Research Topics

Control of queuing networks

One of the classical problems in queuing theory is to schedule customers/jobs in an optimal way. These problems are known as the scheduling problems. They arise in a wide variety of applications, in particular, whenever there are different customer classes present competing for the same resources. In a recent work “Ergodic control of multi-class M/M/N+M queues in the Halfin-Whitt regime”, Ari Arapostathis, Anup Biswas and Guodong Pang solved an ergodic control problem for multi-class many server queuing networks. The optimal solution of the queuing control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. The proof technique introduces a new method of spatial truncation for the diffusion control problem.

Coverage processes in wireless networks

Stochastic geometry provides a natural way of defining and computing macroscopic properties of classical channels of multiuser information theory. These macroscopic properties are obtained by some averaging over all node patterns found in a large random network of the Euclidean plane. One of the most important geometric objects are the coverage regions of a transmitter or a set of transmitters. This domain of research is jointly studied by Jeffrey Andrews, François Baccelli, Gustavo de Veciana, Robert Heath and Sanjay Shakkottai. Most of the initial steps are based on Poisson point processes. Lately, this continued with Yingzhe Li (Simons PhD student, ECE, UT Austin) to the case of determinantal point processes. Another line of work on studying cell-association problems in multi-technology cellular networks was carried out in this paper by Abishek Sankararaman, Jeong-woo Cho and François Baccelli.

Detecting planted communities in random graphs

The stochastic block model (aka. planted partition model) is a popular model for representing networks with communities. Elchanan Mossel, Joe Neeman, and Allan Sly have been investigating algorithms and fundamental limits for detecting and recovering these communities. They established sharp transitions for the problem of extracting non-trivial information and the problem of exactly recovering communities. They also gave a new algorithm that obtains provably optimal accuracy for the problem of detecting communities in “Consistency thresholds for the planted bisection model” and “Belief propagation, robust reconstruction, and optimal recovery of block models“.

Entropy of point processes

François Baccelli and Jae Oh Woo initiated a study on the entropy and mutual information of point processes (On the Entropy and Mutual Information of Point Processes). The main new mathematical objects are the relative entropy rate and the mutual information rate of two stationary point processes. They also derived expression of the mutual information rate in the case of a homogeneous point process and its displacement. This machinery is used to revisit the Gaussian noise channel in the Shannon-Poltyrev regime recently introduced in Capacity and error exponents of stationary point processes under random additive displacements.

Extremes of spatial shot noise processes

Anup Biswas and François Baccelli studied the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, they showed that the shot-noise field can be scaled suitably to have a non degenerate alpha-stable limit, as the intensity of the underlying point process goes to infinity. More precisely, finite dimensional distributions  converge and the finite dimensional distributions of the limiting random field have i.i.d. stable random components. This limit is hence called the alpha- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fréchet white noise field.

Information theory and high dimensional stochastic geometry

The most basic capacity and error exponent questions of information theory can be expressed in terms of random geometric objects living in Euclidean spaces with dimensions tending to infinity. This approach was introduced by Venkat Anantharam and François Baccelli to evaluate random coding error exponents in channels with additive stationary and ergodic noise. More generally, the analysis of stochastic geometry in the Shannon regime leads to new high dimension stochastic geometry questions that are currently investigated.

Mathematical problems in neuroscience

Recent advances in neuroscience provide theoretical neuroscientists with a vast wealth of new data and open questions. The Fiete Lab and Ngoc Mai Tran have made several major contributions to the theory of grid cells in mammals and learning capacity of neural networks (see in particular “A binary Hopfield network with 1/\log(n) information rate and applications to grid cell decoding” and “Robust exponential memory in Hopfield networks“). Some questions have promising connections to the work  on information theory and high dimensional stochastic geometry mentioned above.

Mathematics of social networks

Avhishek Chatterjee, François Baccelli and Sriram Vishwanath proposed a stochastic extension of the bounded confidence model where opinions take their values in the Euclidean space and where friendship and interactions are dynamically defined through time varying and random neighborhoods. Two basic sub-models are defined: the influencing model where each agent is an attractor to the opinions of its neighbors and the listening model where each agent gathers information from others to update its own opinions. The general model contains a rich set of variants for which they proposed a classification. They analyzed the stability of its dynamics. The analysis highlights the need of certain leaders with heavy tailed neighborhoods for stability to hold.

