We examined protein residue networks (PRNs) from a local search perspective to understand why PRNs are highly clustered when having short paths is important for protein functionality. We found that by adopting a local search perspective, this conflict between form and function is resolved as increased clustering actually helps to reduce path length in PRNs. Further, the paths found via our EDS local search algorithm are more congruent with the characteristics of intra-protein communication. EDS identifies a subset of PRN edges called short-cuts that are distinct, have high usage, impacts EDS path length, diversity and stretch, and are dominated by short-range contacts. The short-cuts form a network (SCN) that increases in size and transitivity as a protein folds. The structure of a SCN supports its function and formation, and the function of a SCN influences its formation. Several significant differences in terms of SCN structure, function and formation is found between successful and unsuccessful MD trajectories. By connecting the static and the dynamic aspects of PRNs, the protein folding process becomes a problem of graph formation with the purpose of forming suitable pathways within proteins.