With the advent of mass higher education and the consequent absorption of significant national resources, both public and private, it is inevitable that universities are increasingly expected to meet a range of societal needs. They are expected to "connect" with society at large. In this paper, we argue that connectivity is best integrated with research, teaching and scholarship and should not be relegated to a "third stream". We compare degrees of connectivity of 50 national systems of higher education using ten indicators, making a distinction between domestic and international connectivity. The strongest finding is that smaller countries exhibit the highest level of international connectivity. The higher education systems in countries with large absolute numbers of researchers such as the USA, China and Japan are relatively self-contained compared with countries such as Ireland, Switzerland and Singapore. Another finding is the relative insularity of the education sector in Eastern Europe, including the Russian Federation. When differences in levels of economic development are allowed for, among lower-income countries South Africa stands out as having a well-connected higher education sector.