Structural Analysis of Sequential Total Network Interactions: Connectivity and Proximity Perspectives
Rosalio G. Artes Jr. *
Mindanao State University - Tawi-Tawi College of Technology and Oceanography, Sanga-Sanga, 7500 Bongao, Tawi-Tawi, Philippines.
Roxanne A. Anunciado
Philippine Science High School SOCCSKSARGEN Region Campus, 9506 Koronadal City, South Cotabato, Philippines.
Radana E. Madalim
Mindanao State University - Tawi-Tawi College of Technology and Oceanography, Sanga-Sanga, 7500 Bongao, Tawi-Tawi, Philippines.
Jeffrey Imer C. Salim
Mindanao State University - Tawi-Tawi College of Technology and Oceanography, Sanga-Sanga, 7500 Bongao, Tawi-Tawi, Philippines.
Bayah J. Amiruddin-Rajik
Mindanao State University - Tawi-Tawi College of Technology and Oceanography, Sanga-Sanga, 7500 Bongao, Tawi-Tawi, Philippines.
*Author to whom correspondence should be addressed.
Abstract
This manuscript examines a network structure obtained through sequential total interactions among an independent sequence of finite networks. The study is framed within simple and undirected network theory, where nodes, links, adjacency, connectedness, independent sets, induced subnetworks, connectivity, and proximity are defined in terms of the underlying node-set and link-set. The construction joins each network in the sequence to its immediate successor by adding links between all nodes of two consecutive networks while retaining the internal links of each component network. From this definition, the manuscript describes the order and size of the resulting network in terms of the orders and sizes of the component networks. The main results determine the connectivity of each node according to its position within the sequence. Nodes in the first and last networks gain additional connections only from the adjacent network, whereas nodes in intermediate networks gain additional connections from both neighbouring networks. The manuscript also establishes proximity results for pairs of nodes. Nodes belonging to different component networks have proximity determined by the distance between their network indices, while non-adjacent nodes within the same component network have proximity two in the resulting structure. These results provide a concise structural description of connectivity and pairwise proximity in sequential total network interactions.
Keywords: Sequential total network interactions, finite networks, independent networks, connectivity, proximity, geodetic distance, graph join, node adjacency, induced subnetwork, shortest path, network structure