Section: New Results

Algorithms and estimation for graph data

Modelling of networks of multiagent systems

Participant: A. Muller-Gueudin

External collaborators: A. Girard, S. Martin, I.C. Morarescu (CRAN, Nancy)

We relate here a starting of collaboration with researchers in Automatics in Nancy. We consider here networks, modeled as a graph with nodes and edges representing the agents and their interconnections, respectively. The objective is to study the evolution of the opinion of all the agents. The connectivity of the network, persistence of links and interactions reciprocity influence the convergence speed towards a consensus. The problem of consensus or synchronization is motivated by different applications as communication networks, power and transport grids, decentralized computing networks, and social or biological networks. We then consider networks of interconnected dynamical systems, called agents, that are partitioned into several clusters. Most of the agents can only update their state in a continuous way using only inner-cluster agent states. On top of this, few agents also have the peculiarity to rarely update their states in a discrete way by reseting it using states from agents outside their clusters. In social networks, the opinion of each individual evolves by taking into account the opinions of the members belonging to its community. Nevertheless, one or several individuals can change their opinions by interacting with individuals outside its community. These inter-cluster interactions can be seen as resets of the opinions. This leads us to a network dynamics that is expressed in term of reset systems. We suppose that the reset instants arrive stochastically following a Poisson renewal process. We have an accepted paper in the journal IEEE Transactions on Automatic Control [10] .

Microbial interaction inference by network analysis

Participants: A. Gégout-Petit, A. Muller-Gueudin

External collaborators: A. Deveau (INRA Nancy), C. Raïssy (Inria Orpailleur)

The objective is to characterize microbial interactions in a particular environment: the truffles.

The truffle provides a habitat for complex bacterial communities. The role for bacteria in the development of truffles has been suggested but very little is known regarding the structure and the functional potential of the truffle's bacterial communities along truffle maturation. In a mathematical point of view, two micro-organisms are connected if they are not independent, conditionally to the other micro-organisms. Several models fit into this setting, especially the gaussian graphical models, the bayesians networks, and the graphical log-linear models. But the data, which can be zeros inflated, need developments and we have to proposed new models. Moreover, we are confronted to the problem that $n\ll p$, that is the sample size is much smaller that the number of variables ($n=30,p=200$). Last year, thanks to a financially supported project (PEPS), we have began a collaboration between statisticians and data-miners. The first approches have been notified in a report [31] . The statistical methodologies developed for this project could also be applied to human health (for instance identification of network between bacteria inside colon).

Lossy compression of unordered trees

Participant: R. Azaïs

External collaborators: J-B. Durand, C. Godin

A classical compression method for trees is to represent them by directed acyclic graphs. This approach exploits subtree repeats in the structure and is efficient only for trees with a high level of redundancy. The class of self-nested trees presents remarkable compression properties by this method because of the systematic repetition of subtrees. In particular, the compressed version of a self-nested tree $T$ is a linear directed acyclic graph with only $1+\text{height}\left(T\right)$ nodes. Unfortunately, it should be noted that trees without a high level of redundancy are often insufficiently compressed by this procedure. In a paper recently submitted for publication in an international conference [32] , we introduce a lossy compression method that consists in computing in polynomial time for trees with bounded outdegree the reduction of a self-nested structure that closely approximates the initial data. We prove on a simulated dataset that the error rate of this lossy compression method is always better than the loss involved in a previous algorithm of the literature, while the compression rates are equivalent.

Inference for critical Galton-Watson trees from their Harris process

Participant: R. Azaïs

External collaborator: A. Genadot (Inria CQFD Bordeaux)

Galton-Watson trees are an elementary model for the genealogy of a branching population and thus play a central role in biology. Critical Galton-Watson trees are generated from a sibling distribution $\mu$ whose theoretical expectation $\sum k\mu \left(k\right)$ is equal to 1. Under this assumption, the well-known Harris process of a tree conditioned on having $n$ nodes converges to a Brownian excursion characterized by the variance ${\sigma }^2=\sum {(k-1)}^2\mu \left(k\right)$ of $\mu$. We propose to exploit this asymptotic approximation to define a new estimate of the unknown parameter of interest ${\sigma }^2$ based on a least-square method. In particular, this new technique allows us to take into account the behavior of the Harris path with respect to its asymptotic theoretical expectation. In certain cases, we obtain a better confidence interval than the classical approach. A paper on this work is in preparation.