The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network's degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes.

A quantitative description of a complex system is inherently limited by our ability to estimate the system's internal state from experimentally accessible outputs. Although the simultaneous measurement of all internal variables, like all metabolite concentrations in a cell, offers a complete description of a system's state, in practice experimental access is limited to only a subset of variables, or sensors. A system is called observable if we can reconstruct the system's complete internal state from its outputs. Here, we adopt a graphical approach derived from the dynamical laws that govern a system to determine the sensors that are necessary to reconstruct the full internal state of a complex system. We apply this approach to biochemical reaction systems, finding that the identified sensors are not only necessary but also sufficient for observability. The developed approach can also identify the optimal sensors for target or partial observability, helping us reconstruct selected state variables from appropriately chosen outputs, a prerequisite for optimal biomarker design. Given the fundamental role observability plays in complex systems, these results offer avenues to systematically explore the dynamics of a wide range of natural, technological and socioeconomic systems.algebraic observability | biochemical reactions | control theory T he internal variables of a complex system are rarely independent of each other, as the interactions between the system's components induce systematic interdependencies between them. Hence, a well-selected subset of variables can contain sufficient information about the rest of the variables, allowing us to reconstruct the system's complete internal state, making the system observable. To address observability in quantitative terms, we focus on systems whose dynamics can be described by the generic state-space form _ xðtÞ = fðt; xðtÞ; uðtÞÞ;[1]where xðtÞ ∈ R N represents the complete internal state of the system (e.g., the concentrations of all metabolites in a cell), and the input vector uðtÞ ∈ R K captures the influence of the environment. Observing the system means that we monitor a set of variables yðtÞ ∈ R M that depend on the time t, the system's internal state xðtÞ, and the external input uðtÞ, yðtÞ = hðt; xðtÞ; uðtÞÞ:[2]Observability requires us to establish a relationship between the outputs yðtÞ, the state vector xðtÞ, and the inputs uðtÞ in a manner that we can uniquely infer the system's complete initial state xð0Þ. The observability criteria can be formulated algebraically for dynamical systems consisting of polynomial or rational expressions (1, 2) stating that [1] is observable if the Jacobian matrix J = ½J ij NM × N has full rank, rank J = N;[3]where

Controlling large natural and technological networks is an outstanding challenge. It is typically neither feasible nor necessary to control the entire network, prompting us to explore target control: the efficient control of a preselected subset of nodes. We show that the structural controllability approach used for full control overestimates the minimum number of driver nodes needed for target control. Here we develop an alternate ‘k-walk’ theory for directed tree networks, and we rigorously prove that one node can control a set of target nodes if the path length to each target node is unique. For more general cases, we develop a greedy algorithm to approximate the minimum set of driver nodes sufficient for target control. We find that degree heterogeneous networks are target controllable with higher efficiency than homogeneous networks and that the structure of many real-world networks are suitable for efficient target control.

A reflection of our ultimate understanding of a complex system is our ability to control its behavior. Typically, control has multiple prerequisites: it requires an accurate map of the network that governs the interactions between the system's components, a quantitative description of the dynamical laws that govern the temporal behavior of each component, and an ability to influence the state and temporal behavior of a selected subset of the components. With deep roots in nonlinear dynamics and control theory, notions of control and controllability have taken a new life recently in the study of complex networks, inspiring several fundamental questions: What are the control principles of complex systems? How do networks organize themselves to balance control with functionality? To address these here we review recent advances on the controllability and the control of complex networks, exploring the intricate interplay between a system's structure, captured by its network topology, and the dynamical laws that govern the interactions between the components. We match the pertinent mathematical results with empirical findings and applications. We show that uncovering the control principles of complex systems can help us explore and ultimately understand the fundamental laws that govern their behavior.

