Innovation–Distress Coupling in Multiplex Industrial Networks: Knowledge Diffusion, Failure Risk, and Spectral Thresholds

Abstract

Innovation can strengthen industrial resilience, but in interconnected networks, it may also create financial fragility. This paper develops a coupled nonlinear multiplex-network model in which financial distress and effective knowledge evolve on risk-propagation and knowledge-diffusion layers. Innovation success depends on firm capability, industry conditions, and accessible knowledge; success lowers susceptibility and strengthens recovery, whereas failure adds distress pressure. We establish positive invariance and well-posedness. For the source-free knowledge and conditional unforced distress subsystems, global threshold results show extinction at or below the corresponding spectral threshold and a unique positive equilibrium above it. For the fully coupled unforced system, comparison arguments provide sufficient extinction and persistence bounds. We further show that improved knowledge weakly lowers the conditional financial distress persistence threshold and we derive a local innovation–failure boundary. Extended simulations examine directed asymmetry, rank alignment, edge overlap, and numerical and parameter uncertainty. Capability and supportive industry conditions reduce long-run distress, failure losses attenuate innovation benefits, and favorable knowledge-capability allocation among structurally influential firms is associated with greater resilience. These findings are model-generated mechanisms rather than empirically identified policy optima.

1. Introduction

Industrial innovation creates a resilience paradox. Interfirm collaboration can broaden firms’ knowledge bases and strengthen their adaptive capabilities [1,2,3,4], yet innovation projects require financial and organizational commitments before their outcomes become observable. Empirical evidence shows that difficulties in R&D cooperation can delay or terminate innovation projects, while internal and external financial constraints increase the probability of project abandonment [5,6]. Once unsuccessful innovation weakens liquidity or disrupts production, its consequences may spread beyond the innovating firm: supplier-specific shocks can reduce customers’ output and market value, and production linkages can transmit disruptions across firms and national borders [7,8]. The central question is therefore not simply whether firms should innovate, but when innovation capability reduces persistent financial distress and when innovation–failure risk erodes this stabilizing effect.

This question is inherently multiplex. Industrial firms are connected through a knowledge-diffusion layer formed by R&D collaboration, technical exchange, personnel mobility, and supplier learning, and through a risk-propagation layer formed by production dependence, trade credit, contractual obligations, and financing exposure. The knowledge layer can improve innovation performance and adaptive capacity by expanding access to external expertise [1,2,3,4], whereas the risk layer can transform local operating or liquidity pressure into network-wide distress [7,8,9,10]. Because dynamics on interacting layers cannot generally be inferred from either layer in isolation [11,12], systemic resilience depends on how knowledge access and financial exposure are distributed across the same population of firms.

The innovation literature explains why the benefits of networked knowledge are heterogeneous. Open-innovation research shows that firms benefit from external technological sources and collaborative search [1,2], while absorptive capacity and dynamic capability studies emphasize that the value of external knowledge depends on firms’ ability to recognize, assimilate, and apply it [3,4]. Research on unsuccessful innovation further demonstrates that cooperation difficulties and financial barriers can prevent firms from completing planned projects [5,6]. Recent empirical work also links innovation-network participation to cross-industry knowledge diffusion [13]. However, this literature primarily evaluates innovation at the project or firm level and pays limited attention to how capability heterogeneity, capability–industry mismatch, project failure, and liquidity pressure affect the stability of an interconnected industrial system.

A parallel literature examines how operational and financial shocks propagate through networks. Production-network studies show how firm-level disruptions generate aggregate fluctuations through input dependence and supplier–customer linkages [7,8,9,10]. Financial-network research highlights exposure concentration, structural centrality, and interconnectedness [14,15,16,17]. More recent work studies bank–firm–asset multilayer systems, cross-market spillovers, and cyclic amplification across financial layers [18,19,20,21,22], while two-layer supply-chain models couple disruption risk with information or awareness processes [23]. Endogenous network formation research further shows that systemic risk can depend on the links strategically formed before shocks occur [24]. These studies demonstrate why multiple transmission channels, network architecture, and cross-layer structures matter, but they generally do not model an effective knowledge stock that simultaneously changes innovation success, financial susceptibility, recovery, and failure-induced liquidity pressure.

Coupled awareness–epidemic and resource–epidemic models provide the closest mathematical analogs because a second endogenous process modifies a primary contagion process [25,26,27,28,29]. Cooperative and interacting multilayer spreading models further show that cross-layer reinforcement can alter persistence thresholds and equilibrium regimes [30]. These frameworks provide the mathematical baseline for the present study rather than a novelty claim in themselves. The coupling considered here differs in its economic and dynamical role: effective knowledge is capability-dependent and simultaneously affects innovation success, effective susceptibility, and recovery; financial distress erodes subsequent knowledge accumulation; and unsuccessful innovation generates an additional distress pressure channel. Global-value-chain analysis also highlights the distinct meanings of incoming and outgoing structural positions in production systems [31]. Table 1 therefore compares the present model with the closest contagion frameworks and identifies which results rely on established theory and which arise from the innovation–knowledge coupling.

Table 1. Mathematical comparison with related coupled contagion frameworks.

Framework	States/Layers	Coupling Mechanism	Threshold/Stability Result	Incremental Distinction from the Present Model
Classical network and network-spreading models [32,33,34,35]	One contagion state; one layer	Fixed susceptibility and recovery	Spectral extinction–persistence threshold	No endogenous knowledge or innovation–failure channel
Awareness–contagion models [25,26,27,28]	Infection and awareness; two layers	Awareness modifies susceptibility or behavior	Awareness-conditioned epidemic thresholds and coexistence regimes	Awareness is not capability-dependent knowledge and creates no failure loss
Resource–epidemic models [29]	Infection and resources; interacting layers	Resources affect recovery; infection affects resources	Resource-dependent thresholds and equilibria	No capability–industry conversion or innovation success–failure trade-off
Cooperative multilayer spreading [30]	Multiple spreading states; multilayer	Cross-layer reinforcement or transmission	Coupling-dependent extinction, persistence, or coexistence	Both states spread; neither is an effective knowledge stock
Multilayer financial/supply-chain models [18,19,20,21,22,23]	Defaults, losses, or disruptions; multiple exposure layers	Cross-layer risk transmission, feedback, and overlap	Contagion measures, stress tests, or layer-specific conditions	No endogenous knowledge accumulation or innovation success/failure
Endogenous network formation [24]	Endogenous links and systemic risk states	Link choices alter subsequent exposure	Systemic risk depends on equilibrium network structure	Endogenizes links, but not knowledge–distress dynamics
Present study	Financial distress and effective knowledge; non-identical layers	Knowledge affects success, susceptibility, and recovery; distress erodes learning; failure adds pressure	Subsystem global thresholds; full-system sufficient bounds; forced equilibrium existence and local stability	Knowledge-dependent distress threshold, bidirectional feedback, innovation–failure boundary, and controlled cross-layer tests

Table 1 also clarifies the mathematical status of the results developed below. The generic extinction–persistence structure of the source-free knowledge subsystem and the fixed-knowledge unforced distress subsystem follows established theory for irreducible cooperative and strictly sublinear network systems; the paper does not claim this generic threshold machinery as new. The model-specific analytical contribution begins when knowledge is allowed to change the financial-risk operators: the conditional financial distress persistence threshold becomes an endogenous function of the knowledge state, innovation participation generates a success–failure boundary, and monotonicity of the conditional system yields sufficient comparison bounds for the fully coupled unforced model.

To address this gap, this paper develops a coupled nonlinear multiplex-network model for heterogeneous industrial systems. Each firm is characterized by financial distress intensity and an effective knowledge stock. Knowledge accumulates through internal innovation input and interfirm diffusion on a collaboration layer, while financial distress evolves through recurrent healthy–distressed transitions on a risk exposure layer. Innovation success is modeled as a bounded function of firm capability, industry conditions, and accessible knowledge. Successful innovation and accumulated knowledge reduce effective contagion and enhance recovery, whereas unsuccessful innovation generates additional liquidity pressure. Unlike recent stabilization-oriented network models that focus on rescue strategies or support allocation after distress emerges [36,37], the proposed framework endogenizes how innovation outcomes connect the knowledge and risk layers.

The paper makes three contributions. First, it formulates a bidirectionally coupled risk–knowledge dynamical system in which effective knowledge is more than a protective auxiliary state: knowledge changes innovation success, effective susceptibility, and recovery simultaneously, innovation–failure adds a distinct distress hazard, and financial distress erodes subsequent learning. Second, the paper explicitly separates established threshold machinery from model-specific analytical results. Theorems 3 and 4 use standard cooperative and strictly sublinear-system arguments to characterize two deliberately isolated subsystems. Building on that baseline, Proposition 1 proves that the conditional financial distress persistence threshold $R_D\left(y\right)$ is ordered by the knowledge state, while Theorem 6 converts this monotonicity into sufficient extinction and persistence bounds for the fully coupled unforced system without claiming a sharp threshold for the forced benchmark. Third, Proposition 3 derives a local innovation–failure boundary that balances failure-induced liquidity pressure against the preventive and recovery benefits of successful innovation. The numerical design complements these results by separating rank alignment from edge overlap, testing genuinely directed asymmetric networks, and quantifying numerical and parameter uncertainty. These features distinguish the present paper from the authors’ earlier single-layer stabilization and performance-propagation studies [37,38].

The remainder of the paper is organized as follows. Section 2 formulates the coupled multiplex-network model. Section 3 establishes subsystem thresholds, full-system comparison bounds, forced equilibrium existence, and the local innovation–failure boundary. Section 4 explains the calibration strategy, presents the numerical experiments and extended validation, and discusses the model-generated implications. Section 5 concludes the paper.

2. Model Formulation and Coupled Mechanism

This section formulates the coupled model used in the remainder of the paper. The model is built on the idea that knowledge diffusion and financial distress evolve on two non-identical but interacting network layers. Innovation capability and industry conditions affect knowledge absorption and innovation success, while successful and failed innovation alter the transition between healthy and distressed financial states. The formulation therefore combines a multiplex industrial network, heterogeneous firm attributes, bounded innovation success probabilities, and recurrent distress–knowledge dynamics.

Figure 1 provides a conceptual roadmap of the model. The upper part distinguishes the risk-propagation layer from the knowledge-diffusion layer, while the middle part shows how innovation capability, industry conditions, and accessible knowledge jointly affect innovation success and failure. The lower part summarizes how these mechanisms feed back into financial distress dynamics and network stability.

Figure 1. Integrated mechanism of coupled knowledge diffusion and financial distress contagion in heterogeneous industrial networks.

Guided by this mechanism, the following subsections define the multiplex-network structure, firm-level states, innovation success mechanism, and coupled distress–knowledge dynamics.

