Parameter Recovery in Erds-Rnyi Graphs via Maximum Likelihood Estimations

This study investigates the feasibility of using maximum likelihood estimation to recover the connection probability parameter in the Erdos Renyi graph model. In this model every possible edge between nodes is assumed to appear with the same probability, providing a simplified mathematical description of network structure. By deriving the probability model for the occurrence of edges in a random graph and then develop an estimation formula based on the maximum likelihood principle. A series of simulations are conducted on graphs with a fixed number of nodes and various connection probabilities. The connection probability is estimated by comparing the actual number of observed edges with the maximum number of possible edges. The simulation results indicate that when the connection probability is high the network tends to be dense and the estimated value closely matches the true value. In contrast when the connection probability is low the estimated value is more affected by random fluctuations although it still reflects the basic characteristics of the network. The findings provide theoretical support for parameter estimation in both social and computer networks and lay the foundation for further studies on parameter recovery in more complex network models.