A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided.
翻译:使用 PageRank 和 Max-Linear 模型作为随机图中节点影响指数的预测尾部和末端指数的网络演化演变。 尾端指数显示分布尾的重度。 极端指数是随机过程的组合( 或局部依赖性) 。 集群意味着过程在足够高的阈值上的一系列连续超值。 我们最近关于经常变化随机变量的非静止随机序列的数值和最大随机序列的峰值和最大值的结果, 扩展到随机图。 从一组连接的固定种子群开始, 作为热点, 按其尾部指数排列, 并排列其尾部指数的尾部和极限指数 。 这个程序可以让我们预测尾部和末端指数的时联网演变。 极端指数决定限制PageRank 和 Max- Linear 模式中新连接节点的最大值的分布。 该解析由算和示例提供。 为了验证我们的理论结果, 我们关于线性偏好的网络的模拟和真实数据研究, 以工具的形式提供。