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Network Motion

Periodic Reporting for period 2 - NEMO (Network Motion)

Reporting period: 2020-07-01 to 2021-12-31

NEMO is at the interface of mathematics and networking. Its objective is to improve the mathematical understanding of dynamics in large random networks and to use this understanding to analyze concrete network problems in, e.g. communication networks, social networks or neural networks.

The mathematical part of the project is centered on the theory of unimodular random graphs [15] (this was the topic of the inaugural workshop of NEMO in March 2019 in Paris), of point processes [7] and of stochastic geometry, which are appropriate frameworks for analyzing processes on large networks. Instances of dynamics considered are navigations on graphs [7,9,14,15], optimizations on graphs [3,5], random walks and migrations on graphs [15], epidemics on graphs [10], scheduling and queuing on graphs [16]. Instances of mathematical tools involved, beyond unimodularity, are mean-fields [19], spectral analysis [4,11,13,17], coupling from the past. This mathematical work is based on national (C. Bordenave [4], M. Karray [7], J. Salez [13]) and international (V. Anantharam [3], M.O. Haji-Mirsadeghi [15], C. Hisrch [5]) collaborations. The mathematical understanding of these dynamics is important for the mastering of concrete large and sparse random networks.

The application part of the project has a main component in communication networks. It is focused on the modeling of cellular wireless networks using stochastic geometry [1,2,6,8,9,12,21]. It is based on international collaborations and on a strong collaboration which was initiated in 2019 by the PI with Nokia Bell Laboratories (NBL) Paris and thanks to the LINCS joint laboratory. One of the problems considered with NBL is that of the interplay between beam management and user motion [6,8,9]. This work, which is based on stochastic geometry, answers core questions in 5/6G networking and is leveraged by Nokia for standardization purposes. Another question considered in this domain is that of queuing type dynamics in wireless networks [16], which is an important open question. We address this problem through mean field techniques for spatial point process birth-and-death dynamics. Another application domain is that of neural network dynamics [19], which is also addressed by mean field techniques. This networking part is also the object of active national (NBL France) and international (USA [3,10,14,18,19], India [4], South Korea [12,21]) collaborations.

The preprint of a new research book on the theory of point processes and stochastic geometry was posted in 2020 [7]. It contains several new results on e.g. determinantal and permanantal point processes, on unimodularity, etc. The main achievement in the domain of unimodular graphs is the completion of the mathematical definition of the unimodular Hausdorff and the Minkowski dimensions of random networks (by the PI, A. Khezeli and M.-0 Hadji-Mirsadeghi) [15]. This can for instance be used to evaluate the dimensions of the trees that are associated with vertex-shift type dynamics (M1 in the proposal).
Various specific graph and point process dynamics were also studied: point process based space occupation games [3] (M2), optimal marking on point processes [5] (M2), contact processes on point processes [10] (M2), contention processes on point processes [16] (M2,M3).

Using the framework of stationary point processes in Euclidean space, Blaszczyszyn and Hirsch [5] proposed a general approach to specific problems, ranging from theoretical probability to applications in statistical physics, combinatorial optimization and communication, with optimal tuning of local parameters in large interacting particle systems (M2-M3).

Several results were obtained by S. Coste on spectral properties of random directed networks. When a specific structure is planted in a network, this structure can often be read in the spectrum of the network. In [4,11] (linked to M3), various results were obtained on when the recovery of this structure is feasible based on spectral algorithms. In [13], it is shown how unimodularity of a network is linked with the nature (continuous or pure-point) in the spectrum, around the origin.


Communication Networks:
A stochastic geometry analysis of 5G networking was conducted in collaboration with Nokia Bell Labs. The main effort has been on beam management. The main achievement is a system-level stochastic geometry model encompassing major aspects of the beam management problem [6] (N* in the proposal). This model leads to an analytical expression for the effective spectral efficiency that a user gets under motion. This determines the number of beams per cell that offers the best effective spectral efficiency. This paper received a best paper award at IEEE Globecom 2020 [8]. Other important results obtained by stochastic geometry on cellular networks bear on the gains obtained by bandwidth partitioning [1,2] (N2), the reliability of safety messages in vehicular networks [12,21,22] (N2), and new motion aware association policies [20] (N*).

