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We investigate the effects of modularity, antimodularity, and multiscale influence on random Boolean networks (RBNs). On the one hand, we produced modular, antimodular, and standard RBNs and compared them to identify how antimodularity affects the dynamical behaviors of RBNs. We found that the antimodular networks showed similar dynamics to the standard networks. Confirming previous results, modular networks had more complex dynamics. On the other hand, we generated multilayer RBNs where there are different RBNs in the nodes of a higher scale RBN. We observed the dynamics of micro- and macronetworks by adjusting parameters at each scale to reveal how the behavior of lower layers affects the behavior of higher layers and vice versa. We found that the statistical properties of macro-RBNs were changed by the parameters of micro-RBNs, but not the other way around. However, the precise patterns of networks were dominated by the macro-RBNs. In other words, for statistical properties only upward causation was relevant, while for the detailed dynamics downward causation was prevalent.

The structure of a network can significantly alter its function or behavior. Modularity is a structural property prevalent in many systems, where elements within modules have more connections among themselves than with other elements [

In addition to modularity, another relevant property of complex systems is the causality between scales in a multilayer structure. It has attracted much attention in systems and computational biology how changes at lower layers affect system properties at upper layers and vice versa [

In this study, we aim to investigate the effects of antimodularity and causality between layers in random Boolean networks (RBNs). Because RBNs have been used as models to represent dynamics of gene regulatory networks [

RBNs were suggested as gene regulatory network models by Kauffman [

An example RBN (

A module is a set of nodes that are more densely connected to each other than to nodes from other modules [

Adjacency matrixes and networks of standard, modular, and antimodular RBNs. All the networks consist of 6 nodes (

The three types of RBNs are as follows:

Standard RBNs: regardless of modules, links are randomly distributed in the adjacency matrix. Figure

Modular RBNs: a greater number of links are randomly placed inside the modules than outside the modules with probability

Antimodular RBNs: contrary to modular RBNs, a larger number of links are distributed outside the defined modules than inside the modules with probability

In this study, we consider arbitrarily

We quantitatively measure modularity

Examples showing how to calculate the modularity

To study the causality between upper and lower levels, we developed multilayer RBN model where there are different RBNs in the nodes of a higher scale RBN. We call the meta RBN

Random initial states are assigned to micronetworks (Figure

The node states of the micronetworks are updated at the microlevel during

The node states of the macronetwork are determined based on the node states of micronetworks at time

The node states of the macronetwork are updated during

Schematic diagrams illustrating the updates of the node states in a multilayer RBN model.

To investigate the effect of modularity/antimodularity, we performed 100 independent simulation runs for each parameter combination. For each simulation, we generated modular, antimodular, and standard RBNs composed of 240 nodes. Because the number of nodes is 240, all the possible numbers of modules to be defined in the network are 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240. Varying

Here complexity was calculated based on our previous approach [

Figure

Complexity for different number of nodes, varying average connectivity

To examine the causality between micro- and macronetworks, we conducted 55 independent simulation runs. For each simulation, we produced a multilayer RBN model where each micronetwork has 125 nodes, and the macronetwork has 55 nodes. In the simulation,

Figure

Modularity, numbers of attractors, average length of attractors, and complexity for modular RBNs.

To illustrate how links are distributed in modular networks, we present a few simple examples in Figure

Link distributions depending on

In a second set of experiments, the number of attractors is limited to a maximum number of 1,000. In other words, if 1,000 attractors are found, the search for attractors is halted. As seen in the figure, the numbers of attractors found were zero at

In a third set of experiments, the complexity gradually increased until reaching their peaks at

In another set of experiments, the lengths of attractors were the highest at

Figure

State transitions of networks with

Figure

Numbers of attractors, complexity, and lengths of attractors for antimodular RBNs.

For the complexity, the values had little variation against the number of modules (Figure

Complexity of standard and antimodular RBNs. (a) Standard network. (b) Modular networks. (c) Overlapped plots.