Metastability of Queuing Networks with Mobile Servers

In a paper entitled Metastability of Queuing Networks with Mobile Servers, A. Rybko, S. Shlosman, A. Vladimorov and F. Baccelli study symmetric queuing networks with moving servers and FIFO service discipline. The mean-field limit dynamics demonstrates unexpected behavior which is attributed to the meta-stability phenomenon. Large enough finite symmetric networks on regular graphs are proved to be transient for arbitrarily small inflow rates. However, the limiting non-linear Markov process possesses at least two stationary solutions. The mean-field analysis is based on the Non Linear Markov Process developed for this type of queuing networks in Queuing Networks with Varying Topology – A Mean-Field Approach.

Moving User Time Series in Wireless Stochastic Geometry

With Pranav Madadi, F. Baccelli, and G. de Veciana analyzed the temporal variations in the Shannon rate experienced by a user moving along a straight line in a cellular network represented by a Poisson-Voronoi tessellation. We consider a network that is shared by static users distributed as a Poisson point process and analyzed the time series of the final shared rate and the number of users sharing the network.  The first paper On Shared Rate Time Series for Mobile Users in Poisson Networks was focused on the noise limited case. The ongoing research is focused on the general case, with both interference and thermal noise taken into account.

Phases and phase transitions in complex networks

In the phenomenon of emergence, a system of many interacting objects
exhibits the collective behavior of one or more “phases”, which are
only detectable or even meaningful for the system as a whole. This is
an organizing principle widely used in biology, physics and indeed all
the sciences: crystals, hurricanes, animal flocking etc. One wants to
understand the spontaneous appearance of phases in systems of large
size, in particular to determine a mechanism of some generality. A
convenient framework for such an analysis is large networks with
constraints. Such an analysis has been undertaken by the group of
Richard Kenyon, Charles Radin, Kui Ren and Lorenzo Sadun, on entropy singularities, the edge/triangle system, multipodal structure,
order-disorder transitions and oversaturated networks. There is also a related asymptotic analysis of large permutations undertaken by the group of Richard Kenyon, Daniel Kral, Charles Radin and Peter Winkler: permutations with fixed pattern densities, and a review of phases in general combinatorial systems.

Point maps on point processes

A compatible point-shift f maps, in a translation invariant way, each point of a stationary point process N to some point of N. It is fully determined by its associated point-map, g, which gives the image of the origin by f. The initial question studied by Mir-Omid Haji-Mirsadeghi and François Baccelli is whether there exist probability measures which are left invariant by the translation of -g. The point map probabilities of N are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to the initial question. Point-map probabilities are shown to be a strict generalization of Palm probabilities: when the considered point-shift f is bijective, the point-map probability of N boils down to the Palm probability of N. When it is not bijective, there exist cases where the point-map probability of N is absolutely continuous with respect to its Palm probability, but there also exist cases where it is singular with respect to the latter.

Each such point-shift defines a random graph on the points of the point process. The connected components of this graph can be split into a collection of foils, which are the analogue of the stable manifold of the point-shift dynamics.
The same authors give a general classification of point-shifts in terms of the cardinality of the foils of these connected components. There are three types: F/F, I/F and I/I as shown in the paper Point-Shift Foliation of a Point Process. Using the framework of Günter Last, James Murphy has extended the cardinality classification to the case of point processes on unimodular groups. Murphy has studied point-shifts of point processes on topological groups at length.

Rigidity of dense phases

Dense phases of emergent systems, such as constrained complex networks,
exhibit distinct characteristics, the most studied being broken symmetry.
However for practical purposes “rigidity”, the resistance to change, is
also of wide interest. There are difficulties analyzing rigidity since
when perturbed a system can easily move out of its phase. A new approach to
overcome this contradiction has been initiated by David Aristoff and
Charles Radin: see the discussion in Quanta Magazine.

