We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.

中文翻译：

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通过具有扩展颜色的 Pólya urn 生成优先附件图

我们引入了一种新颖的优先附件模型，使用修改后的 Pólya 瓮的绘制变量，颜色数量不断增加，特别是能够随着图的演变对有影响力的意见（就高度顶点而言）进行建模。与 Barabási-Albert 模型类似，生成的图在每个时间实例的大小都会增加一个顶点；然而，相比之下，图的每个顶点都有唯一的颜色特征，该颜色由 Pólya 瓮中的球颜色表示。更具体地说，在每个时间步骤，我们从瓮中取出一个球，并将其与许多相同颜色的增强球一起放回瓮中；我们还在瓮中添加了另一个新颜色的球。然后，我们在新顶点（对应于新颜色）和绘制色球的现有顶点之间构造一条边。将颜色编码的顶点与时变强化参数结合使用，可以让稍后添加（生成）的顶点有可能以 Barabási-Albert 模型中未捕获的方式获得较高的程度。我们通过分析底层随机过程的绘制向量来研究顶点的度数。特别是，我们建立了计算给定颜色的绘制次数的随机变量的概率分布，该随机变量决定了图中与该颜色相对应的顶点的度数。我们进一步提供模拟结果，展示我们的模型与 Barabási-Albert 网络之间的比较。