We introduce random directed acyclic graph and use it to model the information diffusion network. Subsequently, we analyze the cascade generation model (CGM) introduced by [19]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.
One of the greatest issue in analyzing the cascade size distribution is the lack of a good theoretical background for this process. Recently, researchers @cite_16 has presented new models, designed to fit real distribution of cascade sizes. Moreover, goodness of fit test against CGM model has been conducted. It turns out, that CGM works well for small cascades, however it is unreliable for the large ones. The introduction of time dependent parameters significantly improves predictions for the large cascades @cite_16 .
@cite_16
1988927463
The probability distribution of the cascade generators in a random multiplicative cascade represents a hidden parameter which is reflected in the fine scale limiting behavior of the scaling exponents (sample moments) of a single sample cascade realization as a.s. constants. We identify a large class of cascade generators uniquely determined by these scaling exponents. For this class we provide both asymptotic consistency and confidence intervals for two different estimators of the cumulant generating function (log Laplace transform) of the cascade generator distribution. These results are derived from investigation of the convergence properties of the fine scale sample moments of a single cascade realization.