Steering the dynamics by controlling the temporal interaction network

Melvyn Tyloo

REVIEW 2 major objections 5 minor 23 references

You can steer every node of a networked system onto prescribed trajectories by tuning only the time-varying links between them.

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2026-07-11 10:19 UTC pith:5VVYG5NL

load-bearing objection Clean adjoint pipeline for multiplicative network control; works on the linear model, limited by the usual fidelity gap. the 2 major comments →

arxiv 2607.04998 v1 pith:5VVYG5NL submitted 2026-07-06 nlin.AO physics.soc-ph

Steering the dynamics by controlling the temporal interaction network

Melvyn Tyloo This is my paper

classification nlin.AO physics.soc-ph

keywords temporal networksoptimal controladjoint methodnetwork controllinear time-variant systemscoupling matrixsteering dynamics

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Many real systems are coupled by links whose strengths change with time. This paper shows that those links themselves can be used as the control knobs. Using nonlinear optimal control and the adjoint method, one obtains the gradient of a cost that penalizes both tracking error and control effort; gradient descent then produces a time-dependent coupling matrix that forces the node states onto chosen target paths. On a simple linear model the method succeeds even when only eight percent of the possible links are allowed to vary and the nodes’ intrinsic rest states are heterogeneous. The same workflow is offered for any engineered or artificial network whose interactions can be tuned, so that one no longer needs to inject additive control signals at every node.

Core claim

By treating the off-diagonal entries of a time-dependent coupling matrix as the only control variables and computing their gradient via the adjoint state of a quadratic tracking-plus-effort Lagrangian, a linear networked system can be driven so that every node follows a prescribed target trajectory on a chosen interval, even when only a small fraction of the possible links are free to vary.

What carries the argument

The adjoint-derived gradient of the Lagrangian: after a forward integration of the state and a backward integration of the adjoint, the gradient with respect to each free coupling entry is simply the integral of (control weight times that entry minus the product of the corresponding adjoint and state components).

Load-bearing premise

That the continuous-time linear model with vanishing diagonal and free real-valued off-diagonal links is already a faithful enough description of the engineered or biological systems one ultimately wants to control.

What would settle it

Apply the same adjoint pipeline to a network of nonlinear oscillators or to hardware whose links are constrained to be non-negative or fixed in sparsity pattern; if the nodes systematically fail to track the targets, the central claim does not extend beyond the linear unconstrained setting.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Clean adjoint pipeline for multiplicative network control; works on the linear model, limited by the usual fidelity gap.

read the letter

Melvyn has a short, self-contained methods paper that does one thing cleanly: derive the adjoint gradient when the control lives in the coupling matrix W(t) rather than as an additive input, then show that even 8 % of the off-diagonal entries can drive a linear system onto discontinuous multi-frequency targets.

What is new is the explicit gradient for multiplicative network control (Eqs. 4–10, 16–18) and the numerical demonstration that sparse time-varying edges suffice for tracking. The derivation is standard adjoint optimal control applied correctly to ẋ = α(x* − x) + W(t)x with W_ii = 0. Adam consistently lowers the quadratic tracking-plus-effort cost; the figures and Table I are readable and the targets are prescribed a priori, so there is no circularity. Citations are appropriate and self-citations stay in the background.

Soft spots are ordinary for a short nlin.AO note. There is no additive-control baseline, no code, no error bars, and free parameters (w_c, α, edge fraction, Adam schedule) are simply fixed. The outlook correctly flags that non-negativity, fixed hardware sparsity, or nonlinear nodes are left for later; the paper never claims otherwise. That model-fidelity gap is real but does not break the internal result.

This is for people who already work on temporal networks or programmable engineered links and want a reusable adjoint recipe. It will not re-open a theoretical question, but it is formally grounded enough that a serious editor should send it to referees rather than desk-reject. I would skim the figures and keep the gradient formulas; I would not rush to cite it unless I am already building a similar controller. Worth a look, not a must-read.

