Community structure and scale-free collections of Erdös-Rényi graphs (1112.3644v1)

Abstract: Community structure plays a significant role in the analysis of social networks and similar graphs, yet this structure is little understood and not well captured by most models. We formally define a community to be a subgraph that is internally highly connected and has no deeper substructure. We use tools of combinatorics to show that any such community must contain a dense Erd\"os-R\'enyi (ER) subgraph. Based on mathematical arguments, we hypothesize that any graph with a heavy-tailed degree distribution and community structure must contain a scale free collection of dense ER subgraphs. These theoretical observations corroborate well with empirical evidence. From this, we propose the Block Two-Level Erd\"os-R\'enyi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks.

• The paper introduces the BTER model to capture dense ER subgraphs that form the foundational communities in scale-free networks.
• It employs a two-phase generative process that constructs ER subgraphs respecting heavy-tailed degree distributions and high clustering coefficients.
• Empirical comparisons show that BTER more accurately replicates real-world network metrics, outperforming models like Chung–Lu.

Community Structure and Scale-Free Collections of Erdős–Rényi Graphs

This paper provides a comprehensive mathematical analysis of community structures within social and similar interaction networks and introduces a new generative model called the Block Two-Level Erdős–Rényi (BTER) model. The authors, C. Seshadhri, Tamara G. Kolda, and Ali Pinar from Sandia National Laboratories, explore the dense structure of Erdős–Rényi (ER) subgraphs as fundamental building blocks of such networks, leading to a detailed discussion on their theoretical and practical implications.

Theoretical Insights

At the core of this paper is a mathematical characterization of community structures, which are defined as subgraphs that are internally well-connected and devoid of further substructure. This implies that communities exhibit a significant number of internal edges compared to those expected by standard graph models. The authors leverage combinatorial techniques to demonstrate that these communities contain dense ER subgraphs. Their hypothesis is that typical networks with heavy-tailed degree distributions inherently comprise a scale-free collection of such ER subgraphs.

The paper highlights an important theorem indicating that a constant fraction of the edges in a community are contained within a dense ER subgraph. This result is linked to the notion of modularity and high clustering coefficients observable in real-world networks. These outcomes are bolstered by empirical verification and suggest a structural framework where a network can be viewed as an ensemble of such dense ER components.

Block Two-Level Erdős–Rényi (BTER) Model

The BTER model is introduced as a novel generative model that effectively replicates the key properties of real-world networks, particularly the heavy-tailed degree distribution and high clustering coefficients. The model operates in two phases:

1. Phase 1: Constructs ER subgraphs while respecting a target degree distribution, thereby forming the basic community structures.
2. Phase 2: Connects these ER blocks using a configuration model that considers the excess degree of each node, leading to inter-community connections.

The model not only accommodates any degree distribution but is also scalable, offering advantages for simulating large-scale networks such as those required by algorithms like the Graph 500 Benchmark.

Experimental Results and Comparisons

The paper provides results from experiments that demonstrate the BTER model's efficacy in matching the properties of various real-world networks, such as co-authorship and citation networks. These results are juxtaposed with those derived from the Chung–Lu (CL) model, with BTER consistently achieving closer matches in clustering coefficients and eigenvalue distributions of adjacency matrices.

BTER's ability to emulate the intricate structure of real networks, including the formation of ER-like communities and scale-free topology, positions it as a potent tool for modeling and analyzing the structure and function of complex social networks.

Implications and Future Directions

The proposed model and theoretical insights contribute significantly to understanding the microstructure of networks, offering implications for community detection, network evolution, and dynamics. Furthermore, the BTER model's scalability and configurability make it apt for parallel computation, an essential aspect for processing massive-scale networks.

Future research may extend the framework to encompass more diverse interaction types and further explore overlapping or hierarchical community structures. The potential integration with other models addressing aspects such as temporal dynamics or node attributes could vastly enrich the understanding and predictive capabilities of network models.

In conclusion, this paper lays a foundational framework that rigorously interlinks Erdős–Rényi subgraphs with the complex community structures observed in social networks, presenting a robust model both in theory and application.