2.1. Multiplex Industrial Network

Consider an industrial system consisting of a finite set of firms $V=\left\{1,\dots ,N\right\}$. Firms are connected through two weighted directed layers. The first layer captures risk propagation through trade credit, supplier–customer dependence, contractual obligations, or financing exposure. The second layer captures knowledge diffusion through R&D collaboration, technical exchange, personnel mobility, supplier learning, or other channels of inter-organizational learning. The multiplex network is denoted by

$$\mathcal{G}=\left(V,A^R,A^K\right),\hspace{1em}\hspace{1em}{A}^R=\left(a_{ij}^R\right),\hspace{1em}{A}^K=\left(a_{ij}^K\right).$$

(1)

The convention is that $a_{ij}^R>0$ means that firm $i$ receives financial or operational pressure from firm $j$, whereas $a_{ij}^K>0$ means that firm $i$ can receive knowledge from firm $j$. The two layers need not coincide. A supplier may be strongly dependent on a focal customer without engaging in joint R&D, while two technologically related firms may exchange knowledge even when direct financial exposure is limited. Unless otherwise stated, $a_{ii}^R=a_{ii}^K=0$.

Each firm is characterized by two normalized state variables. The first is financial distress intensity, $x_i\left(t\right)\in \left[0, 1\right]$. It is a normalized continuous distress state rather than an observed default indicator and summarizes the intensity of short-term liquidity and operating pressure; larger values indicate greater financial vulnerability. The second is effective knowledge stock, $y_i\left(t\right)\in \left[0, 1\right]$, which represents knowledge that has been sufficiently absorbed to improve innovation or operational adjustment. Define the state vectors $x\left(t\right)={\left(x_1\left(t\right),\dots ,x_N\left(t\right)\right)}^{\top }$ and $y\left(t\right)={\left(y_1\left(t\right),\dots ,y_N\left(t\right)\right)}^{\top }$.

The local exposure of firm $i$ to financial distress and knowledge is given by

$$R_i\left(t\right)={\left(A^{R}x\left(t\right)\right)}_i={\displaystyle {\displaystyle \sum }_{j=1}^N}{a}_{ij}^R{x}_j\left(t\right),\hspace{1em}\hspace{1em}{K}_i\left(t\right)={\left(A^{K}y\left(t\right)\right)}_i={\displaystyle {\displaystyle \sum }_{j=1}^N}{a}_{ij}^K{y}_j\left(t\right).$$

(2)

Thus, exposure depends jointly on network position and the current state of connected counterparties. A firm connected to highly distressed or heavily weighted counterparties faces greater risk exposure, whereas a firm connected to knowledge-rich partners receives more potential learning input.

2.2. Capability, Industry Conditions, and Innovation Success

Firm $i$ has innovation capability $L_i\in \left[0, 1\right]$ and belongs to industry $s\left(i\right)$, whose normalized innovation environment is denoted by $c_{s\left(i\right)}\in \left[-1, 1\right]$. Innovation capability summarizes the firm’s ability to identify technological opportunities, combine internal and external knowledge, manage innovation projects, and convert R&D inputs into commercially viable outcomes. Industry conditions capture technological maturity, regulatory requirements, policy support, market demand, infrastructure, and the availability of complementary resources.

The joint configuration of $L_i$ and $c_{s\left(i\right)}$ determines how effectively a firm absorbs external knowledge and converts innovation inputs into successful outcomes. High-capability firms operating under supportive industry conditions are more likely to transform accessible knowledge into successful innovation. By contrast, innovation projects undertaken by low-capability firms in restrictive environments may consume liquidity without generating sufficient technological or commercial returns. To capture this heterogeneity, the effective knowledge absorption intensity of firm $i$ is specified as the bounded function

$${\tau }_i={\tau }_0\sigma \left(b_0+b_L{L}_i+b_c{c}_{s\left(i\right)}\right),\hspace{1em}\hspace{1em}\sigma \left(z\right)={\displaystyle \frac{1}{1+e^{-z}}}.$$

(3)

Here, ${\tau }_0>0$ is the maximum absorption scale, $b_L$ measures the contribution of capability, and $b_c$ captures the effect of industry conditions. The logistic mapping guarantees bounded absorption and avoids treating industry conditions themselves as unrestricted probabilities.

Let $u_i\in 0, 1$ indicate whether firm $i$ participates in innovation. The aggregate innovation participation rate is defined by

$$\lambda ={\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{i=1}^N}{u}_i.$$

(4)

Thus, $\lambda \in \left[0, 1\right]$ summarizes the population share of participating firms. The finite-network assignment procedure is described in Section 4.3. For the local comparative-static analysis in Section 3.6, $u_i$ is relaxed to [0, 1], where it represents firm-level innovation intensity.

Innovation success depends on firm capability, industry conditions, and accessible knowledge:

$$p_i\left(t\right)=\sigma \left({\theta }_0+{\theta }_L{L}_i+{\theta }_c{c}_{s\left(i\right)}+{\theta }_K{K}_i\left(t\right)\right).$$

(5)

The logistic form ensures $0<p_i\left(t\right)<1$ and permits diminishing marginal effects when capability or accessible knowledge is already high. Innovation therefore affects financial stability through two opposing channels: successful innovation improves recovery and reduces susceptibility, whereas failed innovation generates liquidity pressure and may increase distress.

Capability and industry conditions operate through two conceptually distinct channels. They affect the rate at which external knowledge is absorbed through ${\tau }_i$, and they also affect the conversion of a given knowledge exposure into a successful innovation outcome through $p_i$. The first channel governs learning intensity, whereas the second governs innovation conversion conditional on current knowledge access.

2.3. Coupled Distress and Knowledge Dynamics

Financial distress arises from persistent external pressure, contagion from distressed counterparties, and unsuccessful innovation. Recovery is supported by baseline adjustment capacity, accumulated knowledge, and successful innovation.

The local exposure $R_i\left(t\right)$ captures contemporaneous distress received from directly connected counterparties. To represent a firm’s time-invariant capacity to transmit pressure through the risk layer, let $s_i^{R,out}$ denote its weighted outgoing strength under the convention $a_{ji}^R>0$ for a link from firm $i$ to recipient firm $j.$ The normalized transmission-based risk centrality is defined by

$$s_i^{R,\mathrm{out}}={\displaystyle {\displaystyle \sum }_{j=1}^N}{a}_{ji}^R,\hspace{1em}\hspace{1em}{C}_i^R={\displaystyle \frac{s_i^{R,\mathrm{out}}-\underset{k}{\min }{s}_k^{R,\mathrm{out}}}{\underset{k}{\max }{s}_k^{R,\mathrm{out}}-\underset{k}{\min }{s}_k^{R,\mathrm{out}}}}$$

(6)

with $C_i^R$ = 0 when all nodes have identical outgoing strength. Thus,$R_i\left(t\right)$ measures current state-dependent pressure received by firm i, whereas $C_i^R$ captures its time-invariant structural influence as a source of risk transmission. The baseline contagion intensity is specified as

$${\beta }_i^0={\beta }_0\left(1+{\kappa }_C{C}_i^R\right),$$

(7)

where ${\beta }_0>0$ is the common baseline contagion intensity and ${\kappa }_C\ge 0$ measures the strength of centrality amplification. When ${\kappa }_C=0$, all firms have the same baseline contagion intensity.

Define the effective contagion rate of firm $i$ as

$${\beta }_i^{eff}\left(t\right)={\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left[1-{\varphi }_y{y}_i\left(t\right)\right]\left[1-{\varphi }_p{u}_i{p}_i\left(t\right)\right],$$

(8)

where ${\varphi }_y,{\varphi }_p\in \left[0, 1\right]$ measure the protective effects of accumulated knowledge and successful innovation, respectively. The innovation–failure hazard and effective recovery rate are

$$h_i\left(t\right)={\alpha }_i+{\xi }_i{u}_i\left[1-p_i\left(t\right)\right],\hspace{1em}\hspace{1em}{d}_i\left(t\right)={\gamma }_i+{\eta }_i{y}_i\left(t\right)+{\omega }_i{u}_i{p}_i\left(t\right),$$

(9)

where ${\alpha }_i$ is the external distress rate, ${\xi }_i$ is the liquidity loss caused by unsuccessful innovation, ${\gamma }_i$ is the baseline recovery rate, ${\eta }_i$ is the recovery contribution of knowledge, and ${\omega }_i$ is the gain from successful innovation.

The financial distress dynamics are

$${\dot{x}}_i\left(t\right)=\left[1-x_i\left(t\right)\right]\left\{h_i\left(t\right)+{\beta }_i^{eff}\left(t\right)R_i\left(t\right)\right\}-x_i\left(t\right)d_i\left(t\right).$$

(10)

The first term describes entry into distress. The factor $1-x_i\left(t\right)$ is the remaining non-distressed component of the firm, and the expression in braces combines external pressure, innovation-related failure pressure, and counterparty risk exposure. The second term describes recovery.

A firm’s effective knowledge stock increases through internal innovation activity and absorption from connected counterparties. It decreases through ordinary depreciation and the erosion of learning capacity caused by financial distress. Let ${\ell }_i$ denote internal knowledge input, ${\delta }_i$ the depreciation rate, and ${\zeta }_i$ the extent to which distress weakens R&D expenditure, employee retention, alliance participation, and absorptive capacity. The knowledge dynamics are

$${\dot{y}}_i\left(t\right)=\left[1-y_i\left(t\right)\right]\left\{{\ell }_i{u}_i+{\tau }_i{K}_i\left(t\right)\right\}-y_i\left(t\right)\left[{\delta }_i+{\zeta }_i{x}_i\left(t\right)\right].$$

(11)

Equations (10) and (11) form the coupled feedback system. Knowledge lowers future contagion and raises recovery, whereas distress weakens future learning and innovation capacity. This feature distinguishes the model from contagion models that treat recovery and susceptibility as fixed parameters.

2.4. Vector Form, Assumptions, and Notation

Let $p\left(y\right)={\left(p_1\left(y\right),\dots ,p_N\left(y\right)\right)}^{\top }$ and $h\left(y\right)={\left(h_1\left(y\right),\dots ,h_N\left(y\right)\right)}^{\top }$. Define $B\left(y\right)=\mathrm{diag}\left({\beta }_1^{eff}\left(y\right),\dots ,{\beta }_N^{eff}\left(y\right)\right)$, $D\left(y\right)=\mathrm{diag}\left(d_1\left(y\right),\dots ,d_N\left(y\right)\right)$, and $\mathcal{T}=\mathrm{diag}\left({\tau }_1,\dots ,{\tau }_N\right)$. Let $\ell ={\left({\ell }_1,\dots ,{\ell }_N\right)}^{\top }$, $\delta ={\left({\delta }_1,\dots ,{\delta }_N\right)}^{\top }$, and $\zeta ={\left({\zeta }_1,\dots ,{\zeta }_N\right)}^{\top }$. The complete node-level system is

$$\dot{x}=\left(\mathbf{1}-x\right)\odot \left[h\left(y\right)+B\left(y\right)A^{R}x\right]-D\left(y\right)x,$$

(12a)

$$\dot{y}=\left(\mathbf{1}-y\right)\odot \left[\ell \odot u+\mathcal{T}{A}^{K}y\right]-y\odot \left(\delta +\zeta \odot x\right).$$

(12b)

Here, $\odot$ denotes the Hadamard product and $\mathbf{1}$ is the $N$-dimensional vector of ones. The unforced benchmark system sets ${\alpha }_i={\xi }_i=0$ for all $i$ and is used for threshold analysis. The forced system allows ${\alpha }_i>0$ and/or ${\xi }_i>0$; in this case, the zero-distress state is generally not an equilibrium and the relevant object is a positive forced equilibrium whose level depends on the forcing intensity and network feedback.

