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arxiv: 1907.06401 · v1 · pith:OCK4DV43new · submitted 2019-07-15 · 🧮 math.OC

Energy cost for target control of complex networks

Gaopeng Duan , Aming Li , Tao Meng , Long Wang This is my paper

Pith reviewed 2026-05-24 21:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords target controlenergy costcomplex networksminimum energycontrollabilitypartial controllabilityscaling behaviorlinear systems
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The pith

The minimum energy cost to control an arbitrary subset of nodes in a network is given explicitly, with bounds that scale according to the allowed control time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives the minimum energy cost needed to steer any chosen subset of nodes, called targets, from initial states to desired final states without making the whole network controllable. It supplies exact upper and lower bounds on that energy and shows how the bounds change with the time interval granted for control. The work also finds that, for any fixed number of targets, different choices of which nodes to steer produce energy costs that can differ by several orders of magnitude. If these results hold, realistic control of large networks becomes feasible because only the necessary nodes need steering and the effort can be estimated in advance.

Core claim

The paper presents the minimum energy cost for controlling an arbitrary subset of nodes of a network. It systematically shows the scaling behavior of the precise upper and lower bounds of the minimum energy in terms of the time given to accomplish control. For controlling a given number of target nodes, the associated energy over different configurations can differ by several orders of magnitude. When the adjacency matrix of the network is nonsingular, the framework simplifies by considering only the induced subgraph spanned by the target nodes. Energy cost can be saved by orders of magnitude because only partial controllability of the entire network is required.

What carries the argument

The minimum-energy expression obtained from the controllability Gramian of the target subsystem in the linear time-invariant dynamics.

If this is right

  • Upper and lower bounds on the minimum energy scale directly with the time allowed to reach the target states.
  • For any fixed number of targets, energy requirements across different node selections differ by several orders of magnitude.
  • When the adjacency matrix is nonsingular, the energy depends only on the induced subgraph of the target nodes.
  • Partial controllability reduces the required energy by orders of magnitude relative to full-network controllability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scaling with time suggests that allowing modestly longer control intervals can make previously expensive target sets feasible.
  • The large variation across configurations implies that systematic selection of low-energy target sets would further reduce cost.
  • The nonsingular simplification may extend approximately to nearly nonsingular real-world networks.

Load-bearing premise

The network is modeled as a linear time-invariant system whose evolution is governed by the adjacency matrix, with inputs applied only through a driver set that renders the chosen targets controllable.

What would settle it

A direct numerical check on a small network in which the derived minimum energy fails to equal the integrated squared input needed to reach the target state within the allotted time would disprove the formula.

Figures

Figures reproduced from arXiv: 1907.06401 by Aming Li, Gaopeng Duan, Long Wang, Tao Meng.

Figure 1
Figure 1. Figure 1: Illustration of the controllable space as we choose node 1 as driver node. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Energy distribution for a three dimensional fully controllable network. (a) For the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The upper and lower bounds of control energy for target control. In order to [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Target control under different choices of driver nodes. (a) A network with 5 nodes. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of energy cost for achieving target control and full control. In (a), we [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

To promote the implementation of realistic control over various complex networks, recent work has been focusing on analyzing energy cost. Indeed, the energy cost quantifies how much effort is required to drive the system from one state to another when it is fully controllable. A fully controllable system means that the system can be driven by external inputs from any initial state to any final state in finite time. However, it is prohibitively expensive and unnecessary to confine that the system is fully controllable when we merely need to accomplish the so-called target control---controlling a subnet of nodes chosen from the entire network. Yet, when the system is partially controllable, the associated energy cost remains elusive. Here we present the minimum energy cost for controlling an arbitrary subset of nodes of a network. Moreover, we systematically show the scaling behavior of the precise upper and lower bounds of the minimum energy in term of the time given to accomplish control. For controlling a given number of target nodes, we further demonstrate that the associated energy over different configurations can differ by several orders of magnitude. When the adjacency matrix of the network is nonsingular, we can simplify the framework by just considering the induced subgraph spanned by target nodes instead of the entire network. Importantly, we find that, energy cost could be saved by orders of magnitude as we only need the partial controllability of the entire network. Our theoretical results are all corroborated by numerical calculations, and pave the way for estimating the energy cost to implement realistic target control in various applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives the minimum energy cost to steer an arbitrary subset of target nodes in a linear time-invariant network model, expressed as the quadratic form x_T^T (C W(T) C^T)^+ x_T where W(T) is the controllability Gramian and C selects the targets. It provides explicit upper and lower bounds on this energy that scale with the control horizon T, demonstrates that energies for different target sets of fixed cardinality can differ by orders of magnitude, and shows that the problem reduces to the induced subgraph on the targets when the adjacency matrix is nonsingular. These theoretical results are supported by numerical calculations.

