Multifractal Measures on Small-World Networks

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. The multifractals can be obtained from the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D0 = 0.920 ~ 0.930 for the different network sizes and random rewiring fractions. In particular, the multifractal structure breaks down into a small-world network, when the rewiring fraction p is larger than the critical value pc = 0.3. Our simulation results are compared with the numerical computations for regular networks.