Raphael Jungers - Academia.edu (original) (raw)

2014 by Raphael Jungers

Network Science, 2014

Peer-to-peer (p2p) systems have driven a lot of attention in the past decade as they have become ... more Peer-to-peer (p2p) systems have driven a lot of attention in the past decade as they have become a major source of Internet traffic. The amount of data flowing through the p2p network is huge and hence challenging both to comprehend and to control. In this work, we take advantage of a new and rich dataset recording p2p activity at a remarkable scale to address these difficult problems. After extracting the relevant and measurable properties of the network from the data, we develop two models that aim to make the link between the low-level properties of the network, such as the proportion of peers that do not share content (i.e., free riders) or the distribution of the files among the peers, and its high-level properties, such as the Quality of Service or the diffusion of content, which are of interest for supervision and control purposes. We observe a significant agreement between the high-level properties measured on the real data and on the synthetic data generated by our models, which is encouraging for our models to be used in practice as large-scale prediction tools. Relying on them, we demonstrate that spending efforts to reduce the amount of free-riders indeed helps to improve the availability of files on the network. We observe however a saturation of this phenomenon after 65% of free-riders. *

Papers by Raphael Jungers

Nonlinear Analysis: Hybrid Systems, 2016

We show that for any positive integer d, there are families of switched linear systemsin fixed di... more We show that for any positive integer d, there are families of switched linear systemsin fixed dimension and defined by two matrices only-that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of an extremal piecewise algebraic Lyapunov function implies the finiteness property of the optimal product, generalizing a result of Lagarias and Wang. As a corollary, we prove that the finiteness property holds for sets of matrices with an extremal Lyapunov function belonging to some of the most popular function classes in controls. Index terms: stability of switched systems, linear difference inclusions, the finiteness conjecture of the joint spectral radius, convex optimization for Lyapunov analysis.

Operations Research Letters, 2016

Solving Markov Decision Processes (MDPs) is a recurrent task in engineering. Even though it is kn... more Solving Markov Decision Processes (MDPs) is a recurrent task in engineering. Even though it is known that solutions for minimizing the infinite horizon expected reward can be found in polynomial time using Linear Programming techniques, iterative methods like the Policy Iteration algorithm (PI) remain usually the most efficient in practice. This method is guaranteed to converge in a finite number of steps. Unfortunately, it is known that it may require an exponential number of steps in the size of the problem to converge. On the other hand, many open questions remain considering the actual worst case complexity. In this work, we provide the first improvement over the fifteen years old upper bound from by showing that PI requires at most k k−1 · k n n + o k n n iterations to converge, where n is the number of states of the MDP and k is the maximum number of actions per state. Perhaps more importantly, we also show that this bound is optimal for an important relaxation of the problem.

Lecture Notes in Computer Science, 2015

53rd IEEE Conference on Decision and Control, 2014

Computing Research Repository, 2007

An edge-colored directed graph is observable if an agent that moves along its edges is able to de... more An edge-colored directed graph is observable if an agent that moves along its edges is able to determine his position in the graph after a sufficiently long observation of the edge colors. When the agent is able to determine his position only from time to time, the graph is said to be partly observable. Observability in graphs is desirable in

Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization proble... more Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization problems of applied mathematics such as for instance Linear Programming (LP), Markov Decision Processes (MDPs) or 2-player Turn Based Stochastic Games (2TBSGs). A polynomial time algorithm to find the sink of a USO would translate into a strongly polynomial time algorithm to solve the aforementioned problems-a major quest for all three cases. In addition, we may translate MDPs and 2TBSGs into the problem of finding the sink of an acyclic USO of a cube, which can be done using the well-known Policy Iteration algorithm (PI). The study of its complexity is the object of this work. Despite its exponential worst case complexity, the principle of PI is a powerful source of inspiration for other methods. As our first contribution, we disprove Hansen and Zwick's conjecture claiming that the number of steps of PI should follow the Fibonacci sequence in the worst case. Our analysis relies on a new combinatorial formulation of the problem-the so-called Order-Regularity formulation (OR). Then, for our second contribution, we (exponentially) improve the Ω(1.4142 n) lower bound on the number of steps of PI from Schurr and Szabó in the case of the OR formulation and obtain an Ω(1.4269 n) bound.

