Steering the dynamics by controlling the temporal interaction network

arXiv:2607.04998 · nlin.AO, physics.soc-ph · Submitted 2026-07-06 · Read on arXiv

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Introduction to the show: ident: Nonlinear Sciences. The day in nonlinear science: chaos, patterns, networks and adaptation.

Kai: We're looking at the paper: "Steering the dynamics by controlling the temporal interaction network".

Mira: Here's the gist of the paper. Many real-world coupled dynamical systems have the interaction structure and strength that evolve or adapt over time <ref:2607.04998#pg2> Here, we investigate how one can control the state of a system by tuning its temporal interaction network <ref:2607.04998#pg2>.

Kai: Theoretical Framework The paper introduces a framework based on nonlinear optimal control to steer the dynamics of a system by controlling its coupling matrix <ref:2607.04998#pg2> This framework uses the adjoint method to derive the gradient of the Lagrangian function, which is essential for designing control strategies <ref:2607.04998#pg2>.

Mira: The general form of nodal dynamics is given by x˙ i = Fi(xi) +X N j=1 Wij (t)Hi(xi, xj), i = 1,, (1) <ref:2607.04998#pg3> The control input is assumed to be Wij (t) = Uij (t), meaning both self-interaction and coupling are controlled <ref:2607.04998#pg4>.

Kai: The total performance metric is defined as C(x, U) = Cu(U) + Ce(x), which accounts for control effort and task achievement <ref:2607.04998#pg5>.

Title and authors: Kai: So, let's talk about the title itself, "Steering the dynamics by controlling the temporal interaction network." It sounds a bit abstract, doesn't it?

Mira: It is definitely abstract because it focuses on controlling the entire structure of how nodes interact over time rather than just nudging individual nodes.

Lev: Basically, it means you’re not just adjusting one parameter in node i; you’re changing all the connections W ij(t) simultaneously to get the desired behavior.

Kai: And they go on to show that this is done using a nonlinear optimal control framework, which is a big step because traditional additive control doesn't handle this multiplicative nature as cleanly.

Mira: They use the adjoint method specifically to derive the gradient of the Lagrangian function, which gives them that necessary expression for designing those control strategies.

Lev: I always like seeing it applied to linear time-variant dynamics first, which simplifies things enough to actually simulate and test before moving into more complex flows.

Kai: The paper focuses on this linear case, where the dynamics are i = alpha (x* i - x i) + sum j=one N W ij(t)x j <ref:2607.04998#pg1>.

Mira: And they define the coupling function H i(x i, x j) as simply the state of node j, so the dynamics become i = alpha (x* i - x i) + sum j=one N W ij(t)x j <ref:2607.04998#pg1>.

Lev: So they assume both self-interaction and coupling are controllable, meaning W ij(t) is the input we manipulate. That's a strong assumption for real hardware, though.

Kai: Well, it’s what they assume to show the framework works in principle before we worry about physical constraints like energy or bandwidth limitations on those controls.

Mira: And their performance metric C* e is that integral of the squared difference between the actual state x j and a desired trajectory j, summed over all N nodes.

Lev: That means they're trying to minimize the total squared error across every single node simultaneously, which is a very demanding task for control algorithms.

Kai: It’s interesting because it moves beyond just trying to fix one node; they are steering the entire network toward a collective goal defined by those targets j.

Mira: And the core idea is that you can tune these temporal coupling weights W ij(t) to achieve this, which is where the real control comes from.

Lev: It’s about finding a time-dependent function for all those weights that makes the system follow the target trajectory as closely as possible.

Kai: So, we're moving from static network design to dynamic network steering through optimal control of those time-varying interactions.

The paper's summary: Mira: Now, let’s look at what they actually summarize in this paper. They introduce the nonlinear optimal control framework for controlling the coupling matrix of a dynamical system.

Kai: In essence, they set up the problem by defining a performance metric C(x, U) that combines control effort with task achievement.

