13 research outputs found
Scalable and Robust Community Detection with Randomized Sketching
This paper explores and analyzes the unsupervised clustering of large
partially observed graphs. We propose a scalable and provable randomized
framework for clustering graphs generated from the stochastic block model. The
clustering is first applied to a sub-matrix of the graph's adjacency matrix
associated with a reduced graph sketch constructed using random sampling. Then,
the clusters of the full graph are inferred based on the clusters extracted
from the sketch using a correlation-based retrieval step. Uniform random node
sampling is shown to improve the computational complexity over clustering of
the full graph when the cluster sizes are balanced. A new random degree-based
node sampling algorithm is presented which significantly improves upon the
performance of the clustering algorithm even when clusters are unbalanced. This
algorithm improves the phase transitions for matrix-decomposition-based
clustering with regard to computational complexity and minimum cluster size,
which are shown to be nearly dimension-free in the low inter-cluster
connectivity regime. A third sampling technique is shown to improve balance by
randomly sampling nodes based on spatial distribution. We provide analysis and
numerical results using a convex clustering algorithm based on matrix
completion
Mean Field Analysis of Personalized PageRank with Implications for Local Graph Clustering
We analyse a mean-field model of Personalized PageRank (PPR) on the Erdős–Rényi (ER) random graph containing a denser planted ER subgraph. We investigate the regimes where the values of PPR concentrate around the mean-field value. We also study the optimization of the damping factor, the only parameter in PPR. Our theoretical results help to understand the applicability of PPR and its limitations for local graph clustering
Inference in the Stochastic Block Model with a Markovian assignment of the communities
We tackle the community detection problem in the Stochastic Block Model (SBM)
when the communities of the nodes of the graph are assigned with a Markovian
dynamic. To recover the partition of the nodes, we adapt the relaxed K-means
SDP program presented in [11]. We identify the relevant signal-to-noise ratio
(SNR) in our framework and we prove that the misclassification error decays
exponentially fast with respect to this SNR. We provide infinity norm
consistent estimation of the parameters of our model and we discuss our results
through the prism of classical degree regimes of the SBMs' literature. MSC 2010
subject classifications: Primary 68Q32; secondary 68R10, 90C35
Partial recovery bounds for clustering with the relaxed means
We investigate the clustering performances of the relaxed means in the
setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM).
After identifying the appropriate signal-to-noise ratio (SNR), we prove that
the misclassification error decay exponentially fast with respect to this SNR.
These partial recovery bounds for the relaxed means improve upon results
currently known in the sGMM setting. In the SBM setting, applying the relaxed
means SDP allows to handle general connection probabilities whereas other
SDPs investigated in the literature are restricted to the assortative case
(where within group probabilities are larger than between group probabilities).
Again, this partial recovery bound complements the state-of-the-art results.
All together, these results put forward the versatility of the relaxed
means.Comment: 39 page