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We propose a scale-invariant derivative of the h-index as “h-dimension”, which is analogous to the fractal dimension of the h-index for institutional performance analysis. The design of h-dimension comes from the self-similar characteristics of the citation structure. We applied this h-dimension to data of 134 Japanese national universities and research institutes, and found well-performing medium-sized research institutes, where we identified multiple organizations related to natural disasters. This result is reasonable considering that Japan is frequently hit by earthquakes, typhoons, volcanoes and other natural disasters. However, these characteristic institutes are screened by larger universities if we depend on the existing h-index. The scale-invariant property of the proposed method helps to understand the nature of academic activities, which must promote fair and objective evaluation of research activities to maximize intellectual, and eventually economic opportunity.