Connectivity Determination Algorithm for Complex Directed Networks

Connectivity characterizes the ability of information transmission in systems modeled by complex networks. It is essential to develop an efficient connectivity determination algorithm with low time complexity and minimal storage requirements. To fulfill this need, a connectivity determination algorithm is designed by incorporating Tarjan's algorithm to identify strongly connected components and leveraging a depth-first search idea to traverse the reachability. This algorithm can ascertain strong connectivity, unilateral connectivity, and weak connectivity of complex directed networks. Besides, the accessibility matrix of complex directed networks is computed and visualized through an interface. As this algorithm relies on only two depth-first searches to accomplish connectivity determination tasks, its computational complexity does not exceed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(n^{2})$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> denotes the number of network nodes. Experiments carried out on some specific networks reveal that the probability of network connections decreases with the increasing number of nodes in directed injective graphs, while in Erdős–Rényi graphs, the likelihood of connections increases as the number of nodes increases. Finally, a comparative example and an application example are provided to demonstrate the effectiveness of the algorithm program.