Lilian Salinas - Academia.edu (original) (raw)

Papers by Lilian Salinas

SSRN Electronic Journal

The asynchronous automaton associated with a Boolean network f : {0, 1} n → {0, 1} n , considered... more The asynchronous automaton associated with a Boolean network f : {0, 1} n → {0, 1} n , considered in many applications, is the finite deterministic automaton where the set of states is {0, 1} n , the alphabet is [n], and the action of letter i on a state x consists in either switching the ith component if f i (x) = x i or doing nothing otherwise. These actions are extended to words in the natural way. A word is then synchronizing if the result of its action is the same for every state. In this paper, we ask for the existence of synchronizing words, and their minimal length, for a basic class of Boolean networks called and-or-nets: given an arc-signed digraph G on [n], we say that f is an and-or-net on G if, for every i ∈ [n], there is a such that, for all state x, f i (x) = a if and only if x j = a (x j = a) for every positive (negative) arc from j to i; so if a = 1 (a = 0) then f i is a conjunction (disjunction) of positive or negative literals. Our main result is that if G is strongly connected and has no positive cycles, then either every and-or-net on G has a synchronizing word of length at most 10(√ 5 + 1) n , much smaller than the bound (2 n − 1) 2 given by the well knownČerný's conjecture, or G is a cycle and no and-or-net on G has a synchronizing word. This contrasts with the following complexity result: it is coNP-hard to decide if every and-or-net on G has a synchronizing word, even if G is strongly connected or has no positive cycles.

ArXiv, 2022

In this paper, we deal the following decision problem: given a conjunctive Boolean network define... more In this paper, we deal the following decision problem: given a conjunctive Boolean network defined by its interaction digraph, does it have a limit cycle of a given length k? We prove that this problem is NP-complete in general if k is a parameter of the problem and in P if the interaction digraph is strongly connected. The case where k is a constant, but the interaction digraph is not strongly connected remains open. Furthermore, we study the variation of the decision problem: given a conjunctive Boolean network, does there exist a block-sequential (resp. sequential) update schedule such that there exists a limit cycle of length k? We prove that this problem is NP-complete for any constant k ≥ 2.

Bioinformatics, 2020

Motivation In the modeling of biological systems by Boolean networks, a key problem is finding th... more Motivation In the modeling of biological systems by Boolean networks, a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the regulatory graph like those proposed by Akutsu et al. and Zhang et al., which consider a feedback vertex set of the graph. However, these methods do not take into account the type of action (activation and inhibition) between its components. Results In this article, we propose a new algorithm for finding the set of fixed points of a Boolean network, based on a positive feedback vertex set P of its regulatory graph and which works, by applying a sequential update schedule, in time O(2|P|·n2+k), where n is the number of components and the regulatory functions of the network can be evaluated in time O(nk), k≥0. The theoretical foundation of this algorithm is due a nice characterization, that we give, of the dynamical behavior of the Boolean networks without positive cycles and...

Information Sciences, 2021

Compressing real-world graphs has many benefits such as improving or enabling the visualization i... more Compressing real-world graphs has many benefits such as improving or enabling the visualization in small memory devices, graph query processing, community search, and mining algorithms. This work proposes a novel compact representation for real sparse and clustered undirected graphs. The approach lists all the maximal cliques by using a fast algorithm and defines a clique graph based on its maximal cliques. Further, the method defines a fast and effective heuristic for finding a clique graph partition that avoids the construction of the clique graph. Finally, this partition is used to define a compact representation of the input graph. The experimental evaluation shows that this approach is competitive with the state-of-the-art methods in terms of compression efficiency and access times for neighbor queries, and that it recovers all the maximal cliques faster than using the original graph. Moreover, the approach makes it possible to query maximal cliques, which is useful for community detection.

Information and Computation, 2020

The asynchronous automaton associated with a Boolean network f : {0, 1} n → {0, 1} n is considere... more The asynchronous automaton associated with a Boolean network f : {0, 1} n → {0, 1} n is considered in many applications. It is the finite deterministic automaton with set of states {0, 1} n , alphabet {1,. .. , n}, where the action of letter i on a state x consists in either switching the ith component if f i (x) = x i or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word w fixes f if, for all states x, the result of the action of w on x is a fixed point of f. In this paper, we ask for the existence of fixing words, and their minimal length. Firstly, our main results concern the minimal length of words that fix monotone networks. We prove that, for n sufficiently large, there exists a monotone network f with n components such that any word fixing f has length Ω(n 2). For this first result we prove, using Baranyai's theorem, a property about shortest supersequences that could be of independent interest: there exists a set of permutations of {1,. .. , n} of size 2 o(n) such that any sequence containing all these permutations as subsequences is of length Ω(n 2). Conversely, we construct a word of length O(n 3) that fixes all monotone networks with n components. Secondly, we refine and extend our results to different classes of fixable networks, including networks with an acyclic interaction graph, increasing networks, conjunctive networks, monotone networks whose interaction graphs are contained in a given graph, and balanced networks.

