Breakdown of Modularity in Complex Networks

The presence of modular organization is a common property of a wide range of complex systems, from cellular or brain networks to technological graphs. Modularity allows some degree of segregation between different parts of the network and has been suggested to be a prerequisite for the evolvability of biological systems. In technology, modularity defines a clear division of tasks and it is an explicit design target. However, many natural and artificial systems experience a breakdown in their modular pattern of connections, which has been associated with failures in hub nodes or the activation of global stress responses. In spite of its importance, no general theory of the breakdown of modularity and its implications has been advanced yet. Here we propose a new, simple model of network landscape where it is possible to exhaustively characterize the breakdown of modularity in a well-defined way. Specifically, by considering the space of minimal Boolean feed-forward networks implementing the 256 Boolean functions with 3 inputs, we were able to relate functional characteristics with the breakdown of modularity. We found that evolution cannot reach maximally modular networks under the presence of functional and cost constraints, implying the breakdown of modularity is an adaptive feature.