Referee Report

2 major / 5 minor

Summary. The paper develops a nonlinear optimal-control framework for steering networked dynamical systems by treating the time-dependent coupling matrix W(t) as the control input. Starting from a general form of coupled ODEs, it derives the adjoint equation and the gradient of a Lagrangian that combines a quadratic tracking cost with a quadratic control-effort cost (Eqs. 2–7). The framework is specialized to the linear time-variant system ẋ = α(x* − x) + W(t)x with vanishing diagonal (Eqs. 8–18). Numerical experiments with N = 50 nodes and Adam gradient descent show that controlling all, 10 %, or only 8 % of the off-diagonal entries can drive the node states onto a common prescribed trajectory or onto three distinct amplitude-scaled trajectories (Figs. 1–2, Table I).

Significance. If the result holds, the work supplies a clean, adjoint-based pipeline for multiplicative network control that is immediately usable for engineered systems whose interaction strengths are tunable (robotic swarms, artificial synapses, etc.). The derivation is standard and correct for the linear model; the numerical illustrations consistently reduce the tracking cost even under severe sparsity. The paper does not claim novelty for the adjoint method itself, but rather for its systematic application to temporal interaction networks with structural constraints. The outlook correctly flags the need for non-negativity, fixed sparsity patterns and nonlinear node dynamics—limitations that do not invalidate the linear demonstration but bound its present scope.

major comments (2)

Sec. III and Table I report only single-run cost values after a fixed 10 000 Adam steps; no comparison is given against an additive-control baseline of comparable effort, nor against random or static W(t). Without such a baseline it is impossible to quantify how much of the observed tracking performance is due to the adjoint-derived gradient versus the mere presence of free parameters in W.
The central claim that “only 8 % of the edges suffice” (Fig. 1 aiii,biii and Fig. 2) rests on an unconstrained real-valued control set. The manuscript never tests whether the same sparse support remains effective once non-negativity or other hardware-imposed sign constraints are enforced—constraints that the outlook itself identifies as essential for the intended applications.

minor comments (5)

Eq. (5) writes the adjoint evolution with an undefined function p; the subsequent sentence clarifies p = ∂ce/∂x, but the notation should be introduced before the equation.
In Sec. II C the indicator functions 1[tu,T] and 1[te,T] appear in the Lagrangian but are never used consistently in the numerical section; the text simply states the intervals [0,10] and [2,10]. Aligning the notation would improve readability.
Figure 1 panels are labeled (ai)–(biii) yet the caption refers only to “each panel”; adding explicit sub-captions would help the reader match text and graphics.
The weight wc = 0.005 is fixed throughout; a brief sensitivity check (or at least a statement that the qualitative conclusions are robust) would strengthen the numerical claims.
A few typographical slips remain: “constrain” → “constraints” (abstract), “a a common” (Sec. III B), and missing spaces around some equation numbers.

Circularity Check

0 steps flagged

No circularity: adjoint gradient of an a-priori quadratic cost is derived and then numerically minimized; targets and dynamics are independent inputs.

full rationale

The paper defines a general controlled network ODE (Eq. 1), forms a Lagrangian with free cost functionals Cu and Ce (Eqs. 2–3), and obtains the gradient of L with respect to the coupling entries via the adjoint equation (Eqs. 4–7). Specializing to the linear time-variant system (Eq. 8) yields an explicit adjoint ODE (Eq. 16) and gradient (Eq. 18). The numerical section then minimizes that gradient (Adam) for prescribed target trajectories (Eqs. 19–20) under explicit sparsity constraints on W. Nothing is fitted to data and later re-presented as a prediction; the targets are chosen a priori and the optimizer is free to fail. Self-citations appear only as background (opinion dynamics, temporal networks, prior control literature) and are not load-bearing for the derivation. The demonstration is therefore self-contained against its own stated model and cost; no circular step exists.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central numerical claim rests on the linear time-variant model, the quadratic costs, the vanishing-diagonal constraint, and a handful of hand-chosen scalars (w_c, α, fraction of controlled edges). No new physical entities are postulated; the free parameters are ordinary optimization and model knobs.

