Dynamics of synchronous Boolean networks with non-binary states Open Access

Boolean networks (BNs) model the behavior of related entities by assigning to each of the entities a binary (on–off) value that evolves over time. Yet, real-world phenomena often present entities that can have more than two possible on–off states. The necessity to model such phenomena using BNs tools is the main motivation of this research. This work extends theoretical results for the dynamics of synchronous BNs with binary states by introducing a framework for BNs with non-binary states, in which the entities take values in a general Boolean algebra of $2 p$ elements, $p>1$⁠. Through this setting, it is possible to infer dynamical properties of a BN with non-binary states from dynamical properties of BNs with binary values. In particular, this work characterizes the main dynamical elements of synchronous BNs with non-binary states: (i) fixed points and periodic orbits, (ii) predecessors and garden-of-eden configurations, and (iii) attractiveness and transient. Overall, this work is a first step in describing the dynamics of distinct classes of BNs with non-binary states.

Mathematical modeling of real-world phenomena and artificial (computational) processes is a key challenge for applied mathematics. To this end, graph dynamical systems (GDSs) are increasingly being used to study the behavior of related entities that change over time. A GDS is the mathematical formalization of such interrelated entities modelled by a (dependency) graph whose states evolve following certain rules through an updating scheme. This tool has been applied to model various phenomena across multiple domains, such as biosocial systems,¹ biology,^2–7 genetics,^8–10 cryptography,^11,12 physics,^13–15 mathematics,^16–18 ecology,¹⁹ etc.

When modeling phenomena or artificial processes by GDSs, the vertices correspond to the entities and the arcs correspond to the interactions among them. Then, the (state) variables associated with the vertices, the (local) functions describing the interactions, and the schedule in which the state variables are updated along time allows us to formulate the GDS.

Deterministic Boolean networks (BNs), also referred to as Boolean finite GDSs, or, for simplicity, Boolean finite dynamical systems (BFDSs),^20–22 are GDSs where both the state sets and the local functions are Boolean. In the literature, one can also find probabilistic BNs.^9,10 For BNs, the case in which the states of the entities are either 0 or 1 have been extensively studied.^23–26 Regarding the local functions, when the entities evolve according to the same global evolution operator, which acts locally over each entity and its related ones, the BN is said to be homogeneous. Other approaches consider local independent functions to update each entity.^27,28

Depending on the updating scheme, two main types of deterministic BNs can be identified: synchronous BNs or parallel dynamical systems (PDSs), and asynchronous BNs or sequential dynamical systems (SDSs).^26,29 Synchronous BNs or PDSs are GDSs whose states values evolve simultaneously in time. On the other hand, in an asynchronous BN or SDS, the state values of the entities evolve in a sequential manner, following a pre-established (permutation) order. Another conception admitted for (deterministic) asynchronous BNs can be found in Ref. 25. In this work, we deal with deterministic BNs, which are homogeneous and synchronously updated.

The study of the dynamics of BNs is usually performed by the algebraic analysis of periodic orbits and the convergence of eventually periodic ones to them. Such convergence is based on the analysis of the predecessors of a state of the system and on the garden-of-eden (GOE) configurations (i.e., states of the system without predecessors). Specifically, for periodic orbits, the following problems have been solved for binary-state variables synchronous BN: existence^28,30,31 and coexistence^32,33 of periodic orbits; uniqueness of periodic orbits;^32,34 number of periodic orbits;³⁵ bounds for this number;^34–36 and an exact formula for the number of fixed points.³⁵ For eventually periodic orbits, the following problems have been solved: existence, coexistence, and bounds for the predecessors and GOE;³⁷ and convergence to periodic orbits.³⁸ As consequences, the existence of global attractors, the basin of attraction of a periodic orbit, and the transient (or width) of a system have been determined in these works.

In real-world problems, it has been observed that the entities of a phenomena can have more than two possible on–off states.^2,6,39,40 The feasible application of this kind of model calls for an extension of the possible states of the vertices to values that are not restricted to the set ${0,1}$⁠. This necessity is the main motivation of the research performed in this work.