Mira: And then they use nonlinear optimal control to find an optimal control signal U(t) by deriving the gradient of this Lagrangian function using the adjoint method.

Lev: The adjoint state evolution itself simplifies to lambda = p(x,) + lambda d g over d x, which is a crucial simplification for solving it iteratively.

Kai: The workflow they outline is a four-step loop: first solve the state dynamics forward from x(zero), then evolve the adjoint state backward from lambda(T)=zero then compute the gradient, and finally update U before repeating until convergence <ref:2607.04998#pg3>.

Mira: That iterative process is what allows them to use that gradient to adjust the control input U ij in a way that minimizes their total cost.

Lev: For their linear time-variant system, they simplify the Lagrangian further, and the gradient of the Lagrangian with respect to U ij becomes Z T zero

t, T: U ij - lambda i x j dt <ref:2607.04998#pg3,the gradient of the Lagrangian with respect to>.

Kai: So they’re using this simplified gradient expression to drive the control updates in that specific linear setting.

Mira: The paper then shows how these nodes can be steered to follow specific trajectories by controlling the coupling matrix structure, imposing various constraints on how W(t) can behave.

Lev: They explore steering nodes toward target signals, and they look at what happens when you constrain the structure of that coupling matrix in different ways.

Kai: This shows that you don't just need total control over every single connection; structuring the control input smartly can achieve the steering goal efficiently.

The paper's improvements: Mira: The paper suggests a few things to improve or extend this work, starting with how we look at the system. They suggest leveraging this framework to build more complex dynamical systems where you control the interaction instead of directly influencing the nodal dynamics.

Kai: That’s interesting because it means we could design artificial neural models with tunable synaptic connectivity that exhibit desired emergent behaviors based on network structure rather than just setting node inputs.

Lev: From a hardware perspective, I think controlling the interaction is way more powerful because it lets you control the emergent behavior of the whole system.

Mira: And they also suggest extending this network control framework to include additional constraints coming from specific applications, like those found in neuroscience, specifically controlling connectivity between neurons.

Kai: That’s a big direction because those biological systems have incredibly complex interaction rules that we can only guess at right now.

Lev: If you apply this to real neural networks, the overhead of calculating that optimal control signal U(t) would be huge, but it's worth exploring for those specific applications.

Mira: They also suggest exploring more complex systems both in terms of their dynamics and the types of inputs they can actually control.

Kai: So they are pointing toward making the system dynamics themselves more flexible, not just focusing on a fixed set of nodes and fixed interaction types.

Conclusion: Kai: So to wrap up, this paper, "Steering the dynamics by controlling the temporal interaction network," it shows how to steer coupled systems by tuning their interaction structure over time.

Mira: The main implication is that we can use nonlinear optimal control with adjoint methods to find a control signal U(t) that achieves a performance metric balancing effort and task achievement.

Lev: For practical implementation, the results show that even when you restrict the number of controlled edges to just ten percent of the total possible pair-wise couplings, precision and cost still improve if your natural states are already somewhat distributed.

Kai: So we can use this to design tunable interaction networks for AI agents that need to synchronize behaviors toward a target signal, or maybe even create multi-modal systems where different subsets converge on specialized patterns.

Mira: It’s about achieving synchronization or specialized coordination by controlling the network structure dynamically instead of just brute-forcing the nodal dynamics.

Lev: My final thought is that this paper lays a solid mathematical foundation for applying this to real hardware, even if the overhead of calculating that optimal control signal U(t) would be substantial when you move to something like a neural network connectivity problem.

Kai: It’s a lot of complex math, but the ability to steer dynamics by controlling the temporal interaction network is a concept worth keeping in mind as we build more sophisticated coupled systems.

