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We consider a large class of random geometric graphs constructed from samples $mathcal{X}_n = {X_1,X_2,ldots,X_n}$ of independent, identically distributed observations of an underlying probability measure $ u$ on a bounded domain $Dsubset mathbb{R}^d$. The popular `modularity clustering method specifies a partition $mathcal{U}_n$ of the set $mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $ u$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $mathcal{U}_n$ converge in a certain sense to a continuum partition $mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvins shape optimization problem.
In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak i
Given a `cost functional $F$ on paths $gamma$ in a domain $Dsubsetmathbb{R}^d$, in the form $F(gamma) = int_0^1 f(gamma(t),dotgamma(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,ldots, X_n$ be points drawn ind
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variab
We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate pha
Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of cu