Multi-agent networks are often modeled as interaction graphs, where the nodes represent the agents and the edges denote some direct interactions. The robustness of a multi-agent network to perturbations such as failures, noise, or malicious attacks largely depends on the corresponding graph. In many applications, networks are desired to have well-connected interaction graphs with relatively small number of links. One family of such graphs is the random regular graphs. In this paper, we present a decentralized scheme for transforming any connected interaction graph with a possibly non-integer average degree of @math into a connected random @math -regular graph for some @math . Accordingly, the agents improve the robustness of the network while maintaining a similar number of links as the initial configuration by locally adding or removing some edges.
A random @math -regular graph with @math nodes can be constructed by generating @math copies for each node, picking a uniform random perfect matching on the @math copies, and connecting any two nodes if the matching contains an edge between their copies (e.g., @cite_16 @cite_17 ). In @cite_29 , the authors present a distributed scheme for incrementally building random @math -regular multi-graphs with @math Hamiltonian cycles. Alternatively, some graph processes may be designed to transform an initial @math -regular graph into a random @math -regular graph by inducing a Markov chain with a uniform limiting distribution over the set of @math -regular graphs (e.g., @cite_5 @cite_25 ). The method in this paper is also based on designing a graph process with a uniform limiting distribution over the set of @math -regular graphs. Compared to the similar works in the literature, the proposed scheme is applicable to the most generic case, and it is decentralized. The initial graph is not required to satisfy some strong properties such as being regular or having an integer average degree. Furthermore, the global transformation is achieved via only some local graph operations.
@cite_29 @cite_5 @cite_16 @cite_25 @cite_17
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We develop a new approach for uniform generation of combinatorial objects, and apply it to derive a uniform sampler REG for @math -regular graphs. REG can be implemented such that each graph is generated in expected time @math , provided that @math . Our result significantly improves the previously best uniform sampler, which works efficiently only when @math , with essentially the same running time for the same @math . We also give a linear-time approximate sampler REG*, which generates a random @math -regular graph whose distribution differs from the uniform by @math in total variation distance, when @math . Let @math be a @math -regular triangle-free graph with @math edges. We present an algorithm which finds a cut in @math with at least @math edges in expectation, improving upon Shearer's classic result. In particular, this implies that any @math -regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of @math . Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm : each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques . We show that for any fixed @math , there exists a weighted neighbourhood graph @math such that there is a one-to-one correspondence between heavy cuts of @math and randomised local algorithms that find large cuts in any @math -regular input graph. This turns out to be a useful tool for analysing the existence of cuts in @math -regular graphs: we can compute the optimal cut of @math to attain a lower bound on the maximum cut size of any @math -regular triangle-free graph. Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process con Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, at each step every vertex chooses a neighbour uniformly at random and adopts its opinion. The voting is completed when all vertices hold the same opinion. In the simplest case, each vertex initially holds one of two different opinions. This partitions the vertices into arbitrary sets A and B. For many graphs, including regular graphs and irrespective of their expansion properties, if both A and B are sufficiently large sets, then pull voting requires @math expected steps, where n is the number of vertices of the graph. In this paper we consider a related class of voting processes based on sampling two opinions. In the simplest case, every vertex v chooses two random neighbours at each step. If both these neighbours have the same opinion, then v adopts this opinion. Otherwise, v keeps its own opinion. Let G be a connected graph with n vertices and m edges. Let P be the transition matrix of a simple random walk on G with second largest eigenvalue @math . We show that if the initial imbalance in degree between the two opinions satisfies @math , then with high probability voting completes in @math steps, and the opinion with the larger initial degree wins. The condition that @math , or only a bound on the conductance of the graph is known, the sampling process can be modified so that voting still provably completes in @math steps with high probability. The modification uses two sampling based on probing to a fixed depth @math from any vertex. In its most general form our voting process allows vertices to bias their sampling of opinions among their neighbours to achieve a desired outcom In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some standard regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of @math when the size of the graph @math tends to infinity. We also derive the exact value for the prefactor. These results allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average.