Hidden Order of Boolean Networks

It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using the semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of the Boolean networks, this article reveals that in addition to this explicit order, there is a certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, the hidden order provides a global horizon to describe the evolution of the overall network. We conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. Then these results about the Boolean networks are further extended to the k -valued case.

Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product

This paper investigates the Sylvester-transpose matrix equation A⋉X+XT⋉B=C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A⋉X=C, the equation X⋉B=C, and the classical Sylvester-transpose matrix equation.

Memory and communication efficient algorithm for decentralized counting of nodes in networks

Node counting on a graph is subject to some fundamental theoretical limitations, yet a solution to such problems is necessary in many applications of graph theory to real-world systems, such as collective robotics and distributed sensor networks. Thus several stochastic and naïve deterministic algorithms for distributed graph size estimation or calculation have been provided. Here we present a deterministic and distributed algorithm that allows every node of a connected graph to determine the graph size in finite time, if an upper bound on the graph size is provided. The algorithm consists in the iterative aggregation of information in local hubs which then broadcast it throughout the whole graph. The proposed node-counting algorithm is on average more efficient in terms of node memory and communication cost than its previous deterministic counterpart for node counting, and appears comparable or more efficient in terms of average-case time complexity. As well as node counting, the algorithm is more broadly applicable to problems such as summation over graphs, quorum sensing, and spontaneous hierarchy creation.

Exploring Observability of Attractor Cycles in Boolean Networks for Biomarker Detection

Boolean Network (BN) is a simple and popular mathematical model that has attracted significant attention from systems biology due to its capacity to reveal genetic regulatory network behavior. In addition, observability, as an important network feature, plays a vital role in deciphering the underlying mechanisms driving a genetic regulatory network and has been widely investigated. Prior studies examined observability of BNs and other complex networks. That said, observability of attractor, which can serve as a biomarker for disease, has not been fully examined in the literature. In this study, we formulated a new definition for singleton or cyclic attractor observability in BNs and developed an effective methodology to resolve the captured problem. We also showed complexity is of O(Pmn), when the maximal period of cyclic attractor is P, the number of attractor is m and the number of genes is n. Importantly, we have confirmed our method can faithfully predict the expression pattern of segment polarity genes in Drosophila melanogaster and showed it can effectively and efficiently deal with the captured observability problem.

Observability of Boolean Control Networks Using Parallel Extension and Set Reachability

This brief reviews various definitions of observability for Boolean control networks (BCNs) and proposes a new one: output-feedback observability. This new definition applies to all BCNs whose initial states can be identified from the history of output measurements. A technique called parallel extension is then proposed to facilitate observability analysis. Furthermore, a technique called state transition graph reconstruction is proposed for analyzing the set reachability of BCNs, based on which new criteria for observability, single-input sequence observability, and arbitrary-input observability, are obtained. Using the proposed techniques, this brief proves that the problem of output-feedback observability can be recast as that of stabilizing a logic dynamical system with output feedback. Then, a necessary and sufficient condition for static output feedback observability is proposed. The relationships between the different definitions of observability are discussed, and the main results are illustrated with examples.

Policy Iteration Algorithm for Optimal Control of Stochastic Logical Dynamical Systems

This brief investigates the infinite horizon optimal control problem for stochastic multivalued logical dynamical systems with discounted cost. Applying the equivalent descriptions of stochastic logical dynamics in term of Markov decision process, the discounted infinite horizon optimal control problem is presented in an algebraic form. Then, employing the method of semitensor product of matrices and the increasing-dimension technique, a succinct algebraic form of the policy iteration algorithm is derived to solve the optimal control problem. To show the effectiveness of the proposed policy iteration algorithm, an optimization problem of p53-Mdm2 gene network is investigated.

Integer programming-based method for observability of singleton attractors in Boolean networks

Boolean network (BN) is a popular mathematical model for revealing the behaviour of a genetic regulatory network. Furthermore, observability, an important network feature, plays a significant role in understanding the underlying network. Several studies have been done on analysis of observability of BNs and complex networks. However, the observability of attractor cycles, which can serve as biomarker detection, has not yet been addressed in the literature. This is an important, interesting and challenging problem that deserves a detailed study. In this study, a novel problem was first proposed on attractor observability in BNs. Identification of the minimum set of consecutive nodes can be used to discriminate different attractors. Furthermore, it can serve as a biomarker for different disease types (represented as different attractor cycles). Then a novel integer programming method was developed to identify the desired set of nodes. The proposed approach is demonstrated and verified by numerical examples. The computational results further illustrates that the proposed model is effective and efficient.

