Pólya Thresholds Graphs

2026-03-18

AI Summary

This work proposes a novel dynamic stochastic threshold graph model based on a two-color Pólya urn process, which explicitly controls graph structure by sequentially generating nodes and determining—via urn draws—whether each new node becomes fully connected or isolated. For the first time, the Pólya urn mechanism is integrated into threshold graph generation, enabling rigorous analysis through the synthesis of random graph theory and algebraic graph theory. The authors derive exact closed-form expressions for the mean and variance of the degree distribution, the expected distance-decay centrality, and the Laplacian spectrum together with its eigenbasis. These analytical results are successfully applied to discrete-time consensus dynamics, revealing an intrinsic connection between structural evolution and network dynamical behavior.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce the Pólya threshold graph model and derive its stochastic and algebraic properties. This random threshold graph is generated sequentially via a two-color Pólya urn process. Starting from an empty graph, each time step involves a draw from the urn that produces an indicator variable, determining whether a newly added node is universal (connected to all existing nodes and itself) or isolated (connected to no existing nodes). This construction yields a random threshold graph with an adjacency matrix that admits an explicit representation in terms of the draw sequence. Using the structure of the Pólya draw process, we derive the exact degree distribution for any arbitrary node, including its mean and variance. Furthermore, we evaluate a distance-based decay centrality score and provide an explicit expression for its expectation. On the algebraic side, we explicitly characterize the Laplacian matrix of the random threshold graph, obtaining a closed-form description of its spectrum and corresponding eigenbasis. Finally, as an application of these structural results, we analyze discrete-time consensus dynamics on Pólya threshold graphs.

Problem

• Pólya urn
• threshold graphs
• random graphs
• Laplacian spectrum
• consensus dynamics

Innovation

• Pólya urn process
• threshold graphs
• Laplacian spectrum
• degree distribution
• consensus dynamics