The human brain is organized in functional modules. Such an organization presents a 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. In a recent study, we present a possible solution to this puzzle. We first show that critical percolation theory unambiguously defines a set of modules made of strong links in functional brain networks. Contrary to the common view, these modules are 'large-world' fractal structures and, therefore, are very far from being small-world. This means that information transfer in and out the modules is very inefficient; all distances are too far. However, incorporating the weak ties to the network below a critical percolation threshold 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 costs. This trade-off arquitecture is reminiscent of the 'strength of weak ties' crucial concept of social networks. Such a design provides a natural solution to the paradox of efficient information flow in the highly modular structure of the brain.