The continuous-time adaptive susceptible-infected-susceptible (ASIS) epidemic model and the adaptive information diffusion (AID) model are two adaptive spreading processes on networks, in which a link in the network changes depending on the infectious state of its end nodes, but in opposite ways: (i) In the ASIS model a link is removed between two nodes if exactly one of the nodes is infected to suppress the epidemic, while a link is created in the AID model to speed up the information diffusion; (ii) a link is created between two susceptible nodes in the ASIS model to strengthen the healthy part of the network, while a link is broken in the AID model due to the lack of interest in informationless nodes. The ASIS and AID models may be considered as first-order models for cascades in real-world networks. While the ASIS model has been exploited in the literature, we show that the AID model is realistic by obtaining a good fit with Facebook data. Contrary to the common belief and intuition for such similar models, we show that the ASIS and AID models exhibit different but not opposite properties. Most remarkably, a unique metastable state always exists in the ASIS model, while there an hourglass-shaped region of instability in the AID model. Moreover, the epidemic threshold is a linear function in the effective link-breaking rate in the AID model, while it is almost constant but noisy in the AID model. Over the past decade, many real-world networks have been characterized via graph metrics [1-3] such as clustering, assortativity, modularity, degree distribution, and spectral properties. Recently, robustness characteristics and complex dependences have been analyzed in networks of networks [4], while a parallel track in network science has focused on relatively simple dynamic processes on networks such as epidemics [5,6], synchronization [7], and opinion diffusion [8][9][10]. In most studies so far, the networks are considered fixed or independent of the dynamic process. After the seminal work of Gross et al. [11], the coupling between epidemic processes and the underlying network topology has been extensively studied [12][13][14][15].The coupling between process and topology is natural in many cases. In epidemics [16], for example, after the observation of an infectious relative, one may either avoid him or her (by changing the social contact network) or increase the protection against the virus (without altering the topology). In human brain networks, Hebbian learning alters the connectivity between brain regions that are trained or neurally excited. Although self-adaptation naturally occurs in biology, adaptive networks, in which the process interacts with the topology, are unfortunately difficult to analyze and we barely understand the interplay between process and topology. Twitter measurements [17,18] show that the topology of the network adaptively changes connectivity towards users with high popularity and the ordinary users tend to directly follow the popular ones to get the information faster, instead of a...