  Communication networks, power grids, and transportation networks are all
examples of networks whose performance depends on reliable connectivity of
their underlying network components even in the presence of usual network
dynamics due to mobility, node or edge failures, and varying traffic loads.
Percolation theory quantifies the threshold value of a local control parameter
such as a node occupation (resp., deletion) probability or an edge activation
(resp., removal) probability above (resp., below) which there exists a giant
connected component (GCC), a connected component comprising of a number of
occupied nodes and active edges whose size is proportional to the size of the
network itself. Any pair of occupied nodes in the GCC is connected via at least
one path comprised of active edges and occupied nodes. The mere existence of
the GCC itself does not guarantee that the long-range connectivity would be
robust, e.g., to random link or node failures due to network dynamics. In this
paper, we explore new percolation thresholds that guarantee not only spanning
network connectivity, but also robustness. We define and analyze four measures
of robust network connectivity, explore their interrelationships, and
numerically evaluate the respective robust percolation thresholds for the 2D
square lattice.
