國立清華大學 數學系

Abstract： The edge-triangle exponential random graph model has been a topic of continued research interest. We review recent developments in the study of this classic model and concentrate on the phenomenon of phase transitions. We first describe the asymptotic feature of the model along general straight lines. We show that as we continuously vary the slopes of these lines, a typical graph exhibits quantized behavior, jumping from one complete multipartite structure to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. We then turn to exponential models where certain constraints are imposed and capture another interesting type of jump discontinuity. Joint work with Alessandro Rinaldo (Carnegie Mellon University), Sukhada Fadnavis (Harvard University), and with Richard Kenyon (Brown University). Tea Time: 10:30AM, R707