Journal of Computer and System Sciences, 2017

Given a graph G, viewed as a loop-less symmetric digraph, we study the maximum number of fixed po... more Given a graph G, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with G as interaction graph. We prove that if G has no induced C 4 , then this quantity equals both the number of maximal independent sets in G and the maximum number of maximal independent sets among all the graphs obtained from G by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if G has a unique induced C 4 .

Theoretical Computer Science, 2008

In this article we study some aspects about the graph associated with parallel and serial behavio... more In this article we study some aspects about the graph associated with parallel and serial behavior of a Boolean network. We conclude that the structure of the associated graph can give some information about the attractors of the network. We show that the length of the attractors of Boolean networks with a graph by layers is a power of two and under certain conditions the only attractors are fixed points. Also, we show that, under certain conditions, dynamical cycles are not the same for parallel and serial updates of the same Boolean network.

Journal of Computer and System Sciences, 2014

We are interested in the number of fixed points in AND-OR-NOT networks, i.e. Boolean networks in ... more We are interested in the number of fixed points in AND-OR-NOT networks, i.e. Boolean networks in which the update function of each component is either a conjunction or a disjunction of positive or negative literals. As main result, we prove that the maximum number of fixed points in a loop-less connected AND-OR-NOT network with n components is at most the maximum number of maximal independent sets in a loop-less connected graph with n vertices, a quantity already known.

Discrete Applied Mathematics, 2013

Boolean networks have been used as models of gene regulation and other biological networks, as we... more Boolean networks have been used as models of gene regulation and other biological networks, as well as for other kinds of distributed dynamical systems. One key element in these models is the update schedule, which indicates the order in which states have to be updated. In Salinas (2008) [22] and Aracena et al. (2009) [1], equivalence classes of deterministic update schedules according to the labeled digraph associated to a Boolean network (update digraph) were defined and it was proved that two schedules in the same class yield the same dynamical behavior. In this paper, we study the relations between the update digraphs and the preservation of limit cycles of Boolean networks iterated under non-equivalent update schedules. We show that the related problems lie in the class of NPhard problems and we prove that the information provided by the update digraphs is not sufficient to determine whether two Boolean networks share limit cycles or not. Besides, we exhibit a polynomial algorithm that works as a necessary condition for two Boolean networks to share limit cycles. Finally, we construct some update schedule classes whose elements share a given limit cycle under certain conditions on the frozen nodes of it.

Biosystems, 2009

Deterministic Boolean networks have been used as models of gene regulation and other biological n... more Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems.

Advances in Applied Mathematics, 2010

Given a Boolean network without negative circuits, we propose a polynomial algorithm to build ano... more Given a Boolean network without negative circuits, we propose a polynomial algorithm to build another network such that, when updated in parallel, it has the same fixed points than the original one, but it does not have any dynamical cycle. To achieve that, we apply a network transformation related to the sequential update. As a corollary, we can find a fixed point in polynomial time for this kind of networks.

SIAM Journal on Discrete Mathematics, 2017

Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed point... more Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} n → {0, 1} n with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound φ(G) ≤ 2 τ , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number φ m (G) of fixed points in a monotone Boolean network with interaction graph G. We establish new upper and lower bounds on φ m (G) that depends on the cycle structure of G. In addition to τ , the involved parameters are the maximum number ν of vertex-disjoint cycles, and the maximum number ν * of vertex-disjoint cycles verifying some additional technical conditions. We improve the feedback bound 2 τ by proving that φ m (G) is at most the largest sub-lattice of {0, 1} τ without chain of size ν + 2, and without another forbidden pattern described by two disjoint antichains of size ν * + 1. Then, we prove two optimal lower bounds: φ m (G) ≥ ν + 1 and φ m (G) ≥ 2 ν *. As a consequence, we get the following characterization: φ m (G) = 2 τ if and only if ν * = τ. As another consequence, we get that if c is the maximum length of a chordless cycle of G then 2 ν/3 c ≤ φ m (G) ≤ 2 cν. Finally, with the techniques introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.