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Adrien Richard

Publications and source records attributed to Adrien Richard.

28 records · Page 2Linked to original sources
arXiv2016

Asynchronous simulation of Boolean networks by monotone Boolean networks

We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known property: if $f:\{0,1\}^n\to\{0,1\}^n$ is monotone, then, for every initial configuration $x$, there exists a fixed point $y$ such that $y$ can be reached from $x$ with a fully asynchronous strategy that contains at most $n$ updates.

cs.DM↗
arXiv2016

Simple dynamics on graphs

Does the interaction graph of a finite dynamical system can force this system to have a "complex" dynamics ? In other words, given a finite interval of integers $A$, which are the signed digraphs $G$ such that every finite dynamical system $f:A^n\to A^n$ with $G$ as interaction graph has a "complex" dynamics ? If $|A|\geq 3$ we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph $G$ there exists a system $f:A^n\to A^n$ with $G$ as interaction graph that converges toward a unique fixed point in at most $\lfloor\log_2 n\rfloor+2$ steps. The boolean case $|A|=2$ is more difficult, and we provide partial answers instead. We exhibit large classes of unsigned digraphs which admit boolean dynamical systems which converge toward a unique fixed point in polynomial, linear or constant time.

cs.DM↗
arXiv2015

A Genetically Modified Hoare Logic

An important problem when modeling gene networks lies in the identification of parameters, even if we consider a purely discrete framework as the one of René Thomas. Here we are interested in the exhaustive search of all parameter values that are consistent with observed behaviors of the gene network. We present in this article a new approach based on Hoare Logic and on a weakest precondition calculus to generate constraints on possible parameter values. Observed behaviors play the role of "programs" for the classical Hoare logic, and computed weakest preconditions represent the sets of all compatible parameterizations expressed as constraints on parameters. Finally we give a proof of correctness of our Hoare logic for gene networks as well as a proof of completeness based on the computation of the weakest precondition.

cs.CE↗
arXiv2014

Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability

Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this problem to determining the number of fixed points of coding functions $f:A^n\to A^n$ over a finite alphabet $A$ (usually referred to as Boolean networks if $A = \{0,1\}$) with a given interaction graph, that describes which local functions depend on which variables. In this paper, we generalise the so-called reduction of coding functions in order to eliminate variables. We then determine the maximum number of fixed points of a fully reduced coding function, whose interaction graph has a loop on every vertex. Since the reduction preserves the number of fixed points, we then apply these ideas and results to obtain four main results on the linear network coding solvability problem. First, we prove that non-decreasing coding functions cannot solve any more instances than routing already does. Second, we show that triangle-free undirected graphs are linearly solvable if and only if they are solvable by routing. This is the first classification result for the linear network coding solvability problem. Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we determine large classes of strictly linearly solvable graphs.

cs.IT↗
arXiv2014

Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks

We are interested in fixed points in Boolean networks, {\em i.e.} functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{0,1\}^n$, and we characterizes a class $\mathcal{F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every $x$ in $\{0,1\}^n$ there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. Then, denoting by $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{F}$ are exactly the networks of $\mathcal{C}^+\cup \mathcal{C}^-$; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in $\mathcal{C}^+$ (resp. $\mathcal{C}^-$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetwork in $\mathcal{C}^+$.

cs.DM↗
arXiv2014

Fixed points of Boolean networks, guessing graphs, and coding theory

In this paper, we are interested in the number of fixed points of functions $f:A^n\to A^n$ over a finite alphabet $A$ defined on a given signed digraph $D$. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on $D$. We then discover relationships between the number of fixed points of $f$ and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.

cs.DM↗
arXiv2009

Local negative circuits and fixed points in Boolean networks

To each Boolean function F from {0,1}^n to itself and each point x in {0,1}^n, we associate the signed directed graph G_F(x) of order n that contains a positive (resp. negative) arc from j to i if the partial derivative of f_i with respect of x_j is positive (resp. negative) at point x. We then focus on the following open problem: Is the absence of a negative circuit in G_F(x) for all x in {0,1}^n a sufficient condition for F to have at least one fixed point? As main result, we settle this problem under the additional condition that, for all x in {0,1}^n, the out-degree of each vertex of G_F(x) is at most one.

cs.DM↗
arXiv2009

Negative circuits and sustained oscillations in asynchronous automata networks

The biologist René Thomas conjectured, twenty years ago, that the presence of a negative feedback circuit in the interaction graph of a dynamical system is a necessary condition for this system to produce sustained oscillations. In this paper, we state and prove this conjecture for asynchronous automata networks, a class of discrete dynamical systems extensively used to model the behaviors of gene networks. As a corollary, we obtain the following fixed point theorem: given a product $X$ of $n$ finite intervals of integers, and a map $F$ from $X$ to itself, if the interaction graph associated with $F$ has no negative circuit, then $F$ has at least one fixed point.

cs.DM↗
arXiv2008

Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive circuits of the interaction graph associated with F. The proof uses and strongly generalizes a theorem of Richard and Comet which corresponds to a discrete version of the Thomas' conjecture: if the interaction graph associated with F has no positive circuit, then F has at most one fixed point. The obtained upper bound on the number of fixed points also strongly generalizes the one established by Aracena et al for a particular class of Boolean networks.

cs.DM↗