  Building resilience into today's complex infrastructures is critical to the
daily functioning of society and its ability to withstand and recover from
natural disasters, epidemics, and cyber-threats. This study proposes
quantitative measures that implement the definition of engineering resilience
advanced by the National Academy of Sciences. The approach is applicable across
physical, information, and social domains. It evaluates the critical
functionality, defined as a performance function of time set by the
stakeholders. Critical functionality is a source of valuable information, such
as the integrated system resilience over a time interval, and its robustness.
The paper demonstrates the formulation on two classes of models: 1) multi-level
directed acyclic graphs, and 2) interdependent coupled networks. For both
models synthetic case studies are used to explore trends. For the first class,
the approach is also applied to the Linux operating system. Results indicate
that desired resilience and robustness levels are achievable by trading off
different design parameters, such as redundancy, node recovery time, and backup
supply available. The nonlinear relationship between network parameters and
resilience levels confirms the utility of the proposed approach, which is of
benefit to analysts and designers of complex systems and networks.
