Stochastic blockmodels and community structure in networks (1008.3926v1)

Abstract: Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly distort the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

• The paper introduces the degree-corrected stochastic blockmodel to improve community detection by integrating vertex-specific degree parameters.
• It demonstrates that the DC-SBM outperforms traditional SBMs in both synthetic and real-world networks, including the Karate Club and political blog networks.
• The study provides exact maximum likelihood estimates for model parameters and discusses future strategies for optimizing community group determination.

Stochastic Blockmodels and Community Structure in Networks

In their seminal work, Karrer and Newman present a significant analytical extension of stochastic blockmodels (SBMs) to include degree heterogeneity, thereby addressing an important limitation of conventional SBMs in community detection tasks. The model proposed, referred to as the degree-corrected stochastic blockmodel (DC-SBM), enhances the ability to detect community structures in networks with broad degree distributions, which are characteristic of many real-world networks.

Introduction to Stochastic Blockmodels

Stochastic blockmodels serve as a foundational framework for modeling networks with community structure. In the traditional SBM, each vertex in a network belongs to one of $K$ groups, and edges between vertices are determined by group memberships according to a probability matrix $\boldsymbol{\psi}$. While SBMs are analytically tractable and versatile, they inherently assume a homogeneous degree distribution within groups. This assumption leads to suboptimal performance when applied to real-world networks where vertex degree distributions are typically heterogeneous.

Degree-Corrected Stochastic Blockmodel

To address this limitation, the authors introduce a degree-corrected variant of the SBM. The DC-SBM integrates vertex-specific degree parameters, thereby allowing the model to account for variations in vertex degree while maintaining the generative nature of the blockmodel. This is achieved by defining the expected number of edges between vertices $i$ and $j$ as $\theta_i \theta_j \omega_{g_i g_j}$, where $\theta_i$ and $\theta_j$ are parameters related to the degrees of vertices $i$ and $j$, and $\omega_{g_i g_j}$ reflects the inter-group edge probabilities.

The authors derive the maximum likelihood estimates for the parameters $\theta_i$ and $\omega_{g_i g_j}$, which provide an exact formulation of the DC-SBM in a closed form. This allows for efficient computation and fitting of the model to empirical data.

Empirical Evaluation

The performance of the DC-SBM is contrasted against the traditional SBM through applications to both synthetic and real-world networks. Key findings include:

1. Karate Club Network: The DC-SBM successfully identifies the known factions in Zachary's karate club network, while the traditional SBM fails by incorrectly partitioning vertices based on degree rather than community affiliation.
2. Political Blog Network: The DC-SBM demonstrates a stronger alignment with the actual political segmentation (liberal vs. conservative) than the uncorrected model, showcasing its robustness in networks with significant degree disparities.

Synthetic Network Benchmarks

The authors further validate their model using synthetic networks with known structures. Networks are generated using a DC-SBM, ensuring the planted structures test the models effectively. Different structural patterns—core-periphery, hierarchical, and simple community—are explored. The DC-SBM consistently outperforms the traditional SBM, accurately uncovering the planted community structures even when initial conditions are randomized.

Theoretical Implications and Future Work

The introduction of the DC-SBM has substantial implications for the theoretical understanding of community detection in heterogeneous networks. By incorporating degree correction, the DC-SBM mitigates biases associated with degree-related edge formation, thus providing a more accurate representation of the underlying structure.

However, the DC-SBM is not without its limitations. These include possible overrepresentation of zero-degree vertices and challenges in maintaining consistent statistical properties when scaling network sizes. Moreover, the model's complexity scales with network size, posing difficulties in extending fits to different-sized networks or in scenarios with unbounded group numbers.

To advance the field further, the authors suggest exploring advanced techniques for determining the optimal number of groups $K$. Potential strategies include cross-validation, minimum description length methods, and nonparametric Bayesian approaches. Additionally, applying the DC-SBM framework to more sophisticated models, such as those allowing for overlapping communities or mixed memberships, could provide valuable insights and enhanced performance in diverse network scenarios.

Conclusion

Karrer and Newman's work on the degree-corrected stochastic blockmodel marks a critical advance in community detection methodologies. By accounting for degree heterogeneity, the DC-SBM significantly improves the detection of community structures in real-world networks, addressing a key limitation of traditional SBMs. Future research will undoubtedly build on these foundations, further refining our understanding and modeling of complex network structures.