. In this article we perform a network analysis of Game of Thrones to make sense of the intricate character relationships and their bearing on the future plot (but we promise: no spoilers!). Driven by cause or circumstance, characters from the many noble families launch into arduous and intertwined journeys. Among these houses are the honorable Stark family (Eddard, Catelyn, Robb, Sansa, Arya, Bran, and Jon Snow), the pompous Lannisters (Tywin, Jaime, Cersei, Tyrion, and Joff rey), the slighted Baratheons (led by Robert’s brother Stannis) and the exiled Daenerys, the last of the oncepowerful House Targaryen.

Product rule wins A competitive game

We consider a game that can be viewed as a random graph process. The game has two players and begins with the empty graph on vertex set [n]. During each turn a pair of random edges is generated and one of the players chooses one of these edges to be an edge in the graph. Thus the players guide the evolution of the graph as the game is played. One player controls the even rounds with the goal of creating a so-called giant component as quickly as possible. The other player controls the odd rounds and has the goal of keeping the giant from forming for as long as possible. We show that the product rule is an asymptotically optimal strategy for both players.

Centers for Random Walks on Trees

We consider two distinct centers which arise in measuring how quickly a random walk on a tree mixes. Lovasz and Winkler [Efficient stopping rules for Markov chains, in Proceedings of the 27th ACM Symposium on the Theory of Computing, 1995, pp. 76-82] point out that stopping rules which “look where they are going” (rather than simply walking a fixed number of steps) can achieve a desired distribution exactly and efficiently. Considering an optimal stopping rule that reflects some aspect of mixing, we can use the expected length of this rule as a mixing measure. On trees, a number of these mixing measures identify particular nodes with central properties. In this context, we study a variety of natural notions of centrality. Each of these criteria identifies the barycenter of the tree as the “average” center and the newly defined focus as the “extremal” center.

A Leapfrog Strategy for Pursuit-Evasion in a Polygonal Environment

We study pursuit-evasion in a polygonal environment with polygonal obstacles. In this turn based game, an evader e is chased by pursuers p1,p2,…,pl. The players have full information about the environment and the location of the other players. The pursuers are allowed to coordinate their actions. On the pursuer turn, each pi can move to any point at distance at most 1 from his current location. On the evader turn, he moves similarly. The pursuers win if some pursuer becomes co-located with the evader in finite time. The evader wins if he can evade capture forever. It is known that one pursuer can capture the evader in any simply-connected polygonal environment, and that three pursuers are always sufficient in any polygonal environment P (possibly with polygonal obstacles). We contribute two new results to this field. First, we fully characterize when an environment with a single obstacle is one-pursuerwin or two-pursuer-win. Second, we give sufficient (but not necessary) conditions for an environment to have a winning strategy for two pursuers. Such environments can be swept by a leapfrog strategy in which the two cops alternately guard/increase the currently controlled area. The running time of this algorithm is O(n⋅h⋅diam(P)) where n is the number of vertices, h is the number of obstacles and diam(P)) is the diameter of the polygonal environment P. More concretely, for an environment with n vertices, we describe an O(n2) algorithm that (1) determines whether the obstacles are well-separated, and if so, (2) constructs the required partition for a leapfrog strategy.

Visibility Number of Directed Graphs

A k-bar visibility representation of a digraph

Exact Mixing Times for Random Walks on Trees

G

Memoryless Rules for Achlioptas Processes

assigns each vertex at most

Random walks and the regeneration time

k

Recent Trends in Combinatorics

horizontal segments in the plane so that