In this paper, we introduce a mathematical framework for the characterization of network interference in wireless systems. We consider a network in which the interferers are scattered according to a spatial Poisson process and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We start by determining the statistical distribution of the aggregate network interference. We then investigate four applications of the proposed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spectrum of the aggregate radio-frequency emission of wireless networks; and 4) coexistence between ultrawideband and narrowband systems. Our framework accounts for all the essential physical parameters that affect network interference, such as the wireless propagation effects, the transmission technology, the spatial density of interferers, and the transmitted power of the interferers.

How can we localize the source of diffusion in a complex network? Because of the tremendous size of many real networks-such as the internet or the human social graph-it is usually unfeasible to observe the state of all nodes in a network. We show that it is fundamentally possible to estimate the location of the source from measurements collected by sparsely placed observers. We present a strategy that is optimal for arbitrary trees, achieving maximum probability of correct localization. We describe efficient implementations with complexity O(N(α)), where α=1 for arbitrary trees and α=3 for arbitrary graphs. In the context of several case studies, we determine how localization accuracy is affected by various system parameters, including the structure of the network, the density of observers, and the number of observed cascades.

Information-theoretic security -widely accepted as the strictest notion of security -relies on channel coding techniques that exploit the inherent randomness of the propagation channels to significantly strengthen the security of digital communications systems. Motivated by recent developments in the field, this paper aims at a characterization of the fundamental secrecy limits of wireless networks. Based on a general model in which legitimate nodes and potential eavesdroppers are randomly scattered in space, the intrinsically secure communications graph (iS-graph) is defined from the point of view of informationtheoretic security. Conclusive results are provided for the local connectivity of the Poisson iS-graph, in terms of node degrees and isolation probabilities. It is shown how the secure connectivity of the network varies with the wireless propagation effects, the secrecy rate threshold of each link, and the noise powers of legitimate nodes and eavesdroppers. Sectorized transmission and eavesdropper neutralization are explored as viable strategies for improving the secure connectivity. Lastly, the maximum secrecy rate between a node and each of its neighbours is characterized, and the case of colluding eavesdroppers is studied. The results help clarify how the spatial density of eavesdroppers can compromise the intrinsic security of wireless networks. Index TermsPhysical-layer security, wireless networks, stochastic geometry, secure connectivity, node degree, secrecy capacity, colluding eavesdroppers. I. INTRODUCTIONContemporary security systems for wireless networks are based on cryptographic primitives that generally ignore two key factors: (a) the physical properties of the wireless medium, and (b) the spatial configuration of both the legitimate and malicious nodes. These two factors are important since they affect the communication channels between the nodes, which in turn determine the fundamental secrecy limits of a wireless network. In fact, the inherent randomness of the physics of the wireless medium and the spatial location of the nodes can be leveraged to provide intrinsic security of the communications infrastructure at the physical-layer level. 1 1 In the literature, the term "security" typically encompasses 3 different characteristics: secrecy (or privacy), integrity, and authenticity. This paper does not consider the issues of integrity or authenticity, and the terms "secrecy and "security" are used interchangeably. 4 The basis for information-theoretic security, which builds on the notion of perfect secrecy [1], was laid in [2] and later in [3]. Moreover, almost at the same time, the basic principles of publickey cryptography, which lead to the predominance of computational security, were published in [4]. More recently, there has been a renewed interest in information-theoretic security over wireless channels. Space-time signal processing techniques for secure communication over wireless links are introduced in [5]. The secrecy of cooperative relay broadcast channels is considered in [6]. ...

We present a mathematical model for communication subject to both network interference and noise. We introduce a framework where the interferers are scattered according to a spatial Poisson process, and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We consider both cases of slow and fastvarying interferer positions. The paper is comprised of two separate parts. In Part I, we determine the distribution of the aggregate network interference at the output of a linear receiver. We characterize the error performance of the link, in terms of average and outage probabilities. The proposed model is valid for any linear modulation scheme (e.g., M -ary phase shift keying or M -ary quadrature amplitude modulation), and captures all the essential physical parameters that affect network interference. Our work generalizes the conventional analysis of communication in the presence of additive white Gaussian noise and fast fading, allowing the traditional results to be extended to include the effect of network interference. In Part II of the paper, we derive the capacity of the link when subject to network interference and noise, and characterize the spectrum of the aggregate interference.Index Terms-Spatial distribution, Poisson field, aggregate network interference, error probability, stable laws.