Fractality in complex networks: Critical and supercritical skeletons

摘要: Fractal scaling\char22{}a power-law behavior of the number boxes needed to tile a given network with respect lateral size box\char22{}is studied. We introduce box-covering algorithm that is modified version original introduced by Song et al. [Nature (London) 433, 392 (2005)]; this enables easy implementation. networks are viewed as comprising skeleton and shortcuts. The skeleton, embedded underneath network, special type spanning tree based on edge betweenness centrality; it provides scaffold for fractality network. When regarded branching tree, exhibits plateau in mean function distance from root. For nonfractal networks, other hand, decays zero without forming plateau. Based these observations, we construct fractal model combining random local can be either critical or supercritical, depending small worldness constructed (supercritical) average vertices within box grows according (an exponential) form cluster-growing method. supercritical skeletons observed protein interaction World Wide Web, respectively. distribution masses, i.e., each box, follows power law ${P}_{m}(M)\ensuremath{\sim}{M}^{\ensuremath{-}\ensuremath{\eta}}$. exponent $\ensuremath{\eta}$ depends ${\ensuremath{\ell}}_{B}$. values ${\ensuremath{\ell}}_{B}$, equal degree $\ensuremath{\gamma}$ scale-free whereas approaches $\ensuremath{\tau}=\ensuremath{\gamma}∕(\ensuremath{\gamma}\ensuremath{-}1)$ ${\ensuremath{\ell}}_{B}$ increases, which cluster-size tree. Finally, study perimeter ${H}_{\ensuremath{\alpha}}$ $\ensuremath{\alpha}$, edges connected different $\ensuremath{\alpha}$ mass ${M}_{B,\ensuremath{\alpha}}$. It obtained over ${M}_{B}$ likely scale $⟨H({M}_{B})⟩\ensuremath{\sim}{M}_{B}$, irrespective