Liquid-Graph Time-Constant Network (LGTC)

• LGTC is a continuous-time graph neural network that adaptively modulates each agent’s dynamics using input-driven, liquid time constants.
• Its closed-form update approximates stiff ODE dynamics in a single communication round, ensuring computational efficiency and stability.
• LGTC achieves communication efficiency by selectively broadcasting only critical hidden features, leading to improved performance in large-scale flocking control.

The Liquid-Graph Time-Constant (LGTC) Network is a continuous-time graph neural network (GNN) architecture designed for distributed control of multi-agent systems. Extending the single-agent Liquid Time-Constant (LTC) network to communication graphs, LGTC introduces agent-specific, input-driven time constants via graph-filtered gating, and provides both ODE-based and closed-form state evolution. Its core innovations include a stable, contractive update rule, communication-efficient message-passing, and empirical validation in large-scale flocking control tasks.

1. Mathematical Formulation of the LGTC Layer

Given an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ of $N$ agents, each maintains a hidden state $x_i(t)\in\mathbb{R}^F$ and receives local input $u_i(t)\in\mathbb{R}^G$. The support matrix $S\in\mathbb{R}^{N\times N}$ encodes the (possibly time-varying) communication topology.

The LGTC layer is governed by the following continuous-time ODE:

$\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$

where:

• $\rho(\cdot)$ is the pointwise ReLU,
• $\sigma_c(\cdot)$ is the pointwise $\tanh$,
• “$\circ$” denotes the Hadamard product,
• $N$0 and $N$1 are graph filters of length $N$2 with learned weight matrices,
• $N$3 are positive agent-wise bias maps.

Every hidden component $N$4 is modulated by an adaptive “liquid time constant” $N$5 determined by graph-aggregated states and inputs.

2. Closed-Form LGTC Update

Integrating the stiff ODE over $N$6 at each step is computationally expensive. A closed-form, single-step approximation, preserving the contraction rate, is constructed as follows: $N$7 where:

• $N$8 denotes the logistic sigmoid,
• $N$9 is the derivative of the ReLU term,
• $x_i(t)\in\mathbb{R}^F$0 prevents division by zero,
• $x_i(t)\in\mathbb{R}^F$1 is a small stabilizing constant.

This update yields $x_i(t)\in\mathbb{R}^F$2 in a single communication round, with a contraction rate matching that of the ODE.

3. Stability via Contraction Analysis

The stability of the LGTC system is grounded in contraction theory. The induced $x_i(t)\in\mathbb{R}^F$3-log-norm is defined as

$x_i(t)\in\mathbb{R}^F$4

A vector field $x_i(t)\in\mathbb{R}^F$5 is $x_i(t)\in\mathbb{R}^F$6-contractive if

$x_i(t)\in\mathbb{R}^F$7

Theorem (δISS of LGTC–ODE):

Under bounded $x_i(t)\in\mathbb{R}^F$8-norm and $x_i(t)\in\mathbb{R}^F$9, if

$u_i(t)\in\mathbb{R}^G$0

then, for all solutions $u_i(t)\in\mathbb{R}^G$1, $u_i(t)\in\mathbb{R}^G$2 with the same $u_i(t)\in\mathbb{R}^G$3 but different inputs or initial states,

$u_i(t)\in\mathbb{R}^G$4

Thus, the LGTC dynamics are incrementally input-to-state stable (δISS) under suitable norm bounds on graph filter weights and biases.

A supporting lemma states: if $u_i(t)\in\mathbb{R}^G$5 and all $u_i(t)\in\mathbb{R}^G$6, then $u_i(t)\in\mathbb{R}^G$7 whenever $u_i(t)\in\mathbb{R}^G$8.

