Flow cores and the problem of vulnerability of complex network systems

In this article, the vulnerability of complex network systems (NSs) to the action of simultaneous group negative impacts is studied. Types of such impacts are analysed, among which targeted attacks and nontargeted disruptions of the system are primarily distinguished. They can have much in common both in terms of the methods of action and the consequences of damage, which allows the development of universal strategies for protecting and restoring NS. The structural model of the network system is considered, and to determine the k-core, a modified method of the pruning process is proposed, devoid of such shortcomings of the classical algorithm as the possibility of degeneracy into an empty set and sensitivity to the removal of elements. An effective scenario of a targeted attack on the k-core is developed, which takes into account the potential capabilities of the intruder. The flow model of the NS is studied, and a method for determining its flow core as the functionally most important component of the system is proposed. The flow core is used as a target for developing an effective scenario of a simultaneous group attack on the process of NS functioning, taking into account the attacker's capabilities. The implementation of attack scenarios on structural and flow cores of real-world network systems is compared, and it is shown that flow-based scenarios are more optimal than structural scenarios in terms of the number of attack targets. The use of these scenarios for successfully counteracting the spread of nontargeted disruption of the system is investigated. To assess the consequences of simultaneous group negative impacts, structural and flow-based approaches are used, and the flow-based approach allows for a much more accurate evaluation of these consequences, especially for disassortative networks. Methods for determining structural and flow aggregate networks and their cores for monoflow multilayer network systems are proposed. This makes it possible to significantly reduce the dimensionality of the vulnerability problem and simplify the development of strategies for protecting the structure and process of intersystem interactions from simultaneous negative group influences, reducing it to a corresponding problem for a single-layer network system.