Abstract
The control of networked dynamical systems opens the possibility for new discoveries and therapies in systems biology and neuroscience. Recent theoretical advances provide candidate mechanisms by which a system can be driven from one pre-specified state to another, and computational approaches provide tools to test those mechanisms in real-world systems. Despite already having been applied to study network systems in biology and neuroscience, the practical performance of these tools and associated measures on simple networks with pre-specified structure has yet to be assessed. Here, we study the behavior of four control metrics (global, average, modal, and boundary controllability) on eight canonical graphs (including Erdős–Rényi, regular, small-world, random geometric, Barábasi–Albert preferential attachment, and several modular networks) with different edge weighting schemes (Gaussian, power-law, and two nonparametric distributions from brain networks, as examples of real-world systems). We observe that differences in global controllability across graph models are more salient when edge weight distributions are heavy-tailed as opposed to normal. In contrast, differences in average, modal, and boundary controllability across graph models (as well as across nodes in the graph) are more salient when edge weight distributions are less heavy-tailed. Across graph models and edge weighting schemes, average and modal controllability are negatively correlated with one another across nodes; yet, across graph instances, the relation between average and modal controllability can be positive, negative, or nonsignificant. Collectively, these findings demonstrate that controllability statistics (and their relations) differ across graphs with different topologies and that these differences can be muted or accentuated by differences in the edge weight distributions. More generally, our numerical studies motivate future analytical efforts to better understand the mathematical underpinnings of the relationship between graph topology and control, as well as efforts to design networks with specific control profiles.
Highlights
Complex systems can be modeled as networks in which the system’s elements and their pairwise interactions are represented, respectively, as nodes and edges in a graph (Newman et al 2011)
We begin by examining how network controllability statistics vary over graphs within a given ensemble and whether that variation differs as a function of graph model
To assess the reliability and reproducibility of our results, we examined the impact of graph size (n = 128, 256, or 512) on network controllability statistics, and their modulation by graph model for the Gaussian edge weight distribution
Summary
Complex systems can be modeled as networks in which the system’s elements and their pairwise interactions are represented, respectively, as nodes and edges in a graph (Newman et al 2011). Network models and associated graph representations have been adopted in a range of disciplines to successfully investigate the structure and function of social, economic, and biological systems (Carrington et al 2005; Schweitzer et al 2009; Goh et al 2007). Complex systems are dynamic, meaning that their elements can be associated with internal states that evolve and fluctuate over time (Boccaletti et al 2006; Porter and Gleeson 2014). The evolution of a system’s state over time depends upon the organization of its underlying network. In neural systems, activity propagates from neuron-to-neuron or region-to-region along axonal projections and fiber bundles (Muldoon et al 2016)
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