Probabilistic Domination in Erdős–Rényi and Random Geometric Graphs: Threshold Analysis and Energy-Efficient Applications in Wireless Sensor Networks - Athraa Talib Breesam; Hussein Jameel Mutashar; Mustafa Adil Hussein; AlSeddiq Oday (Journal of Honai Math, 2026)
Wireless sensor networks (WSNs) require efficient strategies that preserve network coverage and connectivity while minimizing communication overhead and energy consumption. However, theoretical domination properties derived from random graph models are rarely integrated with finite-network simulations and practical network-lifetime measures. This study develops a probabilistic domination framework for Erdős–Rényi and random geometric graphs and applies the resulting dominating sets to cluster-head selection in WSNs. Analytical calculations were performed to estimate domination thresholds and dominating-set benchmarks for Erdős–Rényi graphs, whereas 10,000 Monte Carlo simulations were conducted for each parameter configuration to compare greedy and randomized selection algorithms. Performance was evaluated using dominating-set size, variance, coverage ratio, normalized energy cost, first node death, and last node death. The analytical results showed that the domination threshold decreased from 0.08 for a 50-node graph to 0.03 for a 200-node graph, while the relative dominating-set size declined from 24% to 15% of the network. For an Erdős–Rényi graph with 100 nodes and an edge probability of 0.30, the greedy algorithm reduced the mean dominating-set size by 4.8% and its variance by 38.2% compared with randomized selection. In the corresponding WSN scenario, greedy cluster-head selection increased coverage from 0.93 to 0.95, reduced normalized energy cost by 12%, and improved first node death and last node death by 12.5% and 7.1%, respectively. These findings demonstrate that probabilistic domination effectively connects random graph theory with energy-aware WSN design, although broader validation across diverse parameter settings and comprehensive energy models remains necessary.