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Pólya Thresholds Graphs

Jinghan Yu, Fady Alajaji, Bahman Gharesifard

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.

Pólya Thresholds Graphs

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.
Paper Structure (11 sections, 15 theorems, 86 equations, 5 figures)

This paper contains 11 sections, 15 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pólya urn model
  4. Random threshold graphs
  5. Pólya threshold graph model and properties
  6. Application to consensus
  7. Consensus limit
  8. Simulation results
  9. Pólya threshold graph
  10. Finite-memory Pólya threshold graph
  11. Conclusion

Key Result

Lemma 1

A graph $\mathbf{G}_n=(\mathbf{V}_n,\mathbf{E}_n)$ is a threshold graph if and only if there exists a binary $n$-tuple $z^n= (z_1,\ldots, z_n)$, where $z_t \in\{0,1\}$ for all $t\in \{1,\ldots,n\}$, such that $\mathbf{G}_n$ can be constructed sequentially, up to node relabeling, using $z^n$ from an

Figures (5)

  • Figure 1: A threshold graph of size $5$.
  • Figure 2: Consensus trajectories for one realization of a Pólya threshold graph with $n=10$, urn parameters $(R,B,\Delta)=(5,5,2)$, and draw vector $z^{10}=(0,0,1,0,1,1,0,1,0,1)$. The initial opinions are fixed as $\bar{x}(0)=(0.1,0.6,0.3,1,0.5,3,10,2,9,0.2)^\intercal$. Dashed curves and white nodes correspond to isolated nodes; solid curves and dark nodes correspond to universal nodes.
  • Figure 3: Histogram of consensus values at time $t=100$ over $200$ independent simulations with $n=10$ and urn parameters $(R,B,\Delta)=(5,5,2)$, with $z_{10}$ fixed to be $1$. The initial opinion vector is fixed as $\bar{x}(0)=(0.1,0.6,0.3,1,0.5,3,10,2,9,0.2)^\intercal$. The dashed vertical line indicates the sample mean of the consensus values, and the solid vertical line indicates the theoretical prediction $\pi^{(E)} \bar{x}(0)$.
  • Figure 4: Simulation results for a Pólya threshold graph with $n =100$, urn parameters $(R,B,\Delta)=(5,5,2)$, and draw vector $z^{100}$ (fixed $z_{100}=1$).
  • Figure 5: $\pi^{(E)}\bar{x}(0)$ versus memory length $M$ for $n=10$ and $n=100$ (solid: finite-memory; dashed: infinite-memory).

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:thresholdgraph']}
  • Example 1
  • Lemma 2
  • Definition 3: Beta Function
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 27 more