Networks naturally appear in many high-impact applications. The simplest model of networks is single-layered networks, where the nodes are from the same domain and the links are of the same type. However, as the world is highly coupled, nodes from different application domains tend to be interdependent on each other, forming a more complex network model called multi-layered networks.

Among the various aspects of network studies, network connectivity plays an important role in a myriad of applications. The diversified application areas have spurred numerous connectivity measures, each designed for some specific tasks. Although effective in their own fields, none of the connectivity measures is generally applicable to all the tasks. Moreover, existing connectivity measures are predominantly based on single-layered networks, with few attempts made on multi-layered networks.

Most connectivity analyzing methods assume that the input network is static and accurate, which is not realistic in many applications. As real-world networks are evolving, their connectivity scores would vary by time as well, making it imperative to keep track of those changing parameters in a timely manner. Furthermore, as the observed links in the input network may be inaccurate due to noise and incomplete data sources, it is crucial to infer a more accurate network structure to better approximate its connectivity scores.

The ultimate goal of connectivity studies is to optimize the connectivity scores via manipulating the network structures. For most complex measures, the hardness of the optimization problem still remains unknown. Meanwhile, current optimization methods are mainly ad-hoc solutions for specific types of connectivity measures on single-layered networks. No optimization framework has ever been proposed to tackle a wider range of connectivity measures on complex networks.

In this thesis, an in-depth study of connectivity measures, inference, and optimization problems will be proposed. Specifically, a unified connectivity measure model will be introduced to unveil the commonality among existing connectivity measures. For the connectivity inference aspect, an effective network inference method and connectivity tracking framework will be described. Last, a generalized optimization framework will be built to address the connectivity minimization/maximization problems on both single-layered and multi-layered networks.