This article presents a novel dynamic programming approach to determine the robust controllability of Boolean control networks (BCNs) subject to stochastic disturbances. By applying Bellman's optimality principle, we derive the recurrence relation for computing the optimal time matrix, a crucial concept characterizing robust reachability between two arbitrary states. We develop a finite-termination dynamic programming algorithm to calculate the optimal time matrix exactly and efficiently, with a rigorously certified iteration count. Sufficient and necessary conditions for robust controllability are then established based on the optimal time matrix. Furthermore, for any pair of reachable states, we construct time-optimal state feedback control laws to steer the system from the initial state to the target state, regardless of disturbances. Finally, extensive numerical experiments with biological networks validate the effectiveness of the proposed approach, showing significant improvements in computational efficiency. Additionally, we introduce a Q-learning-based algorithm and compare its performance, highlighting the advantages of our dynamic programming approach in terms of both efficiency and solution quality.

The dissemination of information and the development of public opinion are essential elements of most social networks and are often described as distinct, man-made occurrences. However, what is often disregarded is the interdependence between these two phenomena. Information dissemination serves as the foundation for the formation of public opinion, while public opinion, in turn, drives the spread of information. In our study, we model the coevolutionary relationship between information and public opinion on heterogeneous multiplex networks. This model takes into account a minority of individuals with steadfast opinions and a majority of individuals with fluctuating views. Our findings reveal the equilibrium state of public opinion in this model and examine the consistency between opinions and the network structure. Additionally, by identifying a linear relationship between mainstream public opinion and extreme individuals, we propose a strategy for regulating public opinion by adjusting the positions of extreme groups.

The lack of studying the complex organization of directed network usually limits the understanding of the underlying relationship between network structures and functions. Structural controllability and structural predictability, two seemingly unrelated subjects, are revealed in this article to be both highly dependent on the critical links previously thought to only be able to influence the number of driver nodes in controllable directed networks. Here, we show that critical links can not only contribute to structural controllability but can also have a significant impact on the structural predictability of networks, suggesting the universal pattern of structural reciprocity in directed networks. In addition, it is shown that the fraction and location of critical links have a strong influence on the performance of prediction algorithms. Moreover, these empirical results are interpreted by introducing the link centrality based on corresponding line graphs. This work bridges the gap between the two independent research fields, and it provides indications of developing advanced control strategies and prediction algorithms from a microscopic perspective.

The linear arboricity of a graph G is the minimum number of linear forests which partition the edges of G. In the present, it is proved that if a graph G can be embedded in a surface of Euler characteristic $\varepsilon<0$ and $\Delta(G)\geq\sqrt{46-54\varepsilon}+19$, then its linear arboricity is $\lceil\frac{\Delta(G)}{2}\rceil$. Some related results on the girth and maximum average degree are also obtained.