  Complex networks have recently attracted much interest due to their
prevalence in nature and our daily lives [1, 2]. A critical property of a
network is its resilience to random breakdown and failure [3-6], typically
studied as a percolation problem [7-9] or by modeling cascading failures
[10-12]. Many complex systems, from power grids and the Internet to the brain
and society [13-15], can be modeled using modular networks comprised of small,
densely connected groups of nodes [16, 17]. These modules often overlap, with
network elements belonging to multiple modules [18, 19]. Yet existing work on
robustness has not considered the role of overlapping, modular structure. Here
we study the robustness of these systems to the failure of elements. We show
analytically and empirically that it is possible for the modules themselves to
become uncoupled or non-overlapping well before the network disintegrates. If
overlapping modular organization plays a role in overall functionality,
networks may be far more vulnerable than predicted by conventional percolation
theory.
