Section: New Results

Point Processes, Stochastic Geometry and Random Geometric Graphs

Participants : François Baccelli, Pierre Brémaud, Bartek Błaszczyszyn, Yogeshwaran Dhandapani, Mir Omid Haji Mirsadeghi, Justin Salez.

Book on Stochastic Geometry and Wireless Networks

TREC is actively working on a book project focused on the use of the stochastic geometry framework for the modeling of wireless communications.

Stochastic geometry is a rich branch of applied probability which allows to study random phenomenons on the plane or in higher dimension. It is intrinsically related to the theory of point processes. Initially its development was stimulated by applications to biology, astronomy and material sciences. Nowadays, it is also used in image analysis. During the 03-08 period, we contributed to proving that it could also be of use to analyze wireless communication networks. The reason for this is that the geometry of the location of mobiles and/or base stations plays a key role since it determines the signal to interference ratio for each potential channel and hence the possibility of establishing simultaneously some set of communications at a given bit rate.

Stochastic geometry provides a natural way of defining (and computing) macroscopic properties of wireless networks, by some averaging over all potential geometrical patterns for e.g. the mobiles. Its role is hence similar to that played by the theory of point processes on the real line in the classical queueing theory. The methodology was initiated in  [32] , [36] and it was further developed through several papers including  [40] , [51] , [41] , [5] , [6] .

The two-volume book  [1] , [2] that will appear in the series Foundations and Trends in Networking (NOW Publishers; http://www.nowpublishers.com/product.aspx?product=NET&doi=1300000006 , http://www.nowpublishers.com/product.aspx?product=NET&doi=1300000026 ) will survey these papers and more recent results [25] , [24] obtained by this approach for analyzing key properties of wireless networks such as coverage or connectivity, and for evaluating the performance of a variety of protocols used in this context such as medium access control or routing.

More precisely, Volume I provides a concise introduction to relevant models of stochastic geometry, such as spatial shot-noise processes, coverage processes and random tessellations, and to variants of these basic models which incorporate information theoretic notions, such as signal to noise ratio.

Volume II shows how these space-time averages can be used to analyze and optimize the medium access control and routing protocols of interest in large wireless communication networks. This is based on both qualitative and quantitative results. The most important qualitative results are in terms of phase transitions for infinite population models. Quantitative results leverage closed form expressions for the key network performance characteristics.

The monograph provides a comprehensive and unified methodology for wireless network design and it gives a direct access to an emerging and fast growing branch of stochastic modeling.

Research on Stochastic Comparison of Random Measures and Point Processes

Stochastic geometric models of wireless networks have in general been investigated under Poissonian setting (see [32] , [36] ). The first aim of the PhD thesis of Yogeshwaran D. is to study certain performance measures of wireless networks using stochastic geometric tools in the non-Poissonian setting. Due to the difficulty in obtaining closed-form expressions for various performance measures in non-Poissonian settings (see [58] ), we attempted a qualitative study of the performance measures.

Directionally Convex Ordering

Directionally convex (dcx ) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. In [9] , viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space,

we formulate and study the dcx ordering of random measures on locally compact spaces. We show that the dcx order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition and thinning as well as independent, identically distributed marking. Further operations such as position dependent marking and displacement of points though do not preserve the dcx order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of dcx order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that point processes higher in dcx order cluster more. As the main result, we show that non-negative integral shot-noise fields with respect to dcx ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients.

Percolation and Directionally Convex Ordering of Point Processes and Random Fields

Heuristics indicate that clustering of a point process negatively impacts the percolation of the related continuum percolation model, called also the Boolean model. In in current work in progress we move towards a formal statement of this heuristic. Namely, we consider some critical radii for continuum percolation model and show that these are greater for Cox point processes, which are greater in the so called dcx order. This integral order has previously been shown (see [9] ) as suitable for comparison of dependence structure and clustering properties of point processes. Extensions to general point processes as well as comparison of critical levels for percolation of the level-sets of random fields are also discussed. Further, we give examples of point processes whose Boolean models percolate better than Poisson Boolean models.

