An Algorithm Generating Scale Free Graphs

We propose a simple random process inducing various types of random graphs and the scale free random graphs among others. The model is of a threshold nature and differs from the preferential attachment approach discussed in the literature before. The degree statistics of a random graph in our model is governed by the control parameter $\eta$ stirring the pure exponential statistics for the degree distribution (at $\eta=0,$ when a threshold is changed each time a new edge added to the graph) to a power law (at $\eta=1$, when the threshold is frozen). The exponent $\gamma$ characterizing the power law can vary in the wide range $\gamma\in(1,\infty)$ and can be tuned in different values $\gamma_{\mathrm{in}}$ and $\gamma_{\mathrm{out}}$ for in-degrees and out-degrees probability distributions independently. For the intermediate values of $\eta$, the decay rate is mixed. Taking different statistics for the threshold changes, one obtains dissimilar asymptotic profiles for the degree distribution having, in general, nothing to do with power laws at $\eta=1$, but still uniformly exponential at $\eta=0.$