We study two random processes on an n-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. These processes can also be regarded as protocols for allocating jobs in a distributed network of servers. In both processes n particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until it first encounters an unoccupied vertex, at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called Sequential-IDLA, a single particle moves until settling and only then does the next particle start whereas in the second process, called Parallel-IDLA, all unsettled particles move simultaneously. The second process is akin to running the first in parallel. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the n particles. In order to compare the two processes, we develop a coupling which shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative łog n factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, d-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m=n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m>=n.

We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G.Each ball first chooses a vertex u in G uniformly at random.Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on.In our main result, we prove that this process yields a maximum load of only Θ(log log n) on expander graphs.In addition, we show that for d-dimensional grids the maximum load is Θ log n log log n 1 d+1 .Finally, for almost regular graphs with minimum degree Ω(log n), we prove that the maximum load is constant and also reveal a fundamental difference between random and arbitrary tie-breaking rules.

We consider sequential balls-into-bins processes that randomly allocate m balls into n bins. We analyze two allocation schemes that achieve a close to optimal maximum load of ⌈m/n⌉ + 1 and require only O(m) (expected) allocation time. These parameters should be compared with the classic d-choice-process which achieves a maximum load of m/n + log log n/d + O(1) and requires m • d allocation time.

The paper concerns lattice triangulations, i.e., triangulations of the integer points in a polygon in R2 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). 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 spatial mixing properties and dynamics of random lattice triangulations.

Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks. In this paper we consider a natural extension of the random geometric graph model to the mobile setting by allowing nodes to move in space according to Brownian motion. We study three fundamental questions in this model: detection (the time until a given target point—which may be either fixed or moving—is detected by the network), coverage (the time until all points inside a finite box are detected by the network), and percolation (the time until a given node is able to communicate with the giant component of the network). We derive precise asymptotics for these problems by combining ideas from stochastic geometry, coupling and multi-scale analysis. We also give an application of our results to analyze the time to broadcast a message in a mobile network.

We study the relation between the vertex expansion of a graph and the performance of randomized rumor spreading (push model). We prove that randomized rumor spreading takes O((1/α) · polylog(n)) time on any regular n-vertex graph with vertex expansion α. This bound extends previously known upper bounds by replacing conductance by vertex expansion. Our result is almost tight in the sense that the dependency on (1/α) is optimal (up to logarithmic factors) and that on non-regular graphs with constant vertex expansion, the runtime can be polynomial in n. Our upper bound also implies that randomized rumor spreading is “fast” on every vertex-transitive graph and yields a new upper bound on the cover time of random walks. We also exhibit a subtle difference between the impact of vertex expansion and conductance on rumor spreading. We show that there are regular graphs with constant vertex expansion for which randomized rumor spreading takes considerably longer than on any regular graph with constant conductance. Finally, we also prove a more general, but weaker result for the push & pull model which also covers non-regular graphs.

A Random Geometric Graph (RGG) in two dimensions is constructed by distributing n nodes independently and uniformly at random in and creating edges between every pair of nodes having Euclidean distance at most r, for some prescribed r. We 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 with probability 1 – (n−1) this algorithm informs every node in the largest connected component of an RGG within rounds. This holds for any value of r larger than the critical value for the emergence of a connected component with Ω(n) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in , the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is , which surprisingly has been an open problem so far.