The analysis uses the following assumptions.

Assumption 1 (Network regularity).

$A^R$ and $A^K$ are finite, non-negative, and time-invariant over the analytical horizon. When strict persistence or instability statements are made, the relevant layer is assumed to be irreducible.

Assumption 2 (Parameter admissibility).

The parameters  ${\beta }_0$, ${\gamma }_i$, ${\delta }_i$, and ${\tau }_0$ are strictly positive. All remaining rate parameters are non-negative, with ${\varphi }_y,{\varphi }_p\in \left[0, 1\right]$. The coefficients in the absorption and success functions satisfy $b_L, b_c, {\theta }_L,{\theta }_c,{\theta }_K\ge 0$.

Assumption 3 (Bounded behavioral mappings).

Knowledge absorption and innovation success follow logistic functions, so  $0<{\tau }_i<{\tau }_0$ and $0<p_i\left(t\right)<1$.

Assumption 4 (Recurrent distress).

Financial distress is reversible rather than absorbing. Bankruptcy, permanent exit, and endogenous link rewiring are outside the benchmark model.

For clarity, Table 2 summarizes the main notation used to describe the multiplex-network structure, firm-level states, innovation mechanism, and threshold quantities in the coupled model.

Table 2. Main notation used in the coupled network model.

Symbol	Meaning
$A^R$	Weighted adjacency matrix of the risk-propagation layer
$A^K$	Weighted adjacency matrix of the knowledge-diffusion layer
$x_i\left(t\right)$	Financial distress intensity of firm $i$
$y_i\left(t\right)$	Effective knowledge stock of firm $i$
$L_i$	Innovation capability of firm $i$
$c_{s\left(i\right)}$	Industry environment of firm $i$
$u_i$	Innovation participation indicator or intensity
$\lambda$	Target innovation participation rate
$C_i^R$	Normalized transmission-based risk centrality
$p_i\left(t\right)$	Innovation success probability
$R_i\left(t\right)$	Local financial-risk exposure received by firm $i$
$K_i\left(t\right)$	Local knowledge exposure received by firm $i$
${ℛ}_K$	Knowledge-diffusion threshold
${ℛ}_D$	Conditional financial distress persistence threshold

3. Mathematical Analysis

This section distinguishes mathematical properties inherited from established cooperative network systems from results generated by the specific innovation–knowledge coupling proposed here. Positive invariance and well-posedness establish basic feasibility. Theorems 3 and 4 recover the global extinction–persistence structure of two deliberately isolated subsystems—the source-free knowledge subsystem and the fixed-knowledge unforced distress subsystem—using established cooperative and strictly sublinear-system arguments; these results are analytical baselines rather than the principal mathematical novelty. The model-specific results begin with the endogenous dependence of financial fragility on knowledge: Proposition 1 orders $R_D\left(y\right)$ by the knowledge state, Theorem 5 characterizes local stability of the coupled distress-free equilibrium, Theorem 6 supplies sufficient comparison bounds for the fully coupled unforced system, and Proposition 3 derives the local innovation–failure boundary generated by the competing success and failure channels.

3.1. Positive Invariance and Well-Posedness

Define the feasible state space

$$\Omega ={\left[0,1\right]}^N\times {\left[0,1\right]}^N.$$

(13)

Theorem 1 (Positive invariance).

For every initial condition$\left(x\left(0\right),y\left(0\right)\right)\in \Omega$, the solution of system (12a)–(12b) remains in $\Omega$ for all $t\ge 0$.

Proof. 

Consider the vector field on the boundary of the state space. If $x_i=0$, then ${\dot{x}}_i=h_i+{\beta }_i^{\mathrm{eff}}{R}_i\ge 0$; if $x_i=1$, then ${\dot{x}}_i=-d_i\le 0$. Similarly, if $y_i=0$, then ${\dot{y}}_i={\ell }_i{u}_i+{\tau }_i{K}_i\ge 0$; if $y_i=1$, then ${\dot{y}}_i=-\left({\delta }_i+{\zeta }_i{x}_i\right)\le 0$. Hence, the vector field points inward or is tangent on every boundary face of $\Omega$. No trajectory starting in $\Omega$ can leave the feasible region. □

Theorem 2 (Existence and uniqueness).

For every initial condition in $\Omega$, system (12a)–(12b) admits a unique solution defined for all $t\ge 0$.

Proof. 

The logistic functions in (3) and (5) are continuously differentiable, and the remaining components of the vector field are finite sums and products of continuously differentiable functions. The vector field is therefore locally Lipschitz on an open neighborhood of $\Omega$, which guarantees a unique local solution. By Theorem 1, the solution remains in the compact set $\Omega$ and cannot diverge in finite time. Hence, the local solution extends uniquely to all $t\ge 0$. □

3.2. Global Knowledge-Diffusion Threshold

The first threshold concerns whether knowledge obtained through interfirm links can persist without a continuing internal source. To isolate this mechanism, set ${\ell }_i=0$ and $x_i=0$. The source-free knowledge subsystem is

$${\dot{y}}_i=\left(\mathbf{1}-y_i\right){\tau }_i{\left(A^{K}y\right)}_i-{\delta }_i{y}_i.$$

(14)

The knowledge-free state $y=0$ is an equilibrium. Linearization at $y=0$ gives

$$\dot{y}=\left(\mathcal{T}{A}^K-\Delta \right)y,\hspace{1em}\hspace{1em}\mathcal{T}=\mathrm{diag}\left({\tau }_i\right),\hspace{1em}\hspace{1em}\Delta =\mathrm{diag}\left({\delta }_i\right).$$

(15)

Define the knowledge-diffusion threshold as

$${ℛ}_K=\rho \left({\Delta }^{-1}\mathcal{T}{A}^K\right),$$

(16)

where $\rho (\cdot )$ denotes the spectral radius.

The source-free knowledge subsystem has the cooperative SIS-type structure studied in established network-spreading theory. Theorem 3 therefore records the resulting global threshold property as a baseline result for the coupled model rather than as a new generic threshold theorem.

Theorem 3 (Global knowledge-diffusion threshold).

Suppose that  $A^K$ is irreducible and that  ${\tau }_i>0$  and  ${\delta }_i>0$  for all  $i$. If  ${ℛ}_K\le 1$, the knowledge-free equilibrium  $y=0$  is globally asymptotically stable in  ${\left[0,1\right]}^N$. If  ${ℛ}_K>1$, the knowledge-free equilibrium is unstable, and the source-free subsystem admits a unique positive equilibrium  $y^K\gg 0$  that attracts every nonzero initial state in the feasible cube.

Proof. 

Let $M_K=\mathcal{T}{A}^K-\Delta$. The next-generation matrix argument [39] implies that the sign of the spectral abscissa $s\left(M_K\right)$ is determined by ${ℛ}_K-1$. When ${ℛ}_K<1$,

$$\dot{y}=\left(\mathbf{1}-y\right)\odot \mathcal{T}{A}^{K}y-\Delta y\le M_{K}y.$$

(17)

Because $M_K$ is Hurwitz, the comparison principle gives $y\left(t\right)\to 0$ for every feasible initial condition. At ${ℛ}_K=1$, let $q\gg 0$ be a left Perron vector satisfying $q^{\top }{M}_K=0$ and define $V\left(y\right)=q^{\top }y$. Then,

$$\dot{V}=-q^{\top }\mathrm{diag}\left(y\right)\mathcal{T}{A}^{K}y\le 0.$$

(18)

Irreducibility implies that the largest invariant subset of $\left\{y:\dot{V}=0\right\}$ is $\left\{0\right\}$, so LaSalle’s invariance principle yields $y\left(t\right)\to 0$. When ${ℛ}_K>1$, the zero equilibrium is unstable. Moreover, the vector field is strongly cooperative in the interior and strictly sublinear because, for every $0<a<1$ and $y\gg 0$,

$$f_K\left(ay\right)-a{f}_K\left(y\right)=a\left(1-a\right)\mathrm{diag}\left(y\right)\mathcal{T}{A}^{K}y\gg 0.$$

(19)

A positive equilibrium $y^K\gg 0$ exists by the sub- and supersolution argument detailed in Appendix A.2. Positive invariance and irreducibility, together with strong cooperativity and strict sublinearity, then imply its uniqueness and global attraction for every nonzero feasible initial condition [40,41].

At the critical value ${ℛ}_K=1$, convergence to the knowledge-free equilibrium is generally slower than below the threshold because the linearized system has zero spectral abscissa; extinction is secured by the nonlinear saturation term. □

3.3. Conditional Financial Distress Persistence Threshold

For any feasible knowledge state $y\in {\left[0,1\right]}^N$, define the diagonal susceptibility and recovery matrices by

$$b_i\left(y\right)={\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right)\left[1-{\varphi }_p{u}_i{p}_i\left(y\right)\right],$$

(20a)

$$B\left(y\right)=\mathrm{diag}\left(b_1\left(y\right),\dots ,b_N\left(y\right)\right),$$

(20b)

$$d_i\left(y\right)={\gamma }_i+{\eta }_i{y}_i+{\omega }_i{u}_i{p}_i\left(y\right),$$

(21a)

$$D\left(y\right)=\mathrm{diag}\left(d_1\left(y\right),\dots ,d_N\left(y\right)\right).$$

(21b)

Fix a feasible knowledge state $\overline{y}$ and impose the unforced conditions ${\alpha }_i={\xi }_i=0$. Holding $\overline{y}$ fixed, the complete conditional distress subsystem is

$$\dot{x}=\left(1-x\right)\odot B\left(\overline{y}\right)A^{R}x-D\left(\overline{y}\right)x.$$

(22)

Its linearization at the distress-free equilibrium $x=0$ is

$$\dot{x}=\left[B\left(\overline{y}\right)A^R-D\left(\overline{y}\right)\right]x+o\left(\parallel x\parallel \right).$$

(23)

For every feasible knowledge state $y$, define the conditional financial distress persistence threshold as

$${ℛ}_D\left(y\right)=\rho \left(D{\left(y\right)}^{-1}B\left(y\right)A^R\right).$$

(24)

For fixed knowledge, the conditional distress dynamics likewise reduce to an irreducible cooperative and strictly sublinear SIS-type system. Theorem 4 therefore applies established threshold machinery to the knowledge-conditioned susceptibility and recovery operators defined above.

Theorem 4 (Global conditional distress threshold).

Fix a feasible knowledge state  $\overline{y}$. Suppose that  $A^R$  is irreducible and that  $B\left(\overline{y}\right)$  and  $D\left(\overline{y}\right)$  are positive diagonal matrices. If  ${ℛ}_D\left(\overline{y}\right)\le 1$, the distress-free equilibrium  $x=0$  is globally asymptotically stable in  ${\left[0,1\right]}^N$. If  ${ℛ}_D\left(\overline{y}\right)>1$, the distress-free equilibrium is unstable, and the conditional subsystem admits a unique positive equilibrium  $x^D\left(\overline{y}\right)\gg 0$  that attracts every nonzero feasible initial state.

Proof. 

