The main purpose of this paper is to document a holistic modeling background and set up a corresponding mathematical theory in order to provide a rigorous description of cyber-attacks and cyber-security. The starting point is to determine the concepts of valuations and vulnerabilities of parts of a node constituent. Based on these two concepts, one may be led to consider the fundamental concept of node supervision and subsequently to give the definition of cyber-effects and from this the definition of cyber-interaction. As we shall see a germ of cyber-attack can be viewed as a family of cyber-interactions with coherence properties and depending strongly on subjective purposes, information and/or estimates on the valuations and the vulnerabilities of parts of the involved nodes. In general the germs of cyber-attacks can be distinguished in three types: the germs of correlated cyber-attacks, the germs of absolute cyber-attacks and the germs of partial cyber-attacks. This approach provides immediate possibility of rigorous determination of the concepts of proactive cyber defense and proactive cyber protection.
Mathematics Subject Classifications: 00A71; 68M11, 28A05, 28A12, 32C15
Keywords: Mathematical modeling (models of systems); internet topics; measure theory; complex spaces; valuation of a part of node constituent; vulnerability of a part of node constituent; node supervision, cyber-effect; cyber-interaction; germ of cyber attack; cyber defense; proactive cyber protection.