To compare the results acquired from our modular/antimodular RBNs and the properties of biological networks, we collected seven biological networks from CellCollective.org [

Adjacency matrix and modularity of a biological network related to infection treatment.

Figure

Modularity and complexity of seven biological networks.

Complexity versus modularity for seven biological networks.

Figure

Heat maps of the complexity of micro- and macro-RBNs.

On the contrary, the complexity of the micronetworks was not influenced by macro-

As another experiment for multiscale influence, an analysis of the attractors was performed. What can be seen in Figure

Heat maps showing the numbers of attractors depending on micro- and macro-

Let us recall that an attractor is a set of states that are repeated. If a timestep of the macronetwork does not produce a change in a particular node, this will force its micronetwork to back into a state of only ones or zeroes. Thus, a regular pattern will be observed even if the micronetwork is chaotic. In other words, without the macroinfluence, it would be actually in a transient.

Even if there is a change in the node of the macronetwork, this would push its micronetwork to back into one of two possible states. Thus, on average micronetworks’ attractors are shorter because macronodes that do not change force the same initial state over the micronetwork, while the macronetwork can have longer attractors composed of the combinations of the microattractors, as shown in Figure

Temporal dynamics (time moves downward, columns represent nodes, and colors represent states) showing how an attractor of the macronetwork generates an attractor in an chaotic micronetwork.

Our computational experiments showed that modularity increases the “diversity” of the network dynamics. That is, there are more and longer attractors as the number of modules increases. For the antimodularity, the increase of the modules does not influence the length or number of attractors.

Networks with a median number of modules extend the complexity of RBNs. All networks have a “critical” region where complexity is high (around 2.3 for standard networks due to a finite size effects, as the theoretical phase transition occurs at

Antimodular RBNs showed similar dynamics as standard RBNs. We can infer the reason from Figure

Adjacency matrixes of standard, modular, and antimodular RBN with

For the causality between micro- and macro-RBNs, the complexity of the macronetwork was varied by

However, the study of the attractors for the multiscale networks showed that the attractors in the macronetwork affect how the micronetwork behaves. Our complexity measure is based on Shannon’s information, and thus it is a statistical approximation of the dynamics of a network. In other words, it does not distinguish the precise arrangement of bits and just focuses on their probability distribution. For example, a random bit sequence has the same information (maximal) and complexity (minimal) than an ordered string which has precisely one-half of all zeroes and another half of all ones. In our experiments, the statistical properties of the network dynamics (i.e., the complexity) were determined mainly by the lower scale. However, the precise order of the dynamics (i.e., the attractors) was determined mainly by the higher scale.

Our results suggest that studying only the lower scale of systems would be meaningful only if we are interested in certain statistical properties. Nevertheless, if we want to understand and attempt to predict the precise dynamics, we need to study both scales and how they interact. This can be illustrated with the classical example of different arrangements of carbon atoms, which are the same at the lower scale, but depending on their structure at the higher scale (charcoal, diamond, graphene, etc.) they can have very different properties at both scales. Still, studying whether our results for RBNs can be generalized to all phenomena is an ambitious task which is beyond the scope of this paper.

For further study, we plan to investigate a relationship between modularity/antimodularity and diverse topologies such as cyclic or star shapes and multiscale effects of micro- and macronetworks with different structures. Also, we will study how to control the states of modular, antimodular, and multilayer networks so that our research can be applied to more fields. Specifically, using mathematical approaches like the technique of semitensor product (STP) [

Our simulator and data are available at

The authors declare that there are no conflicts of interest regarding the publication of this paper.

We are grateful to Darío Alatorre, Ewan Colman, Omar Karim Pineda, José Luis Mateos, Dante Pérez, and Fernanda Sánchez-Puig for useful comments and discussions. This research was partially supported by CONACYT and DGAPA, UNAM.