free parameters (4)

w_c (control-cost weight) = 0.005
Hand-set to 0.005 in all reported runs; trades off tracking accuracy against control effort and therefore shapes the recovered networks.
α (intrinsic relaxation rate) = 1
Fixed to 1; sets the time scale on which uncoupled nodes return to their natural states and therefore the amount of coupling needed.
fraction of controllable edges = 0.08–1.0
Explored at 100 %, 10 %, 8 %; the claim that sparse control still works depends on these discrete choices.
Adam step count / learning-rate schedule = 10000 steps
Fixed at 10 000 steps with unspecified learning rate; convergence of the non-convex problem is therefore not fully characterized.

axioms (4)

domain assumption The networked system obeys the linear ODE ẋ = α(x* − x) + W(t)x with real-valued, possibly dense W(t).
Stated in Sec. II.C and used for all analytic and numerical results; more realistic nonlinear or constrained dynamics are left to future work.
ad hoc to paper Only off-diagonal entries of W may be controlled (W_ii = 0).
Imposed throughout Sec. III so that control acts purely on interactions; not derived from first principles.
domain assumption The performance metric is the sum of a quadratic tracking cost and a quadratic control-effort cost.
Standard LQR-style choice (Eqs. 11, 13); other costs (sparsity-promoting, hard constraints) are mentioned but not used.
domain assumption Gradient descent with the adjoint-derived gradient converges to a useful local minimum of the non-convex cost.
Implicit in the numerical procedure of Sec. III; no global-optimality guarantee is claimed.

pith-pipeline@v1.1.0-grok45 · 12512 in / 2865 out tokens · 21058 ms · 2026-07-11T10:19:01.487550+00:00 · methodology

0 comments

read the original abstract

Many real-world coupled dynamical systems have the interaction structure and strength that evolve or adapt over time. Here, we investigate how one can control the state of a system by tuning its temporal interaction network. We present a framework based on nonlinear optimal control, where one has control over the coupling matrix of a dynamical system. We show how to obtain the gradient of the Lagrangian function of the system using the adjoint method. We then focus on a linear time-variant system for which we illustrate the framework. Finally, we explore how the states at the nodes can be steered to target trajectories, by controlling the coupling matrix, imposing various constraint on its structure. The workflow presented here can be leveraged to steer the dynamics of systems with artificial or engineered interaction that is tunable.

Figures

Figures reproduced from arXiv: 2607.04998 by Melvyn Tyloo.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 2 linked inside Pith

[1]

(1) for- ward from the initial conditionx(t= 0) =x (0)

Solve the state dynamics of the system Eq. (1) for- ward from the initial conditionx(t= 0) =x (0)
[2]

(7) backward in time starting from the terminal conditionλ(T) = 0

Use the forward state solution to solve the evolu- tion of the adjoint state Eq. (7) backward in time starting from the terminal conditionλ(T) = 0
[3]

(4) using the adjoint state and update the control signalU

Compute the gradient Eq. (4) using the adjoint state and update the control signalU
[4]

This procedure is usually called theadjoint method[13, 14] , and has been widely used in recent application such as machine learning

Go back to step 1 and repeat until convergence. This procedure is usually called theadjoint method[13, 14] , and has been widely used in recent application such as machine learning. See Ref. [15] for an overview of its application for additive control inputs in neural popula- tion models. Many different algorithms exist to perform the opti- mization of th...
[5]

Pikovsky, M

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- tion, Cambridge university press12(2001)

2001
[6]

Newman,Networks(Oxford university press, 2018)

M. Newman,Networks(Oxford university press, 2018)

2018
[7]

Holme and J

P. Holme and J. Saram¨ aki, Temporal networks, Physics reports519, 97 (2012)

2012
[8]

Starnini, F

M. Starnini, F. Baumann, T. Galla, D. Garcia, G. I˜ niguez, M. Karsai, J. Lorenz, and K. Sznajd-Weron, Opinion dynamics: Statistical physics and beyond, arXiv preprint arXiv:2507.11521 (2025)

Pith/arXiv arXiv 2025
[9]