Let $M_D\left(\overline{y}\right)=B\left(\overline{y}\right)A^R-D\left(\overline{y}\right)$. The next-generation matrix construction [39] implies that the sign of $s\left(M_D\left(\overline{y}\right)\right)$ is determined by ${ℛ}_D\left(\overline{y}\right)-1$. When ${ℛ}_D\left(\overline{y}\right)<1$,

$$\dot{x}=\left(1-x\right)\odot B\left(\overline{y}\right)A^{R}x-D\left(\overline{y}\right)x\le M_D\left(\overline{y}\right)x,$$

(25)

and the comparison principle gives $x\left(t\right)\to 0$. At ${ℛ}_D\left(\overline{y}\right)=1$, a positive left Perron vector yields a Lyapunov function whose derivative is nonpositive, and irreducibility identifies $\left\{0\right\}$ as the largest invariant subset of the zero-derivative set. When ${ℛ}_D\left(\overline{y}\right)>1$, the zero equilibrium is unstable, while strong cooperativity and strict sublinearity imply the existence, uniqueness, and global attraction of $x^D\left(\overline{y}\right)\gg 0$ [40,41]. Detailed arguments are given in Appendix A.3. □

The global statement applies only to the conditional unforced distress subsystem with the knowledge state held fixed. It does not imply global stability of the fully coupled or forced system.

The model-specific implication begins when the fixed-knowledge state is allowed to vary. Unlike the generic SIS threshold result above, the present coupling makes the persistence threshold itself an endogenous function of knowledge.

Proposition 1 (Knowledge-induced reduction in the distress threshold).

Let  $y^{\prime },y\in {\left[0,1\right]}^N$ satisfy  $y^{\prime }\ge y$  componentwise. Because  $A^K$  is non-negative and  ${\theta }_K\ge 0$,

$$A^K{y}^{\prime }\ge A^{K}y,\hspace{1em}\hspace{1em}{p}_i\left(y^{\prime }\right)\ge p_i\left(y\right),\hspace{1em}\hspace{1em}B\left(y^{\prime }\right)\le B\left(y\right),$$

$$D\left(y^{\prime }\right)\ge D\left(y\right),{ℛ}_D\left(y^{\prime }\right)\le {ℛ}_D\left(y\right)$$

(26)

If $A^R$ is irreducible and at least one diagonal entry of $D{\left(y^{\prime }\right)}^{-1}B\left(y^{\prime }\right)$ is strictly smaller than the corresponding entry of $D{\left(y\right)}^{-1}B\left(y\right)$, then ${ℛ}_D\left(y^{\prime }\right)<{ℛ}_D\left(y\right)$.

Proof. 

Since $A^K\ge 0$, $y^{\prime }\ge y$ implies $K_i\left(y^{\prime }\right)\ge K_i\left(y\right)$. The logistic success function is increasing in $K_i$ when ${\theta }_K\ge 0$, so $p_i\left(y^{\prime }\right)\ge p_i\left(y\right)$. The susceptibility factor therefore weakly decreases, whereas the recovery rate weakly increases. Hence, $D{\left(y^{\prime }\right)}^{-1}B\left(y^{\prime }\right)A^R\le D{\left(y\right)}^{-1}B\left(y\right)A^R$. Perron–Frobenius monotonicity for non-negative matrices yields the weak inequality in (26). If $A^R$ is irreducible and at least one diagonal entry is strictly reduced, strict spectral radius monotonicity gives ${ℛ}_D\left(y^{\prime }\right)<{ℛ}_D\left(y\right)$. □

3.4. Local Stability of the Coupled Distress-Free Equilibrium

Let $\overline{y}$ be a locally asymptotically stable equilibrium of the knowledge subsystem at $x=0$, including the internal knowledge source $\ell \odot u$. Under the unforced benchmark, the coupled system admits the equilibrium

$${ℰ}_0=\left(0,\overline{y}\right).$$

(27)

The Jacobian at ${ℰ}_0$ has the block lower triangular form

$$J\left({ℰ}_0\right)=\left(\begin{array}{cc}B\left(\overline{y}\right)A^R-D\left(\overline{y}\right)& 0\\ J_{yx}& J_{yy}\end{array}\right).$$

(28)

The zero upper-right block arises because, at $x=0$ and in the absence of forcing, a marginal change in knowledge affects the distress equation only through terms multiplied by current distress. The lower-left block is generally nonzero because financial distress directly erodes knowledge accumulation.

Theorem 5 (Local stability of the coupled equilibrium).

Suppose that  $\overline{y}$ is locally asymptotically stable in the knowledge subsystem. If  ${ℛ}_D\left(\overline{y}\right)<1$, the coupled equilibrium  ${ℰ}_0$  is locally asymptotically stable. If  ${ℛ}_D\left(\overline{y}\right)>1$,  ${ℰ}_0$  is unstable.

Proof. 

Because $J\left({ℰ}_0\right)$ is block lower triangular, its eigenvalues are the union of the eigenvalues of $B\left(\overline{y}\right)A^R-D\left(\overline{y}\right)$ and $J_{yy}$. The second block is stable by assumption. The first block has negative spectral abscissa when ${ℛ}_D\left(\overline{y}\right)<1$ and positive spectral abscissa when ${ℛ}_D\left(\overline{y}\right)>1$. The result follows. □

No general stability conclusion is asserted for the full coupled system at ${ℛ}_D\left(\overline{y}\right)=1$. At this critical value, the Jacobian has zero spectral abscissa, and higher-order cross-layer feedback terms may determine the local behavior. The equality result in Theorem 4 applies only to the conditional subsystem with knowledge held fixed.

3.5. Comparison Bounds for the Fully Coupled Unforced System

Although the subsystem thresholds are not a sharp global threshold for the full coupled model, monotonicity of the conditional persistence threshold yields informative sufficient bounds. Consider the fully coupled unforced system with ${\alpha }_i$ = ${\xi }_i$ = 0. Because every feasible knowledge trajectory satisfies 0 ≤ y(t) ≤ 1 and Proposition 1 implies that susceptibility decreases while recovery increases with knowledge,

$$B\left(\mathbf{1}\right)\preccurlyeq B\left(y\left(t\right)\right)\preccurlyeq B\left(\mathbf{0}\right),\hspace{1em}\hspace{1em}D\left(\mathbf{0}\right)\preccurlyeq D\left(y\left(t\right)\right)\preccurlyeq D\left(1\right).$$

(29)

Define the worst- and best-protection conditional persistence thresholds by

$${ℛ}_D^{max}={ℛ}_D\left(\mathbf{0}\right),\hspace{1em}\hspace{1em}{ℛ}_D^{min}={ℛ}_D\left(1\right).$$

(30)

Theorem 6 (Comparison bounds for the fully coupled unforced system).

Suppose that  $A^R$ is irreducible and the effective susceptibility and recovery matrices at y = 0 and y = 1 satisfy the conditions of Theorem 4. (i) If  $R_D^{max}$, then  $x\left(t\right)$→0 for every feasible initial state of the fully coupled unforced system. (ii) If  $R_D^{min}$, then every solution with  $x\left(0\right)$  ≠ 0 is uniformly persistent: it is bounded below asymptotically by the unique positive equilibrium $x^{-}$ of the conditional system frozen at y = 1.

$${ℛ}_D^{max}\le 1\hspace{1em}\Rightarrow \hspace{1em}x\left(t\right)\to \mathbf{0}.$$

(31)

$${ℛ}_D^{min}>1\hspace{1em}\Rightarrow \hspace{1em}\underset{t\to \infty }{liminf}{x}_i\left(t\right)\ge x_i^{-}>0,\hspace{1em}\hspace{1em}i=1,\dots ,N.$$

(32)

Proof. 

The distress vector field is order-preserving in its conditional susceptibility and recovery matrices. The bounds above imply the comparison inequality

$$\left(\mathbf{1}-x\right)\circ B\left(\mathbf{1}\right)A^{R}x-D\left(\mathbf{1}\right)x\preccurlyeq \dot{x}\preccurlyeq \left(\mathbf{1}-x\right)\circ B\left(\mathbf{0}\right)A^{R}x-D\left(\mathbf{0}\right)x.$$

(33)

The upper comparison system is the conditional model at y = 0; Theorem 4 gives global extinction when $R_D^{max}$. The lower comparison system is the conditional model at y = 1; when $R_D^{min}$, Theorem 4 gives a unique globally attracting positive equilibrium, and comparison yields the stated uniform lower bound. The complete argument is provided in Appendix A.6. These are sufficient, not necessary, conditions; the intermediate region $R_D^{\min }\le 1<R_D^{max}$ remains genuinely state-dependent. □

3.6. Forced Equilibrium and Innovation–Failure Boundary

When ${\alpha }_i>0$ or ${\xi }_i>0$, the state $x=0$ is generally not an equilibrium. At an equilibrium, Equations (10) and (11) imply

$$x_i={\displaystyle \frac{h_i+{\beta }_i^{\mathrm{eff}}{R}_i}{h_i+{\beta }_i^{\mathrm{eff}}{R}_i+d_i}},$$

(34)

$$y_i={\displaystyle \frac{{\ell }_i{u}_i+{\tau }_i{K}_i}{{\ell }_i{u}_i+{\tau }_i{K}_i+{\delta }_i+{\zeta }_i{x}_i}}.$$

(35)

Proposition 2 (Existence of a forced equilibrium).

Under Assumptions 1–4, the forced system admits at least one feasible equilibrium in  $\Omega$.

Proof. 

The right-hand sides of (34)–(35) define a continuous map from the compact convex set $\Omega$ into itself. The numerators are non-negative, and all denominators are strictly positive because $d_i\ge {\gamma }_i>0$ and ${\delta }_i>0$. Brouwer’s fixed point theorem therefore guarantees at least one fixed point. □

Innovation participation can be locally stabilizing or destabilizing. Relax $u_i$ to $\left[0,1\right]$ and hold the current states $x_i$, $y_i$, $R_i$, and the associated success probability $p_i$ fixed. This treatment is consistent with the benchmark specification because $p_i$ depends on capability, industry conditions, and current knowledge exposure, but not directly on $u_i$. Define the total marginal stabilization intensity

$${\Psi }_i={\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right){\varphi }_p{R}_i+{\displaystyle \frac{x_i}{1-x_i}}{\omega }_i.$$

(36)

The first term in ${\Psi }_i$ is the preventive benefit of successful innovation, and the second is the curative benefit when the firm is already distressed. Differentiating the distress drift with respect to $u_i$ gives

$${\displaystyle \frac{\partial {\dot{x}}_i}{\partial u_i}}=\left(1-x_i\right)\left[{\xi }_i\left(1-p_i\right)-p_i{\Psi }_i\right].$$

(37)

Proposition 3 (Local innovation–failure boundary).

Let  $0<x_i<1$. If  ${\xi }_i>0$, an increase in innovation intensity raises the instantaneous distress drift if and only if $p_i<p_i^{\mathrm{crit}}$, where

$$p_i^{\mathrm{crit}}={\displaystyle \frac{{\xi }_i}{{\xi }_i+{\Psi }_i}}.$$

(38)

Equivalently,

$$p_i^{\mathrm{crit}}={\displaystyle \frac{{\xi }_i}{{\xi }_i+{\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right){\varphi }_p{R}_i+\frac{x_i}{1-x_i}{\omega }_i}}.$$

(39)

If  ${\xi }_i=0$, then 

$${\displaystyle \frac{\partial {\dot{x}}_i}{\partial u_i}}=-\left(1-x_i\right)p_i{\Psi }_i\le 0,$$

(40)

and no local innovation–failure region exists.