Recent literature shows several approaches to study some types of GDSs whose entities take values in other (non-binary) finite sets.

In Ref. 6, the authors introduce GDSs whose entities have multiple states, as a natural generalization of synchronous (random) Boolean networks (RBNs), with the restriction of fixed connectivity $K$⁠. Later, in Ref. 39, a special kind of such (random) networks, where the local functions take outputs in ${−1,+1}$⁠, which represent the modification of the state value corresponding to every local function, is studied.

In both previous works, the authors loose the Boolean framework. In contrast with them, in Ref. 40, the author maintains the Boolean framework in multiple state networks by choosing the least $p$ such that $2 p$ is an upper bound for the number of possible states. Then, classifying the binary representation vectors in suitable groups allows us to treat the states of the elements of the system in a Boolean way (see Ref. 25). The simplest extension corresponds to a finite field with three elements, in which the state of an entity can be interpreted as “on,” “intermediate,” and “off.” Then, we can use ${ 0 , 1 } 2={(0,0),(0,1),(1,0),(1,1)}$ to model this framework by identifying $(0,1)$ and $(1,0)$ to represent the intermediate state. Although this embedding allows us to carry out the usual study of the corresponding Boolean dynamical system with $2 p$ elements, the identification process may give rise to some issues. For instance, the number of elements that have to be identified can be quite high for finite fields with $2 p+d$ elements, being $d>1$ a small integer.

Regardless of these inconveniences, the study of BFDSs on general Boolean algebras with $2 p$ elements, $p≥1$⁠, requires no additional features to be introduced and, likewise, becomes basic for the study of previous networks. A specific algebraic generalization of binary BNs was introduced in Ref. 41, considering that all the state values of the entities belong to the same (arbitrary) Boolean algebra $B$ with $2 p$ elements, $p∈ N,p≥1$⁠, and allowing any arbitrary connectivity of the entities, in contrast to the fixed $K$-connectivity of other approaches. Thanks to the Stone representation theorem of Boolean algebras,⁴² each element $x∈ B$ can be identified with a $p$-tuple $y∈ { 0 , 1 } p$⁠, which allows us to study the dynamics of BNs with non-binary states on general Boolean algebras, in an intuitive and direct way.

The identification of elements of any given general Boolean algebra with p-tuples of binary values allows us to establish a systematic method to understand the dynamics of BNs in which the entities can have more than two states. This constitutes the key novelty of this work and its main worth since it opens the door to study the dynamics of BFDSs in a more general setting.

In the setting proposed in this paper, the entities take value on a Boolean algebra with $2 p$ elements, $p>1$⁠. We apply the proposed method to the (fundamental) class of synchronous BNs induced by maxterm (minterm) functions on a general Boolean algebra with $2 p$ elements, $p>1$⁠. Consequently, we can study the fundamental dynamical elements in this class of BNs: fixed points, periodic orbits, predecessors, GOE points, and transient of the system.

Indeed, with this identification in mind, the results for binary-state variables BNs become a particular case of the ones for BNs with variables taking values on a general Boolean algebra. The question of knowing whether the theoretical results for $p>1$ follow the same pattern as the ones for $p=1$ naturally arises. In this sense, it was seen that the theory is similar regarding the existence⁴¹ and coexistence of periodic orbits.³² In particular, for synchronous BNs with non-binary states, it was shown that the only periodic orbits are fixed points and two-periodic orbits,⁴¹ and they cannot coexist.³² Furthermore, a fixed point theorem was proved (Ref. 32), which determines the conditions for the uniqueness of fixed points in a synchronous BN with variables taking values on a general Boolean algebra. However, many problems usually assessed for the theoretical study of GDSs have still not been solved. In view of this, the aim of this paper is to continue the study of the dynamics of synchronous BNs with non-binary states.

In some cases, the results are similar to those obtained for the binary framework. Nevertheless, other results are significantly different. In particular, the existence and coexistence of periodic orbits are the same in both cases.

We prove that the uniqueness of two-periodic orbits is not possible when $p>1$⁠, thus breaking the pattern found for synchronous BNs with binary-state values.

