Information-communication technology provides an increasingly accurate picture of human interactions, allowing a detailed mapping of the underlying network structures that mediate contagion processes. In social contagion, characteristic for spreading of innovations or rumours, transmission is a collective phenomenon where all peers of an individual may be involved. Threshold models of social contagion on unweighted, static, single-layer networks predict large-scale cascades of adoption only for relatively sparse networks. Empirical social networks, however, indicate that individuals can maintain hundreds of ties varying in time and strength across a range of social contexts yet they exhibit frequent system-wide cascades of social contagion. We address this issue through a serious of studies [1-4] incorporating some of the most relevant features of empirical social networks into a conventional threshold model [1]. First, we consider that network ties appear with non-uniform weights and show that the time of cascade emergence depends non-monotonously on weight heterogeneities, which accelerate or decelerate the dynamics [2]. Further we consider edge "types" modelled as weighted multiplex structures and demonstrate that multiplexity can lead to global cascades in networks with average degree in the hundreds or thousands, perturbed only by a single initial adoption [3]. As a novel observation, we also show that in a multiplex network with increasing link density a sequence of re-entrant phase transitions occurs, resulting in alternating phases of stability and instability to global cascades. Finally, we address how time-varying interactions with bursty dynamics can lead to the acceleration and deceleration of threshold driven dynamics [4].