Proof. 

For ${\xi }_i>0$, Equation (37) is positive if and only if ${\xi }_i\left(1-p_i\right)>p_i{\Psi }_i$. Rearranging gives $p_i<{\xi }_i/\left({\xi }_i+{\Psi }_i\right)$, which proves (38)–(39). If ${\xi }_i=0$, Equation (37) reduces to (40). □

The boundary in Proposition 3 is local and instantaneous. It does not, by itself, determine the long-run systemic equilibrium, because knowledge accumulation, recovery reinforcement, and network feedback continue to evolve after a marginal change in innovation intensity.

3.7. Analytical Implications

The analytical results have three distinct levels of interpretation. First, Theorems 3 and 4 establish the baseline extinction–persistence behavior of the two isolated cooperative subsystems. The source-free knowledge subsystem is characterized by the knowledge-diffusion threshold ${ℛ}_K$, whereas the fixed-knowledge unforced distress subsystem is characterized by the conditional financial distress persistence threshold ${ℛ}_D\left(y\right)$. These results provide the analytical baseline and do not constitute a sharp global threshold for the fully coupled forced system.

Second, Proposition 1 provides the endogenous link between the two subsystem thresholds. A componentwise improvement in the knowledge state weakly lowers ${ℛ}_D\left(y\right)$, and the reduction is strict under irreducibility and a nontrivial change in the effective susceptibility-to-recovery ratios. Thus, knowledge affects systemic resilience through two reinforcing channels: it reduces effective contagion and increases recovery capacity.

Third, Theorem 6 converts this knowledge-dependent ordering into sufficient comparison bounds for the fully coupled unforced system. If distress disappears even under the worst feasible knowledge protection, extinction follows for the fully coupled unforced system. If distress persists even under maximal knowledge protection, uniform persistence follows. Between these bounds, the endogenous knowledge trajectory and cross-layer feedback determine the outcome. Theorem 6 therefore provides a full-system statement without converting the subsystem thresholds into a single sharp threshold for the coupled model.

Theorem 5 further clarifies the distinction between conditional and coupled stability. The global result in Theorem 4 applies when the knowledge state is held fixed. For the full coupled system, local stability of the distress-free equilibrium follows when the corresponding conditional financial distress persistence threshold is below one, while instability follows when it is above one. At the critical value, higher-order cross-layer feedback may determine the local behavior. This distinction prevents the conditional threshold from being interpreted as a global threshold for the fully forced multiplex system.

Finally, Proposition 3 characterizes a different object: the state-dependent local effect of innovation participation. Its instantaneous effect depends on the balance between failure-induced liquidity pressure and the preventive and recovery benefits of successful innovation. This local innovation–failure boundary does not imply a long-run backfire region. In the numerical range examined below, participation reduces long-run distress, although its stabilizing benefit weakens as failure losses rise. The relevant question is therefore not simply whether more firms should innovate, but which firms should innovate, under what industry conditions, and with what access to complementary knowledge.

4. Numerical Experiments

This section examines the subsystem threshold patterns, capability–industry effects, innovation–failure losses, and cross-layer organization. It also adds four reviewer-motivated validation blocks: directed asymmetric networks; separate rank alignment and edge overlap experiments; convergence diagnostics using longer horizons and an adaptive solver without ex post clipping; and parameter uncertainty. The reported 95% intervals are Monte Carlo sampling intervals conditional on each maintained design, not empirical or structural confidence intervals.

4.1. Parameter Interpretation and Calibration Strategy

The numerical analysis is a stylized, mechanism-oriented calibration rather than an empirical estimation exercise. Parameters are organized into three groups. Recurrent contagion and recovery magnitudes are anchored to the normalized ranges used in related financial-network simulations [36,37]. Evidence on failed R&D cooperation and financially constrained innovation [5,6] motivates the direction and economic role of the success/failure channels, but does not provide directly portable continuous-time rates; the innovation success coefficients and failure loss parameter are therefore treated as normalized mechanism intensities rather than empirical estimates. Classical network-generation settings provide structural benchmarks, while the remaining innovation–knowledge coefficients are chosen to keep the forced system away from mechanical extinction or explosion and are evaluated through structured scans, directed-network alternatives, and joint parameter uncertainty.

Table 3 provides an empirical interpretation map. It clarifies which observable firm- or network-level measures could be used in a future calibration while preserving the present paper’s claim as a theoretical and computational study.

Table 3. Mapping from abstract model parameters to potential measurable constructs.

Parameter/Object	Model Interpretation	Potential Empirical Proxy	Current Status
$L_i$	Innovation and absorptive capability	R&D intensity, R&D personnel, patent quality, prior innovation success	Normalized firm attribute
$c_{s\left(i\right)}$	Industry innovation environment	Demand growth, policy support, regulatory uncertainty, complementary infrastructure	Normalized industry scenario
$a_{ij}^R$	Directional financial/operational exposure	Trade credit, supplier dependence, contract value, financing exposure	Synthetic weighted network
$a_{ij}^K$ and $y_i$	Knowledge access and effective stock	R&D alliances, patent citations, inventor mobility, alliance-weighted knowledge stock	Synthetic knowledge layer/state
${\xi }_i$	Loss from unsuccessful innovation	Abandoned R&D expenditure or write-offs relative to liquid assets	Normalized loss intensity
${\eta }_i$, ${\omega }_i$	Knowledge- and success-assisted recovery	Post-shock productivity or liquidity recovery conditional on knowledge/innovation outcomes	Normalized recovery intensities
${\phi }_y$, ${\phi }_p$	Protection against network contagion	Reduction in distress sensitivity associated with knowledge stock or successful innovation	Normalized protection intensities
${\beta }_0$, ${\gamma }_i$	Baseline contagion and recovery	Estimated transition intensities from panel distress states or payment delays	Literature-informed normalized range

4.2. Experimental Design and Benchmark Setting

We simulate the continuous-time node-level model developed in Section 2. Let

$$\overline{X}\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{i=1}^N}{x}_i\left(t\right),\hspace{1em}\hspace{1em}\overline{Y}\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{i=1}^N}{y}_i\left(t\right)$$

(41)

denote average financial distress intensity and average effective knowledge stock, respectively.

The benchmark risk layer is generated as a weighted Barabási–Albert network, while the knowledge layer follows a weighted Watts–Strogatz structure. The former captures concentrated exposure, whereas the latter represents clustered collaboration and short learning paths. Alternative Barabási–Albert, Watts–Strogatz, and Erdős–Rényi structures are examined in the robustness analysis [42,43,44]. Edge weights are normalized by the target average degree.

Cross-layer similarity is measured by the directed-edge overlap coefficient

$${\mathcal{O}}_{RK}={\displaystyle \frac{\left|E^R\cap E^K\right|}{min\left\{\left|E^R\right|,\left|E^K\right|\right\}}},$$

(42)

where $E^R$ and $E^K$ denote the edge sets of the risk and knowledge layers. The benchmark value ${\mathcal{O}}_{RK}=0.250$ allows the two layers to be related but non-identical.

The benchmark contagion and recovery parameters are informed by qualitative ranges used in related financial-network simulations [36,37], but they are interpreted as normalized mechanism intensities rather than directly estimated empirical rates. Innovation-related coefficients without direct empirical counterparts are likewise treated as normalized mechanism intensities. Table 4 summarizes the benchmark settings, while Table 3 maps the principal abstract parameters to measurable constructs for future empirical calibration.

Table 4. Benchmark simulation settings and parameter rationale.

Category	Parameter or Setting	Benchmark	Basis
Simulation	$N$	300	Computational benchmark
Simulation	$T_{\max },\Delta t$	120, 0.5	Convergence horizon
Simulation	Replications	1000	Monte Carlo averaging
Risk layer	Topology of $A^R$	BA	Scale-free benchmark [42]
Risk layer	Average degree	4	Sparse exposure scenario
Knowledge layer	Topology of $A^K$	WS	Small-world benchmark [43]
Knowledge layer	Average degree	6	Collaboration scenario
Cross-layer relation	${\mathcal{O}}_{RK}$	0.25	Scenario calibration
Initial states	$\overline{X}\left(0\right),\overline{Y}\left(0\right)$	0.10, 0.10	Non-degenerate baseline
Innovation participation	$\lambda$	0.50	Intermediate participation
Capability	$L_i$	Beta (3,3)	Normalized capability distribution
Industry condition	$c$	$\left\{-0.5, 0, 0.5\right\}$	Normalized industry scenarios
Distress	${\beta }_0,\gamma$	0.12, 0.05	Normalized ranges informed by [36,37]
Distress	$\alpha ,\xi$	0.002, 0.08	Forcing and failure loss calibration
Recovery	$\eta ,\omega$	0.10, 0.10	Normalized mechanism intensity
Protection	${\varphi }_y,{\varphi }_p$	0.45, 0.35	Normalized protection effects
Knowledge	$\ell ,{\tau }_0,\delta ,\zeta$	0.02, 0.06, 0.06, 0.06	Threshold-oriented calibration
Success function	${\theta }_0,{\theta }_L,{\theta }_c,{\theta }_K$	−1.2, 3.0, 0.9, 1.5	Logistic anchor calibration
Network centrality	${\kappa }_C$	0.60	Scenario calibration

The absorption function uses b0 = −1.0, bL = 1.5, and bc = 0.8 as logistic anchor coefficients.

4.3. Numerical Implementation and Reproducibility

The primary simulations use the second-order Heun method with Δt = 0.5. Projection to [0, 1] is retained only as a floating-point safeguard, and the implementation records every prospective bound violation before projection. In the 50-seed convergence audit reported below, no Heun projection was triggered. As an independent check, the same trajectories are solved with adaptive DOP853 (relative tolerance ${10}^{-9}$ and absolute tolerance ${10}^{-11}$) without clipping.

For each of M = 1000 replications, the network layers, firm attributes, innovation assignments, and initially distressed firms are regenerated from seeds derived from master seed 20,260,713. In the reciprocal benchmark, connected BA, WS, and ER skeletons are represented by weighted arcs. Edge weights are drawn from [0.8, 1.2] and normalized by target average degree; the WS rewiring probability is 0.10. Industry conditions are assigned with equal probability, and round ($\lambda N$) firms are selected uniformly without replacement for innovation. Section 4.10 separately evaluates genuinely directed asymmetric layers with distinct incoming and outgoing strength distributions. Reported intervals quantify Monte Carlo variation across replications under the specified model and parameter design.

$$\overline{z}\left(t\right)\pm 1.96{\displaystyle \frac{s_z\left(t\right)}{\sqrt{M}}}.$$

(43)

where $s_z\left(t\right)$ is the sample standard deviation across replications. For threshold validation, the relevant absorption or contagion scale is rescaled for each realized network to obtain target threshold values of 0.8, 1.0, and 1.2.

Numerical convergence is evaluated more stringently in Section 4.11 and Appendix C. Across T $\in$ {120, 240, 480}, we report terminal vector field norms, successive-window differences, bound violations, and agreement between Heun and adaptive DOP853. Threshold patterns are also checked on individual-network realizations and multiple initial conditions, rather than only through averages over rescaled networks.

