  Self-organization of robust and efficient networks is important for a future
design 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 $\beta\geq 0$ 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,
when $\beta$ is large enough, a small amount of random attachment is necessary
for the network efficiency. Such attachment models indicate a prospective
direction to the future growth of our network infrastructures.
