Massimo Franceschetti

Stochastic geometry and random graphs for the analysis and design of wireless networks

Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.

Foundations of Control and Estimation Over Lossy Networks

This paper considers control and estimation problems where the sensor signals and the actuator signals are transmitted to various subsystems over a network. In contrast to traditional control and estimation problems, here the observation and control packets may be lost or delayed. The unreliability of the underlying communication network is modeled stochastically by assigning probabilities to the successful transmission of packets. This requires a novel theory which generalizes classical control/estimation paradigms. The paper offers the foundations of such a novel theory. The central contribution is to characterize the impact of the network reliability on the performance of the feedback loop. Specifically, it is shown that for network protocols where successful transmissions of packets is acknowledged at the receiver (e.g., TCP-like protocols), there exists a critical threshold of network reliability (i.e., critical probabilities for the successful delivery of packets), below which the optimal controller fails to stabilize the system. Further, for these protocols, the separation principle holds and the optimal LQG controller is a linear function of the estimated state. In stark contrast, it is shown that when there is no acknowledgement of successful delivery of control packets (e.g., UDP-like protocols), the LQG optimal controller is in general nonlinear. Consequently, the separation principle does not hold in this circumstance

Closing the Gap in the Capacity of Wireless Networks Via Percolation Theory

An achievable bit rate per source-destination pair in a wireless network of n randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same 1/radicn transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a 1/radicnlogn reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme

The Capacity of Wireless Networks: Information-Theoretic and Physical Limits

It is shown that the capacity scaling of wireless networks is subject to a fundamental limitation which is independent of power attenuation and fading models. It is a degrees of freedom limitation which is due to the laws of physics. By distributing uniformly an order of n users wishing to establish pairwise independent communications at fixed wavelength inside a two-dimensional domain of size of the order of n, there are an order of n communication requests originating from the central half of the domain to its outer half. Physics dictates that the number of independent information channels across these two regions is only of the order of radicn, so the per-user information capacity must follow an inverse square-root of n law. This result shows that information-theoretic limits of wireless communication problems can be rigorously obtained without relying on stochastic fading channel models, but studying their physical geometric structure.

Data Rate Theorem for Stabilization Over Time-Varying Feedback Channels

A data rate theorem for stabilization of a linear, discrete-time, dynamical system with arbitrarily large disturbances, over a rate-limited, time-varying communication channel is presented. Necessary and sufficient conditions for stabilization are derived, their implications and relationships with related results in the literature are discussed. The proof techniques rely on both information-theoretic and control-theoretic tools.

Detecting emotional contagion in massive social networks

Happiness and other emotions spread between people in direct contact, but it is unclear whether massive online social networks also contribute to this spread. Here, we elaborate a novel method for measuring the contagion of emotional expression. With data from millions of Facebook users, we show that rainfall directly influences the emotional content of their status messages, and it also affects the status messages of friends in other cities who are not experiencing rainfall. For every one person affected directly, rainfall alters the emotional expression of about one to two other people, suggesting that online social networks may magnify the intensity of global emotional synchrony.

A random walk model of wave propagation

This paper shows that a reasonably accurate description of propagation loss in small urban cells can be obtained with a simple stochastic model based on the theory of random walks, that accounts for only two parameters: the amount of clutter and the amount of absorption in the environment. Despite the simplifications of the model, the derived analytical solution correctly describes the smooth transition of power attenuation from an inverse square law with the distance to the transmitter, to an exponential attenuation as this distance is increased - as it is observed in practice. Our analysis suggests using a simple exponential path loss formula as an alternative to the empirical formulas that are often used for prediction. Results are validated by comparison with experimental data collected in a small urban cell.

Random networks for communication : from statistical physics to information systems

Preface 1. Introduction 2. Phase transitions in infinite networks 3. Connectivity of finite networks 4. More on phase transitions 5. Information flow in random networks 6. Navigation in random networks Appendix References Index.

On the throughput scaling of wireless relay networks

The throughput of wireless networks is known to scale poorly when the number of users grows. The rate at which an arbitrary pair of nodes can communicate must decrease to zero as the number of users tends to infinity, under various assumptions. One of them is the requirement that the network is fully connected: the computed rate must hold for any pair of nodes of the network. We show that this requirement can be responsible for the lack of throughput scalability. We consider a two-dimensional (2-D) network of extending area with only one active source-destination pair at any given time, and all remaining nodes acting only as possible relays. Allowing an arbitrary small fraction of the nodes to be disconnected, we show that the per-node throughput remains constant as the network size increases. As a converse bound, we show that communications occurring at a fixed nonzero rate imply a fraction of the nodes to be disconnected. Our results are of information theoretic flavor, as they hold without assumptions on the communication strategies employed by the network nodes.

Lower bounds on data collection time in sensory networks

Data collection, i.e., the aggregation at the user location of information gathered by sensor nodes, is a fundamental function of sensory networks. Indeed, most sensor network applications rely on data collection capabilities, and consequently, an inefficient data collection process may adversely affect the performance of the network. In this paper, we study via simple discrete mathematical models, the time performance of the data collection and data distribution tasks in sensory networks. Specifically, we derive the minimum delay in collecting sensor data for networks of various topologies such as line, multiline, and tree and give corresponding optimal scheduling strategies. Furthermore, we bound the data collection time on general graph networks. Our analyses apply to networks equipped with directional or omnidirectional antennas and simple comparative results of the two systems are presented.