4.4. Outcome Measures

The numerical analysis uses peak distress, cumulative distress, long-run distress, and long-run knowledge:

$$X_{\max }=\underset{0\le t\le T_{\max }}{\max }\overline{X}\left(t\right),\hspace{1em}\hspace{1em}{A}_X={\displaystyle {{\displaystyle \int }}_0^{T_{\max }}}\overline{X}\left(t\right) dt,$$

(44)

$$X_{\infty }={\displaystyle \frac{1}{T_{\max }-T_0}}{\displaystyle {{\displaystyle \int }}_{T_0}^{T_{\max }}}\overline{X}\left(t\right) dt,\hspace{1em}\hspace{1em}{Y}_{\infty }={\displaystyle \frac{1}{T_{\max }-T_0}}{\displaystyle {{\displaystyle \int }}_{T_0}^{T_{\max }}}\overline{Y}\left(t\right) dt,$$

(45)

where $T_0=0.8T_{max}$. We also report the terminal share of innovating firms below the local failure boundary:

$$S_{FB}={\displaystyle \frac{{\sum }_{i=1}^N{u}_i\mathbf{1}\{p_i\left(T_{\max }\right)<p_i^{crit}\left(T_{\max }\right)\}}{{\sum }_{i=1}^N{u}_i}},$$

(46)

where 1{·} is the indicator function.

4.5. Benchmark Dynamics and Threshold Validation

Because the benchmark contains external forcing and innovation–failure losses, the distress-free state is not an equilibrium. Under the benchmark initialization, the simulation approaches a positive numerically observed forced equilibrium. The benchmark produces $X_{\max }=0.42630,A_X=46.365,X_{\infty }=0.42615$, and $Y_{\infty }=0.13082$, with a 95% Monte Carlo interval of $\left[0.42588,0.42672\right]$ for peak distress. The similarity between $X_{\max }$ and $X_{\infty }$ reflects the near-monotonic convergence toward this numerically observed steady state. The estimated local failure boundary share is $S_{\mathrm{FB}}\approx 0.280$, and the benchmark edge overlap is $O_{RK}=0.250$.

Threshold validation is conducted using the two corresponding unforced subsystems. For the knowledge subsystem, internal knowledge input is suppressed and ${ℛ}_K$ is rescaled to 0.8, 1.0, and 1.2. For the distress subsystem, knowledge is fixed at its conditional equilibrium and both external forcing and innovation–failure losses are removed before rescaling ${ℛ}_D$.

Figure 2a shows rapid knowledge extinction when ${ℛ}_K=0.8$, slower decay at the critical value ${ℛ}_K=1.0$, and persistence when ${ℛ}_K=1.2$. Figure 2b displays the corresponding distress pattern: distress disappears below the threshold at ${ℛ}_D=0.8$, converges more slowly at the critical value ${ℛ}_D=1$, and approaches a positive level above it at ${ℛ}_D=1.2$. These results numerically corroborate the subsystem threshold patterns established in Section 3.2 and Section 3.3; they are not presented as a validation of a sharp threshold for the fully coupled forced benchmark.

Figure 2. Threshold validation for the coupled network model. Panel (a) reports knowledge-diffusion dynamics under target values of ${ℛ}_K$. Panel (b) reports financial distress dynamics under target values of ${ℛ}_D$.

4.6. Joint Effects of Capability and Industry Conditions

We vary mean innovation capability $\overline{L}\in \left\{0.1, 0.3, 0.5, 0.7, 0.9\right\}$ and industry condition $c\in \left\{-1, -0.5, 0, 0.5, 1\right\}$, while holding the remaining parameters at their benchmark values. Figure 3 presents the resulting capability–industry regime map, where the horizontal axis represents mean innovation capability, the vertical axis represents industry condition, and each cell reports the corresponding long-run distress level $X_{\infty }$.

Figure 3. Capability–industry regime map. The heatmap reports the long-run distress level $X_{\infty }$ under different combinations of mean innovation capability $\overline{L}$ and industry condition $c$. Lower values indicate stronger systemic resilience.

Long-run distress decreases monotonically with both capability and industry support. It falls from 0.584 at $\left(\overline{L},c\right)=\left(0.1, -1\right)$ to 0.278 at (0.9, 1). Thus, higher capability and more supportive industry conditions jointly strengthen resilience by improving firms’ ability to convert accessible knowledge into successful innovation.

4.7. Innovation Participation Under Failure Losses

We next vary the innovation participation rate λ and failure loss ξ. The incremental long-run effect of innovation is

$$\Delta X_{\infty }\left(\lambda ,\xi \right)=X_{\infty }\left(\lambda ,\xi \right)-X_{\infty }\left(0,\xi \right).$$

(47)

Negative values indicate that innovation participation reduces long-run distress. Figure 4 summarizes these results.

Figure 4. Attenuation of innovation benefits under failure losses. Panel (a) reports the long-run distress level $X_{\infty }$. Panel (b) reports $\Delta X_{\infty }$, where negative values indicate a stabilizing effect of innovation participation.

Innovation remains stabilizing throughout the examined parameter range, but its benefit decreases as failure losses rise. When ξ = 0.03, increasing participation from zero to one reduces long-run distress from 0.570 to 0.193. When ξ = 0.15, the corresponding reduction is from 0.570 to 0.395. The results therefore show attenuation rather than a long-run backfire region, consistent with the distinction between the local drift condition in Proposition 3 and the system’s long-run equilibrium.

4.8. Joint Cross-Layer Configurations of Structural Influence, Knowledge Access, and Capability

We examine how the joint allocation of knowledge access and innovation capability across structurally influential firms is associated with persistent financial distress. Three configurations—aligned, random, and mismatched—are compared. Within each Monte Carlo replication, the risk layer realization, the marginal topology and weight distribution of the knowledge layer, the capability distribution, and the random seeds are preserved. The configurations differ in how knowledge layer nodes and capability values are assigned to firms ranked by transmission-based risk centrality $C^R$. Because the reassignment of knowledge layer nodes also changes the realized edge overlap between the two layers, the experiment compares joint cross-layer configurations rather than isolating a ceteris paribus rank alignment effect.

In the aligned configuration, knowledge layer weighted degree $d^K$ and innovation capability $L$ are assigned in the same rank direction as $C^R$. In the mismatched configuration, both attributes are assigned in the reverse rank direction. In the random configuration, knowledge layer node labels and capability values are independently permuted while their marginal distributions are retained. To make the resulting cross-layer structures transparent, Figure 5 reports the realized Spearman rank correlations $\rho \left(C^R,d^K\right)$ and $\rho \left(C^R,L\right)$, together with the realized edge overlap coefficient ${\mathcal{O}}_{RK}$.

Figure 5. Joint cross-layer configurations and persistent financial distress. Panel (a) reports the mean long-run distress level, with error bars denoting 95% Monte Carlo intervals across 1000 Monte Carlo replications. Panel (b) reports the realized Spearman rank correlations between transmission-based risk centrality and knowledge layer weighted degree, and between transmission-based risk centrality and innovation capability. Panel (c) reports the realized edge overlap coefficient. Because both node-level rank associations and realized edge overlap vary across configurations, the figure should be interpreted as a comparison of joint cross-layer configurations rather than as a pure marginal alignment experiment.

Figure 5a shows a consistent ordering of long-run distress. The aligned configuration yields the lowest mean long-run distress, =0.3863, followed by the random configuration at 0.3980 and the mismatched configuration at 0.4063. Relative to the random configuration, the aligned configuration reduces long-run distress by approximately 2.95%; relative to the mismatched configuration, the reduction is approximately 4.94%. The narrow and non-overlapping Monte Carlo intervals indicate high sampling precision conditional on this synthetic design; they do not represent empirical or structural uncertainty.

The realized structural measures clarify what differs across the three cases. In the aligned configuration, $\rho \left(C^R,d^K\right)$ = 0.8408 and $\rho \left(C^R,L\right)$ = 0.6530, indicating that firms with greater structural risk influence tend to receive stronger knowledge access and possess higher innovation capability. In the random configuration, the capability–risk association is approximately zero, $\rho \left(C^R,L\right)$ = −0.0023, whereas a residual positive association remains between risk exposure and knowledge layer degree, $\rho \left(C^R,d^K\right)$ = 0.5735. In the mismatched configuration, both associations are negative, with $\rho \left(C^R,d^K\right)$ = −0.5911 and $\rho \left(C^R,L\right)$ = −0.6516.

The realized edge overlap coefficients are 0.0322, 0.2500, and 0.0155 for the aligned, random, and mismatched configurations, respectively. Consequently, the observed differences cannot be attributed exclusively to rank alignment. Nevertheless, the aligned and mismatched configurations both exhibit relatively low edge overlap but produce distinctly different distress outcomes. This contrast is consistent with the mechanism that concentrating knowledge access and innovation capability among firms with high structural risk influence is associated with lower persistent financial distress. The result should therefore be interpreted as evidence on favorable joint cross-layer organization, rather than as a causal decomposition of a single alignment parameter.

4.9. Robustness Checks

Robustness is examined with respect to capability dispersion, risk layer topology, and the size and location of the initial shock. The corresponding robustness results are summarized in Table 5.

Table 5. Robustness results based on long-run distress X∞.

Dimension	Scenario	Mean	95% Monte Carlo Interval	Interpretation
Capability dispersion	Low dispersion	0.4247	[0.4244, 0.4251]	Lower capability inequality
Capability dispersion	Moderate dispersion	0.4270	[0.4266, 0.4274]	Benchmark dispersion
Capability dispersion	High dispersion	0.4299	[0.4294, 0.4304]	Greater capability heterogeneity
Risk layer topology	ER risk layer [44]	0.475	[0.474, 0.476]	Random exposure structure
Risk layer topology	WS risk layer [43]	0.488	[0.487, 0.489]	Clustered exposure structure
Risk layer topology	BA risk layer [42]	0.4261	[0.4257, 0.4266]	Scale-free benchmark
Initial shock	5% random	0.4269	[0.4265, 0.4274]	Smaller initial shock
Initial shock	10% random	0.4270	[0.4266, 0.4274]	Benchmark initial shock
Initial shock	15% random	0.4271	[0.4266, 0.4275]	Larger initial shock
Initial shock	10% high influence	0.4271	[0.4266, 0.4275]	Shock to high-influence firms
Initial shock	10% low capability	0.4270	[0.4266, 0.4274]	Shock to low-capability firms

Greater capability dispersion slightly increases long-run distress, although the magnitude is limited. The topology results depend on the normalized-exposure and cross-layer settings: the WS and ER layers produce higher long-run distress than the BA benchmark, but this should not be interpreted as a universal topology ranking.

Long-run distress is nearly invariant to the initial shock, whereas cumulative distress is not. For example, A_X increases from 45.416 under a 5% random shock to 47.368 under a 15% shock. Initial conditions therefore affect the transition burden more strongly than the forced long-run equilibrium.

