We discuss measuring and detecting influential observations and outliers in the context of exponential family random graph (ERG) models for social networks. We focus on the level of the nodes of the network and consider those nodes whose removal would result in changes to the model as extreme or “central” with respect to the structural features that “matter”. We construe removal in terms of two case-deletion strategies: the tie-variables of an actor are assumed to be unobserved, or the node is removed resulting in the induced subgraph. We define the difference in inferred model resulting from case deletion from the perspective of information theory and difference in estimates, in both the natural and mean-value parameterisation, representing varying degrees of approximation. We arrive at several measures of influence and propose the use of two that do not require refitting of the model and lend themselves to routine application in the ERGM fitting procedure. MCMC p values are obtained for testing how extreme each node is with respect to the network structure. The influence measures are applied to two well-known data sets to illustrate the information they provide. From a network perspective, the proposed statistics offer an indication of which actors are most distinctive in the network structure, in terms of not abiding by the structural norms present across other actors.

This paper defines the ‘Case Deleted’ Deviance - a new objective function for evaluating Generalised Linear Models, and applies this to a number of practical examples in the pricing of general insurance. The paper details practical approximations to enable the efficient calculation of the objective, and derives modifications to the standard Generalised Linear Modelling algorithm to allow the derivation of scaled parameters from this measure to reduce potential over fitting to historical data. These scaled parameters improve the predictiveness of the model when applied to previously unseen data points, the most likely being related to future business written. The potential for over fitting has increased due to number of factors now used, particularly in pricing personal lines business and the advent of price comparison sites which has increased the penalties of mis-estimation. New material in this paper has been included in a UK patent application No. 1020091.3.