Spatial birth and death processes

Spatial point processes involving birth and death dynamics are ubiquitous in networks. Such dynamics are particularly important in peer-to-peer networks and in wireless networks. In the paper “Mutual Service Processes in R^d, Existence and Ergodicity”, Fabien Mathieu (Bell Laboratories), Ilkka Norros (VTT) and François Baccelli proposed a way to analyze the long term behavior of such dynamics on Euclidean spaces using coupling techniques. This line of though is continued by Mayank Manjrekar on other classes of spatial birth and death processes.

Vertex Shifts on Random Graphs

Ali Khezeli, Mir-Omid Haji-Mirsadeghi and François Baccelli studied dynamics on the vertices of a random graph in the paper Dynamics on Unimodular Graphs. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils, as in the case of point-shifts on point processes.
The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three types of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils).
An infinite connected component of the graph of a vertex-shift on a random network forms an infinite directed tree with one selected end which is referred to as an Eternal Family Tree. An Eternal Family Tree contains a subtree which is a stochastic generalization of a branching process. In a unimodular Eternal Family Tree, the subtree in question is a generalization of a critical branching process. In a $\sigma$-invariant Eternal Family Tree, the subtree is a generalizisation of a non-necessarily critical branching process. The latter trees allow one to analyze dynamics on networks which are not necessarily unimodular.

Zeros of Random Tropical Polynomials

Ngoc Mai Tran and François Baccelli derived a tropical version of the result of Kac on the zeros of polynomials with random coefficients (Zeros of Random Tropical Polynomials, Random Polygons and Stick-Breaking).

The common roots of tropical of a class of random polynomials in two variables is analyzed in a recent work by the same authors in Iterated Gilbert Mosaics and Poisson Tropical Plane Curves using a stochastic geometry approach based on iterated random tessellations.

Publications

2017_1000829

On solutions of mean field games with ergodic cost

Ari Arapostathis, Anup Biswas, and Johnson CarrollJournal de Mathematiques Pures et Appliqueesvol. 107, no. 5, pp. 205-251, 2017
2017-04-14_1000789

Metastability of Queuing Networks with Mobile Servers

F. Baccelli, A. Rybko, S. Shlosman, A. Vladimirovhttps://arxiv.org/abs/1704.02521
2017-04-01_1000817

2016_1000824

2016_1000818

2016_1000816

2016_1000621

2016_1000586

End-to-End Optimization of High Throughput DNA Sequencing

Eliza O'Reilly, Francois Baccelli, Gustavo de Veciana, Haris VikaloTo appear in Journal of Computational Biology2016
2016-10-16_1000749

Iterated Gilbert Mosaics and Poisson Tropical Plane Curves

Francois Baccelli and Ngoc Tranhttps://arxiv.org/abs/1610.08533
2016-09-25_1000612

A 3-D Spatial Model for In-building Wireless Networks with Correlated Shadowing

Junse Lee, Xinchen Zhang, Francois BaccelliAccepted to IEEE Transactions on Wireless Communications
2016-09-18_1000754

On Shared Rate Time Series for Mobile Users in Poisson Networks

2016-08-22_1000655

Scaling Laws for Ergodic Spectral Efficiency in MIMO Poisson Networks

Junse Lee, Namyoon Lee, Francois BaccelliSubmitted to IEEE Transactions on Information Theory
2016-08-21_1000762

Dynamics on Unimodular Random Graphs

François Baccelli, Mir-Omid Haji-Mirsadeghi, Ali Khezelihttps://arxiv.org/abs/1608.05940
2016-08-11_1000660

On the Entropy and Mutual Information of Point Processes

François Baccelli, Jae Oh WooIEEE International Symposium on Information Theory2016/695-699
2016-04-27_1000640

A symmetry breaking transition in the edge/triangle network model

2016-01-19_1000533

Phases in large combinatorial systems

2016-01-14_1000623

Point-Shift Foliation of a Point Process

Francois Baccelli and Mir-Omid Haji-MirsadeghiArXiv 2016
2015-10-21_1000150

Ergodic Control of Multi-class M/M/N+M Queues in the Halfin-Whitt Regime

Ari Arapostathis, Anup Biswas and Guodong PangAnnals of Applied Probabilityvol. 25, no. 6, pp. 3511-3570, 2015
2015-09-18_1000522

