Disassortativity of random critical branching trees

Random critical branching trees (CBTs) are generated by the multiplicative branching process, where the branching number is determined stochastically, independent of the degree of their ancestor. Here we show analytically that despite this stochastic independence, there exists the degree-degree correlation (DDC) in the CBT and it is disassortative. Moreover, the skeletons of fractal networks, the maximum spanning trees formed by the edge betweenness centrality, behave similarly to the CBT in the DDC. This analytic solution and observation support the argument that the fractal scaling in complex networks originates from the disassortativity in the DDC.Comment: 3 pages, 2 figure

Spectral dimensions of hierarchical scale-free networks with shortcuts

The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractals or non-fractals depending on the weight of shortcuts. Applying the renormalization group (RG) approach to the Gaussian model, we obtain the spectral dimension exactly. While the spectral dimension varies between $1$ and $2$ for the fractal case, it remains at $2$, independent of the variation of network structure for the non-fractal case. The crossover behavior between the two cases is studied through the RG flow analysis. The analytic results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.Comment: 10 pages, 3 figure

Finite-size scaling theory for explosive percolation transitions

The finite-size scaling (FSS) theory for continuous phase transitions has been useful in determining the critical behavior from the size dependent behaviors of thermodynamic quantities. When the phase transition is discontinuous, however, FSS approach has not been well established yet. Here, we develop a FSS theory for the explosive percolation transition arising in the Erd\H{o}s and R\'enyi model under the Achlioptas process. A scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point $t_c$ diverges with system size in a power-law manner, which is different from the conventional one based on the divergence of the correlation length at $t_c$. We show that the susceptibility is also described in the same scaling form. Numerical simulation data for different system sizes are well collapsed on the respective scaling functions.Comment: 5 pages, 5 figure