Для случая, когда орграф зависимостей не содержит остовного входящего дерева, указана область сходимости дифференциальной модели поиска консенсуса и показано, что консенсус может быть достигнут использованием метода ортогональной проекции, который ранее был предложен для дискретной модели согласования.

The paper studies the problem of achieving consensus in multi-agent systems in the case where the dependency digraph Г has no spanning in-tree. We consider the regularization protocol that amounts to the addition of a dummy agent (hub) uniformly connected to the agents. The presence of such a hub guarantees the achievement of an asymptotic consensus. For the "evaporation" of the dummy agent, the strength of its influences on the other agents vanishes, which leads to the concept of latent consensus. We obtain a closed-form expression for the consensus when the connections of the hub are symmetric; in this case, the impact of the hub upon the consensus remains fixed. On the other hand, if the hub is essentially influenced by the agents, whereas its influence on them tends to zero, then the consensus is expressed by the scalar product of the vector of column means of the Laplacian eigenprojection of Г and the initial state vector of the system. Another protocol, which assumes the presence of vanishingly weak uniform background links between the agents, leads to the same latent consensus.

This paper solves the problem of usefull random signal filtering in a discrete-time nonlinear observation model. As such model, we consider the multiplicative model with nonnegative signals and noises. In contrast to standard filtering problems, our statement assumes that the distribution and equation of the useful signal are unknown. To solve the nonlinear filtering problem, the idea is to employ a generalized optimal filtering equation with a feature that the optimal estimator is expressed only through characteristics of the observed process. The role of such characteristics in the equation belongs to the logarithmic derivative of the conditional multidimensional density of observations. We find the solution of the equation using nonparametric kernel estimation methods with nonsymmetrical gamma kernel functions defined on the positive semiaxis. Moreover, we establish convergence conditions for the kernel estimator of the logarithmic derivative of the multidimensional density by dependent observations, as well as derive explicit formulas for optimal bandwidths and construct a stable (regularized) filtering estimator.

This paper suggests a new approach to the constructive definition of user influence levels in social networks—the so-called actional model. According to the approach, the influence level of a user is calculated on the basis of their actions taking into account the goals of a control subject (a Principal). An example illustrates how the actional model can be applied to find the influence level of users in a real social network on the assumption of available initial data.

A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states.