We develop an empirical network model to characterize the density of bilateral sovereign credit risk spillovers during the European debt crisis. We show that the spillover density is often asymmetric with heavy tails and that its location and shape vary strongly and systematically in relation to published indicators of systemic stress. Using auxiliary panel data models, we show that the intensity of bilateral spillovers is related to the portfolio investment exposures among country pairs. Because our spillover statistics can be updated daily, they represent a valuable supplement to existing weekly and monthly measures of systemic stress.