General fixed-mixing spectral minimax characterization

Characterize the fixed-mixing minimax complexity for general graph–matrix pair classes with independently prescribed node count, graph diameter, and inverse spectral-gap upper bound, under an algorithm class that specifies how a designated mixing matrix may be used.

Background

The paper distinguishes diameter-based minimax complexity, which is defined over graph classes, from spectral-gap-based complexity, which requires graph–matrix pair classes and an explicit restriction on how algorithms access the designated mixing matrix. It formulates a possible fixed-mixing minimax model using the class of graph–matrix pairs with prescribed parameters (n, diameter, and inverse spectral-gap bound) and a single uniformly specified algorithmic template whose cross-agent aggregation is restricted to applications of the designated matrix.

The paper establishes lower bounds only for coupled weighted-path specializations, where the inverse-gap exponent is determined by the path construction. It explicitly leaves unresolved the characterization for general, independently prescribed parameters and emphasizes that informative nonempty pair classes and an appropriate algorithmic interface are necessary.