In everyday life, we often observe unusually frequent interactions among people before or during important events, e.g., we receive/send more greetings from/to our friends on Christmas Day, than usual. We also observe that some videos suddenly go viral through people's sharing in online social networks (OSNs). Do these seemingly different phenomena share a common structure? All these phenomena are associated with sudden surges of user activities in networks, which we call "bursts" in this work. We find that the emergence of a burst is accompanied with the formation of triangles in networks. This finding motivates us to propose a new method to detect bursts in OSNs. We first introduce a new measure, "triadic cardinality distribution", corresponding to the fractions of nodes with different numbers of triangles, i.e., triadic cardinalities, within a network. We demonstrate that this distribution changes when a burst occurs, and is naturally immunized against spamming social-bot attacks. Hence, by tracking triadic cardinality distributions, we can reliably detect bursts in OSNs. To avoid handling massive activity data generated by OSN users, we design an efficient sample-estimate solution to estimate the triadic cardinality distribution from sampled data. Extensive experiments conducted on real data demonstrate the usefulness of this triadic cardinality distribution and the effectiveness of our sample-estimate solution.