## 1 Introduction: The Use of Simple Graphs in the Description of the Fundamental Entinties
* Continuously monitor and update the attack graph as new vulnerabilities are discovered or as the network configuration changes. By modeling threats using attack graphs, organizations can gain insights into potential attack scenarios, prioritize security efforts, and enhance the resilience of their networks against cyber threats. The computational complexity of using graphs in real-life applications can vary widely depending on the specific problem being addressed, the size and complexity of the graph, and the algorithms and techniques employed. Here are some considerations:  
1. Graph representation: The complexity of representing a graph in memory depends on the number of vertices (nodes) and edges, as well as the data structures used. For sparse graphs, where the number of edges is relatively low compared to the number of vertices, adjacency lists may be more efficient than adjacency matrices. 2. Graph traversal: Algorithms for traversing graphs, such as depth-first search (DFS) or breadth-first search (BFS), have time complexities of O(V + E), where V is the number of vertices, and E is the number of edges. Traversal can become computationally expensive for large graphs or graphs with dense connectivity. 3. Shortest paths: Finding the shortest path between two vertices in a graph, using algorithms like [PERSON]'s algorithm or the Bellman-Ford algorithm, typically has a time complexity of O((V + E)logV) or O(VE), depending on the algorithm used and whether negative edge weights are allowed. 4. Graph connectivity: Determining graph connectivity, such as whether the graph is strongly connected or whether there exists a path between every pair of vertices, can be computationally intensive. Algorithms like [PERSON]'s strongly connected components algorithm have a time complexity of O(V + E). 5. Graph coloring: Problems such as graph coloring, where vertices must be assigned colors such that no adjacent vertices share the same color, are known to be NP-hard. Finding an optimal coloring for an arbitrary graph can be computationally intractable, and heuristic approaches are often used in practice. 6. Maximum flow: Computing maximum flow in a graph, such as in network flow problems, can be solved efficiently using algorithms like the Ford-Fulkerson algorithm or the Edmonds-Karp algorithm, with a time complexity of O(V E2). 7. Graph clustering and community detection: Identifying clusters or communities within a graph involves partitioning the vertices into groups based on their connectivity patterns. Various algorithms exist for this task, such as the Louvain method or spectral clustering, with varying computational complexities depending on the approach. In summary, the computational complexity of using graphs in real-life applications can range from linear time complexity for simple traversal algorithms to exponential or NP-hard complexity for more complex problems such as graph coloring or finding optimal solutions. Efficient algorithms, careful problem formulation, and leveraging problem-specific properties of the graph are essential for managing computational complexity in practical applications. In previous works [1; 32; 33; 34; 35; 36], a theory for the description of the fundamental entities, named apriorics, was suggested and elaborated. The predictions of this theory were compatible with experimental findings in elementary particle physics. In this theory, the fundamental entities (FEs) are the vertices in a simple undirected graph [37; 38; 39; 40] called an ontological structure (OS), and they can be described by their internal connections defining the graph, without any needed reference to anything else. These nodes must satisfy the following five axioms (which are explained and justified in [32; 33]):  
1. Every node (vertex) must be connected to one other vertex (at least). 2.