  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.
