Modeling Overlapping Communities with Node Popularities
We develop a probabilistic approach for accurate network modeling using node popularities within the framework of the mixed-membership stochastic blockmodel (MMSB). Our model integrates two basic properties of nodes in social networks: homophily and preferential connection to popular nodes. We develop a scalable algorithm for posterior inference, based on a novel nonconjugate variant of stochastic variational inference. We evaluate the link prediction accuracy of our algorithm on nine real-world networks with up to 60,000 nodes, and on simulated networks with degree distributions that follow a power law. We demonstrate that the AMP predicts significantly better than the MMSB.
This paper proposes a new generative model and 
      associated link inference method based on both node popularity and 
      similarity. The starting point for the model is the prior work in [11] 
      where the assortative mixed-membership stochastic blockmodel (AMMSB) 
      was presented. In the prior model, link structure is generated via 
      community strength (via a blockmodel) and community membership. In the 
      new work, link structure is generated by using the prior model and 
      adding "popularity" to the generative model. After the model 
      is presented, the authors then derive an optimization criterion based 
      upon a variational method (since exact inference is impossible). The 
      resulting criterion is optimized using a stochastic gradient ascent 
      technique. The new model is then applied to real networks and the 
      "standard" LFR benchmarks for performance evaluation. Precision/recall 
      curves show that the new model is better at low-recall/high-precision 
      operating points. Additionally, Figure 5b shows a strong relation 
      between node popularity and node degree. Quality The 
      paper is technically sound and the paper has extensive detail on 
      assumptions, methods, and algorithms. The results should be readily 
      reproducible. Clarity The paper could use improvement 
      in the area of clarity. There is a strong reliance on past work for 
      notation and assumptions which make this paper difficult to be read 
      independently. A few comments: - Some terms are not defined when they 
      are first mentioned. E.g., (z_{a->b}, z_(b->a)) is called a 
      community assignment (line 090). A simple mention that this is a 
      vector (or function on i=1,..,K) assuming values 0 or 1 would be 
      helpful. Also, \phi_{ab}^{ij) (line 160) is never really defined. 
      - What is the relation between community strength, beta, in the new 
      work and in [11]? In the current work, beta is normal, but in the [11] 
      it is a Beta distribution. Originality This paper has 
      a good result that builds off of previous work in [11]. A comparison 
      with [11] shows similarity in approach and methodology. The methods 
      are new, but they certainly build off of prior work. 
      Significance Results presented by the authors are significant. 
      Methods to generate and fit networks for link prediction continue to 
      be a key task since many real world networks are not observable. The 
      fact that the algorithm can produce high precision links (albeit with 
      low recall) is of interest in some applications. A drawback of the 
      methods, possibly for future work, is to continue to increase the 
      scalability of the method to larger networks.