Living Systems Institute, University of Exeter · Department of Mathematics and Statistics, Faculty of Environment, Science, and Economy, University of Exeter

nlin.AO, physics.soc-ph

Submitted: 2026-07-06

Updated: 2026-10-29

Importance score: 76/100

The gist: Many real-world coupled dynamical systems have the interaction structure and strength that evolve or adapt over time <ref:2607.04998#pg2> Here, we investigate how one can control the state of a

Key concepts

Nodal Dynamics: This describes how each individual unit (node) in the system changes its state over time. It is modeled by a differential equation that includes both internal dynamics and the influence it receives from other connected units through their interactions.
Coupling Matrix Control: Instead of directly changing how a node behaves, this method allows control over the interaction structure itself—the weights or strengths of the connections between nodes. This tuning is achieved by treating these interaction strengths as time-dependent control inputs.
Adjoint Method: This mathematical technique is used to efficiently calculate the gradient of a complex performance metric (cost function) with respect to the control inputs. It simplifies finding the optimal way to adjust the coupling matrix over time.

Terminology

Steering — 14× in this paper · explained in 2 episodes
Adjoint Method — 10× in this paper · explained in 3 episodes
Adam — 2× in this paper · explained in 3 episodes
AI Agents — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes
Sparsity — 1× in this paper · explained in 5 episodes

Summary

Many real-world coupled dynamical systems have the interaction structure and strength that evolve or adapt over time <ref:2607.04998#pg2> Here, we investigate how one can control the state of a system by tuning its temporal interaction network <ref:2607.04998#pg2>.

Theoretical Framework

The paper introduces a framework based on nonlinear optimal control to steer the dynamics of a system by controlling its coupling matrix <ref:2607.04998#pg2> This framework uses the adjoint method to derive the gradient of the Lagrangian function, which is essential for designing control strategies <ref:2607.04998#pg2>.

The general form of nodal dynamics is given by x˙ i = Fi(xi) +X N j=1 Wij (t)Hi(xi, xj), i = 1,, (1) <ref:2607.04998#pg3> The control input is assumed to be Wij (t) = Uij (t), meaning both self-interaction and coupling are controlled <ref:2607.04998#pg4>. The total performance metric is defined as C(x, U) = Cu(U) + Ce(x), which accounts for control effort and task achievement <ref:2607.04998#pg5>.

Control Signal Derivation

To obtain the optimal control signal U(t), the Lagrangian of the controlled system is defined as L(x, x˙, U) = Z T 0 1[tu,T]cu(t) + 1[te,T]ce(t) (3) + λ ⊤g(x, x˙, U) dt ′ <ref:2607.04998#pg6>. The gradient of the Lagrangian with respect to the Uij is given by dL dUij = Z T 0 1[tu,T] ∂cu/∂Uij − λ⊤ i Hi(xi, xj) dt, (4).

The adjoint state evolution is simplified to λ˙ ⊤ = p(x, x˙) + λ⊤ ∂g/∂x (7). The process for obtaining the optimal control involves a sequential procedure:

Solve the state dynamics of the system Eq. (1) forward from the initial condition x(t = 0) = x(0).
Use the forward state solution to solve the evolution of the adjoint state Eq. (7) backward in time starting from the terminal condition λ(T) = 0.
Compute the gradient Eq. (4) using the adjoint state and update the control signal U.
Go back to step 1 and repeat until convergence.

Steering to Target Trajectories

For a linear time-variant dynamics, the system is described by x˙ i = α (x∗ i − xi) +X N j=1 Wij (t)xj, i = 1,, (8). The performance metric C∗e is defined as C∗e = 1/2 Z T te X N j=1 (xj − x˜j) squared dt, (11).

The control cost is defined as Cu = wc / 2 Z T tu X N i,j=1 U squared ij (t) dt, (13). The gradient of the Lagrangian with respect to Uij simplifies to dL dUij = Z T 0 1[tu,T]Uij − λixj dt, i, j = 1,, (18).

Numerical Results and Findings

The numerical simulations demonstrate the framework's capability to steer nodes with heterogeneous intrinsic states to a common trajectory <ref:2607.04998#pg2>. When all edges weights are controlled, the common target signal is reached by all the nodes <ref:2607.04998#pg2> However, when natural states are more broadly distributed, both precision and control costs are lower compared to cases where natural states are closer to each other <ref:2607.04998#pg2>.

