## Section: New Results

### Stochastic Modeling

Participants : Sara Alouf, Eitan Altman, Konstantin Avrachenkov, Alain Jean-Marie, Giovanni Neglia.

#### Network growth models

Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In the framework of joint team with Brazil (Thanes), G. Neglia, together with G. Iacobelli and D. Figueiredo (both from UFRJ, Brazil), has studied a simple model where network growth and a random walker are coupled [23]. In particular, they consider the No Restart Random Walk model where a walker builds its graph (tree) while moving around. The walker takes $s$ steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another $s$ steps, and the process repeats. They have analyzed this process from the perspective of the walker and the network, showing a fundamental dichotomy between transience and recurrence for the walker as well as power law and exponential degree distribution for the network.

#### Controlled Markov chains

E. Altman in collaboration with D. Josselin and S. Boularouk (CERI/LIA, Univ Avignon) study in [26] a multiobjective dynamic program where all the criteria are in the form of total expected sum of costs till absorption in some set of states. They assume that instantaneous costs are strictly positive and make no assumption on the ergodic structure of the Markov Decision Process. Their main result is to extend the linear program solution approach that was previously derived for transient Constrained Markov Decision Processes to the general ergodic structure. Several (additive) cost metrics are defined and (possibly randomized) routing policies are sought which minimize one of the costs subject to constraints over the other objectives.

#### Escape probability estimation in large graphs

Consider large graphs as the object of study and specifically the problem of escape probability estimation. Generally, this characteristic cannot be calculated analytically nor even numerically due to the complexity and large size of the investigation object. In [32], K. Avrachenkov and A. Borodina (Karelian Institute of Applied Mathematical Research, Russia) have presented an effective method for estimating the probability that the random walk on graph first enters a node $b$ before returning into the starting node $a$. Regenerative properties of the random walk allow the use of an accelerated method for the simulation of cycles based on the splitting technique. The results of numerical experiments confirm the advantages of the proposed method.

#### Random surfers and prefetching

Prefetching is a basic technique used to reduce the latency of diverse computer services. Deciding what to prefetch amounts to make a compromise between latency and the waste of resources (network bandwidth, storage, energy) if contents is mistakenly prefetched. Modeling the problem in case of web/video/gaming navigation, is done by identifying a graph of “documents” connected by links representing the possible chaining. A surfer, either random or strategic, browses this graph. The prefetching controller must make it sure that the documents browsed are always available locally. In the case where the surfer is random and/or the graph is not completely known in advance, the question is largely unexplored. Q. Petitjean, under the supervision of S. Alouf and A. Jean-Marie, has determined through extensive simulations that when the graph is a tree, neither the greedy strategy, nor the one optimal when the tree is completely known, are optimal when the tree is discovered progressively.

#### The `marmoteCore` platform

The development of marmoteCore (see Section 6.1) has been pursued. Its numerical features for computing stationary distributions, average hitting times and absorption probabilities have been used in a joint work with F. Cazals, D. Mazauric and G. Santa Cruz (Abs team) and J. Roux (Univ Cote d'Azur) [52]. The software has been presented to young researchers in networking at the ResCom 2019 summer school.