The main aim of this paper is to compare the stability, in terms of systemic risk, of conventional and Islamic banking systems. To this aim, we propose correlation network models for stock market returns based on graphical Gaussian distributions, which allows us to capture the contagion effects that move along countries. We also consider Bayesian graphical models, to account for model uncertainty in the measurement of financial systems interconnectedness. Our proposed model is applied to the Middle East and North Africa (MENA) region banking sector, characterized by the presence of both conventional and Islamic banks, for the period from 2007 to the beginning of 2014. Our empirical findings show that there are differences in the systemic risk and stability of the two banking systems during crisis times. In addition, the differences are subject to country specific effects that are amplified during crisis period.