The whole is greater than the sum of its parts. But can we discover the mechanisms leading to such observations? And if yes, can we take advantage of these mechanisms to exert some control over the whole? These questions are central in a plethora of contexts—ecology, neurology, epidemiology,
engineering, economics, etc.—which can be unified under the banner of"complex systems". In many cases, the challenge lies in determining how the interactions among the simple entities forming a complex system can give rise to the properties of the whole. Complex networks are a convenient way to represent these interactions, and the dynamical evolution of the system can be conceived as a stochastic process (i.e., a process involving randomness) taking place on a complex network.
Powerful analytical tools are available to study such processes, but these tools are not always adequate to address the challenges of modern society. Indeed, innovations are often required to model these problems within an acceptable level of quantitative and/or qualitative "realism": a recurring theme of my research is to perform such innovations. In this talk, I will use examples and motivations from epidemiology, self organization, and cascading failures to discuss different ways to "divide and conquer" complex networks. In particular, I will present two recent developments that I judge particularly promising: the first one is a generic method to account for information bouncing back-and-forth in the network structure, while the second allows for random graphs containing cycles of arbitrary length with intricate overlaps. In addition to their direct impact on the modeling and characterization of complex systems, these innovations provide new perspectives on topics such as Bayesian networks, belief propagation algorithms, and tensor networks.