In this paper, we study crucial elements of a complex network, namely its nodes and connections, which play a key role in maintaining the network's structure and function under unexpected structural perturbations of nodes and edges removal. Specifically, we want to identify vital nodes and edges whose failure (either random or intentional) will break the most number of connected triples (or triangles) in the network. This problem is extremely important because connected triples form the foundation of strong connections in many real-world systems, such as mutual relationships in social networks, reliable data transmission in communication networks, and stable routing strategies in mobile networks. Disconnected triples, analog to broken mutual connections, can greatly affect the network's structure and disrupt its normal function, which can further lead to the corruption of the entire system. The analysis of such crucial elements will shed light on key factors behind the resilience and robustness of many complex systems in practice. We formulate the analysis under multiple optimization problems and show their intractability. We next propose efficient approximation algorithms, namely DAK-n and DAK-e, which guarantee an $(1-1/e)$-approximate ratio (compared to the overall optimal solutions) while having the same time complexity as the best triangle counting and listing algorithm on power-law networks. This advantage makes our algorithms scale extremely well even for very large networks. In an application perspective, we perform comprehensive experiments on real social traces with millions of nodes and billions of edges. These empirical experiments indicate that our approaches achieve comparably better results while are up to 100x faster than current state-of-the-art methods.