4. Communication-Efficient Message Passing

LGTC achieves communication efficiency through selective message broadcasting: each agent communicates only a subset $u_i(t)\in\mathbb{R}^G$9 of its hidden features, and only $S\in\mathbb{R}^{N\times N}$0 input channels if required. The graph filter $S\in\mathbb{R}^{N\times N}$1 is computed with $S\in\mathbb{R}^{N\times N}$2 successive 1-hop exchanges; lowering $S\in\mathbb{R}^{N\times N}$3 reduces per-edge payload. The adaptive time-constant term $S\in\mathbb{R}^{N\times N}$4 is locally computable without exchanging additional gating variables, unlike in standard GNNs (e.g., GGNN) where all hidden and gate vectors are broadcast. This design constrains per-step communication to $S\in\mathbb{R}^{N\times N}$5.

5. Empirical Evaluation in Flocking Control

The LGTC network is evaluated in decentralized flocking, modeling agents as double-integrators in $S\in\mathbb{R}^{N\times N}$6, updated discretely as $S\in\mathbb{R}^{N\times N}$7, $S\in\mathbb{R}^{N\times N}$8 with $S\in\mathbb{R}^{N\times N}$9 s. Communication links are determined by proximity ($\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$0), with $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$1 and team size $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$2 varied across experiments. The centralized expert implements a leader-follower control policy based on global velocity averaging and collision avoidance.

Each agent receives a $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$3-dimensional input vector, processes it through a single LGTC (or alternative) layer (hidden size $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$4, filter length $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$5, $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$6 communicated dims), and outputs control via a readout MLP. All models are regularized for contraction using a Softplus penalty.

Training follows the DAGGER paradigm over 60 expert trajectories, with the Adam optimizer, and mean-squared error loss between predicted and expert controls.

Results:

• Scalability: For $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$7 m, LGTC and its closed-form variant (CfGC) reduce flocking error by $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$830–40% and leader error by $\begin{cases} f(x,u,S) = \rho(\hat{A}_S(x) + b_x) + \rho(\hat{B}_S(u) + b_u), \[6pt] \dot{x} = -\left(b + f(x,u,S)\right)\circ x - \sum_{k=1}^K S^k x A_k + f(x,u,S)\circ \sigma_c(B_S(u)), \end{cases} \tag{1} \label{LGTC-ODE}$910% compared to GGNN, while GraphODE performs worst. LGTC/CfGC performance is near-identical, confirming closed-form fidelity.
• Communication Range Robustness: All methods degrade at $\rho(\cdot)$0 m, flocking improves for $\rho(\cdot)$1 m but leader tracking worsens. LGTC/CfGC maintain closest adherence to expert policy under range variation.
• Communication efficiency: LGTC/CfGC outperform or match baselines with dramatically fewer exchanged features.

Model	Mean Flocking Error	Leader Tracking Error	Comm. Dims per Edge
LGTC/CfGC	Lowest	Lowest	$\rho(\cdot)$2
GGNN	Higher	Higher	$\rho(\cdot)$3
GraphODE	Highest	Highest	$\rho(\cdot)$4

6. Implementation Details and Hyperparameters

One step of the discrete (closed-form) LGTC update is:

$\sigma_c(\cdot)$2

Key hyperparameters include hidden size $\rho(\cdot)$5, communicated dims $\rho(\cdot)$6, filter length $\rho(\cdot)$7, step $\rho(\cdot)$8, bias initializations $\rho(\cdot)$9, $\sigma_c(\cdot)$0, $\sigma_c(\cdot)$1, and contraction margin in the Softplus regularization.

7. Significance and Context

LGTC advances multi-agent control by enabling each agent’s state evolution to depend adaptively on both local and graph-filtered signals, via liquid time constants. The closed-form update achieves the expressivity of continuous-time dynamics with the computational tractability and communication frugality needed for large-scale distributed deployment. LGTC consistently outperforms discrete models in challenging flocking control tasks, with strong theoretical stability guarantees grounded in contraction analysis. These properties position LGTC as a theoretically principled and practically scalable approach to distributed learning and control on graphs (Marino et al., 2024).