Poisson-Voronoi Spanning Trees with Applications to the Optimization of Communication Networks

We define a family of random trees in the plane. Their nodes of level k, k=0\thru m are the points of a homogeneous Poisson point process $\Pi_k$, whereas their arcs connect nodes of level $k$ and $k+1$, according to the least distance principle: if $V$ denotes the Voronoi cell w.r.t. $\Pi_k+1$ with nucleus $x$, where $x$ is a point of $\Pi_k+1$, then there is an arc connecting $x$ to all the points of $\Pi_k$ which belong to $V$. This creates a family of stationary random trees rooted in the points of $\Pi_m$. These random trees are useful to model the spatial organization of several types of hierarchical communication networks. In relation with these communication networks, it is natural to associate various cost functions with such random trees. Using point process techniques, like the exchange formula between two Palm measures, and integral geometry techniques, we show how to compute the average cost in function of the intensity parameters of the Poisson processes. The formulas which are derived for the average value of the cost function are then exploited for parametric optimization purposes

Connected Spatial Networks over Random Points and a Route-Length Statistic

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina

STARR-DCS: Spatio-temporal adaptation of random replication for data-centric storage

This article presents a novel framework for data-centric storage (DCS) in a wireless sensor and actor network (WSAN) that employs a randomly selected set of data replication nodes, which also change over time. This enables reductions in the average network traffic and energy consumption by adapting the number of replicas to applications' traffic, while balancing energy burdens by varying their locations. To that end, we propose and validate a simple model to determine the optimal number of replicas, in terms of minimizing average traffic/energy consumption, based on measurements of applications' production and consumption traffic. Simple mechanisms are proposed to decide when the current set of replication nodes should be changed, to enable new applications and nodes to efficiently bootstrap into a working WSAN, to recover from failing nodes, and to adapt to changing conditions. Extensive simulations demonstrate that our approach can extend a WSAN's lifetime by at least 60%, and up to a factor of 10× depending on the lifetime criterion being considered. The feasibility of the proposed framework has been validated in a prototype with 20 resource-constrained motes, and the results obtained via simulation for large WSANs have been also corroborated in that prototype.The research leading to these results has been partially funded by the Spanish MEC under the CRAMNET project (TEC2012-38362-C03-01) and the FIERRO project (TEC 2010- 12250-E), and by the General Directorate of Universities and Research of the Regional Government of Madrid under the MEDIANET Project (S2009/TIC-1468). G. de Veciana was supported by the National Science Foundation under Award CNS-0915928Publicad