Large Finite Graphs Research Articles

Min–max formulas for nonlocal elliptic operators

In this work, we give a characterization of Lipschitz operators on spaces of $$C^2(M)$$ functions (also $$C^{1,1}$$, $$C^{1,\gamma }$$, $$C^1$$, $$C^\gamma $$) that obey the global comparison property—i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called “elliptic” operators. Here M is a complete Riemannian manifold. In particular, we show that all such operators can be written as a min–max over linear operators that are a combination of drift–diffusion and integro-differential parts. In the linear (and nonlocal) case, these operators had been characterized in the 1960s, and in the local, but nonlinear case—e.g. local Hamilton–Jacobi–Bellman operators—this characterization has also been known and used since approximately since 1960s or 1970s. Our main theorem contains both of these results as special cases. It also shows any nonlinear scalar elliptic equation can be represented as an Isaacs equation for an appropriate differential game. Our approach is to “project” the operator to one acting on functions on large finite graphs that approximate the manifold, use non-smooth analysis to derive a min–max formula on this finite dimensional level, and then pass to the limit in order to lift the formula to the original operator.

$$L^p$$ Norms and Support of Eigenfunctions on Graphs

This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our estimates hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts.

Mathematics of Web science: structure, dynamics and incentives

Dr Chayes' talk described how, to a discrete mathematician, 'all the world's a graph, and all the people and domains merely vertices'. A graph is represented as a set of vertices V and a set of edges E, so that, for instance, in the World Wide Web, V is the set of pages and E the directed hyperlinks; in a social network, V is the people and E the set of relationships; and in the autonomous system Internet, V is the set of autonomous systems (such as AOL, Yahoo! and MSN) and E the set of connections. This means that mathematics can be used to study the Web (and other large graphs in the online world) in the following way: first, we can model online networks as large finite graphs; second, we can sample pieces of these graphs; third, we can understand and then control processes on these graphs; and fourth, we can develop algorithms for these graphs and apply them to improve the online experience.

Sharp threshold for percolation on expanders

We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then for any c in ]0,1[, the property that the random subgraph contains a giant component of size c|V| has a sharp threshold.

Parameter testing in bounded degree graphs of subexponential growth

AbstractParameter testing algorithms are using constant number of queries to estimate the value of a certain parameter of a very large finite graph. It is well‐known that graph parameters such as the independence ratio or the edit‐distance from 3‐colorability are not testable in bounded degree graphs. We prove, however, that these and several other interesting graph parameters are testable in bounded degree graphs of subexponential growth. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Estimation From Relative Measurements: Electrical Analogy and Large Graphs

We examine the problem of estimating vector-valued variables from noisy measurements of the difference between certain pairs of them. This problem, which is naturally posed in terms of a measurement graph, arises in applications such as sensor network localization, time synchronization, and motion consensus. We obtain a characterization on the minimum possible covariance of the estimation error when an arbitrarily large number of measurements are available. This covariance is shown to be equal to a matrix-valued effective resistance in an infinite electrical network. Covariance in large finite graphs converges to this effective resistance as the size of the graphs increases. This convergence result provides the formal justification for regarding large finite graphs as infinite graphs, which can be exploited to determine scaling laws for the estimation error in large finite graphs. Furthermore, these results indicate that in large networks, estimation algorithms that use small subsets of all the available measurements can still obtain accurate estimates.

Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Let (G n ) =1 ∞ be a sequence of finite graphs, and let Y t be the length of a loop-erased random walk on G n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G n is the d-dimensional torus of size-length n for d≥4, the process (Y t ) =0 ∞ , suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous.

Exact and approximate results for deposition and annihilation processes on graphs

We consider random sequential adsorption processes where the initially empty sites of a graph are irreversibly occupied, in random order, either by monomers which block neighboring sites, or by dimers. We also consider a process where initially occupied sites annihilate their neighbors at random times. We verify that these processes are well defined on infinite graphs, and derive forward equations governing joint vacancy/occupation probabilities. Using these, we derive exact formulae for occupation probabilities and pair correlations in Bethe lattices. For the blocking and annihilation processes we also prove positive correlations between sites an even distance apart, and for blocking we derive rigorous lower bounds for the site occupation probability in lattices, including a lower bound of 1/3 for Z2. We also give normal approximation results for the number of occupied sites in a large finite graph.