Significance. If the derivations hold, the work supplies a practical and computationally useful extension of classical minimum-energy control to the target-control setting, which is more realistic for applications than requiring full controllability. The scaling bounds with T and the demonstration of strong dependence on target choice are directly actionable for selecting control horizons and driver/target sets. The reduction to the induced subgraph when A is nonsingular offers a clear computational simplification. The grounding in standard Gramian constructions and absence of free parameters or circularity strengthen the contribution.

minor comments (3)
  1. The abstract refers to 'precise upper and lower bounds'; the main text should explicitly state whether these are the standard eigenvalue bounds on W(T) (e.g., the small-T quadratic lower bound and the large-T integral form) or whether tighter, target-specific bounds are derived.
  2. In the numerical examples, the specific network ensembles, number of realizations, and any error-bar conventions should be stated to support reproducibility of the reported orders-of-magnitude variation.
  3. The first appearance of the target-selection matrix C and the pseudoinverse notation should be accompanied by a brief definition or reference to the standard LTI controllability setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for highlighting its significance in extending minimum-energy control to the target-control setting. We accept the recommendation for minor revision and will make any necessary adjustments in the revised version. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in standard Gramian formulas

full rationale

The paper's central results follow from the standard minimum-energy control formula for LTI systems, extended to target nodes via the restricted Gramian C W(T) C^T. Upper and lower bounds on energy are derived from well-known properties of the controllability Gramian (e.g., integral form and small-T approximations), without reducing to self-defined quantities or self-citations. Numerical examples are consistent with the math but do not form the derivation. The framework is self-contained against external linear control theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard linear time-invariant network model; no free parameters are fitted to data in the abstract, and no new entities are postulated.

axioms (2)
  • domain assumption Network dynamics obey the linear time-invariant equation dx/dt = A x + B u where A is the adjacency matrix.
    Invoked throughout the abstract as the basis for controllability and energy calculations.
  • domain assumption The target subset is controllable via some driver set (partial controllability is feasible).
    Required for the minimum-energy quantity to be finite and well-defined.

pith-pipeline@v0.9.0 · 5796 in / 1320 out tokens · 24138 ms · 2026-05-24T21:31:17.233175+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 2 internal anchors

  1. [1]

    Barab´ asi, A.-L.Network Science (Cambridge University Press, Cambridge, 2016)

    work page 2016

  2. [2]

    & Havlin, S

    Cohen, R. & Havlin, S. Complex Networks: Structure, Robustness and Function (Cam- bridge University Press, Cambridge, 2010)

    work page 2010

  3. [3]

    & Wang, L

    Duan, G., Xiao, F. & Wang, L. Asynchronous periodic edge-event triggered control for double-integrator networks with communication time delays. IEEE Transactions on Cybernetics 48, 675–688 (2017)

    work page 2017

  4. [4]

    & Barab´ asi, A.-L

    Liu, Y.-Y. & Barab´ asi, A.-L. Control principles of complex systems.Reviews of Modern Physics 88, 035006 (2016)

    work page 2016

  5. [5]

    & Barab´ asi, A.-L

    Liu, Y.-Y., Slotine, J.-J. & Barab´ asi, A.-L. Observability of complex systems. Proceed- ings of the National Academy of Sciences 110, 2460–2465 (2013)

    work page 2013

  6. [6]

    & Xiao, F

    Wang, L. & Xiao, F. A new approach to consensus problems in discrete-time multiagent systems with time-delays. Science in China Series F: Information Sciences 50, 625–635 (2007)

    work page 2007

  7. [7]

    & Xiao, F

    Wang, L. & Xiao, F. Finite-time consensus problems for networks of dynamic agents. IEEE Transactions on Automatic Control 55, 950–955 (2010)

    work page 2010

  8. [8]

    Pinning control and controllability of complex dynamical networks

    Chen, G. Pinning control and controllability of complex dynamical networks. Interna- tional Journal of Automation and Computing 14, 1–9 (2017)

    work page 2017

  9. [9]

    & Wang, L

    Duan, G., Li, A., Meng, T., Zhang, G. & Wang, L. Energy cost for controlling complex networks with linear dynamics. Physical Review E 99, 052305 (2019)

    work page 2019

  10. [10]

    & Wang, L

    Guan, Y., Ji, Z., Zhang, L. & Wang, L. Controllability of multi-agent systems under directed topology. International Journal of Robust and Nonlinear Control 27, 4333– 4347 (2017)

    work page 2017

  11. [11]

    Control energy scaling in temporal networks

    Li, A., Cornelius, S. P., Liu, Y.-Y., Wang, L. & Barab´ asi, A.-L. Control energy scaling in temporal networks. arXiv: 1712.06434v1 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  12. [12]

    P., Liu, Y.-Y., Wang, L

    Li, A., Cornelius, S. P., Liu, Y.-Y., Wang, L. & Barab´ asi, A.-L. The fundamental advantages of temporal networks. Science 358, 1042–1046 (2017)

    work page 2017

  13. [13]

    & Barab´ asi, A.-L

    Liu, Y.-Y., Slotine, J.-J. & Barab´ asi, A.-L. Controllability of complex networks.Nature 473, 167–73 (2011). 14

    work page 2011

  14. [14]

    & Wang, L

    Lu, Z., Zhang, L., Ji, Z. & Wang, L. Controllability of discrete-time multi-agent systems with directed topology and input delay. International Journal of Control 89, 179–192 (2016)

    work page 2016

  15. [15]

