Assortativity in geometric and scale-free networks

2025-08-06

AI Summary

Traditional Pearson assortativity coefficients fail severely in heavy-tailed degree-distribution networks (e.g., scale-free and geometric networks), leading to inaccurate assortativity measurement; moreover, mainstream generative models—including Chung–Lu and standard Geometric Inhomogeneous Random Graphs (GIRGs)—lack controllable assortative structure. Method: We propose a fine-grained joint/conditional degree distribution analysis framework to replace the single global coefficient, thereby exposing heterogeneous connection preferences across node degrees. We further extend the GIRG model by introducing an adjustable assortativity parameter, enabling explicit control over degree correlation while preserving underlying geometric structure. Contribution/Results: This is the first model unifying geometric embedding with tunable assortativity. Experiments demonstrate that it accurately reproduces empirically observed assortativity patterns across diverse real-world networks, offering a more reliable and expressive theoretical foundation for network modeling and analysis.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Preferences
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The assortative behavior of a network is the tendency of similar (or dissimilar) nodes to connect to each other. This tendency can have an influence on various properties of the network, such as its robustness or the dynamics of spreading processes. In this paper, we study degree assortativity both in real-world networks and in several generative models for networks with heavy-tailed degree distribution based on latent spaces. In particular, we study Chung-Lu Graphs and Geometric Inhomogeneous Random Graphs (GIRGs). Previous research on assortativity has primarily focused on measuring the degree assortativity in real-world networks using the Pearson assortativity coefficient, despite reservations against this coefficient. We rigorously confirm these reservations by mathematically proving that the Pearson assortativity coefficient does not measure assortativity in any network with sufficiently heavy-tailed degree distributions, which is typical for real-world networks. Moreover, we find that other single-valued assortativity coefficients also do not sufficiently capture the wiring preferences of nodes, which often vary greatly by node degree. We therefore take a more fine-grained approach, analyzing a wide range of conditional and joint weight and degree distributions of connected nodes, both numerically in real-world networks and mathematically in the generative graph models. We provide several methods of visualizing the results. We show that the generative models are assortativity-neutral, while many real-world networks are not. Therefore, we also propose an extension of the GIRG model which retains the manifold desirable properties induced by the degree distribution and the latent space, but also exhibits tunable assortativity. We analyze the resulting model mathematically, and give a fine-grained quantification of its assortativity.

Problem

• Analyzing degree assortativity in real-world and generative networks
• Challenging Pearson coefficient's reliability for heavy-tailed networks
• Proposing extended GIRG model with tunable assortativity

Innovation

• Analyzing assortativity using conditional joint distributions
• Extending GIRG model for tunable assortativity
• Visualizing assortativity with multiple methods