Length-Independent Stability and $L_2$-Gain Certificates for Chains of Delayed Neural Modules

Ömer Faruk İşcanlı

cs.NEOct 18, 2026

Length-Independent Stability and $L_2$ -Gain Certificates for Chains of Delayed Neural Modules

Authors: Ömer Faruk İşcanlı

Organizations: Department of Computer Engineering, Istanbul University-Cerrahpaşa, Istanbul, Türkiye

Abstract

Networks of repeated neural modules with delayed neighbour coupling need stability guarantees for every number of modules, but delay-dependent criteria certify one network at a time and grow with its size. This paper derives stability and $L_2$ -gain certificates of length-independent size for open chains and rings of modules of two alternating types with slope-restricted activations. A local and a transmission delay act independently and are only bounded and continuous. A Lyapunov-Krasovskii functional with fixed integration limits and module-local variables that repeat with the module type reduces the certificate of every chain to one cell of two modules. Local conditions and a single linear matrix inequality of order 96 on this cell, or 104 for the gain, certify global asymptotic stability and a common $L_2$ -gain bound for all open chains and all rings with an even number of at least four modules. Up to a margin, the stability inequality is also necessary for the repeated variables to certify every length. All uniform certificates, including certified coupling regions, are verified in exact rational arithmetic. In the numerical study, the certified coupling reaches 92% of an outer bound given by unstable members of the model class for short delays and 79% at the nominal delay bounds. For 16 modules at unit coupling strength, certificates of the same functional computed for that length alone raise the certified delay scale by 2% to 7% in four cases. A single stability problem with 1298 scalar variables thus replaces per-length certification of long chains at little cost.

Figures & tables

Figure 1: Open chain of modules of types 1 and 2. Loops mark the local feedback through

A_{t}

and

B_{1,t}

, arrows the delayed couplings

F_{f}

and

F_{b}

, and dashed boxes the cells. In cell order, the certificate matrix is block tridiagonal with the cell blocks

\Xi_{0}

and

\Xi_{1}

(Lemma 3 ).

Figure 2: (a) Certified region in the

(\kappa,s)

plane for

(F_{f},F_{b})=(\kappa\hat{G},-\kappa\hat{G}^{\top})

. The shaded set is the union of the exactly verified rectangles

[\kappa_{-},\kappa_{+}]\times[0,s]

of Corollary 1 and holds for every network in

\mathcal{N}

, and its gray part is certified also for

T_{ab}=0

. Markers give the couplings at which unstable members are found for constant delays (

\kappa_{\mathrm{out}}

), and the outer staircase through them bounds

\kappa_{\mathrm{safe}}

from above (Section 4.5 ). (b) Ratio of the certified gain

\hat{\gamma}

to the lower bound

\gamma_{\mathrm{lb}}

on the coupling cells of (a) for three delay scales.

Figure 3: Certified bounds

\bar{\tau}_{1}

on the local delay and

\bar{\tau}_{2}

on the transmission delay for

\kappa=1

. Each marker, including the diamond for the common delay bound

\bar{d}^{\star}=15/16

of Section 5.6 , is an exactly verified certificate of Theorem 2 and covers the box between it and the origin. The shaded set is the union of these boxes, and its gray part is certified also for

T_{ab}=0

Figure 4: (a) Price of uniformity

s^{\star}_{N}/s^{\star}

for block-central and central certificates. (b) Block-central gain bound

\gamma_{N}

for I1 relative to

\hat{\gamma}

. (c) Wall times at

\kappa=1

and

s=1

, median and range of five runs. At

N=64

, the construction of the chordal problem exceeds the address-space limit.

	I1	I2	I3
$\hat{\gamma}^{2}$ , Theorem 2	$241/50$	$77/20$	$667/100$
$\hat{\gamma}$ , Theorem 2	$2.196$	$1.963$	$2.583$
$\hat{\gamma}$ , Theorem 2 with $T_{ab}=0$	$6.849$	$7.881$	$15.621$
$\hat{\gamma}$ , Remark 3	$2.115$	$1.908$	–
$\gamma_{\mathrm{lb}}$	$1.419$	$1.346$	$1.586$
$\gamma_{\mathrm{lb}}^{\mathrm{tv}}$	$1.653$	$1.525$	$1.732$

Table 1:

L_{2}

-gain bounds at

s=1

. Upper bounds are rounded up and lower bounds down. The variant of Remark 3 was not computed for I3.

