The behavior of real systems is often stochastic, and the connections between their elements can be adequately described as correlations. In recent years, there have been trends of increasing and complicating modern networks with the growth of their dependence on each other. We observe how several networks are combined into one interdependent network structure. This leads to an increase in the risks that the failure of nodes in one network may lead to the failure of dependent nodes in other networks. As a result of such failures, catastrophic cascade failures can occur in such interconnected network structures. Given the scale of such structures, which are often critical infrastructures, this problem becomes very relevant. The chapter considers the problem of the influence of correlation between individual nodes on the risk of cascading failures in networks. However, determining the correlation between networks is a difficult task in practice. Therefore, for the risk analysis of real network structures, it is first necessary to conduct a study on models, and only then move on to solving practical problems. Applied to Gaussian model network structures, the influence of the closeness of the relationship between subsystems on the risk of cascading failures has been studied. The value of the risk was estimated as the probability of such failures. As an indicator of the risk of cascading failures in the network structure, it is proposed to use the entropy indicator of the relationship between its subsystems. And to reduce the risk of cascading failures in the network structure, it is necessary to reduce the tightness of correlation between the most interconnected elements of subsystems.