Epidemic spreading or information spreading in a network leads to a broader range of propagation due to the presence of bridge nodes, as defined by the betweenness centrality. The existing literature shows that computing betweenness centrality by considering connection topology is computationally expensive and less efficient. Moreover, exploiting the edge features in the network can determine how strongly the nodes are connected. This work proposes a technique to leverage connection topology and tie strength for modeling Weighted Betweenness Graph Neural Network (WBGNN) for a faster-ranking estimation. This measure is based on shortest path calculations, which take connection strength into account for determining the fastest path for contagion or information spread in the network. A variant of statistical weighted betweenness centrality, namely Stable Betweenness Centrality (C_{SB}) and the proposed GNN-based WBGNN are compared to analyze the effectiveness of the proposed model. Further, the WBGNN model is also compared with different machine learning regressors and a neural network-based model to examine the efficacy of the proposed model. The experimental outcome has revealed that the WBGNN model has achieved a 0.203 to 0.536 improvement in Kendall's \tau score, and also takes less time for node ranking estimation compared to C_{SB} and other machine learning-based regressor models.