This paper proposes lower bounds on a quantity called LpL^pLp-norm joint spectral radius, or in sho... more This paper proposes lower bounds on a quantity called LpL^pLp-norm joint spectral radius, or in short, ppp-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the ppp-radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.

The concept of elementary flux vector is valuable in a number of applications of metabolic engine... more The concept of elementary flux vector is valuable in a number of applications of metabolic engineering. For instance, in metabolic flux analysis, each admissible flux vector can be expressed as a non-negative linear combination of a small number of elementary flux vectors. However a critical issue concerns the number of elementary flux vectors which may be huge because it combinatorially

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Re... more We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node. The evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior. In order to perform our analysis, we introduce a new way of computing the distribution of distances between nodes. Our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations. It allows us to characterize the phase transitions appearing when the connectivity probability varies.

Siam Journal on Computing, May 1, 2010

We consider the problem of partial order production: arrange the elements of an unknown totally o... more We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing IT LB + o(IT LB) + O(n) comparisons in the worst case. Here, n denotes the size of the ground sets, and IT LB denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target partial order. Our approach is to replace the target partial order by a weak order (that is, a partial order with a layered structure) extending it, without increasing the information theoretic lower bound too much. We then solve the problem by applying an efficient multiple selection algorithm. The overall complexity of our algorithm is polynomial. This answers a question of Yao (SIAM J. Comput. 18, 1989). We base our analysis on the entropy of the target partial order, a quantity that can be efficiently computed and provides a good estimate of the information-theoretic lower bound.

We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative ... more We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on R d + , which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of converges is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and language theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessarily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.

2015 54th IEEE Conference on Decision and Control (CDC), 2015

We consider the problem of determining the existence of a sequence of matrices driving a discrete... more We consider the problem of determining the existence of a sequence of matrices driving a discrete-time multi-agent consensus system to consensus. We transform this problem into the problem of the existence of a product of the (stochastic) transition matrices that has a positive column. This allows us to make use of results from automata theory to sets of stochastic matrices. Our main result is a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.

Siam Journal on Control and Optimization, Feb 25, 2014

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joi... more We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimumof-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case perfomance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

Ieee Transactions on Information Theory, Jan 10, 2006

We consider questions related to the computation of the capacity of codes that avoid forbidden di... more We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of nnn-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with nnn. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.

49th IEEE Conference on Decision and Control (CDC), 2010

ABSTRACT The stability of a switching linear dynamical system is ruled by the so-called joint spe... more ABSTRACT The stability of a switching linear dynamical system is ruled by the so-called joint spectral radius of the set of matrices characterizing the dynamical system. In some situations, the system is not stable in the classical sense, but might still be stable in a weaker meaning. We introduce the new notion of weak stability or Lp-stability of a switched dynamical system based on the so-called p-radius of the set of matrices. The p-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in the recent years. We analyze the computability of this quantity, and we describe a series of approximations that converge to the p-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the p-radius computation. We finally show the efficiency of our methods on several practical examples.

52nd IEEE Conference on Decision and Control, 2013

ABSTRACT We analyse the problem of stability of a continuous time linear switching system (LSS) v... more ABSTRACT We analyse the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. It is well-known that the existence of a positive τ for which the corresponding discrete time system with stepsize τ is stable implies the stability of the LSS. Our main goal is to obtain a converse statement, that is to estimate the discretization stepsize τ > 0 up to a given accuracy ε > 0. This would lead to a method for deciding the stability of a continuous time LSS with a guaranteed accuracy. As a first step towards the solution of this problem, we show that for systems of matrices with real spectrum the parameter τ can be effectively estimated. We prove that in this special case, the discretized system is stable if and only if the Lyapunov exponent of the LSS is smaller than - C τ, where C is an effective constant depending on the system. The proofs are based on applying Markov-Bernstein type inequalities for systems of exponents.