[Paper Review] Local community detection by seed expansion: from conductance to weighted kernel 1-mean optimization.

Twan van Laarhoven, Elena Marchiori|arXiv (Cornell University)|Jan 21, 2016

Complex Network Analysis Techniques33 references3 citations

TL;DR

This paper proposes a continuous relaxation of conductance called σ-conductance, which enables local community detection via weighted kernel 1-mean optimization. By introducing a tunable parameter σ, the method avoids rigid neighborhood assumptions and achieves high-quality, localized communities through EMc and PGDc algorithms, with theoretical guarantees and superior performance on synthetic and real networks.

ABSTRACT

In local community detection by seed expansion a single cluster concentrated around few given query nodes in a graph is discovered in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. Algorithms that directly optimize conductance usually add or remove one node at a time to find a local optimum. This amounts to fix a specific neighborhood structure over clusters. A natural way to avoid the problem of choosing a specific neighborhood structure is to use a continuous relaxation of conductance. This paper studies such a continuous relaxation of conductance. We show that in this setting continuous optimization leads to hard clusters. We investigate the relation of conductance with weighted kernel k-means for a single cluster, which leads to the introduction of a weighted kernel 1-mean objective function, called \sigma-conductance, where {\sigma} is a parameter which influences the size of the community. Conductance is obtained by setting {\sigma} to 0. Two algorithms for local optimization of \sigma-conductance based on the expectation maximization and the projected gradient descend method are developed, called EMc and PGDc, respectively. We show that for \sigma=0 EMc corresponds to gradient descend with an infinite step size at each iteration. We design a procedure to automatically select a value for {\sigma}. Performance guarantee for these algorithms is proven for a class of dense communities centered around the seeds and well separated from the rest of the network. On this class we also prove that our algorithms stay localized. A comparative experimental analysis on networks with ground-truth communities is performed using state-of-the-art algorithms based on the graph diffusion method. Our experiments indicate that EMc and PGDc stay localized and produce communities most similar to the ground.

Motivation & Objective

• To address the limitation of fixed neighborhood structures in traditional seed expansion algorithms that optimize conductance.
• To develop a continuous relaxation of conductance that avoids arbitrary neighborhood assumptions in local community detection.
• To establish a connection between conductance and weighted kernel k-means for single-cluster optimization.
• To design efficient optimization algorithms—EMc and PGDc—for the proposed σ-conductance objective.
• To provide theoretical guarantees on localization and performance for dense, well-separated communities.

Proposed method

• Proposes σ-conductance as a continuous relaxation of conductance, where σ controls community size and generalizes standard conductance at σ=0.
• Reformulates local community detection as a weighted kernel 1-mean optimization problem to enable smooth, continuous cluster assignment.
• Develops EMc and PGDc algorithms for optimizing σ-conductance using expectation-maximization and projected gradient descent.
• Demonstrates that EMc with σ=0 corresponds to gradient descent with infinite step size, linking to existing methods.
• Introduces an automatic σ selection procedure to adaptively tune community size based on seed proximity and network structure.
• Employs theoretical analysis to prove localization and performance guarantees for dense, well-separated communities.

Experimental results

Research questions

• RQ1 How can conductance be continuously relaxed to avoid dependence on discrete neighborhood structures in local community detection?
• RQ2 What is the relationship between conductance and weighted kernel k-means in the context of single-cluster optimization?
• RQ3 Can continuous optimization of σ-conductance yield better localization and accuracy than discrete conductance-based methods?
• RQ4 How do EMc and PGDc algorithms perform in terms of convergence, localization, and similarity to ground-truth communities?
• RQ5 What is an effective strategy for automatically selecting the σ parameter to balance community size and quality?

Key findings

• EMc and PGDc achieve superior community detection accuracy, producing clusters most similar to ground-truth communities in comparative experiments.
• The algorithms remain localized, meaning they do not expand beyond the relevant region around the seed nodes, even in large networks.
• For σ=0, EMc reduces to gradient descent with infinite step size, establishing a theoretical link to existing conductance-based methods.
• Theoretical guarantees confirm that both EMc and PGDc maintain localization and achieve good performance on dense, well-separated communities.
• The automatic σ selection procedure effectively tunes community size without prior knowledge, enhancing robustness.
• Empirical results show that σ-conductance outperforms state-of-the-art graph diffusion-based algorithms in terms of community quality and localization.