Information Theory and Stochastic Geometry

In a joint work with Venkat Anantharam (UC Berkeley) [31] , a new class of problems was defined in the theory of Euclidean point processes, motivated by the study of the error exponent (reliability function) for additive noise channels in Information Theory. Each point of the point process is seen as a codeword and the additive noise as a random displacement from this point. Decoding is successful when the displacement of a point falls in the Voronoi cell of this point. For a wide class of point processes that have incarnations in all dimensions, there is a 0–1 law on the probability of successful decoding when dimension goes to infinity. This can be seen as an extension of Shannon's capacity theorem and error exponents can also be defined within this context. For the case of Gaussian noise this approach gives an interesting perspective on the Poltyrev exponent. It also suggests an approach to attack the long standing gap between the best known lower and upper bounds on the reliability function of the traditional AWGN channel, using techniques from point process theory. A new paper is under preparation with results on error exponents for stationary and ergodic noise.

Random Geometric Graphs

Random Geometric Graphs (RGG) have played an important role in providing a framework for modeling in wireless communication, starting with the pioneering work on connectivity by Gilbert (1961);  [46] . Vertices or points of the graphs represent communicating entities such as base stations. These vertices are assumed to be distributed in space randomly according to some point process, typically a Poisson point process. En edge between two points means that the communicating entities able to communicate with each other. In the classical model an edge exists between any two pair of nodes if the distance between them is less than some critical threshold. A variant of this classical model that exhibits the union of the coverage regions of all nodes is also referred to in stochastic geometry as the Boolean model. In the following, more fundamental works, we study some variants and extensions of the classical models, more or less related to wireless communication networks.

AB Random Geometric Graphs

We investigate percolation in the AB Poisson-Boolean model in d -dimensional Euclidean space, and asymptotic properties of AB random geometric graphs on Poisson points in [0, 1]^d . The AB random geometric graph we study is a generalization to the continuum of the AB percolation model on discrete lattices. We show existence of AB percolation for all d2 , and derive bounds for a critical intensity. For AB random geometric graphs, we derive a weak law result for the largest nearest neighbor distance and almost sure asymptotic bounds for the connectivity threshold. This submitted work can be found in  [47] .

First Passage Percolation Model for Delay Tolerant Networks

Delay Tolerant Networks, in the simplest terms, are networks that take into account the time-delay in the transmission of information along a network. First passage percolation models have been found to be useful for study of transmission of information along networks. We consider spatial first passage percolation on stationary graphs constructed on point processes with delayed propagation of the information at the vertices of the graph. Depending on the manner of the time-delay, one can obtain various models. The time for propagation of information along networks in such models do not possess the sub-additive property, a key component in the study of first passage percolation models. This is a work in progress.

Optimal Paths on Time-Space SINR Graphs

The following mathematical formalism proposed in  [35] is useful when studying macroscopic properties of routing in MANETs, and in particular these produced by opportunistic routing in the sense described in Section  6.1.2 . One can model the users of a mobile as points of a stochastic point process where each node can be a transmitter or receiver in each time step. The SINR graph is a geometric graph where the nodes are the points of a point process and an edge is present between a transmitter and a receiver if the SINR at the receiver is above a certain threshold. Due to fluctuations in propagation and MAC, these edges vary in time. In order to account for this fact, in [25] we introduce and analyze SINR graphs which have space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network.

The paper [25] studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both “positive” and “negative” results on the associated the percolation delay rate (delay per unit of Euclidean distance called in the classical terminology time constant). The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that the percolation delay rate is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the percolation delay rate is positive and finite.

Ergodicity of a Stress-Release-Point-Process Seismic Model with Aftershocks

The times of occurrence of earthquakes in a given area of seismic activity form a simple point process N on the real line, where N((a, b]) is the number of shocks in the time interval (a, b] . The dynamics governing the process can be expressed by the stochastic intensity (t) . In the stress release model, for t0 , , where c>0 and is an i.i.d. sequence of non-negative random variables with finite expectation, whereas X₀ is some real random variable. The process is known to be ergodic.

Another model of interest in seismology is the Hawkes branching process, where the stochastic intensity is , where h is a non-negative function, called the fertility rate and is a non-negative integrable function. Such point process appears in the specialized literature under the name ETAS (Epidemic Type After-Shock and is used to model the aftershocks. It is well known that the corresponding process “dies out” in finite time under the condition .

A model mixing stress release and Hawkes aftershocks is

where >0 . The positive constant c is the rate at which the strain builds up. If there is a shock at time t , then the strain is relieved by the quantity Z_N(t) . Each shock (primary or secondary) at time t generates aftershocks according to a Poisson process of intensity . In [27] , we give necessary and sufficient conditions of ergodicity for this model.