We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point-fixed or moving-is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in R 2 whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation σ has weight λ |σ| , where λ is a positive real parameter, and |σ| is the total length of the edges in σ. Empirically, this model exhibits a "phase transition" at λ = 1 (corresponding to the uniform distribution): for λ < 1 distant edges behave essentially independently, while for λ > 1 very large regions of aligned edges appear. We substantiate this picture as follows. For λ < 1 sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for λ > 1 we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.

A random geometric graph (RGG) is defined by placing n points uniformly at random in [0, n 1/d ] d , and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of ω log n r d−1 , their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n 1/d /r) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n 1/d /r + log n) rounds. A conference version with slightly weaker results appeared in the 22nd International Symposium on Algorithms and Computation.

We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density µ c for sustained activity is strictly between 0 and 1. It was known that µ c > 0 on Z d , d ≥ 1, and that µ c < 1 on Z for small enough sleeping rate. We show that µ c → 0 as λ → 0 in all vertex-transitive transient graphs, implying that µ c < 1 for small enough sleeping rate. We also show that µ c < 1 for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that µ c > 0 in any vertex-transitive amenable graph, and that µ c ∈ (0, 1) for any sleeping rate on regular trees.

We study the problem of disseminating a piece of information through all the nodes of a network, given that it is known originally only to a single node. In the absence of any structural knowledge on the network other than the nodes' neighborhoods, this problem is traditionally solved by flooding all the network's edges. We analyze a recently introduced probabilistic algorithm for flooding and give an alternative probabilistic heuristic that can lead to some cost-effective improvements, like better trade-offs between the message and time complexities involved. We analyze the two algorithms both mathematically and by means of simulations, always within a random-graph framework and considering relevant nodedegree distributions.