This paper aims to build novel measures of systemic risk that take the multivariate nature of the problem into account by means of network models. To account for model uncertainty, we also employ a Bayesian approach, which allows model averaging over different network classes. The resulting systemic risk measure, which we call NetMES, is applied to the evaluation of the financial stability of the banking system in the Gulf Cooperation Council countries. Banks are classified as fully-fledged Islamic banks, conventional banks or hybrids: conventional banks with an Islamic window. The empirical findings indicate the presence of a difference between the two banking systems in terms of systemic risk, which can be explained by different levels of capitalization and leverage.