Revolutionaries and spies on random graphs

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of $r$ revolutionaries tries to hold an unguarded meeting consisting of $m$ revolutionaries. A team of $s$ spies wants to prevent this forever. For given $r$ and $m$, the minimum number of spies required to win on a graph $G$ is the spy number $\sigma(G,r,m)$. We present asymptotic results for the game played on random graphs $G(n,p)$ for a large range of $p = p(n), r=r(n)$, and $m=m(n)$. The behaviour of the spy number is analyzed completely for dense graphs (that is, graphs with average degree at least n^{1/2+\eps} for some \eps > 0). For sparser graphs, some bounds are provided

The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time \AC(G) defined for a connected graph $G$. We focus on \G(n,r,p), a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability \AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random geometric graph, provided that $\pi n r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph G \in \G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1} \ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some \eps>0

On the hyperbolicity of random graphs

Let $G=(V,E)$ be a connected graph with the usual (graph) distance metric $d:V \times V \to N \cup \{0 \}$. Introduced by Gromov, $G$ is $\delta$-hyperbolic if for every four vertices $u,v,x,y \in V$, the two largest values of the three sums $d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x)$ differ by at most $2\delta$. In this paper, we determinate the value of this hyperbolicity for most binomial random graphs.Comment: 20 page

Rank-based attachment leads to power law graphs

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment

Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games

On the threshold for k-regular subgraphs of random graphs

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph \G(n,p) asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in \G(n,p) to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$