Communication structure plays a key role in the learning capability of decentralized systems. Structural self-adaptation, by means of self-organization, changes the order as well as the input information of the agents' collective decision-making. This paper studies the role of agents' repositioning on the same communication structure, i.e. a tree, as the means to expand the learning capacity in complex combinatorial optimization problems, for instance, load-balancing power demand to prevent blackouts or efficient utilization of bike sharing stations. The optimality of structural self-adaptations is rigorously studied by constructing a novel large-scale benchmark that consists of 4000 agents with synthetic and real-world data performing 4 million structural self-adaptations during which almost 320 billion learning messages are exchanged. Based on this benchmark dataset, 124 deterministic structural criteria, applied as learning meta-features, are systematically evaluated as well as two online structural self-adaptation strategies designed to expand learning capacity. Experimental evaluation identifies metrics that capture agents with influential information and their optimal positioning. Significant gain in learning performance is observed for the two strategies especially under low-performing initialization. Strikingly, the strategy that triggers structural self-adaptation in a more exploratory fashion is the most cost-effective.