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Abstract. We consider graphs attached to (Z/qZ) n , where q = p r , for an odd prime p, using an analogue of the Euclidean distance. These graphs are shown to be mostly non-Ramanujan, in contrast to the case of Euclidean graphs over finite fields.

We study Community Detection (CD) on a class of sparse spatial random graphs embedded in the Euclidean space. Our random graph is the planted-partition version of the random connection model studied in Stochastic Geometry. Each node has two labels-an i.i.d. uniform {−1, +1} valued community label and a R d valued location label which form the support of a Poisson Point Process of intensity λ on R d. Conditional on the labels, edges are drawn independently at random depending both on the Euclidean distance between the nodes and the community labels on the nodes. The CD problem then consists in estimating the partition of nodes into communities better than at random, based on an observation of the random graph and the spatial location labels on nodes. We establish a non-trivial phase-transition for this problem in terms of λ. We show that for small λ, there exists no algorithm for CD, For large λ, our algorithm solves CD efficiently. We show that for small λ, there exists no algorithm for CD. For large λ, we propose an algorithm which solves CD efficiently.In certain special cases, we establish the exact threshold on λ which separates the existence of an algorithm which solves CD from the impossibility of any algorithm. We also establish a distinguishability result which says that one can always efficiently infer the existence of a partition given the graph and the spatial locations even when one cannot identify the partition better than at random. This is a new phenomenon not observed thus far in any non-spatial Erd˝ os-Rényi based models.