Summary: Network analysis is becoming a fundamental tool in the study of systemic risk and financial contagion in the banking sector. Still, the network structure must typically be estimated from noisy and aggregated data, as micro data on the status quo banking network structure are often unavailable, or the true network is unobservable. Graphical models can help researchers to infer network structures, but they are often criticized for relying too heavily on unrealistic assumptions. They also tend to yield dense structures that are difficult to interpret. Here, we propose the tlasso model for estimating sparse banking networks. The tlasso captures the conditional dependence structure between banks through partial correlations, and estimates sparse networks in which only the relevant links are identified. The model also accounts for the non-Gaussianity of financial data and it is robust to outliers and model misspecification. Our empirical analysis focuses on estimating the dependence structure of a sample of European banks from credit default swap data. We observe that the presence of communities in the banking network plays an important role in terms of systemic risk and contagion dynamics. We also introduce a decomposition of strength centrality that allows us to better characterize the role of each bank in the network and to identify the most relevant channels for the transmission of financial distress.