Analysis of networks that evolve dynamically requires the joint modelling of individual snapshots and time dynamics. This paper proposes a new flexible two-way heterogeneity model towards this goal. The new model equips each node of the network with two heterogeneity parameters, one to characterize the propensity to form ties with other nodes statically and the other to differentiate the tendency to retain existing ties over time. With $n$ observed networks each having $p$ nodes, we develop a new asymptotic theory for the maximum likelihood estimation of $2p$ parameters when $np\rightarrow \infty$. We overcome the global non-convexity of the negative log-likelihood function by the virtue of its local convexity, and propose a novel method of moment estimator as the initial value for a simple algorithm that leads to the consistent local maximum likelihood estimator (MLE). To establish the upper bounds for the estimation error of the MLE, we derive a new uniform deviation bound, which is of independent interest. The theory of the model and its usefulness are further supported by extensive simulation and a data analysis examining social interactions of ants.

翻译：分析动态演变的网络需要对个体快照和时间动态进行联合建模。本文针对这一目标提出了一种新型灵活的双向异质性模型。该模型为网络中的每个节点配备两个异质性参数：一个用于刻画节点静态连接其他节点的倾向性，另一个用于区分节点随时间保留现有连接的趋势。在观测到n个网络、每个网络包含p个节点的情形下，我们针对np→∞时的2p个参数的最大似然估计发展了新的渐近理论。我们利用负对数似然函数的局部凸性克服其全局非凸性，并提出一种新颖的矩估计法作为简单算法的初始值，该算法能够收敛到一致局部最大似然估计。为建立最大似然估计误差的上界，我们推导了新的均匀偏差界，这一结果具有独立的理论价值。通过广泛的模拟实验以及对蚂蚁社会互动的数据分析，进一步验证了模型的有效性和实用性。