246,069 research outputs found
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 and for the
fractal case, it remains at , 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 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 . 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
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