# Probability distributions of the hyperdegree in the KSPH model on a logarithmic scale with the different parameters .

The node number = 1000, the size of local-world = 15, and the mean value = 2. (a) When , the hyperdegree distribution obeys the stretched exponential distribution, as , where is a constant and is the characteristic exponent. If , it is a normal exponential distribution. Using log() as *x*-axis and log(−log) as *y*-axis, if the corresponding curve can be well fitted by a straight line, then the slope equals . In the inset, we plot the linear correlation between log(−log) and log. (b) When , for big values of , the hyperdegree distribution follows a power-law distribution and the exponent is approximately equal to 2.75. (c) When , for big values of , the hyperdegree distribution follows a power-law distribution and the exponent is approximately equal to 2.68. (d) When , the hyperdegree distribution follows a power-law distribution and the exponent is approximately equal to 2.37. Each simulation result is obtained by averaging over 100 independent runs.