Network controllability measures of subnetworks: implications for neurosciences

Abstract

Recent progress in network sciences has made it possible to apply key findings from control theory to the study of networks. Referred to as network control theory, this framework describes how the interactions between interconnected system elements and external energy sources, potentially constrained by different optimality criteria, result in complex network behavior. A typical example is the quantification of the functional role certain brain regions or symptoms play in shaping the temporal dynamics of brain activity or the clinical course of a disease, a property that is quantified in terms of the so-called controllability metrics. Critically though, contrary to the engineering context in which control theory was originally developed, a mathematical understanding of the network nodes and connections in neurosciences cannot be assumed. For instance, in the case of psychological systems such as those studied to understand the psychiatric disorders, a potentially large set variables are unknown. As such, while the measures offered by network control theory would be mathematically correct, in that they can be calculated with high precision, they could have little translational values with respect to their putative role suggested by controllability metrics. It is therefore critical to understand if and how the controllability metrics computer over subnetworks would deviate, if access to the complete set of variables, as in neurosciences, cannot be taken for granted. In this paper, we use a host of simulations based on synthetic as well as structural MRI data to study the potential deviation of controllability metrics in sub- compared to the full networks. Specifically, we estimate average- and modal-controllability, two of the most widely used controllability measures in neurosciences, in a large number of settings where we systematically vary network type, network size, and edge density. We find out, across all network types we test, that average and modal controllability are systematically, either over- or underestimated depending on the number of nodes in the sub- and full network and the edge density. Finally, we provide a formal theoretical proof that our observations generalize to any network type and discuss the ramifications of this systematic bias and potential solutions to alleviate the problem.