# Mathematical Approaches Transform Cybersecurity from Protoscience to Science
## 3 The Literature Review and Related Activities  
The cyber protection of complex integrated information systems and networks is becoming more complicated. Building labs and testbeds to research such scenarios is a high-profile challenge. There is a shortage of cybersecurity expertise. The lack of specialists profiled in cyber defense is particularly noticeable. These are the conditions in which the academic community conducts and publishes research.  
This part consists of a simple statistical analysis of related works. We note the achievements in the establishment if basic standard categorical apparatus with the National Vulnerabilities Data Base, Common Vulnerability Scoring System (CVSS) [46], MITRE ATT@CK taxonomy, STIX, TAXII, CyBOX, OpenIOC [47; 48; 49].  
We observed issues with the terminological discrepancy commented on in [15; 50; 51]. We considered how the developing science of cybersecurity needs a categorical apparatus that is complete, consistent, and correct. The next step was to consolidate a set of appropriate mathematical instruments that would be applicable to suitable cases. Mathematical approaches were our focus. Defining an adequate set of mathematical practices can power and drive intensive research.  
Part of the considered mathematical approaches was presented with applicable specifics in the reliability and availability engineering field in the book [52]. The analogy with its problematics gave rise to the proposed cyber security engineering category at the end of the section results and conclusions. This work presents the analytical and modeling techniques currently used. It contains standard approaches to nascent methods: binary decision diagrams, dynamic fault trees, Bayesian belief networks, stochastic Petri nets, non-homogeneous Markov chains, semi-Markov processes, and phase-type expansions [52].  
Traditional mathematical approaches to cybersecurity problems include elements of set theory, probability theory, statistics, and graph theory [53; 54; 55; 56; 57]. In recent years, researchers have applied ideas from artificial intelligence, neural networks, graphs, game theory, and Markov chains.  
The light research from Scopus with a search string \"cybersecurity_X\", where X is from the set [game theory, probability, Bayesian] shows [228; 41; 99] the sources respectively. The other sources contain applied approaches from Combinatorial mathematics, probabilities, and statistic [58], graphs theory [59], neural networks [60; 61], fuzzy logic and fuzzy sets [62; 63], and group theory in cryptography [64; 65; 66].  
The first preliminary research question was about the publications and authors' use of mathematical approaches. We received the relations between authors using the intersection with the search string \"mathematical model cybersecurity\" and \"mathematical model cyber security\" through Scopus services and VOSviewer. The results are presented in the tables in Figures 1 and 2. Five clusters are highlighted with different colors, in which the families of the leading authors are noted. Their research can be found through sections combining the top authors' surnames with the indicated search string.  
Figure 1: The authors applied mathematics to cybersecurity models.  
Additional proof for this work is a part of articles mentioning mathematical approaches in the cybersecurity area. The search in January 2022 only in Scopus with the terms \"cybersecurity\" and \"cyber security\" showed more than 20,000 results for every one of them. The number of articles with mathematical approaches was \(\sim\)400 for intersections presented in the whole set, which is about 2 percent.