A network min-system is an abstract model comprising a set of min-equations, each representing the values of network links along a path. While these systems have significant practical applications, the current theoretical solutions provide estimates for variable bounds that are often too loose. This paper aims to refine these bounds using additional data derived from link correlations. Specifically, we explore two types of link correlations: fairness constraints and total capacity constraints among a node's adjacent links. We theoretically demonstrate how these correlations can enhance the performance bounds of network min-systems. Our approach systematically addresses two primary challenges in tightening these bounds: the effects of synchronous versus asynchronous updates and the cascading effect. Experimental results in the context of bandwidth tomography indicate that our algorithms advance the current state-of-the-art by significantly tightening the performance bounds of network min-systems.