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Outliers and Influential Observations in Exponential Random Graph Models


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.

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    Ties were interactions that were (1) limited to radical organisation activities; (2) extend beyond radical organisations to include such categories as co-workers and roommates; (3) those that would die for each other. Further detail may be found in Rhodes and Jones (2009) who use a different version of the network.


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Correspondence to Johan Koskinen.

Additional information

Johan Koskinen would like to acknowledge financial support from the Leverhulme Trust Grant RPG-2013-140 and SRG2012.

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Koskinen, J., Wang, P., Robins, G. et al. Outliers and Influential Observations in Exponential Random Graph Models. Psychometrika 83, 809–830 (2018).

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  • statistical analysis of social networks
  • exponential random graph models
  • outliers
  • leverage
  • missing data principle
  • case deletion