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We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time…
The interaction between transitivity and sparsity, two common features in empirical networks, implies that there are local regions of large sparse networks that are dense. We call this the blessing of transitivity and it has consequences…
The evolution of many stochastic systems is accurately described by random walks on graphs. We here explore the close connection between local steady-state fluctuations of random walks and the global structure of the underlying graph.…
We study diffusion (random walks) on recursive scale-free graphs, and contrast the results to similar studies in other analytically soluble media. This allows us to identify ways in which diffusion in scale-free graphs is special. Most…
In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical definition of…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
Nodes are randomly distributed within an annulus (and then a shell) to form a point pattern of communication terminals which are linked stochastically according to the Rayleigh fading of radio-frequency data signals. We then present…
In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit…
An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are…
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning…
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle…
We investigate searching efficiency of different kinds of random walk on complex networks which rely on local information and one-step memory. For the studied navigation strategies we obtained theoretical and numerical values for the graph…
In previous papers, threshold probabilities for the properties of a random distance graph to contain strictly balanced graphs were found. We extend this result to arbitrary graphs and prove that the number of copies of a strictly balanced…
Random walks are studied on disordered cellular networks in 2-and 3-dimensional spaces with arbitrary curvature. The coefficients of the evolution equation are calculated in term of the structural properties of the cellular system. The…
The betweenness centrality of graphs using random walk paths instead of geodesics is studied. A scaling collapse with no adjustable parameters is obtained as the graph size $N$ is varied; the scaling curve depends on the graph model. A…
Random walks are the simplest way to explore or search a graph, and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world, e.g. they have been used to identify the…
In this work we investigate the dynamics of random walk processes on scale-free networks in a short to moderate time scale. We perform extensive simulations for the calculation of the mean squared displacement, the network coverage and the…
Random walks can reveal communities or clusters in networks, because they are more likely to stay within a cluster than leave it. Thus, one family of community detection algorithms uses random walks to measure distance between pairs of…
Motivated by a problem arising from pharmaceutical science [B. Baeumer et al., Discr. Contin. Dyn. Sys. B 12], we study random walks on the contact graph of a bidisperse random sphere packing. For a random walk on the unweighted graph that…
There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a…