The number of fixed points and two-periodic orbits for a synchronous BN on a general Boolean algebra are also different, even when the graph of inter-relations is the same. These numbers can be determined by means of combinatorial calculus from the binary case. We provide lower and upper bounds for these numbers, as well as the exact number of fixed points. The exact number of two-periodic orbits is also obtained when the evolution operator is either NAND or NOR.

We solve the problems of existence, coexistence, and number of predecessors for synchronous BNs with non-binary states and characterize when a configuration is a GOE for one of these systems. Moreover, bounds for the number of predecessors and GOE are also calculated, extending the results for the binary case.

Regarding the convergence to periodic orbits, the characterization of attractive periodic orbits in synchronous BNs with non-binary states coincides with the one for the binary case. Furthermore, as unique fixed points can appear, globally attractive fixed points can also appear. However, the impossibility of uniqueness of two-periodic orbits ensures that globally attractive two-periodic orbits cannot appear in synchronous BNs on general Boolean algebra.

Finally, it has been observed that for such more general BNs, the transient holds the same as in the binary case.

All in all, the results of this paper provide an extension of the ones for synchronous BNs with binary states. Moreover, they open the door to an important widening of the applications of BNs, since conventional binary ones are not suitable for modeling real-world phenomena in which a range of state values for the entities is a necessary attribute, whereas synchronous BNs on general Boolean algebras may provide a suitable model for this type of situations. In fact, applications of multi-valued models are also present in the literature. In Ref. 2, a multi-bit BN is constructed to model biological phenomena in which the multiple state attribute of the entities is compulsory for an adequate representation of the situation. In particular, the theoretical results presented in this work describe relevant features to model real-world situations. Furthermore, these models can also provide a simpler alternative to the usually assessed continuous models.⁴⁰ 

This paper is organized as follows. In Sec. II, we give some preliminaries related to the study of homogeneous synchronous BNs with non-binary states. In particular, we revise some notions regarding the dependency graph that will be useful throughout the paper. Stone representation theorem, which grants the mentioned extension on general Boolean algebras, is also reviewed and used to show how to study synchronous BNs with non-binary states (an introduction to Boolean algebras can be seen, for example, in Ref. 42). In Sec. III, we deal with periodic orbits in synchronous BNs with non-binary states. Specifically, we give a result for the existence, coexistence, and uniqueness of periodic orbits. We also study the number of periodic orbits that can appear in such systems. In Sec. IV, we study the predecessors and GOE configurations in such synchronous BNs with non-binary states, giving conditions for their existence and providing bounds for their number. Finally, the convergence to periodic orbits is also analyzed by means of the attractors, the global attractors, and the transient of the system.

The dynamics of a synchronous BN is determined by the relations and the states of the entities, which are modelled by a graph $G=(V,E)$⁠, where $V={1,…,n}$ is the vertex set and $E$ is the edge set, called the dependency graph of the system. In this paper, we work with synchronous BNs over simple, connected graphs, where the relations between the vertices are bidirectional. The state of a vertex $i∈V$ is represented by $x i$⁠, which belongs to a (general) Boolean algebra $( B,⋏,⋎, ′, O, I)$⁠. Thus, a state of the system can be represented by a configuration $x=( x 1,…, x n)∈ B n$⁠.

We consider, for every entity $i∈V$⁠, the set $A G(i)={j∈V:{i,j}∈E}∪{i}$ of entities that influence its updating, i.e., the set of its neighbors plus the vertex $i$ itself.

In contrast, in generalized dynamical systems (see Refs. 43 and 44), the state of an entity does not necessarily affect its own updating, so making its orbital structure more involved.

Next, we define a synchronous BN on a (general) Boolean algebra $( B,⋏,⋎, ′, O, I)$⁠:

In Theorems 3 and 5 of Ref. 32, the existence and coexistence of periodic orbits in binary BNs $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$] are characterized in terms of the set of vertices $W C ′$⁠. Analogously, as we will see in Remark III.4, the existence and coexistence of periodic orbits in BNs with non-binary states $( B,G, MAX)$ [resp. $( B,G, MIN)$] also rely on this set.