	Zhai et al. (2025)	Thm. 1		Rem. 3	Thm. 2	Rem. 3
$N$	$2$	$2$	$4$	$2$	all	all
$\kappa=1/2$	$\geq 8$	$21/4$	$61/16$	$231/32$	$17/8$	$81/32$
$\kappa=1$	$11/2$	$25/8$	$7/4$	$137/32$	$15/16$	$33/32$
$\kappa=5/4$	$63/16$	$39/16$	$5/4$	$53/16$	$21/32$	$11/16$
variables	2544	110	396	136	1298	4862
LMI order	76	28	56	28	96	96

Table 2: Certified common delay bound

\bar{d}^{\star}

and problem size. Values for one

N

are verified in double precision and values for all

N

exactly. The bound

8

is the search limit. The variant of Remark 3 for all

N

has four LMIs of order 96.

Neural Network Stability

See all 37 papers

Delayed Verification Destabilizes Multi-Agent LLM Belief: Instability Thresholds and Optimal Corrector Placement
Jun 25, 2026Igor ItkinAI Agent Reliability Multi-Agent Coordination
Independent Researcher, Tel Aviv, Israel
Scalable Incremental Robustness Analysis of Neural Network Feedback Systems
Sep 18, 2026Zichen Wang, Peter Seiler, Geir Dullerud +1Neural Network Robustness Feedback Control
Coordinated Science Laboratory and the Department of Electrical and Computer Engineering, University of Illinois Urbana–Champaign · Department of Electrical Engineering and Computer Science, at the University of Michigan · Electrical and Computer Engineering Department at the University of Minnesota
$(\text{DNN})^2$ : Doubly Non-Negative Relaxations for Deep Neural Networks
Aug 25, 2026Hanna Jiamei Zhang, Alan Papalia, Michael Everett +1Neural Network Verification ReLU Neural Networks
Departments of Computer Science, Electrical and Computer Engineering, and Mathematics Northeastern University, Boston, MA. · Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI.

Jun 25, 2026J/K move · Enter open · S save

Delayed Verification Destabilizes Multi-Agent LLM Belief: Instability Thresholds and Optimal Corrector Placement

Igor Itkin

Independent Researcher, Tel Aviv, Israel

AI Agent Reliability Multi-Agent Coordination LLM Hallucination Mitigation LLM Agent Verification Multi-Agent Consensus Multi-Agent LLMs+1

Open paper AskPDF arXiv

Multi-agent large language model (LLM) systems often rely on verifier and critic agents to suppress hallucinations, but verification is delayed. During this delay, false claims can propagate through the agent network. We model this process as delayed consensus on a graph with grounded corrector nodes. Spectral decomposition by the grounded Laplacian yields a closed-form stability threshold for the verification dose: correction that is too strong or too delayed can turn consensus into oscillation. The most unstable regime occurs when the communication and verification delays coincide; for delay two, the threshold is the inverse golden ratio. The same framework gives a supermodular placement objective and a greedy (1-1/e)-approximation rule for assigning a limited corrector budget to influential nodes. Experiments across five open models confirm the predicted dose-delay oscillations. By contrast, grounded factual answering makes truth an absorbing boundary and eliminates the effect, suggesting that the instability is specific to signed-belief tasks while grounded verification remains stabilizing

Abstract

Neural Network Stability

Explore similar work

Delayed Verification Destabilizes Multi-Agent LLM Belief: Instability Thresholds and Optimal Corrector Placement

Scalable Incremental Robustness Analysis of Neural Network Feedback Systems

(DNN)2(\text{DNN})^2(DNN)2: Doubly Non-Negative Relaxations for Deep Neural Networks

$(\text{DNN})^2$ : Doubly Non-Negative Relaxations for Deep Neural Networks