The analysis of the network structure of the functional connectivity data constructed from fMRI images provides basic information about functions and features of the brain activity. We focus on the two features which are considered as relevant to the brain activity, the criticality and the constraint regarding energy consumptions. Within a wide variety of complex systems, the critical state occurs associated with a phase transition between distinct phases, random one and order one. Although the hypothesis that human brain activity is also in a state of criticality is supported by some experimental results, it still remains controversial. One issue is that experimental distributions exhibit deviations from the power law predicted by the criticality. Based on the assumption that constraints on brain from the biological costs cause these deviations, we derive a distribution model. The evaluation using the information criteria indicates an advantage of this model in fitting to experimental data compared to other representative distribution models, the truncated power law and the power law. Our findings also suggest that the mechanism underlying this model is closely related to the cost effective behavior in human brain with maximizing the network efficiency for the given network cost.