Additionally, let $G 1,…, G l$ be the connected components of direct entities, which result from $G$ when we remove all the vertices in $W ′$ and the edges adjacent to those vertices. Last, we call $W 1 ′={i∈ W ′: there exists a unique j,1≤j≤l, such that i is adjacent to G j},$ i.e., the complemented vertices, which are adjacent to a unique connected component $G j$⁠.

The first step in the study of the dynamics of a synchronous BN $( B,G,F)$ is to analyze its periodic orbits. An $m$-periodic orbit is a sequence of $m$ distinct configurations, called $m$-periodic points, which, when reached by the system, is repeated every $m$ iterations. Formally, $x∈ B n$ is an $m$-periodic point if $F m(x)=x$⁠, with $F k(x)≠x, for all k<m$⁠. Note that any configuration in a synchronous BN $( B,G,F)$ either eventually reaches a periodic orbit or belongs to one (i.e., it is repeated after, at most, $| B n |$ iterations). We say that a synchronous BN $( B,G,F)$ is a fixed point system if all the periodic orbits of the system are fixed points. Analogously, we say it is a m-periodic system if all the periodic orbits of the system have period $m$⁠.

Now, we recall the main theorem in Ref. 41, which allows us to translate the known results for the simplest case $(p=1)$ when $B={0,1}$ to the more complex scenario of a general Boolean algebra with $2 p$ elements, $p>1$⁠.

From the function $F: B n→ B$ defined over $( B,⋏,⋎, ′, O, I)$⁠, we can consider $F { 0 , 1 }: { 0 , 1 } n→{0,1}$ the Boolean function resulting from interchanging the inner operators $⋏,⋎$⁠, and $′$ appearing in $F$⁠, by the inner operators $∧,∨$⁠, and $′$ defined in $({0,1},∧,∨, ′, 0, 1)$⁠, respectively. For simplicity, we will omit the subindex and just write $F≡ F { 0 , 1 }$⁠, as the domain becomes evident by the context.

Then, the study of the dynamics of a synchronous BN $( B,G,F)$ can be performed by studying the dynamics of $p$ synchronous ones of the form $({0,1},G,F)$⁠, which we call the fibers and refer by $f k$⁠, $1≤k≤p$⁠, of the global one, whose state configurations are represented as a row of the global state matrix $X$⁠. Note that all the fibers have the same dependency graph $G$ as the general system.

This connection between a synchronous BN on a general Boolean algebra $B$ with $2 p$ elements and the synchronous BN on ${0,1}$ associated with its fibers $f k$⁠, $1≤k≤p$⁠, will be the key for the translation of the results to the case $p>1$⁠.

In this section, we study the dynamics of synchronous BNs of the form $( B,G,F)$⁠, which are determined by their periodic orbits.

Now, we give a preliminary result about the periodic orbits of such systems based on their relationship with their fibers, which will be useful for proving other core results.

Proposition III.2 shows that the existence of periodic orbits of the system is determined by the existence of periodic orbits on each fiber. Thus, theoretical results for the simpler case $({0,1},G,F)$ can be used to establish results in $( B,G,F)$⁠. In particular, in a synchronous BN $({0,1},G, MAX)$ (resp. $({0,1},G, MIN)$⁠), all the periodic orbits of the system are fixed points or two-periodic orbits (Corollaries 3.1 and 3.2 in Ref. 41). Furthermore, the coexistence of periodic orbits of different periods, i.e., of fixed points and two-periodic orbits, is not possible. The proof of such a theorem (Theorems 3 and 5 in Ref. 32) relies only on the structure of the graph and on the evolution operator. In other words, a synchronous BN $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$] is either a fixed point system or a two-periodic system. The same occurs for synchronous BNs with non-binary states (see Corollaries 3.1 and 3.2 in Ref. 41 and Sec. 5 in Ref. 32), as we recall in the next remark:

In Ref. 35, Theorem 6 (see also Corollary 2 in this reference), the (exact) number of fixed points of a synchronous BN $({0,1},G, MAX)$ (resp. a $({0,1},G, MIN)$⁠) is provided. Thanks to this result, we can calculate the (exact) number of fixed points of a synchronous BN with non-binary states.