4.10. Directed Asymmetry and Isolation of Cross-Layer Mechanisms

The original joint configuration experiment changes rank associations and realized edge overlap simultaneously. We therefore add two controlled comparisons. First, edge overlap is fixed at $O_{RK}$ = 0.25 while knowledge-degree and capability ranks are aligned, randomized, or reversed relative to structural risk influence. Second, aligned ranks are held fixed while target overlap is varied. A third experiment replaces reciprocal benchmark layers with moderately and strongly asymmetric directed layers. Risk arcs are oriented using a latent supplier–customer power score, so an arc from source j to recipient i represents pressure transmitted from j and received by i; knowledge arcs preferentially flow from higher to lower capability firms. Reciprocity is reduced while total layer weight is approximately preserved. Appendix C, Table A5 reports realized reciprocity together with mean in- and out-degree and the corresponding coefficients of variation for both layers, while the overall layer weight scale is normalized across designs. This makes the directional structure explicit rather than treating the directed model as a relabeled reciprocal network. The resulting comparisons are summarized in Figure 6.

Figure 6. Mechanism-isolation and directed-network experiments. Panel (a) fixes realized edge overlap at 0.25 and varies rank configuration. Panel (b) fixes aligned knowledge and capability ranks and varies realized edge overlap. Panel (c) compares reciprocal, moderately asymmetric, and strongly asymmetric directed networks. Error bars in panels (a,b) are 95% Monte Carlo intervals across 1000 replications.

With overlap fixed at 0.25, long-run distress is 0.3843 under alignment, 0.3973 under random assignment, and 0.4069 under mismatch. With aligned ranks fixed, increasing realized overlap from approximately 0.033 to 0.600 lowers long-run distress from 0.3862 to 0.3805. The directed-network experiment produces 0.4268, 0.3931, and 0.3228 under reciprocal, moderate-asymmetry, and strong-asymmetry designs, respectively. The last ordering is conditional on the stated directional generator and is not interpreted as a universal benefit of asymmetry; it demonstrates that directionality can materially change the equilibrium and should not be suppressed by a reciprocal benchmark alone.

To check that the threshold evidence is not specific to reciprocal benchmark graphs, Appendix C also reports a small realized network validation on the directed designs. For ten common-seed networks under each reciprocity setting, we first compute the unscaled realized knowledge-diffusion threshold RK and the conditional financial distress persistence threshold RD(y*), where y* is the equilibrium knowledge state of the x = 0 knowledge subsystem with the benchmark internal knowledge source. We then rescale each realized network separately to target RK values of 0.8 and 1.2 for the source-free knowledge subsystem and target RD(y*) values of 0.8 and 1.2 for the conditional unforced distress subsystem, and integrate each isolated subsystem to T = 480. Table A6 shows that directionality changes the realized spectral quantities, but the qualitative threshold pattern remains unchanged: both subsystems approach zero below one and remain positive above one under reciprocal, moderately asymmetric, and strongly asymmetric networks. This validation complements the broader multiple-initial-condition tests in Table A3 and is not interpreted as a threshold for the fully coupled forced system.

4.11. Numerical Convergence, Individual-Network Thresholds, and Parameter Uncertainty

Convergence diagnostics use 50 common network realizations at T = 120, 240, and 480. At T = 120, the mean terminal infinity norm of the Heun vector field is 5.90 × ${10}^{-5}$ and the mean difference between the final two 20-unit windows is 5.77 × ${10}^{-4}$. At T = 240, these values fall to 7.48 × ${10}^{-8}$ and 4.07 × ${10}^{-7}$, and at T = 480 to 1.49 × 1$0^{-13}$ and 3.71 × ${10}^{-13}$. No Heun projection or DOP853 bound violation occurs. The mean Heun–DOP853 difference in terminal long-run distress is 1.40 × ${10}^{-7}$ at T = 120 and below 5 × ${10}^{-12}$ at T = 480.

Individual-realization tests use 10 networks, three initial levels (0.01, 0.10, and 0.30), and T = 480. Below the target threshold, terminal states approach zero for every initial condition; at the critical target, decay is substantially slower over the finite horizon; and above the threshold, trajectories approach a positive level that is nearly independent of the initial condition. These checks corroborate the qualitative threshold patterns without relying solely on cross-network averages.

Parameter uncertainty is assessed with 80 Latin-hypercube designs that vary β0, γ, ξ, τ0, ${\phi }_y$, ${\phi }_p$, η, and ω jointly by ±20%, using 20 network replications per design. The resulting 5th–95th percentile range of X∞ is [0.363, 0.493], with a median of 0.425. Univariate Spearman associations identify β0 (0.657) and γ (−0.494) as the strongest monotone drivers, followed by η (−0.250). These are global rank-sensitivity diagnostics under the specified joint ranges, not empirical parameter estimates.

4.12. Discussion

From a mathematical perspective, the contribution should not be interpreted as a new generic SIS threshold theorem. The cooperative and strictly sublinear structures of the isolated subsystems allow established extinction–persistence results to serve as analytical building blocks. What is specific to the present formulation is that effective knowledge modifies both susceptibility and recovery through innovation success, financial distress feeds back into knowledge accumulation, and innovation–failure creates an opposing distress channel. These features generate the knowledge-dependent threshold mapping, the local innovation–failure boundary, and the full-system comparison bounds. Consistent with this positioning, the global theorems in Section 3.2 and Section 3.3 concern source-free and conditional unforced subsystems, whereas Theorem 6 provides only sufficient bounds for the fully coupled unforced model; no sharp global threshold is claimed for the forced benchmark.

The directed and mechanism-isolation experiments show that topology, directionality, rank association, and edge overlap are distinct design dimensions. At fixed overlap, favorable allocation of knowledge access and capability to structurally influential firms lowers distress. At fixed alignment, greater overlap is also stabilizing in the examined range. Directed asymmetry materially changes long-run outcomes, but its sign depends on the economic direction rule and should not be generalized beyond the maintained generator.

The local innovation–failure boundary and the long-run innovation effect remain different objects. Proposition 3 concerns an instantaneous firm-level drift, whereas long-run outcomes incorporate knowledge accumulation, recovery reinforcement, and network feedback. Parameter scans and Latin-hypercube variation show attenuation of innovation benefits but no long-run backfire region in the examined domain. The 95% intervals reported for fixed designs measure Monte Carlo sampling variation only; the wider parameter uncertainty range in Figure 7 better represents sensitivity to maintained coefficients.

Figure 7. Parameter uncertainty and sensitivity. Panel (a) reports Spearman rank correlations between jointly varied parameters and X∞ across 80 Latin-hypercube designs. Panel (b) reports the median and 5th–95th percentile range of long-run distress across those designs.

Accordingly, the managerial and policy implications are framed as model-generated hypotheses. The framework does not include intervention costs, budget constraints, moral hazard, endogenous participation, firm exit, or endogenous partner formation, and therefore does not establish an optimal or welfare-improving targeting rule. It predicts that knowledge support may be more effective when structurally influential firms possess sufficient absorptive capability; empirical calibration and explicit optimization are required before this mechanism can support prescriptive policy.

5. Conclusions

This paper develops a coupled nonlinear multiplex-network model to examine how effective knowledge, innovation capability, success and failure, and financial distress interact across heterogeneous industrial networks. Its mathematical contribution does not lie in generic SIS threshold machinery; rather, it lies in the asymmetric innovation–knowledge coupling that makes the distress persistence threshold knowledge-dependent, creates an explicit success–failure boundary, and permits comparison bounds for the fully coupled unforced system.

The findings are fourfold. First, the source-free knowledge and fixed-knowledge unforced distress subsystems exhibit the global threshold structure expected for irreducible cooperative systems and therefore provide the analytical baseline. Second, a componentwise improvement in knowledge weakly lowers the conditional financial distress persistence threshold, and comparison systems yield sufficient extinction and persistence bounds for the fully coupled unforced model. Third, the local innovation–failure boundary identifies when failure pressure exceeds the preventive and recovery benefits of successful innovation. Fourth, directed-network, mechanism-isolation, convergence, and uncertainty experiments show that capability–industry conditions, rank allocation, edge overlap, network directionality, and parameter variation jointly shape long-run distress.

These model-based results generate testable hypotheses rather than policy optima. They suggest that the resilience value of innovation may depend on whether knowledge access reaches structurally influential firms with adequate capability, and that failure loss control may preserve innovation benefits. Because costs, resource constraints, endogenous behavior, and welfare are not modeled, prescriptive targeting recommendations require empirical calibration and an explicit intervention problem.

The study has several limitations. The framework is phenomenological and uses stylized parameterization with synthetic network layers. The directed experiments rely on an imposed economic direction rule, and the cross-layer assignments remain exogenous. Monte Carlo intervals are conditional on the maintained design, while the Latin-hypercube analysis covers only specified ±20% ranges. Future research should calibrate the model using firm-level supply-chain, patent, and financial data; estimate directional exposure and knowledge links; introduce endogenous innovation entry, exit, and partner selection; and formulate an optimal-control problem with intervention costs, budgets, and welfare criteria.

Appendix A.1. Mathematical Preliminaries

For vectors $z$ and $w$, $z\ge w$ denotes componentwise inequality, $z>w$ means $z\ge w$ and $z\ne w$, and $z\gg w$ means strict inequality in every component. For a square matrix $M$, $s\left(M\right)$ denotes its spectral abscissa and $\rho \left(M\right)$ its spectral radius. A Metzler matrix has non-negative off-diagonal entries.

Lemma A1 (Spectral sign equivalence).

Let  $F$  be a non-negative irreducible matrix and let  $D$  be a positive diagonal matrix. Then 

$$\mathrm{sgn}s\left(F-D\right)=\mathrm{sgn}\left[\rho \left(D^{-1}F\right)-1\right].$$

(A1)

This is the standard next-generation matrix relation for irreducible Metzler systems [39]. Equivalently, $D-F$ is a nonsingular $M$-matrix when $\rho \left(D^{-1}F\right)<1$, a singular irreducible $M$-matrix when $\rho \left(D^{-1}F\right)=1$, and ceases to be an $M$-matrix when $\rho \left(D^{-1}F\right)>1$.

Lemma A2 (Cooperativity and strict sublinearity).

Let $F$ be non-negative and irreducible and let $D$ be positive diagonal. Consider

$$g\left(z\right)=\left(\mathbf{1}-z\right)\odot Fz-Dz.$$

(A2)

For  $z$  in the interior of  ${\left[0, 1\right]}^N$, the off-diagonal derivatives satisfy

$${\displaystyle \frac{\partial g_i}{\partial z_j}}=\left(1-z_i\right)F_{ij}\ge 0,\hspace{1em}\hspace{1em}i\ne j.$$

(A3)

Hence, the system is cooperative, and irreducibility of  $F$ makes it strongly cooperative in the interior. Moreover, for every  $0<a<1$,

$$g\left(az\right)-ag\left(z\right)=a\left(1-a\right)\mathrm{diag}\left(z\right)Fz.$$

(A4)

For $z\gg 0$, irreducibility implies $Fz\gg 0$, so the right-hand side is strictly positive. Thus, the vector field is strictly sublinear. Standard results for irreducible cooperative and strictly sublinear systems imply uniqueness and global attraction of a positive equilibrium whenever the zero equilibrium is unstable and a positive equilibrium exists [40,41].

