Abstract
Controlling complex networked systems to desired states is a key research goal in contemporary science. Despite recent advances in studying the impact of network topology on controllability, a comprehensive understanding of the synergistic effect of network topology and individual dynamics on controllability is still lacking. Here we offer a theoretical study with particular interest in the diversity of dynamic units characterized by different types of individual dynamics. Interestingly, we find a global symmetry accounting for the invariance of controllability with respect to exchanging the densities of any two different types of dynamic units, irrespective of the network topology. The highest controllability arises at the global symmetry point, at which different types of dynamic units are of the same density. The lowest controllability occurs when all selfloops are either completely absent or present with identical weights. These findings further improve our understanding of network controllability and have implications for devising the optimal control of complex networked systems in a wide range of fields.
Introduction
Complex networks, such as Internet, WWW, powergrid, cellular and ecological networks, have been at the forefront of complex system studies for more than a decade^{1,2}. Universal principles that govern the topology and evolution of complex networks have significantly enriched our understanding of them^{3,4}. Fairly recently, controlling complex networks to desired final states has been a very hot research topic in complex system studies^{5,6,7,8}.
As a key notion in control theory, controllability denotes our ability to drive a dynamical system from any initial state to any desired final state in finite time^{9,10}. For the canonical linear timeinvariant (LTI) system with state vector , state matrix and control matrix , Kalman's rank condition (i.e., ) is sufficient and necessary to assure controllability. Yet, in many cases system parameters are not exactly known, rendering classical controllability tests impossible. By assuming that system parameters are either fixed zeros or freely independent, structural control theory (SCT) helps us overcome this difficulty for LTI systems^{11,12,13,14,15}. Quite recently, many research activities have been devoted to study the structural controllability of linear systems with complex network structure, where system parameters (e.g., the elements in A, representing link weights or interaction strengths between nodes) are typically not precisely known, only the zerononzero pattern of A is known^{5,6,16,17,18,19,20,21}. Network controllability problem can be typically posed as a combinatorial optimization problem, i.e., identify a minimum set of driver nodes, with size denoted by N_{D}, whose control is sufficient to fully control the system's dynamics^{5}. While the intrinsic individual dynamics can be incorporated in the network model, it would be more natural and fruitful to consider their effects separately. Hence, most of the previous studies focused on the impact of network topology, rather than the individual dynamics of nodes, on network controllability^{5,17}. Other control related issues, e.g., energy cost of control, have also been extensively studied for complex networked systems^{22,23,24,25}.
If one explores the impact of individual dynamics on network controllability in the SCT framework, a specious result would be obtained—a single control input can make an arbitrarily large linear system controllable. Although this result as a special case of the minimum inputs theorem has been proved^{5} and its implication was further emphasized in Ref. 26, this result is inconsistent with empirical situations, implying that the SCT is inapplicable in studying network controllability, if individual dynamics of nodes are imperative to be incorporated to capture the collective dynamic behavior of a networked system. To overcome this difficulty and more importantly, to understand the impact of individual dynamics on network controllability, we revisit the key assumption of SCT, i.e., the independency of system parameters. We anticipate that major new insights can be obtained by relaxing this assumption, e.g., considering the natural diversity and similarity of individual dynamics. This also offers a more realistic characterization of many realworld networked systems where not all the system parameters are completely independent.
To solve the network controllability problem with dependent system parameters, we rely on the recently developed exact controllability theory (ECT)^{27}. ECT enables us to systematically explore the role of individual dynamics in controlling linear systems with arbitrary network topology. In particular, we consider prototypical linear forms of individual dynamics (from firstorder to highorders) that can be incorporated within the network representation of the whole system in a unified matrix form. This paradigm leads to the discovery of a striking symmetry in network controllability: if we exchange the fractions of any two types of dynamic units, the system's controllability (quantified by N_{D}) remains the same. This exchangeinvariant property gives rise to a global symmetry point, at which the highest controllability (i.e., lowest number of driver nodes) emerges. This symmetryinduced optimal controllability holds for any network topology and various categories of individual dynamics. We substantiate these findings numerically in a variety of network models.
Results
Controllability measurement
ECT^{27} claims that for arbitrary network topology and link weights characterized by the state matrix A in the LTI system , the minimum number of driver nodes N_{D} required to be controlled by imposing independent signals to fully control the system is given by the maximum geometric multiplicity max_{i}{μ(λ_{i})} of A's eigenvalues {λ_{i}}^{28,29,30,31,32}. Here μ(λ_{i}) ≡ N − rank(λ_{i}I_{N} − A) is the geometric multiplicity of the eigenvalue λ_{i} and I_{N} is the identity matrix. Calculating all the eigenvalues of A and subsequently counting their geometric multiplicities are generally applicable but computationally prohibitive for large networks. If A is symmetric, e.g., in undirected networks, N_{D} is simply given by the maximum algebraic multiplicity max_{i}{δ(λ_{i})}, where δ(λ_{i}) denotes the degeneracy of eigenvalue λ_{i}. Calculating N_{D} in the case of symmetric A is more computationally affordable than in the asymmetric case. Note that for structured systems where the elements in A are either fixed zeros or free independent parameters, ECT offers the same results as that of SCT^{27}.