In particular, a fixed point theorem for synchronous BNs with non-binary states can be stated for $( B,G, MAX)$ [resp. $( B,G, MIN)$] (Theorem 11 in Ref. 32).

Determining the number of exact fixed points of a synchronous BN may be computationally demanding for large graphs. Thus, it is usual to work with bounds of such a number. The following two corollaries provide bounds for the number of fixed points of a synchronous BN $( B,G, MAX)$ [resp. $( B,G, MIN)$]. These results follow from Proposition III.6 and from previous results, which provide bounds for the number of fixed points of a synchronous BN $({0,1},G, MAX)$ (resp. $({0,1},G, MIN)$⁠). Specifically, in Ref. 35, Theorem 4, the minimum number $Ψ$ of fixed points of a synchronous BN $({0,1},G, MAX)$ (resp. $({0,1},G, MIN)$⁠) was obtained. On the other hand, in Ref. 34, Theorem 1, the maximum number of fixed points $Φ$ was given for a synchronous BN $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$]. Next, we generalize such results for synchronous BN on general Boolean algebras.

Let us now explore the issue of uniqueness of two-periodic orbits on general Boolean algebras. In Ref. 34, Theorems 3 and 4, the necessary and sufficient conditions for the existence of a unique two-periodic orbit in a synchronous BN $({0,1},G,F)$ are given. However, when $B$ is a Boolean algebra with $2 p$ elements, $p>1$⁠, such uniqueness is not possible. This is an important difference because it breaks the known pattern for synchronous binary BNs and highlights a significant dynamic change between synchronous BN $({0,1},G,F)$ and $( B,G,F)$⁠. Indeed, we have the following result:

As a consequence, we prove that two-periodic orbits cannot be unique in synchronous BNs on general Boolean algebras:

In Theorem 8 of Ref. 35 and Theorem 5 of Ref. 34, the minimum and maximum numbers of two-periodic points for a synchronous BN $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$] are obtained. Reasoning as in Proposition III.11, we get the following results for synchronous BNs with non-binary states:

In particular, in Ref. 35, Corollaries 3 and 4, the number $α$ of two-periodic points in a synchronous BN $({0,1},G, NAND)$ [resp. $({0,1},G, NOR)$] is given for two special cases that depend on the structure of the dependency graph $G$⁠. Thus, under the same conditions of $G$⁠, the number of two-periodic points of $( B,G, NAND)$ [resp. $( B,G, NOR)$] is $α p$⁠, and therefore, the number of two-periodic orbits is $1 2 α p$⁠.

In this section, we tackle the existence and uniqueness of predecessors in synchronous BNs with non-binary states. The existence of predecessors allows us to characterize the GOE configurations of the system. On the other hand, uniqueness allows us to derive some results regarding the convergence to periodic orbits. Now, we state the core result of this section about the relation between predecessors in $({0,1},G,F)$ and those in $( B,G,F)$⁠.

In Ref. 37, Corollary 5, the structure of a predecessor $x∈ { 0 , 1 } n$ for a configuration $y∈ { 0 , 1 } n$ that is not a GOE in $({0,1},G, MAX)$ is provided. Thus, according to Proposition IV.1, if a configuration $y∈ B n$ is not a GOE, then a predecessor $x∈ B n$ of $y$ can be constructed by taking as the $k$th row $( x k , 1,…, x k , n)∈ { 0 , 1 } n$ of the matrix representation of $x$ the predecessor of the $k$th row $( y k , 1,…, y k , n)∈ { 0 , 1 } n$ of the matrix representation of $y$ on $({0,1},G, MAX)$⁠, $1≤k≤p$⁠. More precisely, for every $1≤k≤p$⁠, $( x k , 1,…, x k , n)$ has the following structure:

• If $y k , i=0$⁠, then for every entity $j∈ A G(i)$⁠, it is

  1. $x k , j=0$ when $x k , j$ is in a direct form in MAX and

  2. $x k , j=1$ when $x k , j$ is in a complemented form in MAX.

