<p>In many decision-making systems, relationship among entities are best represented using directed graphs. They capture asymmetric connections among the entities where influence flows in one direction rather than being reciprocated. In such systems, decisions are significantly impacted by variations in the directed edges, as they determine how influence propagates through the entities of the considered network. This study focuses on developing fusion functions that incorporate these directional influences represented through directed edges into the aggregation process, ensuring that the final outcome reflects the structural dependencies and varying levels of influence within the system. The first step is to analyze how different entities in a network are connected and how influence flows through these directed links. Then, to model the aggregation framework, we propose a fusion function well adapted to incorporate the directional nature of influence. This function ensures that aggregation respects the asymmetric relationships in the network, effectively capturing the impact of influence propagation on the final decision. We also extend our approach by developing a pre-aggregation version of the directed graph-based fusion functions. The pre-aggregation function is designed to incorporate the non-symmetric relationships among entities, where aggregation increment is done in a particular direction. This generalization is adaptable for various decision-making systems with different types of relationships. In such aggregation systems, certain properties of fusion functions such as idempotency and averaging behavior, etc. are often desirable. These properties are incorporated into the proposed framework to ensure its usability across diverse decision-making contexts.</p>