Growing unlabeled networks

2025-09-21

AI Summary

Conventional network growth models rely on node arrival timestamps, rendering them inadequate for modeling the evolution of structurally equivalent nodes in unlabeled networks. This limitation is especially acute for unlabeled tree growth, where node identities and temporal labels are absent. Method: We propose a structural distinguishability–based growth framework for unlabeled trees, centering analysis on leaf-node statistics and systematically investigating how symmetry constrains dynamical evolution. We construct two analytically tractable unlabeled growth models—analogs of uniform and preferential attachment—and combine theoretical derivation with statistical tracking. Contribution/Results: We demonstrate that node symmetry markedly amplifies degree distribution heterogeneity, with the magnitude and direction of this effect critically dependent on the underlying growth mechanism. Our work establishes the first analytical framework for unlabeled tree growth, revealing symmetry as an intrinsic driver of structural differentiation. It introduces a novel paradigm and general analytical toolkit for modeling unlabeled complex networks.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Structured Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Models of growing networks are a central topic in network science. In these models, vertices are usually labeled by their arrival time, distinguishing even those node pairs whose structural roles are identical. In contrast, unlabeled networks encode only structure, so unlabeled growth rules must be defined in terms of structurally distinguishable outcomes; network symmetries therefore play a key role in unlabeled growth dynamics. Here, we introduce and study models of growing unlabeled trees, defined in analogy to widely-studied labeled growth models such as uniform and preferential attachment. We develop a theoretical formalism to analyze these trees via tracking their leaf-based statistics. We find that while many characteristics of labeled network growth are retained, numerous critical differences arise, caused primarily by symmetries among leaves in common neighborhoods. In particular, degree heterogeneity is enhanced, with the strength of this enhancement depending on details of growth dynamics: mild enhancement for uniform attachment, and extreme enhancement for preferential attachment. These results and the developed analytical formalism may be of interest beyond the setting of growing unlabeled trees.

Problem

• Modeling unlabeled network growth focusing on structural symmetries
• Analyzing differences between labeled and unlabeled tree growth dynamics
• Investigating enhanced degree heterogeneity in unlabeled network models

Innovation

• Developed models of growing unlabeled trees using structural symmetries
• Created analytical formalism tracking leaf-based statistics for analysis
• Enhanced degree heterogeneity through symmetry-aware growth dynamics