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A single protein molecule is regarded as a contact network of amino-acidresidues. Some studies have indicated that this network is a small worldnetwork (SWN), while other results have implied that this is a fractal network(FN). However, SWN and FN are essentially different in the dependence of theshortest path length on the number of nodes. In this paper, we investigate thisdependence in the residue contact networks of proteins in native structures,and show that the networks are not SWN but FN. FN is generally characterized byseveral dimensions. Among them, we focus on three dimensions; the networktopological dimension $D_c$, the fractal dimension $D_f$, and the spectraldimension $D_s$. We find that proteins universally yield $D_c \approx 1.9$,$D_f \approx 2.5$ and $Ds \approx 1.3$. These values are in surprisingly goodcoincidence with those in three dimensional critical percolation cluster. Hencethe residue contact networks in the protein native structures belong to theuniversality class of three dimensional percolation cluster. The criticality isrelevant to the ambivalent nature of the protein native structures, i.e., thecoexistence of stability and instability, both of which are necessary for aprotein to function as a molecular machine or an allosteric enzyme.

Residue network in protein native structure belongs to the universality class of three dimensional critical percolation cluster