Controllability associated with firstorder individual dynamics
We first study the simplest case of firstorder individual dynamics . The dynamical equations of an LTI control system associated with firstorder individual dynamics^{33} can be written as
where the vector captures the states of N nodes, is a diagonal matrix representing intrinsic individual dynamics of each node, denotes the coupling matrix or the weighted wiring diagram of the networked system, in which a_{ij} represents the weight of a directed link from node j to i (for undirected networks, a_{ij} = a_{ji}). is the input vector of M independent signals, is the control matrix and Φ ≡ Λ + A is the state matrix. Without loss of generality, we assume Λ is a “constant” matrix over the field (rational numbers) and A is a structured matrix over the field (real numbers). In other words, we assume all the entries in Φ have been rescaled by the individual dynamics parameters. The resulting state matrix Φ is usually called a mixed matrix with respect to ()^{34}. The firstorder individual dynamics in Φ is captured by selfloops in the network representation of Φ (see Fig. 1a). N_{D} can then be determined by calculating the maximum geometric multiplicity max_{i}{μ(λ_{i})} of Φ's eigenvalues.
We study two canonical network models (ErdösRényi and Scalefree) with random edge weights and a ρ_{s} fraction of nodes associated with identical individual dynamics (i.e., selfloops of identical weights). As shown in Fig. 2a, b, the fraction of driver nodes n_{D} ≡ N_{D}/N is symmetric about ρ_{s} = 0.5, regardless of the network topology. (Note that SCT predicts that in case of independent selfloop weights, n_{D} monotonically decreases as ρ_{s} augments and eventually ρ_{s} = 1 leads to n_{D} = 1/N, implying that a single driver node can fully control the whole network^{26}.) The symmetry can be theoretically predicted (see Methods). An immediate but counterintuitive consequence from the symmetry is that n_{D} in the absence of selfloops is exactly the same as the case that each node has a selfloop with identical weight. This is a direct consequence of Kalman's rank condition for controllability^{9}:
where the left and the right hand sides are the rank of controllability matrix in the absence and full of identical selfloops, respectively (see Supplementary Sec.1 for proof).
The presence of two types of nonzero selfloops s_{2} and s_{3} leads to even richer behavior of controllability. If the three types of selfloops (including selfloops of zero weights) are randomly distributed at nodes, the impact of their fractions on n_{D} can be visualized by mapping the three fractions into a 2D triangle, as shown in Fig. 2e. We see that n_{D} exhibits symmetry in the triangle and the minimum n_{D} occurs at the center that represents identical fractions of the three different selfloop types. The symmetryinduced highest controllability can be generalized to arbitrary number of selfloops. Assume there exist n types of selfloops with weights , respectively, we have
for sparse networks with random weights (see Supplementary Sec.2 for detailed derivation and the formula of dense networks). An immediate prediction of Eq. (3) is that N_{D} is primarily determined by the selfloop with the highest density, simplifying Eq. (3) to be , where is the weight of the prevailing selfloop (see Supplementary Sec.2). Using Eq. (3) and the fact that Φ is a mixed matrix, we can predict that N_{D} remains unchanged if we exchange the densities of any two types of selfloops (see Methods), accounting for the symmetry of N_{D} for arbitrary types of selfloops. Due to the dominance of N_{D} by the selfloop with the highest density and the exchangeinvariance of N_{D}, the highest controllability with the lowest value of N_{D} emerges when distinct selfloops are of the same density.
To validate the symmetryinduced highest controllability predicted by our theory, we quantify the density heterogeneity of selfloops as follows:
where N_{s} is the number of different types of selfloops (or the diversity of selfloops). Note that Δ = 0 if and only if all different types of selfloops have the same density, i.e., and the larger value of Δ corresponds to more diverse case. Figure 2e, f shows that n_{D} monotonically increases with Δ and the highest controllability (lowest n_{D}) arises at Δ = 0, in exact agreement with our theoretical prediction. The effect of the heterogeneity of nodal dynamics on the controllability resembles that of the structural heterogeneity discovered in Ref. 5, i.e., more degree heterogeneity leads to larger n_{D} and hence worse controllability. Figure 2g, h display n_{D} as a function of N_{s}. We see that n_{D} decreases as N_{s} increases, suggesting that the diversity of individual dynamics facilitates the control of a networked system. When N_{s} = N (i.e., all the selfloops are independent), n_{D} = 1/N, which is also consistent with the prediction of structural control theory^{5}.
