Thematic team Complex Systems, Networks, and Distributed Computing

INRIA project-team GANG

## Complex systems

#### Day, hour and place

Tuesday at 2pm, room 1007

#### Contact(s)

#### Future talks

Systèmes complexes

mardi 20 mars 2018, 14h00, Salle 1007

**Tba** () *TBA*

TBA

Systèmes complexes

mardi 10 avril 2018, 14h00, Salle 1007

**Tba** () *TBA*

#### Past talks

Systèmes complexes

mardi 13 février 2018, 14h00, Salle 1007

**Nicolas Behr** (ENS) *Combinatorics of stochastic rewriting systems*

Based on recent joint work with G.H.E. Duchamp and K.A. Penson (arXiv:1712.06575, on the combinatorics of chemical reaction systems) and with V. Danos and I. Garnier (in preparation), I will explain how dynamical and statistical aspects of stochastic graph rewriting systems may be computed using combinatorial techniques. The main part of the talk will focus on the special case of discrete graph rewriting (i.e. chemical reactions) for clarity and brevity, but time permitting the generalization to generic graph rewriting in the form of a stochastic mechanics formalism based on the concept of rule algebras will be discussed as well.

Systèmes complexes

mardi 16 janvier 2018, 14h00, Salle 3014

**Nicolas Thiery** (Université Paris Sud) *Computing huge subspaces of diagonal harmonic polynomials: symmetries to the rescue!*

Last spring, I visited François Bergeron and we worked on his favorite objects: the spaces H(n,k) of diagonal harmonic polynomials in k sets of n variables. Those spaces connect to many areas of algebraic combinatorics, representation theory and beyond, and the case H(n,2) became famous a decade ago with the n! conjecture and its proof by Haiman.

To fuel his ongoing studies François needed to compute the structure of H(5,6). This is a space of dimension 6.10^5 made of polynomials in 30 variables of degree up to 15, each having thousands of terms.

In this talk, I'll explain how the calculation can now be completed in 45 minutes with a dozen cores and ~15Go of memory. This exploits a combination of strategies (symmetries, representation theory of the symmetric and general linear group, …), each of which reduces the complexity in time and memory by one or two orders of magnitude.

There will be little prerequisites and it's my hope that some strategies (and maybe the code!) could be used in other contexts.

Systèmes complexes

mardi 12 décembre 2017, 14h00, Salle 3052

**Jean Krivine** (IRIF) *Incremental Update for Graph Rewriting*

Graph rewriting formalisms are well-established models for the representation of biological systems such as protein-protein interaction networks. The combinatorial complexity of these models usually prevents any explicit representation of the variables of the system, and one has to rely on stochastic simulations in order to sample the possible trajectories of the underlying Markov chain. The bottleneck of stochastic simulation algorithms is the update of the propensity function that describes the probability that a given rule is to be applied next. In this talk we present an algorithm based on a data structure, called extension basis, that can be used to update the counts of predefined graph observables after a rule of the model has been applied.

Reference: Boutillier P., Ehrhard T., Krivine J. (2017) Incremental Update for Graph Rewriting. In: Yang H. (eds) Programming Languages and Systems. ESOP 2017. Lecture Notes in Computer Science, vol 10201. Springer, Berlin, Heidelberg

Systèmes complexes

mardi 14 novembre 2017, 14h00, Salle 3052

**Laurent Massoulié** (MSR-Inria) *Rapid Mixing of Local Graph Dynamics*

Graph dynamics arise naturally in many contexts. For instance in peer-to-peer networks, a participating peer may replace an existing connection with one neighbour by a new connection with a neighbour's neighbour. Several such local rewiring rules have been proposed to ensure that peer-to-peer networks achieve good connectivity properties (e.g. high expansion) in equilibrium. However it has remained an open question whether there existed such rules that also led to fast convergence to equilibrium. In this work we provide an affirmative answer: We exhibit a local rewiring rule that converges to equilibrium after each participating node has undergone only a number of rewirings that is poly-logarithmic in the system size. The proof involves consideration of the whole isoperimetric profile of the graph, and may be of independent interest.

This is joint work with Rémi Varloot.

Systèmes complexes

mardi 17 octobre 2017, 14h00, Salle 3052

**Claire Mathieu** (DI - ENS) *Online k-compaction*

Given, at each time t = 1, 2, …, n, a new file of length l(t) and a read rate r(t), an online k-compaction algorithm must maintain a collection of at most k files, choosing (at each time t, without knowing future inputs) some of the files to merge into one, thereby incurring a merge cost equal to the total length of the merged files and a read cost equal to the read rate r(t) times the number of files present at time t. The goal is to minimize the total cost over time. K-compaction algorithms are a key component of log-structured merge trees, the file-based data structure underlying NoSQL databases such as Accumulo, Bigtable, Cassandra, HBase,and others. We initiate the theoretical study of k-compaction algorithms. We formalize the problem, consider worst-case, average-case and competitive analysis (per-instance optimality), and propose new algorithms that are optimal according to these metrics.

This is joint work with Carl Staelin, Neal E. Young, and Arman Yousefi.

#### Former seminar