Product Multicommodity Flow in Wireless Networks

We provide a tight approximate characterization of the $n$-dimensional product multicommodity flow (PMF) region for a wireless network of $n$ nodes. Separate characterizations in terms of the spectral properties of appropriate network graphs are obtained in both an information theoretic sense and for a combinatorial interference model (e.g., Protocol model). These provide an inner approximation to the $n^2$ dimensional capacity region. These results answer the following questions which arise naturally from previous work: (a) What is the significance of $1/\sqrt{n}$ in the scaling laws for the Protocol interference model obtained by Gupta and Kumar (2000)? (b) Can we obtain a tight approximation to the "maximum supportable flow" for node distributions more general than the geometric random distribution, traffic models other than randomly chosen source-destination pairs, and under very general assumptions on the channel fading model? We first establish that the random source-destination model is essentially a one-dimensional approximation to the capacity region, and a special case of product multi-commodity flow. Building on previous results, for a combinatorial interference model given by a network and a conflict graph, we relate the product multicommodity flow to the spectral properties of the underlying graphs resulting in computational upper and lower bounds. For the more interesting random fading model with additive white Gaussian noise (AWGN), we show that the scaling laws for PMF can again be tightly characterized by the spectral properties of appropriately defined graphs. As an implication, we obtain computationally efficient upper and lower bounds on the PMF for any wireless network with a guaranteed approximation factor.Comment: Revised version of "Capacity-Delay Scaling in Arbitrary Wireless Networks" submitted to the IEEE Transactions on Information Theory. Part of this work appeared in the Allerton Conference on Communication, Control, and Computing, Monticello, IL, 2005, and the Internation Symposium on Information Theory (ISIT), 200

On the freezing of variables in random constraint satisfaction problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.Comment: 32 pages, 7 figure

On the scaling limits of planar percolation

We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Energy-Latency Tradeoff for In-Network Function Computation in Random Networks

The problem of designing policies for in-network function computation with minimum energy consumption subject to a latency constraint is considered. The scaling behavior of the energy consumption under the latency constraint is analyzed for random networks, where the nodes are uniformly placed in growing regions and the number of nodes goes to infinity. The special case of sum function computation and its delivery to a designated root node is considered first. A policy which achieves order-optimal average energy consumption in random networks subject to the given latency constraint is proposed. The scaling behavior of the optimal energy consumption depends on the path-loss exponent of wireless transmissions and the dimension of the Euclidean region where the nodes are placed. The policy is then extended to computation of a general class of functions which decompose according to maximal cliques of a proximity graph such as the $k$-nearest neighbor graph or the geometric random graph. The modified policy achieves order-optimal energy consumption albeit for a limited range of latency constraints.Comment: A shorter version appears in Proc. of IEEE INFOCOM 201

SPoT: Representing the Social, Spatial, and Temporal Dimensions of Human Mobility with a Unifying Framework

Modeling human mobility is crucial in the analysis and simulation of opportunistic networks, where contacts are exploited as opportunities for peer-topeer message forwarding. The current approach with human mobility modeling has been based on continuously modifying models, trying to embed in them the mobility properties (e.g., visiting patterns to locations or specific distributions of inter-contact times) as they came up from trace analysis. As a consequence, with these models it is difficult, if not impossible, to modify the features of mobility or to control the exact shape of mobility metrics (e.g., modifying the distribution of inter-contact times). For these reasons, in this paper we propose a mobility framework rather than a mobility model, with the explicit goal of providing a exible and controllable tool for modeling mathematically and generating simulatively different possible features of human mobility. Our framework, named SPoT, is able to incorporate the three dimensions - spatial, social, and temporal - of human mobility. The way SPoT does it is by mapping the different social communities of the network into different locations, whose members visit with a configurable temporal pattern. In order to characterize the temporal patterns of user visits to locations and the relative positioning of locations based on their shared users, we analyze the traces of real user movements extracted from three location-based online social networks (Gowalla, Foursquare, and Altergeo). We observe that a Bernoulli process effectively approximates user visits to locations in the majority of cases and that locations that share many common users visiting them frequently tend to be located close to each other. In addition, we use these traces to test the exibility of the framework, and we show that SPoT is able to accurately reproduce the mobility behavior observed in traces. Finally, relying on the Bernoulli assumption for arrival processes, we provide a throughout mathematical analysis of the controllability of the framework, deriving the conditions under which heavy-tailed and exponentially-tailed aggregate inter-contact times (often observed in real traces) emerge