Consistency of Spectral Clustering on Hierarchical Stochastic Block Models

04/30/2020
by   Lihua Lei, et al.
0

We propose a generic network model, based on the Stochastic Block Model, to study the hierarchy of communities in real-world networks, under which the connection probabilities are structured in a binary tree. Under the network model, we show that the eigenstructure of the expected unnormalized graph Laplacian reveals the community structure of the network as well as the hierarchy of communities in a recursive fashion. Inspired by the nice property of the population eigenstructure, we develop a recursive bi-partitioning algorithm that divides the network into two communities based on the Fiedler vector of the unnormalized graph Laplacian and repeats the split until a stopping rule indicates no further community structures. We prove the weak and strong consistency of our algorithm for sparse networks with the expected node degree in O(log n) order, based on newly developed theory on ℓ_2→∞ eigenspace perturbation, without knowing the total number of communities in advance. Unlike most of existing work, our theory covers multi-scale networks where the connection probabilities may differ in order of magnitude, which comprise an important class of models that are practically relevant but technically challenging to deal with. Finally we demonstrate the performance of our algorithm on synthetic data and real-world examples.

READ FULL TEXT
research
10/02/2018

Hierarchical community detection by recursive bi-partitioning

The problem of community detection in networks is usually formulated as ...
research
09/04/2018

Determining the Number of Communities in Degree-corrected Stochastic Block Models

We propose to estimate the number of communities in degree-corrected sto...
research
06/01/2023

When Does Bottom-up Beat Top-down in Hierarchical Community Detection?

Hierarchical clustering of networks consists in finding a tree of commun...
research
01/12/2022

Multiple Hypothesis Testing To Estimate The Number of Communities in Sparse Stochastic Block Models

Network-based clustering methods frequently require the number of commun...
research
06/15/2022

Sparse Subspace Clustering in Diverse Multiplex Network Model

The paper considers the DIverse MultiPLEx (DIMPLE) network model, introd...
research
04/21/2021

A class of network models recoverable by spectral clustering

Finding communities in networks is a problem that remains difficult, in ...
research
07/11/2017

Unsupervised robust nonparametric learning of hidden community properties

We consider learning of fundamental properties of communities in large n...

Please sign up or login with your details

Forgot password? Click here to reset