Hongsheng Qi

Controllability and observability of Boolean control networks

The controllability and observability of Boolean control networks are investigated. After a brief review on converting a logic dynamics to a discrete-time linear dynamics with a transition matrix, some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix. Based on the discrete-time dynamics, the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely. Then the problem of observability is also solved by giving necessary and sufficient conditions.

A Linear Representation of Dynamics of Boolean Networks

A new matrix product, called semi-tensor product of matrices, is reviewed. Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping. Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system. Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor. The corresponding algorithms are developed and used to some examples.

Input-state incidence matrix of Boolean control networks and its applications☆

The input-state incidence matrix of a control Boolean network is proposed. It is shown that this matrix contains complete information of the input-state mapping. Using it, an easily verifiable necessary and sufficient condition for the controllability of a Boolean control network is obtained. The corresponding control which drives a point to a given reachable point is designed. Moreover, certain topological properties such as the fixed points and cycles of a Boolean control network are investigated. Then, as another application, a sufficient condition for the observability is presented. Finally, the results are extended to mix-valued logical control systems.

Realization of Boolean control networks

Based on the linear expression of the dynamics of Boolean networks, the coordinate transformation of Boolean variables is defined. It follows that the state space coordinate transformation for the dynamics of Boolean networks is revealed. Using it, the invariant subspace for a Boolean control network is defined. Then the structure of a Boolean control network is analyzed, and the controllable and observable normal forms and the Kalman decomposition form are presented. Finally the realization problem, including minimum realization, of Boolean control networks is investigated.

An introduction to semi-tensor product of matrices and its applications

Multi-Dimensional Data Semi-Tensor Product of Matrices Multilinear Mappings Among Vector Spaces Right and General Semi-Tensor Products Rank, Pseudo-Inverse, and Positivity of STP Matrix Expression of Logic Mix-Valued Logic Logical Matrix, Fuzzy Set and Fuzzy Logic Fuzzy Relational Equation Fuzzy Control with Coupled Fuzzy Relations Boolean Function with Galois Field Structure Decomposition of Logical Functions Boolean Calculus Lattice, Graph, and Universal Algebra Boolean Network Boolean Control System Game Theory Multi-Variable Polynomials Some Applications to Differential Geometry and Algebra Morgans Problem Linearization of Nonlinear Control Systems Stability Region of Dynamic System.

Modeling, Analysis and Control of Networked Evolutionary Games

Consider a networked evolutionary game (NEG). According to its strategy updating rule, a fundamental evolutionary equation (FEE) for each node is proposed, which is based on local information. Using FEEs, the network strategy profile dynamics (SPD) is expressed as a k-valued (deterministic or probabilistic) logical dynamic system. The SPD is then used to analyze the network dynamic behaviors, such as the fixed points, the cycles, and the basins of attractions, etc. Particularly, when the homogeneous networked games are considered, a necessary and sufficient condition is presented to verify when a stationary stable profile exists. Then the equivalence of two NEGs is investigated. Finally, after a rigorous definition of controlled NEGs, some control problems, including controllability, stabilization, and network consensus, are considered, and some verifiable conditions are presented. Examples with various games are presented to illustrate the theoretical results. The basic tool for this approach is the semi-tensor product (STP) of matrices, which is a generalization of the conventional matrix product.

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.

Evolutionarily Stable Strategy of Networked Evolutionary Games

The evolutionarily stable strategy (ESS) of networked evolutionary games (NEGs) is studied. Analyzing the ESS of infinite popular evolutionary games and comparing it with networked games, a new verifiable definition of ESS for NEGs is proposed. Then, the fundamental evolutionary equation (FEE) is investigated and used to construct the strategy profile dynamics (SPDs) of homogeneous NEGs. Two ways for verifying the ESS are proposed: 1) using the SPDs to verify it directly. The SPDs provides complete information about the NEGs, and then necessary and sufficient conditions are revealed. It can be used for NEGs with small size and 2) some sufficient conditions are proposed to verify the ESS of NEGs via their FEEs. This method is particularly suitable for large scale networks. Some illustrative examples are included to demonstrate the theoretical results.

Stabilization of random Boolean networks

This paper considers the stabilization of random Boolean networks. We first give a survey on the semi-tensor product approach to Boolean (control) networks, which is a new technique developed by us. Particularly, the stability and stabilization of Boolean (control) networks are reviewed. Then, under the same framework, the topological structure of the random Boolean networks is investigated and revealed. Using it the stabilization of random Boolean networks is considered. A sufficient condition is provided. A stabilizer design method is proposed.

A SURVEY ON SEMI-TENSOR PRODUCT OF MATRICES ⁄?

Semi-tensor product of matrices is a generalization of conventional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It was firstly proposed about ten years ago. Since then it has been developed and applied to several different fields. In this paper we will first give a brief introduction. Then give a survey on its applications to dynamic systems, to logic, to differential geometry, to abstract algebra, respectively.