A stochastic equivalence approach for an Ornstein-Uhlenbeck process driven power system dynamics

This paper develops a stochastic equivalence approach for an Ornstein-Uhlenbeck process-driven power system. The concept of stochastic equivalence coupled with stochastic differential rule plays the important role to develop the stochastic equivalence approach of this paper. This paper also develops the prediction theory of power system dynamics with the OU process as well.

Learning Stochastic Equivalence based on Discrete Ricci Curvature

Role-based network embedding methods aim to preserve node-centric connectivity patterns, which are expressions of node roles, into low-dimensional vectors. However, almost all the existing methods are designed for capturing a relaxation of automorphic equivalence or regular equivalence. They may be good at structure identification but could show poorer performance on role identification. Because automorphic equivalence and regular equivalence strictly tie the role of a node to the identities of all its neighbors. To mitigate this problem, we construct a framework called Curvature-based Network Embedding with Stochastic Equivalence (CNESE) to embed stochastic equivalence. More specifically, we estimate the role distribution of nodes based on discrete Ricci curvature for its excellent ability to concisely representing local topology. We use a Variational Auto-Encoder to generate embeddings while a degree-guided regularizer and a contrastive learning regularizer are leveraged to improving both its robustness and discrimination ability. The effectiveness of our proposed CNESE is demonstrated by extensive experiments on real-world networks.

MULTIPLICATIVE APPROXIMATION OF A RANDOM PROCESS

In this paper we consider the stochastic Ito differential equation in an infinite-dimensional real Hilbert space. Using the method of multiplicative representations of Daletsky - Trotter, its approximate solution is constructed. Under classical conditions on the coefficients, there is a single to the stochastic equivalence of solutions of the stochastic equation, which is a random process. This development generates an evolutionary family of resolving operators by the formula x(t)= S(t, Construct the division of the segment by the points. An equation with time-uniform coefficients is considered on each elementary segment . There is a single solution of this equation on the elementary segment, which generates the resolving operator by the formula The multiplicative expression is constructed. Using the method of Dalecki-Trotter multiplicative representations, it is proved that this multiplicative expression is stochastically equivalent to the representation generated by the solution of the original equation. This means that the specified multiplicative expression is respectively a representation of the solution of the original equation. That is, the probability of one coincides with the solution of the original stochastic equation. It should be noted that this is possible under additional conditions for the coefficients of the equation. These conditions are the time continuity of the coefficients of the equation. Thus, the constructed multiplicative representation can be interpreted as an approximate solution of the original equation. This method of multiplicative approximation makes it possible to simplify the study of the corresponding random process both at the elementary segment and as a whole. It is known, that the solution of a stochastic equation by a known formula generates a solution of the inverse Kolmogorov equation in the corresponding space. This scheme of multiplicative approximation can be transferred to the solution of the parabolic equation, which is the inverse Kolmogorov equation. Thus, the method of multiplicative approximation makes it possible to simplify the study of both stochastic equations and partial differential equations.

On construction of a field of forces along given trajectories in the presence of random perturbations

In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.

Bayesian Testing for Exogenous Partition Structures in Stochastic Block Models

Network data often exhibit block structures characterized by clusters of nodes with similar patterns of edge formation. When such relational data are complemented by additional information on exogenous node partitions, these sources of knowledge are typically included in the model to supervise the cluster assignment mechanism or to improve inference on edge probabilities. Although these solutions are routinely implemented, there is a lack of formal approaches to test if a given external node partition is in line with the endogenous clustering structure encoding stochastic equivalence patterns among the nodes in the network. To fill this gap, we develop a formal Bayesian testing procedure which relies on the calculation of the Bayes factor between a stochastic block model with known grouping structure defined by the exogenous node partition and an infinite relational model that allows the endogenous clustering configurations to be unknown, random and fully revealed by the block–connectivity patterns in the network. A simple Markov chain Monte Carlo method for computing the Bayes factor and quantifying uncertainty in the endogenous groups is proposed. This strategy is evaluated in simulations, and in applications studying brain networks of Alzheimer’s patients.