Most hubs' binding sites are able to transiently interact with numerous proteins. We focus on β-rich hubs with the goal of inferring features toward design. Since they are able to interact with many partners and association of β-conformations may lead to amyloid fibrils, we ask whether there is some property that distinguishes them from low-connectivity β-rich proteins, which may be more interaction specific. Identification of such features should be useful as they can be incorporated in interface design while avoiding polymerization into fibrils. We classify the proteins in the yeast interaction map according to the types of their secondary structures. The small number of the obtained β-rich protein structures in the Protein Data Bank likely reflects their low occurrence in the proteome. Analysis of the obtained structures indicates that highly connected β-rich proteins tend to have clusters of conserved residues in their cores, unlike β-rich structures with low connectivity, suggesting that the highly packed conserved cores are important to the stability of proteins, which have residue composition and sequence prone to β-structure and amyloid formation. The enhanced stability may hinder partial unfolding, which, depending on the conditions, is more likely to lead to polymerization of these sequences.