D. G. Clark and L. F. Abbott, Theory of coupled neuronal-synaptic dynamics, Physical Review X14, 021001 (2024)

2024
[10]

V´ as´ arhelyi, C

G. V´ as´ arhelyi, C. Vir´ agh, G. Somorjai, T. Nepusz, A. E. Eiben, and T. Vicsek, Optimized flocking of autonomous drones in confined environments, Science Robotics3, eaat3536 (2018)

2018
[11]

Baumann, I

F. Baumann, I. M. Sokolov, and M. Tyloo, Periodic cou- pling inhibits second-order consensus on networks, Phys- ical Review E102, 052313 (2020)

2020
[12]

Farotimi, A

O. Farotimi, A. Dembo, and T. Kailath, Neural network weight matrix synthesis using optimal control techniques, Advances in Neural Information Processing Systems2 (1989)

1989
[13]

El Chamie and T

M. El Chamie and T. Ba¸ sar, Optimal strategies for dy- namic weight selection in consensus protocols in the pres- ence of an adversary, in53rd IEEE Conference on Deci- sion and Control(IEEE, 2014) pp. 735–740

2014
[14]

Baumann, I

F. Baumann, I. M. Sokolov, and M. Tyloo, A laplacian approach to stubborn agents and their role in opinion formation on influence networks, Physica A: Statistical Mechanics and its Applications557, 124869 (2020)

2020
[15]

Tyloo, Layered complex networks as fluctuation amplifiers, Journal of Physics: Complexity3, 03LT01 (2022)

M. Tyloo, Layered complex networks as fluctuation amplifiers, Journal of Physics: Complexity3, 03LT01 (2022)

2022
[16]

Marino and P

R. Marino and P. Tomei, Adaptive control of linear time- varying systems, Automatica39, 651 (2003)

2003
[17]

A. D. Belegundu, Lagrangian approach to design sensi- tivity analysis, Journal of Engineering mechanics111, 680 (1985)

1985
[18]

M. B. Giles and N. A. Pierce, An introduction to the adjoint approach to design, Flow, turbulence and com- bustion65, 393 (2000)

2000
[19]

Salfenmoser and K

L. Salfenmoser and K. Obermayer, A framework for opti- mal control of oscillations and synchrony applied to non- linear models of neural population dynamics, Frontiers in Computational Neuroscience18, 1483100 (2024)

2024
[20]

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)

Pith/arXiv arXiv 2014
[21]

R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. Abarbanel, Synchronous be- havior of two coupled biological neurons, Physical Review Letters81, 5692 (1998)

1998
[22]

S. T. Keene, C. Lubrano, S. Kazemzadeh, A. Melianas, Y. Tuchman, G. Polino, P. Scognamiglio, L. Cin` a, A. Salleo, Y. van de Burgt,et al., A biohybrid synapse with neurotransmitter-mediated plasticity, Nature Mate- rials19, 969 (2020)

2020
[23]

S. Fu, H. Gao, S. Wang, X. Wang, T. Woodard, Z. Wang, J. Kong, D. R. Lovley, and J. Yao, Constructing artifi- cial neurons with functional parameters comprehensively matching biological values, Nature Communications16, 8599 (2025)

2025

[1] [1]

(1) for- ward from the initial conditionx(t= 0) =x (0)

Solve the state dynamics of the system Eq. (1) for- ward from the initial conditionx(t= 0) =x (0)

[2] [2]

(7) backward in time starting from the terminal conditionλ(T) = 0

Use the forward state solution to solve the evolu- tion of the adjoint state Eq. (7) backward in time starting from the terminal conditionλ(T) = 0

[3] [3]

(4) using the adjoint state and update the control signalU

Compute the gradient Eq. (4) using the adjoint state and update the control signalU

[4] [4]

This procedure is usually called theadjoint method[13, 14] , and has been widely used in recent application such as machine learning

Go back to step 1 and repeat until convergence. This procedure is usually called theadjoint method[13, 14] , and has been widely used in recent application such as machine learning. See Ref. [15] for an overview of its application for additive control inputs in neural popula- tion models. Many different algorithms exist to perform the opti- mization of th...

