SIAM Journal on Applied Mathematics, Volume 81, Issue 4, Page 1389-1415, January 2021.
Exponential family random graph models (ERGMs) are widely used to model networks by parameterizing graph probability in terms of a set of user-selected sufficient statistics. Equivalently, ERGMs can be viewed as expressing a probability distribution on graphs arising from the action of competing social forces that make ties more or less likely, depending on the state of the rest of the graph. Such forces often lead to a complex pattern of dependence among edges, with non-trivial large-scale structures emerging from relatively simple local mechanisms. While this provides a powerful tool for probing macro-micro connections, much remains to be understood about how local forces shape global outcomes. One very simple question of this type is that of the conditions needed for social forces to stabilize a particular structure: that is, given a specific structure and a set of alternatives (e.g., arising from small perturbations), under what conditions will said structure remain more probable than the alternatives? We refer to this property as local stability and seek a general means of identifying the set of parameters under which a target graph is locally stable with respect to a set of alternatives. Here, we provide a complete characterization of the region of the parameter space inducing local stability, showing it to be the interior of a convex cone whose faces can be derived from the change scores of the sufficient statistics vis-à-vis the alternative structures. As we show, local stability is a necessary but not sufficient condition for more general notions of stability, the latter of which can be explored more efficiently by using the “stable cone” within the parameter space as a starting point. In addition to facilitating the understanding of model behavior, we show how local stability can be used to determine whether a fitted model implies that an observed structure would be expected to arise primarily from the action of social forces, versus by merit of the model permitting a large number of high probability structures, of which the observed structure is one (i.e., entropic effects). We also use our approach to identify the dyads within a given structure that are the least stable, and hence predicted to have the highest probability of changing under the current social forces. The utility of the “stable cone” for ERGM parameter optimization is then demonstrated on a physical model of amyloid fibril formation. This demonstration features a visualization of the stable region, whereby it is shown that the majority of the region of the model's parameter space where ERGM simulations produce the highest fibril yield lies within the “stable cone.”

中文翻译：

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指数族随机图模型中的局部图稳定性

SIAM 应用数学杂志，第 81 卷，第 4 期，第 1389-1415 页，2021 年 1 月。
指数族随机图模型 (ERGM) 被广泛用于通过根据一组用户选择的足够统计量参数化图概率来对网络进行建模。等效地，ERGM 可以被视为表达图上的概率分布，这些分布是由社会竞争力量的作用引起的，这些社会力量或多或少地产生联系，这取决于图的其余部分的状态。这种力通常会导致边缘之间出现复杂的依赖模式，从相对简单的局部机制中产生非平凡的大规模结构。虽然这为探索宏观-微观联系提供了强大的工具，但关于局部力量如何塑造全球结果还有很多需要了解。这种类型的一个非常简单的问题是社会力量稳定特定结构所需的条件：即，给定一个特定的结构和一组替代方案（例如，由小扰动引起），在什么条件下，所述结构比替代方案更有可能？我们将此属性称为局部稳定性，并寻求一种通用方法来识别一组参数，在该组参数下，目标图相对于一组替代方案是局部稳定的。在这里，我们提供了引起局部稳定性的参数空间区域的完整表征，表明它是凸锥的内部，其面可以从相对于替代结构的足够统计的变化分数中得出。正如我们所展示的，局部稳定性是更一般稳定性概念的必要但不是充分条件，通过使用参数空间内的“稳定锥”作为起点，可以更有效地探索后者。除了促进对模型行为的理解之外，我们还展示了如何使用局部稳定性来确定拟合模型是否意味着观察到的结构主要来自社会力量的作用，而不是模型的优点允许大量高概率结构，其中观察到的结构是其中之一（即熵效应）。我们还使用我们的方法来识别给定结构中最不稳定的二元组，因此预计在当前社会力量下发生变化的可能性最高。然后在淀粉样原纤维形成的物理模型上证明了“稳定锥”用于 ERGM 参数优化的效用。