State feedback control design for Boolean networks

Background

Driving Boolean networks to desired states is of paramount significance toward our ultimate goal of controlling the progression of biological pathways and regulatory networks. Despite recent computational development of controllability of general complex networks and structural controllability of Boolean networks, there is still a lack of bridging the mathematical condition on controllability to real boolean operations in a network. Further, no realtime control strategy has been proposed to drive a Boolean network.

Results

In this study, we applied semi-tensor product to represent boolean functions in a network and explored controllability of a boolean network based on the transition matrix and time transition diagram. We determined the necessary and sufficient condition for a controllable Boolean network and mapped this requirement in transition matrix to real boolean functions and structure property of a network. An efficient tool is offered to assess controllability of an arbitrary Boolean network and to determine all reachable and non-reachable states. We found six simplest forms of controllable 2-node Boolean networks and explored the consistency of transition matrices while extending these six forms to controllable networks with more nodes. Importantly, we proposed the first state feedback control strategy to drive the network based on the status of all nodes in the network. Finally, we applied our reachability condition to the major switch of P53 pathway to predict the progression of the pathway and validate the prediction with published experimental results.

Conclusions

This control strategy allowed us to apply realtime control to drive Boolean networks, which could not be achieved by the current control strategy for Boolean networks. Our results enabled a more comprehensive understanding of the evolution of Boolean networks and might be extended to output feedback control design.

Synchronization design of Boolean networks via the semi-tensor product method

We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semi-tensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach.

Complete synchronization of Boolean networks

We examine complete synchronization of two deterministic Boolean networks (BNs) coupled unidirectionally in the drive-response configuration. A necessary and sufficient criterion is presented in terms of algebraic representations of BNs. As a consequence, we show that complete synchronization can occur only between two conditionally identical BNs when the transition matrix of the drive network is nonsingular. Two examples are worked out to illustrate the obtained results.

Binary higher order neural networks for realizing Boolean functions

In order to more efficiently realize Boolean functions by using neural networks, we propose a binary product-unit neural network (BPUNN) and a binary π-ς neural network (BPSNN). The network weights can be determined by one-step training. It is shown that the addition " σ," the multiplication " π," and two kinds of special weighting operations in BPUNN and BPSNN can implement the logical operators " ∨," " ∧," and " ¬" on Boolean algebra 〈Z(2),∨,∧,¬,0,1〉 (Z(2)={0,1}), respectively. The proposed two neural networks enjoy the following advantages over the existing networks: 1) for a complete truth table of N variables with both truth and false assignments, the corresponding Boolean function can be realized by accordingly choosing a BPUNN or a BPSNN such that at most 2(N-1) hidden nodes are needed, while O(2(N)), precisely 2(N) or at most 2(N), hidden nodes are needed by existing networks; 2) a new network BPUPS based on a collaboration of BPUNN and BPSNN can be defined to deal with incomplete truth tables, while the existing networks can only deal with complete truth tables; and 3) the values of the weights are all simply -1 or 1, while the weights of all the existing networks are real numbers. Supporting numerical experiments are provided as well. Finally, we present the risk bounds of BPUNN, BPSNN, and BPUPS, and then analyze their probably approximately correct learnability.

Model construction of Boolean network via observed data

In this paper, a set of data is assumed to be obtained from an experiment that satisfies a Boolean dynamic process. For instance, the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells. With the observed datasets, several methods to construct the dynamic models for such Boolean networks are proposed. Instead of building the logical dynamics of a Boolean network directly, its algebraic form is constructed first and then is converted back to the logical form. Firstly, a general construction technique is proposed. To reduce the size of required data, the model with the known network graph is considered. Motivated by this, the least in-degree model is constructed that can reduce the size of required data set tremendously. Next, the uniform network is investigated. The number of required data points for identification of such networks is independent of the size of the network. Finally, some principles are proposed for dealing with data with errors.