  The human brain is organized in functional modules. Such an organization
presents a basic conundrum: modules ought to be sufficiently independent to
guarantee functional specialization and sufficiently connected to bind multiple
processors for efficient information transfer. It is commonly accepted that
small-world architecture of short lengths and large local clustering may solve
this problem. However, there is intrinsic tension between shortcuts generating
small-worlds and the persistence of modularity; a global property unrelated to
local clustering. Here, we present a possible solution to this puzzle. We first
show that a modified percolation theory can define a set of hierarchically
organized modules made of strong links in functional brain networks. These
modules are "large-world" self-similar structures and, therefore, are far from
being small-world. However, incorporating weaker ties to the network converts
it into a small-world preserving an underlying backbone of well-defined
modules. Remarkably, weak ties are precisely organized as predicted by theory
maximizing information transfer with minimal wiring cost. This trade-off
architecture is reminiscent of the "strength of weak ties" crucial concept of
social networks. Such a design suggests a natural solution to the paradox of
efficient information flow in the highly modular structure of the brain.
