Maximum number of fixed points in AND–OR–NOT networks

Aracena, Julio; Richard, Adrien; Salinas, Lilian

doi:10.1016/j.jcss.2014.04.025

Cited by 23 publications

(31 citation statements)

References 19 publications

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“…Secondly, there is a one-to-one correspondence between conjunctive networks and full-signed digraphs, which make easier the study of the relationships between structure and dynamics, because graph theoretic tools and results can be used in a more straightforward way. In particular, there are many connections with kernels and maximal independent sets [4,38,30,5].…”

Section: Resultsmentioning

confidence: 99%

“…If G is a graph, this is precisely the definition of a maximal independent set. Thus we have the following basic relation between fixed sets and maximal independent sets, already exhibited in [5].…”

Section: Preliminariesmentioning

confidence: 98%

“…The sign σ(j, i) of an arc from j to i then indicates whether the Boolean function f i is an increasing (positive sign), In this situation, it is natural to focus our attention on the influence of negative cycles. A natural framework for this problem is to fix G and study the variation of φ(G σ ) according to labeling function σ, which gives the repartition of signs on the arcs of G. This is, however, a difficult problem, widely open, with very few formal results (see however [5,8]). This comes from the versatility of negative cycles: depending on the way they are connected to positive cycles, that can either be favorable or unfavorable to the presence of many fixed points.…”

Section: Introductionmentioning

confidence: 99%

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Fixed points in conjunctive networks and maximal independent sets in graph contractions

Aracena¹,

Richard

Salinas³

2017

Journal of Computer and System Sciences

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International audienceGiven 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

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Section: Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Fixed points in conjunctive networks and maximal independent sets in graph contractions

Aracena¹,

Richard

Salinas³

2017

Journal of Computer and System Sciences

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“…A large amount of work considers the influence of the signed interaction graph (see [24] for a survey of such work); we shall use some of these results but we will not carry out a comprehensive survey of the signed interaction graph, let alone of its local variants. Secondly, some work also restricts the nature of the local functions f v : it can be linear [19,17], threshold [22,21], conjunctive or disjunctive [1,2], etc. Thirdly, a large body of work is devoted to the study of different update schedules, where instead of applying f to x, for instance only one entity updates its state at each time step, and x becomes (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

On the influence of the interaction graph on a finite dynamical system

Gadouleau

2019

Nat Comput

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A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images (the so-called rank), the number of periodic points (the so-called periodic rank) and the number of fixed points. In particular, we investigate the minimum, average, and maximum number of images (or periodic points, or fixed points) of FDSs with a prescribed interaction graph and a given alphabet size; thus yielding nine quantities to study. The paper is split into two parts. The first part considers the minimum rank, for which we derive the first meaningful results known so far. In particular, we show that the minimum rank decreases with the alphabet size, thus yielding the definition of an absolute minimum rank. We obtain lower and upper bounds on this absolute minimum rank, and we give classification results for graphs with very low (or highest) rank. The second part is a comprehensive survey of the results obtained on the nine quantities described above. We not only give a review of known results, but we also give a list of relevant open questions. *

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“…The study of the number of fixed points and its maximization in particular is the subject of a stream of work, e.g. in [, , , ].…”

Section: Introductionmentioning

confidence: 99%

On the possible values of the entropy of undirected graphs

Gadouleau

2017

Journal of Graph Theory

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The entropy of a digraph is a fundamental measure which relates network coding, information theory, and fixed points of finite dynamical systems. In this paper, we focus on the entropy of undirected graphs. We prove that for any integer k the number of possible values of the entropy of an undirected graph up to k is finite. We also determine all the possible values for the entropy of an undirected graph up to the value of four.

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Maximum number of fixed points in AND–OR–NOT networks

Cited by 23 publications

References 19 publications

Fixed points in conjunctive networks and maximal independent sets in graph contractions

Fixed points in conjunctive networks and maximal independent sets in graph contractions

On the influence of the interaction graph on a finite dynamical system

On the possible values of the entropy of undirected graphs

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