Bipodal structure in oversaturated random graphs

2015-09-13_1000135

Zeros of Random Tropical Polynomials, Random Polygons and Stick-Breaking

François Baccelli and Ngoc Mai TranTo appear in the Transctions of the AMS2015
2015-09-01_1000506

Permutations with fixed pattern densities

R. Kenyon, D. Kral, C. Radin, P. WinklerArXiv 2015
2015-06-18_1000441

On Impact of Fairness and Heterogeneity on Delays in Large-scale Content Delivery Networks

Virag Shah, Gustavo de VecianaACM Sigmetrics 2015
2015-06-16_1000329

Consistency thresholds for the planted bisection model

Elchanan Mossel, Joe Neeman and Allan SlyTo appear in Symposium on the Theory of Computer Science 2015
2015-05-28_1000151

On Scaling Limits of Power Law Shot-noise Fields

François Baccelli and Anup BiswasTo appear in Stochastic Models2015
2015-04-30_1000324

Pairwise Stochastic Bounded Confidence Opinion Dynamics: Heavy Tails and Stability

François Baccelli, Avhishek Chatterjee and Sriram VishwanathProceedings IEEE Infocom 2015
2015-04-28_1000153

Controlled Equilibrium Selection in Stochastically Perturbed Dynamics

Ari Arapostathis, Anup Biswas and Vivek BorkarArxiv2015
2015-04-21_1000233

Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

Venkat Anantharam and François BaccelliAdvances Applied Prob. Volume 47, Number 1, 2015
2015-03-01_1000218

Singularities in the Entropy of Asymptotically Large Simple Graphs

2015-01-22_1000165

2014_1000830

Convergence of the relative value iteration for the ergodic control problem of nondegenerate diffusions under near- monotone costs

Ari Arapostathis, Vivek S. Borkar, and K. Suresh Kumarvol. 52, no. 1, pp. 1–31, 2014
2014-12-05_1000282

Statistical Modeling and Probabilistic Analysis of Cellular Networks with Determinantal Point Processes

Yingzhe Li, François Baccelli, Harpreet S. Dhillon and Jeffrey G. AndrewsArxiv 2014
2014-11-17_1000382

Robust exponential memory in Hopfield networks

Christopher Hillar and Ngoc Mai TranarXiV2014
2014-10-28_1000283

Spectral Efficiency Scaling Laws in Dense Random Wireless Networks with Multiple Receive Antennas

Namyoon Lee, François Baccelli and Robert W. Heath JrArxiv2014
2014-08-06_1000286

The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Venkat Anantharam and François BaccelliArxiv 2014To appear in Advances in Applied Probability
2014-07-22_1000381

A binary Hopfield network with 1/\log(n) information rate and applications to grid cell decoding

Ila Fiete, David J. Schwab and Ngoc M Tran 2nd Workshop on Biological Distributed Algorithms2014
2014-07-19_1000288

Compactification of the Action of a Point-Map on the Palm Probability of a Point Process

François Baccelli and Mir-Omid Haji-MirsadeghiArxiv 2014To appear in the Annals of Probability
2014-07-16_1000167

Spatial Reuse and Fairness in Ad Hoc Networks with Channel-Aware CSMA Protocols

Yuchul Kim, François Baccelli and Gustavo de VecianaIEEE Transactions on Information Theory60(7):4139-57, July 2014
2014-06-18_1000333

Belief propagation, robust reconstruction, and optimal recovery of block models

Elchanan Mossel, Joe Neeman and Allan SlyConference on Learning Theory2014
2014-06-15_1000331

Testing surface area with arbitrary accuracy

Joe NeemanSymposium on the Theory of Computer Science2014
2014-04-19_1000152

On a Class of Stochastic Differential Equations With Jumps and its Properties

Ari Arapostathis, Anup Biswas and Luis CaffarelliArxiv2014
2014-04-13_1000212

The Asymptotics of Large Constrained Graphs

Charles Radin, Kui Ren and Lorenzo SadunJ. Phys. A: Math. Theor. 47 (2014) 175001
2014-04-10_1000209

Multipodal Structure and Phase Transitions in Large Constrained Graphs

2014-04-08_1000146