Social networks often contain dense and overlapping connections that obscure their essential interaction patterns, making analysis and interpretation challenging. Identifying the structural backbone of such networks is crucial for understanding community organization, information flow, and functional relationships. This study introduces a multi-step network pruning framework that leverages principles from information theory to balance structural complexity and task-relevant information. The framework iteratively evaluates and removes edges from the graph based on their contribution to task-relevant mutual information, producing a trajectory of network simplification that preserves most of the inherent semantics. Motivated by gradient boosting, we propose IGPrune, which enables efficient, differentiable optimization to progressively uncover semantically meaningful connections. Extensive experiments on social and biological networks show that IGPrune retains critical structural and functional patterns. Beyond quantitative performance, the pruned networks reveal interpretable backbones, highlighting the method's potential to support scientific discovery and actionable insights in real-world networks.