Social Networks:
Opinion dynamics of the noisy bounded confidence type was investigated in [18]. The main new result is a computational framework allowing the interaction point process rate to depend on the instantaneous geometry of opinions (d.3.3 in the proposal).

Neural Networks:
Replica mean field techniques were investigated in [19] (N*) in relation with a class of point process based dynamical systems that includes the discrete time version of the Galves-Loecherbach neural network model. The main existence result obtained validates the computational neuroscience literature which was relying on the conjecture that this mean-field limit holds.
The work on unimodular dimensions will be continued in several new directions. We will study the connections to other definitions of dimensions for infinite discrete structures. We also expect to solve some of the open problems listed in [15]. We also intend to use this machinery for analyzing the foliation of new classes of vertex-shifts on unimodular random graphs.

In [4,11], the authors discovered new and surprising thresholds on the signal-to-noise ratio, above which recovery is feasible even in very sparse and very random, disordered and inhomogeneous networks. In the course of such works we had to perform a very fine study of concentration properties in random rooted networks. The work in [13] solves a twenty-years old problem which originated in the physics of random networks.

S. Khaniha, ERC grad. student, works on the definition of measure-valued dynamics associated with a null recurrent Markov chain with discrete state space S, the taboo dynamics and the potential dynamics. Her analysis is based on a subgraph of the Doeblin graph called the Bridge Graph. In the irreducible case, the Bridge Graph of a positive recurrent Markov Chain is a unimodularisable tree which contains a unique bi-infinite path. For a null recurrent Markov Chain, the Bridge graph is not unimodularizable. It can either be a tree or a forest which contains no bi-infinite path. Both dynamics have a steady state on the space of random measures on S. A paper on the matter is under preparation (M1).

B. Roy Choudhury, ERC grad. student, studies the asymptotic behavior of vertex shifts on random graphs. Thi behavior can be described by their limit random graphs called f-probabilities. He obtained conditions for periodicity and finiteness of orbits. He also works on the conditions for existence of this limiting probability. His first results concern the parent vertex-shift of a directed unimodular Canopy tree and a unimodular Eternal Galton-Watson tree. This can be used to study the record vertex-shift on simple random walks on the integers. An important result is that the existence of its f-probability depends solely on its f-graph. A paper on the matter is in preparation (M1).

The line of thoughts on the analysis of beam management has several natural continuations. The first one is the development of a methodology allowing one to incorporate load management and user/beam scheduling. This will be particularly important for spatial multiplexing. Another interesting question is the design of new association policies that take beam management into account. A first result on the matter is currently worked out by P. Popineau, ERC PhD student [20] (N*). The work with C.S. Choi (South Korea) on vehicular networks also has several potential continuations of practical interest (N2).

Bartlomiej Blaszczyszyn, Dhandapani Yogeshwaran, and Joseph Yukich have started a new line of research on the limit theory for asymptotically de-correlated dynamic spatial random models (M1-M2). This applies to general interacting particle systems an extends previous works by Penrose (2008) and Ahlberg and Tykesson (2018).

Jointly with Ahmad Alammouri (Chateaubriand Fellow) and Jeffrey Andrews (UT Austin), the PI has made significant progress on the analysis of cellular networks with user dynamics of the birth-and-death type when death rate is determined by information theory. Several mean-field techniques were developed for this problem and an intriguing meta-stability phenomenon was identified. This line of thoughts is central for M2 and M3.

Bharath Roy Choudhury, Simon Coste, and Bartlomiej Blaszczyszyn are currently workin on a new project on the continuity of adjacency operators and scattering coefficients on the space of rooted marked graphs (M1-M2).

The new line of thoughts on the survival of epidemics on infinite point processes [10] (N*) has several developments which are currently investigated jointly with S. Shlosman at Skoltech in Moscow and S. Foss of the Sobolev Institute of Mathematics in Novosibirsk.