Mobile Edge-Cloud Network is a new network structure after fog-cloud computing, where service and data computing are scattered in the most logical, nearby and efficient place. It provides better services than fog-cloud computing with better performance in reasonably low cost way and allows users to eliminate numerous limitations inherent in fog-cloud computing, although it inherits those security-privacy issues from fog-cloud computing. A novel privacy-preserving mutual authentication in TMIS for mobile Edge-Cloud architecture (abbreviated to NPMA) is constructed in this paper. NPMA scheme not only mitigates some weaknesses of fog-cloud computing, but has other advantages. First, NPMA scheme supports patients(edge-servers) anonymity and forward-backward untraceability (traceability, when needed), since their identities are hidden in two distinct dynamic anonyms and a static one and only the trusted center can recover their real identities, when needed. Second, each edge-server shares a secret value, which realizes authentication with extremely low computional cost in authentication phase. Finally, NPMA scheme is proven safely against passive and active attacks under elliptic curve computable Diffie-Hellman problem (ECDHP) assumption in random oracle model. Hence, it achieves the required security properties and outperforms prior approaches in terms of energy and computational costs.