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.

翻译：我们引入Pólya阈值图模型，并推导其随机与代数性质。该随机阈值图通过双色Pólya罐过程依次生成。从空图开始，每一步从罐中抽取一个指示变量，决定新添加节点是通用节点（与所有既有节点及自身相连）还是孤立节点（不与任何既有节点相连）。此构造生成的随机阈值图，其邻接矩阵可基于抽取序列显式表示。利用Pólya抽取过程的结构，我们推导出任意节点的精确度分布，包括其均值与方差。此外，我们评估了基于距离的衰减中心性得分，并给出其期望的显式表达式。在代数方面，我们显式刻画了随机阈值图的拉普拉斯矩阵，获得了其谱与相应特征基的闭式描述。最后，作为这些结构结果的应用，我们分析了Pólya阈值图上的离散时间共识动力学。