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1. Learning coherent clusters in weakly‑connected network systems

   Min, Hancheng; Mallada, Enrique (July 2023, Proceedings of The 5th Annual Learning for Dynamics and Control Conference)

   We propose a structure-preserving model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. We provide an upper bound on the approximation error when the network graph is randomly generated from a weight stochastic block model. Finally, numerical experiments align with and validate our theoretical findings. 

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2. Exact Community Recovery in Correlated Stochastic Block Models

   Gaudio, Julia; Racz, Miklos Z.; Sridhar, Anirudh (July 2022, Proceedings of Machine Learning Research)

   We consider the problem of learning latent community structure from multiple correlated networks. We study edge-correlated stochastic block models with two balanced communities, focusing on the regime where the average degree is logarithmic in the number of vertices. Our main result derives the precise information-theoretic threshold for exact community recovery using multiple correlated graphs. This threshold captures the interplay between the community recovery and graph matching tasks. In particular, we uncover and characterize a region of the parameter space where exact community recovery is possible using multiple correlated graphs, even though (1) this is information-theoretically impossible using a single graph and (2) exact graph matching is also information-theoretically impossible. In this regime, we develop a novel algorithm that carefully synthesizes algorithms from the community recovery and graph matching literatures. 

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3. The decomposition of the higher-order homology embedding constructed from the k-Laplacian

   Chen, Yu-Chia; Meila, Marina (December 2021, Advances in neural information processing systems)

   M. Ranzato; A. Beygelzimer; Y. Dauphin; P.S. Liang; J. Wortman Vaughan (Ed.)

   The null space of the k-th order Laplacian Lk, known as the {\em k-th homology vector space}, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian L0 has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the k-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em connected sum} of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components. The proposed framework is applied to the {\em shortest homologous loop detection} problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images. 

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4. Optimal Bayesian estimation for random dot product graphs

   https://doi.org/10.1093/biomet/asaa031  

   Xie, Fangzheng; Xu, Yanxun (July 2020, Biometrika)

   Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data. 

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5. Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise

   https://doi.org/10.1093/imaiai/iaae026  

   Cheng, Xiuyuan; Landa, Boris (September 2024, Information and Inference: A Journal of the IMA)

   Abstract Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn–Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when $$n$$ data points are i.i.d. sampled from a general $$d$$-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of $$n \to \infty $$ and kernel bandwidth $$\epsilon \to 0$$, the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be $$ O( n^{-1/(d/2+3)})$$ at finite large $$n$$ up to log factors, achieved at the scaling of $$\epsilon \sim n^{-1/(d/2+3)} $$. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise. 

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