P\'olya Thresholds Graphs

TL;DR

This paper introduces the Pólya threshold graph model, analyzing its stochastic, algebraic properties, degree distribution, spectrum, and applications to consensus dynamics, providing explicit formulas and characterizations.

Contribution

It presents a novel Pólya urn-based threshold graph model with explicit structural, spectral, and dynamic properties derived analytically.

Findings

Exact degree distribution derived for any node

Closed-form spectrum of the Laplacian matrix obtained

Application to consensus dynamics analyzed

Abstract

We introduce the P\'olya threshold graph model and derive its stochastic and algebraic properties. This random threshold graph is generated sequentially via a two-color P\'olya 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 P\'olya 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 P\'olya threshold graphs.