When reducing the number of controlled edges to 10% of the total number of potential pair-wise coupling, the precision and control costs go up <ref:2607.04998#pg2>. The activity is well steered towards the target trajectory even when only 8% of total possible pair-wise interaction is controlled <ref:2607.04998#pg2>.

Future Directions

The framework can be leveraged to build more complex dynamical systems where one controls the interaction instead of directly influencing the nodal dynamics <ref:2607.04998#pg2>. Future work should extend the network control framework to include additional constraints coming from specific applications in neuroscience, such as controlling connectivity between neurons <ref:2607.04998#pg2>. Exploring more complex systems, both in terms of their dynamics and their controllable inputs, is also suggested <ref:2607.04998#pg2>.

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Steering the dynamics by controlling the temporal interaction network Melvyn Tyloo1 Living Systems Institute, University of Exeter, Exeter, United Kingdom and Department of Mathematics and Statistics, Faculty of Environment, Science, and Economy, University of Exeter, Exeter, United Kingdom (Dated: July 7, 2026) 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 I. INTRODUCTION The collective behavior displayed by coupled dynamical systems is the result of the interplay between internal parameters of the individual components and the interaction among them [1, 2] Using complex networks, individual units are typically modeled as nodes while the interaction is represented by edges While in some cases or approximations, the interaction can be taken as timeindependent – the structure and the weight of the edges remain constant over time – it is not an accurate modelling for systems in various fields [3] For example, in social interactions, people exchange social cues with different individuals over the course of time [4]; or in neuronal networks, where synaptic connectivity evolves according to plasticity rules that depend on the activity [5] The interaction can also be modulated in engineered systems where individual units share information, e.g. in vehicular platoons or swarming robots [6] In general, the time-dependence of the interaction enables the system to access states that could not be reached on a static network [7] Here, we investigate how one can steer the dynamics of the system to desired states by controlling the interaction network Such network control strategy is typically more challenging than traditional additive control, mostly because of its multiplicative nature Previous works have used a similar approach to design networks that realize specific input-output relationships [8], or select edges weights that will enable a consensus to be reached in discrete systems [9] In this work instead, we want to find the time-dependent coupling weights that will steer the node dynamics to prescribed trajectories Using a nonlinear optimal control framework and a general form of coupled dynamical system, we define the Lagrangian function of the control problem that accounts for costs from both performing the desired task, and applying the control input Then, we derive the adjoint state evolution to obtain an expression for the gradient of the total cost which can be used to design control strategies To apply the method on a concrete example, we focus on a linear time-variant dynamics The latter, together with their time-invariant counter part are used in various fields such as opinion dynamics [4, 10], network diffusion [11], system control [12] We investigate how the activity of such networked system can be steered to desired trajectories by controlling the entire interaction, or only a fraction of it II. THEORETICAL FRAMEWORK In this section, we introduce a framework for controlling the state of a network by controlling its connectivity First we consider a general form of nodal dynamics and performance metrics and derive the gradient of the Lagrangian function Then, we apply the formalism to a linear time-variant system in order to steer its dynamics to prescribed target trajectories A.

Improvements for AI systems

Control of heterogeneous node states to common trajectories by tuning interaction strength allows AI agents to synchronize their internal behaviors, such as in multi-agent reinforcement learning or distributed robotics, toward a desired collective behavior defined by a target signal x˜i(t).
Steering nodes to distinct target signals enables the creation of complex multi-modal AI systems where different subsets of agents converge to specialized functional patterns simultaneously, useful for tasks requiring diverse coordinated outputs.
The framework can be leveraged to build more complex dynamical systems where one controls the interaction instead of directly influencing nodal dynamics, allowing for the design of artificial neural models with tunable synaptic connectivity that exhibit desired emergent behaviors.