Emergence of robust and efficient networks in a family of attachment models

Self-organization of robust and efficient networks is important for the future designs of communication or transportation systems, because both characteristics are not coexisting in many real networks. As one of the candidates for the coexisting, the optimal robustness of onion-like structure with positive degree–degree correlations has recently been found, and it can be generated by incrementally growing methods based on a pair of random and intermediation attachments with the minimum degree selection. In this paper, we introduce a continuous interpolation by a parameter β≥0 between random and the minimum degree attachments to investigate the reason why the minimum degree selection is important. However, we find that the special case of the minimum degree attachment can generate highly robust networks but with low efficiency as a chain structure. Furthermore, we consider two intermediation models modified with the inverse preferential attachment for investigating the effect of distance on the emergence of robust onion-like structure. The inverse preferential attachments in a class of mixed attachment and two intermediation models are effective for the emergence of robust onion-like structure. However, a small amount of random attachment is necessary for the network efficiency, when β is large enough. Such attachment models indicate a prospective direction to the future growth of our network infrastructures.