Efficiently Sampling Multiplicative Attribute Graphs Using a Ball-Dropping Process

We introduce a novel and efficient sampling algorithm for the Multiplicative Attribute Graph Model (MAGM - Kim and Leskovec (2010)}). Our algorithm is \emph{strictly} more efficient than the algorithm proposed by Yun and Vishwanathan (2012), in the sense that our method extends the \emph{best} time complexity guarantee of their algorithm to a larger fraction of parameter space. Both in theory and in empirical evaluation on sparse graphs, our new algorithm outperforms the previous one. To design our algorithm, we first define a stochastic \emph{ball-dropping process} (BDP). Although a special case of this process was introduced as an efficient approximate sampling algorithm for the Kronecker Product Graph Model (KPGM - Leskovec et al. (2010)}), neither \emph{why} such an approximation works nor \emph{what} is the actual distribution this process is sampling from has been addressed so far to the best of our knowledge. Our rigorous treatment of the BDP enables us to clarify the rational behind a BDP approximation of KPGM, and design an efficient sampling algorithm for the MAGM.