[5] [5]

Pikovsky, M

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- tion, Cambridge university press12(2001)

2001

[6] [6]

Newman,Networks(Oxford university press, 2018)

M. Newman,Networks(Oxford university press, 2018)

2018

[7] [7]

Holme and J

P. Holme and J. Saram¨ aki, Temporal networks, Physics reports519, 97 (2012)

2012

[8] [8]

Starnini, F

M. Starnini, F. Baumann, T. Galla, D. Garcia, G. I˜ niguez, M. Karsai, J. Lorenz, and K. Sznajd-Weron, Opinion dynamics: Statistical physics and beyond, arXiv preprint arXiv:2507.11521 (2025)

Pith/arXiv arXiv 2025

[9] [9]

D. G. Clark and L. F. Abbott, Theory of coupled neuronal-synaptic dynamics, Physical Review X14, 021001 (2024)

2024

[10] [10]

V´ as´ arhelyi, C

G. V´ as´ arhelyi, C. Vir´ agh, G. Somorjai, T. Nepusz, A. E. Eiben, and T. Vicsek, Optimized flocking of autonomous drones in confined environments, Science Robotics3, eaat3536 (2018)

2018

[11] [11]

Baumann, I

F. Baumann, I. M. Sokolov, and M. Tyloo, Periodic cou- pling inhibits second-order consensus on networks, Phys- ical Review E102, 052313 (2020)

2020

[12] [12]

Farotimi, A

O. Farotimi, A. Dembo, and T. Kailath, Neural network weight matrix synthesis using optimal control techniques, Advances in Neural Information Processing Systems2 (1989)

1989

[13] [13]

El Chamie and T

M. El Chamie and T. Ba¸ sar, Optimal strategies for dy- namic weight selection in consensus protocols in the pres- ence of an adversary, in53rd IEEE Conference on Deci- sion and Control(IEEE, 2014) pp. 735–740

2014

[14] [14]

Baumann, I

F. Baumann, I. M. Sokolov, and M. Tyloo, A laplacian approach to stubborn agents and their role in opinion formation on influence networks, Physica A: Statistical Mechanics and its Applications557, 124869 (2020)

2020

[15] [15]

Tyloo, Layered complex networks as fluctuation amplifiers, Journal of Physics: Complexity3, 03LT01 (2022)

M. Tyloo, Layered complex networks as fluctuation amplifiers, Journal of Physics: Complexity3, 03LT01 (2022)

2022

[16] [16]

Marino and P

R. Marino and P. Tomei, Adaptive control of linear time- varying systems, Automatica39, 651 (2003)

2003

[17] [17]

A. D. Belegundu, Lagrangian approach to design sensi- tivity analysis, Journal of Engineering mechanics111, 680 (1985)

1985

[18] [18]

M. B. Giles and N. A. Pierce, An introduction to the adjoint approach to design, Flow, turbulence and com- bustion65, 393 (2000)

2000

[19] [19]

Salfenmoser and K

L. Salfenmoser and K. Obermayer, A framework for opti- mal control of oscillations and synchrony applied to non- linear models of neural population dynamics, Frontiers in Computational Neuroscience18, 1483100 (2024)

2024

[20] [20]

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)

Pith/arXiv arXiv 2014

[21] [21]

R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. Abarbanel, Synchronous be- havior of two coupled biological neurons, Physical Review Letters81, 5692 (1998)

1998

[22] [22]

S. T. Keene, C. Lubrano, S. Kazemzadeh, A. Melianas, Y. Tuchman, G. Polino, P. Scognamiglio, L. Cin` a, A. Salleo, Y. van de Burgt,et al., A biohybrid synapse with neurotransmitter-mediated plasticity, Nature Mate- rials19, 969 (2020)

2020

[23] [23]

S. Fu, H. Gao, S. Wang, X. Wang, T. Woodard, Z. Wang, J. Kong, D. R. Lovley, and J. Yao, Constructing artifi- cial neurons with functional parameters comprehensively matching biological values, Nature Communications16, 8599 (2025)

2025