A Uniform Methodology for Ranking Internet Topology Models

In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment, motivated by real-world graphs such as the Web or Internet topology. Typically these models are designed to mimic particular properties observed in the graphs, such as power-law degree distribution or the small-world phenomenon. The mainstream approach to comparing models for these graphs has been somewhat subjective and very application dependent. Comparisons are often based on specific graph properties, without adequate justification for prioritizing some properties over others. We propose to use the Maximum Likelihood Estimation (MLE) principle to compare graph models: models are scored by the probability with which they generate the real data. Our methodology has several advantages. It is uniform, in that its definition does not presuppose any information about the data or the models. It is unambiguous, in that it yields a clearly defined score for each model, and thus an ordering of models. Moreover, it can be used to determine the best values of the parameters for a given model. We demonstrate the feasibility of the approach by designing and implementing algorithms computing the probability for four natural models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We tested our algorithms on three different snapshots of the AS-level Internet topology. We found that the preferential attachment model performed the best, closely followed by the power-law model, with the other two models lagging behind. An interesting aspect of the findings is the fact that the optimal parameters for the power-law models have not changed significantly over time, even though the size of the data has grown by an order of magnitude.