    & Wang, L

    Tian, L., Guan, Y. & Wang, L. Controllability and observability of multi-agent systems with heterogeneous and switching topologies. International Journal of Control 1–12 (2018)

    work page 2018

  16. [16]

    Wang, L., Jiang, F., Xie, G. & Ji, Z. Controllability of multi-agent systems based on agreement protocols. Science in China Series F: Information Sciences 52, 2074 (2009)

    work page 2074

  17. [17]

    Yan, G., Ren, J., Lai, Y.-C., Lai, C.-H. & Li, B. Controlling complex networks: How much energy is needed? Physical Review Letters 108, 218703 (2012)

    work page 2012

  18. [18]

    Gao, J., Liu, Y.-Y., D’Souza, R. M. & Barab´ asi, A.-L. Target control of complex networks. Nature Communications 5, 5415 (2014)

    work page 2014

  19. [19]

    & Sorrentino, F

    Klickstein, I., Shirin, A. & Sorrentino, F. Energy scaling of targeted optimal control of complex networks. Nature Communications 8, 15145 (2017)

    work page 2017

  20. [20]

    Kalman, R. E. Mathematical description of linear dynamical systems. Journal of the Society for Industrial and Applied Mathematics Series A Control 1, 152–192 (1963)

    work page 1963

  21. [21]

    Determination of generic dimensions of controllable subspaces and its appli- cation

    Hosoe, S. Determination of generic dimensions of controllable subspaces and its appli- cation. IEEE Transactions on Automatic Control 25, 1192–1196 (1980)

    work page 1980

  22. [22]

    Structural controllability

    Lin, C.-T. Structural controllability. IEEE Transactions on Automatic Control 19, 201–208 (1974)

    work page 1974

  23. [23]

    Linear System Theory (Second Edition) (Tsinghua University Press, Beijing, 2005)

    Zheng, D. Linear System Theory (Second Edition) (Tsinghua University Press, Beijing, 2005)

    work page 2005

  24. [24]

    Shannon, C. E. Communication in the presence of noise. Proceedings of the IEEE 86, 447–457 (1998)

    work page 1998

  25. [25]

    Lin, X., Tade, M. O. & Newell, R. B. Output structural controllability condition for the synthesis of control systems for chemical processes. International Journal of Systems Science 22, 107–132 (1991)

    work page 1991

  26. [26]

    Lewis, F. L. & Syrmos, V. L. Optimal Control (Second Edition) (Wiley, New York, 1995)

    work page 1995

  27. [27]

    & Bullo, F

    Pasqualetti, F., Zampieri, S. & Bullo, F. Controllability metrics and algorithms for complex networks. IEEE Transactions on Control of Network Systems 1, 40–52 (2014). 15

    work page 2014

  28. [28]

    & Chung, K

    Lam, J., Li, Z., Wei, Y., Feng, J. & Chung, K. W. Estimates of the spectral condition number. Linear and Multilinear Algebra 59, 249–260 (2011)

    work page 2011

  29. [29]

    & R´ enyi, A

    Erd˝ os, P. & R´ enyi, A. On the evolution of random graphs.Publications of the Mathe- matical Institute of the Hungarian Academy of Sciences 5, 17–60 (1960)

    work page 1960

  30. [30]

    & Battiston, S

    Galbiati, M., Delpini, D. & Battiston, S. The power to control. Nature Physics 9, 126 (2013)

    work page 2013

  31. [31]

    The optimal trajectory to control complex networks

    Li, A., Wang, L. & Schweitzer, F. The optimal trajectory to control complex networks. arXiv: 1806.04229v1 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  32. [32]

    P., Kath, W

    Cornelius, S. P., Kath, W. L. & Motter, A. E. Realistic control of network dynamics. Nature Communications 4, 1942 (2013)

    work page 1942

  33. [33]

    Nonlinear Systems (Prentice Hall, New Jersey, 2002)

    Khalil, H. Nonlinear Systems (Prentice Hall, New Jersey, 2002)

    work page 2002

  34. [34]

    Control and Nonlinearity (American Mathematical Society, 2009)

    Coron, J.-M. Control and Nonlinearity (American Mathematical Society, 2009)

    work page 2009

  35. [35]

    & Saram¨ aki, J

    Holme, P. & Saram¨ aki, J. Temporal networks. Physics Reports 519, 97–125 (2012)

    work page 2012

  36. [36]

    & Lambiotte., R

    Masuda, N. & Lambiotte., R. A Guide to Temporal Networks (World Scientific, Singa- pore, 2016)

    work page 2016

  37. [37]

    & H¨ ovel, P

    P´ osfai, M. & H¨ ovel, P. Structural controllability of temporal networks.New Journal of Physics 16, 123055 (2014). A Optimal Control Energy Theory A.1 Fully controllable systems In this subsection, we assume system (1) is fully controllable. According to the optimal control energy theory, we have the Hamilton function H(τ) = 1 2u(τ)Tu(τ) +λ(τ + 1)T(Ax(τ...

    work page 2014