Controllability for highorder individual dynamics
In some real networked systems, dynamic units are captured by highorder individual dynamics, prompting us to check if the symmetryinduced highest controllability still holds for higherorder individual dynamics. The graph representation of dynamic units with 2ndorder dynamics is illustrated in Fig. 1b. In this case, the eigenvalues of the dynamic unit's state matrix plays a dominant role in determining N_{D}. For two different units as distinguished by distinct (a_{0} a_{1}) one can show that their state matrices almost always have different eigenvalues, except for some pathological cases of zero measure that occur when the parameters satisfy certain accidental constraints. The eigenvalues of dynamic unit's state matrix take over the roles of selfloops in the 1storder dynamics, accounting for the following formulas for sparse networks
where λ^{(i)} is either one of the two eigenvalues of typei dynamic unit's state matrix. The formula implies that N_{D} is exclusively determined by the prevailing dynamic unit, (see Supplementary Section 2). The symmetry of N_{D}, i.e., exchanging the densities of any types of dynamic units does not alter N_{D} (see Methods) and the emergence of highest controllability at the global symmetry point can be similarly proved as we did in the case of 1storder individual dynamics.
The 3rdorder individual dynamics are graphically characterized by a dynamic unit composed of three nodes (Fig. 1c), leading to a 3N × 3N state matrix (Fig. 1c). We can generalize Eq. (5) to arbitrary order of individual dynamics:
where d is the order of the dynamic unit, is any one of the d eigenvalues of typei dynamic units and I_{dN} is the identity matrix of dimension dN. In analogy with the simplified formula for the 1storder dynamics, insofar as a type of individual dynamics prevails in the system, Eq. (6) is reduced to , where is one of the eigenvalues of the prevailing dynamic unit's state matrix. The global symmetry of controllability and the highest controllability occurs at the global symmetry point can be proved for individual dynamics of any order and arbitrary network topology. Fig. 3 displays the results for 2nd and 3rdorder individual dynamics, where the density heterogeneity for highorder dynamic units is defined as , N_{u} is the number of different dynamic units and is the density of typei dynamic unit.
We have also explored the mixture of individual dynamics with different orders, finding the symmetry of n_{D} and the highest controllability at the global symmetry point, in agreement with those found in the networks with singleorder individual dynamics (see Fig. 4).
Discussion
In summary, we map individual dynamics into dynamic units that can be integrated into the matrix representation of a networked system, offering a general paradigm to explore the joint effect of individual dynamics and network topology on the system's controllability. The paradigm leads to a striking discovery: the universal symmetry of controllability as reflected by the invariance of controllability with respect to exchanging the fractions of any two different types of individual dynamics and the emergence of highest controllability at the global symmetry point. The global symmetry indicates that the controllability is determined exclusively by the densities of different individual dynamics rather than their specific intrinsic dynamics. These findings generally hold for arbitrary networks and individual dynamics of any order. The symmetryinduced highest controllability has immediate implications for devising and optimizing the control of complex systems by for example, perturbing individual dynamics to approach the symmetry point without the need to adjust network structure.
The theoretical paradigm and tools developed here also allow us to address a number of questions, the answers to which could offer further insights into the control of complex networked systems. For example, similar individuals are often accompanied by dense inner connections among them, accounting for the widely observed communities with relatively sparse connections among them in natural and social systems. How such structural property in combination with the similarity and diversity of individual dynamics impacts control is worthy of exploration. Despite the advantage of our tools compared to the other methods in the literature, the network systems that we can address are still the tip of the iceberg, raising the need of new tools based on network science, statistic physics and control theory. At the present, we are incapable of tackling general nonlinear dynamical systems, which is extremely challenging not only in complex networks but also in the canonical control theory. Nevertheless, our approach, we hope, will inspire further interest from physicists and other scientists towards achieving ultimate control of complex networked systems.
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Acknowledgements
We thank G. Tsekenis for discussions. W.X.W. was supported by NSFC under Grant No. 11105011, CNNSF under Grant No. 61074116 and the Fundamental Research Funds for the Central Universities. Y.Y.L. gratefully acknowledge the support from the John Templeton Foundation (award #51977).
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J.J.S., Y.Y.L. and W.X.W. conceived the research; W.X.W., Y.Y.L. and J.J.S. contributed analytic tools; C.Z. and W.X.W. performed numerical calculations; W.X.W. and Y.Y.L. wrote the paper, J.J.S. edited the paper.
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Zhao, C., Wang, W., Liu, Y. et al. Intrinsic dynamics induce global symmetry in network controllability. Sci Rep 5, 8422 (2015). https://doi.org/10.1038/srep08422
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