• If $y k , i=1$⁠, there exists an entity $j∈ A G(i)$ that fulfills one of the following conditions:

  1. $x k , j=1$ if $x k , j$ is in a direct form in MAX or

  2. $x k , j=0$ if $x k , j$ is in a complemented form in MAX.

• If $y k , i=1$⁠, then for every entity $j∈ A G(i)$⁠, it is

  1. $x k , j=1$ when $x k , j$ is in a direct form in MIN and

  2. $x k , j=0$ when $x k , j$ is in a complemented form in MIN.

• If $y k , i=0$⁠, there exists an entity $j∈ A G(i)$ that fulfills one of the following conditions:

  1. $x k , j=0$ if $x k , j$ is in a direct form in MIN or

  2. $x k , j=1$ if $x k , j$ is in a complemented form in MIN.

Proposition IV.1 and the matrix representation (2) allow us to determine the necessary and sufficient conditions for the existence of GOE in a synchronous BN with non-binary states as follows:

Next, we determine bounds for the number of GOE in a synchronous BN $( B,G, MAX)$ [resp. $( B,G, MIN)$]. To obtain a lower bound, we will use Ref. 37, Corollaries 3 and 4, which allow us to provide a lower bound for the number of GOE in $({0,1},G, MAX)$ and $({0,1},G, MIN)$⁠, respectively. On the other hand, to obtain the upper bound, we will use Ref. 37, Theorems 1 and 2, from which can be easily inferred that the configurations $(0,…,0)$ and $(1,…,1)$ are neither a GOE in a $({0,1},G, MAX)$ nor in a $({0,1},G, MIN)$⁠.

The extension of these results gives rise to the following proposition:

In Ref. 37, Theorems 3 and 4, sufficient and necessary conditions for a configuration in a synchronous BN $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$] to have a unique predecessor are provided. Such conditions, jointly with Proposition IV.1, allow one to determine when a configuration in $( B,G, MAX)$ [resp. $( B,G, MIN)$] has a unique predecessor, so solving the coexistence of a predecessor problem.

Once the conditions for the existence of more than one predecessor are known, the following step is to determine the number of them for any given state. In Ref. 37, Corollaries 7 and 8, the set of all predecessors for a configuration $y∈ { 0 , 1 } n$ is given for a synchronous BN $({0,1},G, MAX)$ (resp. $({0,1},G, MIN)$⁠). Then, the set of all predecessors for $y∈ B n$ in $( B,G, MAX)$ [resp. $( B,G, MIN)$] can be obtained as the combinations of all the predecessors of the rows $( y k , 1,…, y k , n)∈ { 0 , 1 } n$ of the matrix representation of $y$⁠. Hence, we know how to theoretically obtain the set of all the predecessors of $y∈ B n$ and, consequently, its cardinal.

The conditions for the existence of a predecessor for a given configuration $y=( y 1,…, y n)∈ { 0 , 1 } n$ in $({0,1},G, MAX)$ [resp. $({0,1},G, MIN)$] are stated in Ref. 37, Theorems 1 and 2. The proofs of these results rely on the definition of the subsets $V 0(y)={i∈V: y i=0}$⁠, the set of deactivated vertices; and $V 1(y)={i∈V: y i=1}$⁠, the set of activated ones. To state the upcoming result about the number of predecessors of a synchronous BN $( B,G, MAX)$ [resp. $( B,G, MIN)$], it is necessary to define similar sets for each fiber. Given a global configuration $y∈ B n$⁠, with matrix representation $Y$⁠, we define the set $V 0 k(Y)={i∈V: y k , i=0}$ and $V 1 k(Y)={i∈V: y k , i=1}$⁠, for every $1≤k≤p$⁠. For simplicity, and when there is no possible confusion regarding the configuration $Y$⁠, we will represent these sets as $V 0 k$ and $V 1 k$⁠, respectively.