Model Selection for Complex Networks

The Akaike Information Criterion (AIC) is a powerful and highly principled method of model selection. When working with probabilistic models it is typical to fit them to the data by maximum likelihood, using Markov Chain Monte Carlo or Belief Propagation. The maximum likelihood value obtained in this way gives an indication of how good a fit the model is to the data. Unfortunately, models with more parameters tend to fit the data strictly better, simply because they have more degrees of freedom and not necessarily because they describe it better. The AIC counters this by taking the −`(θ̂) maximum likelihood value and adding a penalty term of k = |θ|. The best model according to the AIC is then the one that minimizes −`(θ̂) + k. This simple penalty has a powerful justification, but makes certain assumptions that pose major problems to models used to infer networks. The first of these assumptions, that all parameters in θ are continuous, can be remedied by a monte carlo prodcedure that marginalizes over these variables. The second, which assumes narrow peak around in the model’s likelihood, is normally easy to deal with given enough data to accurately estimate parameters. Unfortunately, highly correlated distributions like the Degree-Corrected Stochastic Block Model tend to violate this assumption. We attempt to create a variant of the TIC that treats pairwise group assignment marginals across edges as independent samples from the distribution over network edges.