Observing and controlling complex networks are of paramount interest for understanding complex physical, biological and technological systems. Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. Yet, the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for different directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving to a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.
A classic measure of ecological stability describes the tendency of a community to return to equilibrium after small perturbation. While many advances show how the network structure of these communities severely constrains such tendencies, few if any of these advances address one of the most fundamental properties of network structure: heterogeneity among nodes with different numbers of links. Here we systematically explore this property of degree heterogeneity and find that its effects on stability systematically vary with different types of interspecific interactions. Degree heterogeneity is always destabilizing in ecological networks with both competitive and mutualistic interactions while its effects on networks of predator-prey interactions such as food webs depend on prey contiguity, i.e., the extent to which the species consume an unbroken sequence of prey in community niche space. Increasing degree heterogeneity stabilizes food webs except those with the most contiguity. These findings help explain previously unexplained observations that food webs are highly but not completely contiguous and, more broadly, deepens our understanding of the stability of complex ecological networks with important implications for other types of dynamical systems.