The recent realization that human-associated microbial communities play a crucial role in determining our health and well-being1,2 has led to the ongoing development of microbiome-based therapies3 such as fecal microbiota transplantation4,5. Thosemicrobial communities are very complex, dynamic6 and highly personalized ecosystems3,7, exhibiting a high degree of inter-individual variability in both species assemblages8 and abundance profiles9. It is not known whether the underlying ecological dynamics, which can be parameterized by growth rates, intra- and inter-species interactions in population dynamics models10, are largely host-independent (i.e. “universal”) or host-specific. If the inter-individual variability reflects host-specific dynamics due to differences in host lifestyle11, physiology12, or genetics13, then generic microbiome manipulations may have unintended consequences, rendering them ineffectual or even detrimental. Alternatively, microbial ecosystems of different subjects may follow a universal dynamics with the inter-individual variability mainly stemming from differences in the sets of colonizing species7,14. Here we developed a novel computational method to characterize human microbial dynamics. Applying this method to cross-sectional data from two large-scale metagenomic studies, the Human Microbiome Project9,15 and the Student Microbiome Project16, we found that both gut and mouth microbiomes display pronounced universal dynamics, whereas communities associated with certain skin sites are likely shaped by differences in the host environment. Interestingly, the universality of gut microbial dynamics is not observed in subjects with recurrent Clostridium difficile infection17 but is observed in the same set of subjects after fecal microbiota transplantation. These results fundamentally improve our understanding of forces and processes shaping human microbial ecosystems, paving the way to design general microbiome-based therapies18.

Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. But 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 di erent 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 at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.M any natural and man-made systems can be represented as networks 1-3 , where nodes are the system's components and links describe the interactions between them. Thanks to these interactions, perturbations of one node can alter the states of the other nodes 4-6 . This property has been exploited to control a network-that is, to move it from an initial state to a desired final state 7-9 -by manipulating the state variables of only a subset of its nodes 10,11 . Such control processes 10-26 play an important role in the regulation of protein expression 27 , the coordination of moving robots 28 , and the inhibition of undesirable social contagions 29 . At the same time the interdependence between nodes means that the states of a small number of sensor nodes contain sufficient information about the rest of the network, so that we can reconstruct the system's full internal state by accessing only a few outputs 30 . This can be utilized for biomarker design in cellular networks, or to monitor in real time the state and functionality of infrastructural 31 and socialecological 32 systems for early warning of failures or disasters 33 .Although recent advances in driver and sensor node identification constitute unavoidable steps towards controlling and observing real networks, in practice we continue to face significant challenges: the control of a large network may require a vast amount of energy 16-18 , and measurement noise 34 causes uncertainties in the observation process. To quantify these issues we formalize the dynamics of a controlled network with N nodes and N D external control inputs as 7-10ẋwhere the vector x(t) = [x 1 (t), x 2 (t), . . . , x N (t)] T describes the states of the N nodes at time t and x i (t) can represent the concentration of a metabolite in a metabolic network 35 , the geometric state of a chromosome in a chromosomal interaction network 14 , or the belief of an individual in opinion dynamics 29,36 . The vect...

Most networked systems of scientific interest are characterized by temporal links, meaning the network's structure changes over time. Link temporality has been shown to hinder many dynamical processes, from information spreading to accessibility, by disrupting network paths. Considering the ubiquity of temporal networks in nature, we ask: Are there any advantages of the networks' temporality? We use an analytical framework to show that temporal networks can, compared to their static counterparts, reach controllability faster, demand orders of magnitude less control energy, and have control trajectories, that are considerably more compact than those characterizing static networks. Thus, temporality ensures a degree of flexibility that would be unattainable in static networks, enhancing our ability to control them.

The protein-protein interaction (PPI) network is crucial for cellular information processing and decision-making. With suitable inputs, PPI networks drive the cells to diverse functional outcomes such as cell proliferation or cell death. Here, we characterize the structural controllability of a large directed human PPI network comprising 6,339 proteins and 34,813 interactions. This network allows us to classify proteins as "indispensable," "neutral," or "dispensable," which correlates to increasing, no effect, or decreasing the number of driver nodes in the network upon removal of that protein. We find that 21% of the proteins in the PPI network are indispensable. Interestingly, these indispensable proteins are the primary targets of disease-causing mutations, human viruses, and drugs, suggesting that altering a network's control property is critical for the transition between healthy and disease states. Furthermore, analyzing copy number alterations data from 1,547 cancer patients reveals that 56 genes that are frequently amplified or deleted in nine different cancers are indispensable. Among the 56 genes, 46 of them have not been previously associated with cancer. This suggests that controllability analysis is very useful in identifying novel disease genes and potential drug targets.network biology | controllability | protein-protein interaction network | disease genes | drug targets