Appendix A.2. Detailed Proof of Theorem 3

For the source-free knowledge subsystem, define $F_K=\mathcal{T}{A}^K$ and $M_K=F_K-\Delta$. The subsystem can be written as

$$\dot{y}=M_{K}y-\mathrm{diag}\left(y\right)F_{K}y.$$

(A5)

Case 1: ${ℛ}_K<1$. By Lemma A1, $s\left(M_K\right)<0$, so $M_K$ is Hurwitz. Since the nonlinear saturation term is non-negative componentwise,

$$\dot{y}\le M_{K}y.$$

(A6)

The semigroup generated by a Metzler matrix is non-negative. The comparison principle therefore gives

$$0\le y\left(t\right)\le e^{M_{K}t}y\left(0\right).$$

(A7)

Because $M_K$ is Hurwitz, $e^{M_{K}t}\to 0$ as $t\to \infty$. Hence, $y\left(t\right)\to 0$ for every feasible initial condition, proving global asymptotic stability below the threshold.

Case 2: ${ℛ}_K=1$. By Lemma A1, $s\left(M_K\right)=0$. Since $M_K$ is irreducible Metzler, there exists $q\gg 0$ satisfying $q^{\top }{M}_K=0$. Define $V\left(y\right)=q^{\top }y$. Along solutions,

$$\dot{V}=-q^{\top }\mathrm{diag}\left(y\right)F_{K}y\le 0.$$

(A8)

Let $E=\left\{y\in {\left[0,1\right]}^N:\dot{V}=0\right\}$. Every term in $\sum _i{q}_i{y}_i{\left(F_{K}y\right)}_i$ is non-negative. If $y\gg 0$, irreducibility gives $F_{K}y\gg 0$, contradicting $\dot{V}=0$. If $y$ is a nonzero boundary point, let $S$ be its nonempty proper support. For every $i\in S$, the equality condition requires ${\left(F_{K}y\right)}_i=0$. Irreducibility ensures that at least one node $k\notin S$ receives a positive link from $S$, so ${\left(F_{K}y\right)}_k>0$. Since $y_k=0$, Equation (A5) gives ${\dot{y}}_k={\left(F_{K}y\right)}_k>0$, and the trajectory immediately leaves $E$. Therefore, the largest invariant subset of $E$ is $\left\{0\right\}$. LaSalle’s invariance principle yields $y\left(t\right)\to 0$ for every feasible initial condition.

Case 3: ${ℛ}_K>1$. By Lemma A1, $s\left(M_K\right)>0$, so $y=0$ is unstable. Let $v\gg 0$ satisfy $M_{K}v=s_{K}v$, where $s_K=s\left(M_K\right)>0$. For sufficiently small $\epsilon >0$,

$$f_K\left(\epsilon v\right)=\epsilon s_{K}v-{\epsilon }^2\mathrm{diag}\left(v\right)F_{K}v\gg 0.$$

(A9)

At the upper boundary, $f^K\left(\mathbf{1}\right)=-\Delta \mathbf{1}\ll 0$. Hence, a positive equilibrium exists between a sufficiently small multiple of $v$ and the upper solution $\mathbf{1}$. By Lemma A2, the subsystem is strongly cooperative and strictly sublinear. The global convergence theorem for irreducible cooperative sublinear systems therefore implies that the positive equilibrium $y^K$ is unique and attracts every nonzero feasible initial condition [40,41]. This completes the proof of Theorem 3.

Appendix A.3. Detailed Proof of Theorem 4

Fix a feasible knowledge state $\overline{y}$ and define

$$F_D\left(\overline{y}\right)=B\left(\overline{y}\right)A^R,\hspace{1em}\hspace{1em}{M}_D\left(\overline{y}\right)=F_D\left(\overline{y}\right)-D\left(\overline{y}\right).$$

(A10)

The complete conditional unforced distress subsystem is

$$\dot{x}=M_D\left(\overline{y}\right)x-\mathrm{diag}\left(x\right)F_D\left(\overline{y}\right)x.$$

(A11)

Under the assumptions of Theorem 4, $F_D\left(\overline{y}\right)$ is non-negative and irreducible, while $D\left(\overline{y}\right)$ is positive diagonal. Lemma A1 therefore applies with

$${ℛ}_D\left(\overline{y}\right)=\rho \left(D{\left(\overline{y}\right)}^{-1}{F}_D\left(\overline{y}\right)\right).$$

(A12)

Case 1: ${ℛ}_D\left(\overline{y}\right)<1$. The matrix $M_D\left(\overline{y}\right)$ is Hurwitz and the nonlinear term in (A11) is non-negative, so

$$0\le x\left(t\right)\le e^{M_D\left(\overline{y}\right)t}x\left(0\right)\to 0.$$

(A13)

Thus, the distress-free equilibrium is globally asymptotically stable below the threshold.

Case 2: ${ℛ}_D\left(\overline{y}\right)=1$. Choose $r\gg 0$ satisfying $r^{\top }{M}_D\left(\overline{y}\right)=0$ and define $W\left(x\right)=r^{\top }x$. Then,

$$\dot{W}=-r^{\top }\mathrm{diag}\left(x\right)F_D\left(\overline{y}\right)x\le 0.$$

(A14)

The support argument used in the proof of Theorem 3 shows that the largest invariant subset of $\left\{x:\dot{W}=0\right\}$ is $\left\{0\right\}$. Any nonzero boundary point satisfying $\dot{W}=0$ immediately generates a positive derivative in a zero component along a path of the irreducible matrix $F_D\left(\overline{y}\right)$. LaSalle’s invariance principle therefore gives $x\left(t\right)\to 0$.

Case 3: ${ℛ}_D\left(\overline{y}\right)>1$. The zero equilibrium is unstable. Let $s_D=s\left(M_D\left(\overline{y}\right)\right)>0$, and let $v\gg 0$ be the corresponding Perron eigenvector. For sufficiently small $\epsilon >0$,

$$f_D\left(\epsilon v;\overline{y}\right)=\epsilon s_{D}v-{\epsilon }^2\mathrm{diag}\left(v\right)F_D\left(\overline{y}\right)v\gg 0.$$

(A15)

Moreover, $f_D\left(\mathbf{1};\overline{y}\right)=-D\left(\overline{y}\right)\mathbf{1}\ll 0$. A positive equilibrium therefore exists. Lemma A2 gives strong cooperativity and strict sublinearity, so the positive equilibrium $x^D\left(\overline{y}\right)$ is unique and attracts every nonzero feasible initial condition [40,41]. This completes the proof of Theorem 4.

Appendix A.4. Detailed Proof of Proposition 1

Let $y^{\prime }\ge y$. Since $A^K\ge 0$,

$$K\left(y^{\prime }\right)=A^K{y}^{\prime }\ge A^{K}y=K\left(y\right).$$

(A16)

Because ${\theta }_K\ge 0$ and the logistic map is increasing,

$$p_i\left(y^{\prime }\right)\ge p_i\left(y\right),\hspace{1em}\hspace{1em}i=1,\dots ,N.$$

(A17)

Consequently, each diagonal susceptibility term weakly decreases and each recovery term weakly increases:

$$B\left(y^{\prime }\right)\le B\left(y\right),\hspace{1em}\hspace{1em}D{\left(y^{\prime }\right)}^{-1}\le D{\left(y\right)}^{-1}.$$

(A18)

Since all matrices involved are non-negative,

$$D{\left(y^{\prime }\right)}^{-1}B\left(y^{\prime }\right)A^R\le D{\left(y\right)}^{-1}B\left(y\right)A^R.$$

(A19)

Perron–Frobenius monotonicity gives ${ℛ}_D\left(y^{\prime }\right)\le {ℛ}_D\left(y\right)$. If $A^R$ is irreducible and $D{\left(y^{\prime }\right)}^{-1}B\left(y^{\prime }\right)\ne D{\left(y\right)}^{-1}B\left(y\right)$, then both scaled matrices in (A19) are irreducible and the inequality is strict in at least one entry. Strict Perron–Frobenius monotonicity therefore gives ${ℛ}_D\left(y^{\prime }\right)<{ℛ}_D\left(y\right)$. This proves Proposition 1.

Appendix A.5. Detailed Proof of Theorem 5

Let the full vector field be written as $\dot{x}=f_x\left(x,y\right)$ and $\dot{y}=f_y\left(x,y\right)$. At the unforced equilibrium ${ℰ}_0=\left(0,\overline{y}\right)$, the derivative of the distress vector field with respect to $x$ is

$${{\displaystyle \frac{\partial f_x}{\partial x}}|}_{{ℰ}_0}=B\left(\overline{y}\right)A^R-D\left(\overline{y}\right).$$

(A20)

Because $a_i={\xi }_i=0$ and every explicit dependence of $f_x$ on y is multiplied by $x$ or $A^{R}x$, the upper-right derivative block vanishes:

$${{\displaystyle \frac{\partial f_x}{\partial y}}|}_{{ℰ}_0}=0.$$

(A21)

The knowledge vector field depends directly on $x$ through the erosion term $-y\odot \left(\zeta \odot x\right)$, so its lower-left derivative block is generally nonzero. Consequently,

$$J\left({ℰ}_0\right)=\left(\begin{array}{cc}B\left(\overline{y}\right)A^R-D\left(\overline{y}\right)& 0\\ J_{yx}& J_{yy}\end{array}\right).$$

(A22)

The spectrum of a block triangular matrix is the union of the spectra of its diagonal blocks. By assumption, all eigenvalues of $J_{yy}$ have negative real parts. Lemma A1 shows that the upper-left block has negative spectral abscissa when ${ℛ}_D\left(\overline{y}\right)<1$ and positive spectral abscissa when ${ℛ}_D\left(\overline{y}\right)>1$. The coupled equilibrium is therefore locally asymptotically stable in the first case and unstable in the second. At ${ℛ}_D\left(\overline{y}\right)=1$, the upper-left block has zero spectral abscissa, so linearization alone is inconclusive for the full coupled system. This proves Theorem 5.

Appendix A.6. Detailed Proof of Theorem 6

For every feasible knowledge trajectory, 0 ≤ y(t) ≤ 1. By the monotonicity established in Proposition 1, B(1) ≼ B(y(t)) ≼ B(0) and D(0) ≼ D(y(t)) ≼ D(1). Therefore, the distress component of the fully coupled unforced vector field is bounded between the conditional vector fields frozen at y = 1 and y = 0.

$$\left(\mathbf{1}-x\right)\circ B\left(\mathbf{1}\right)A^{R}x-D\left(\mathbf{1}\right)x\preccurlyeq \dot{x}\preccurlyeq \left(\mathbf{1}-x\right)\circ B\left(\mathbf{0}\right)A^{R}x-D\left(\mathbf{0}\right)x.$$

(A23)

Let $x^{+}$(t) solve the upper comparison system at y = 0 with $x^{+}\left(0\right)=x\left(0\right)$. Cooperativity gives x(t) ≤ $x^{+}$(t). If $R_D\left(0\right)\le 1,$ Theorem 4 implies $x^{+}$(t)→0, and hence x(t)→0. This proves part (i).

For part (ii), let $x^{-}$(t) solve the lower comparison system at y = 1 with the same nonzero initial condition. Then, $x^{-}\left(t\right)$≤ x(t). If $R_D$(1) > 1, Theorem 4 gives a unique positive equilibrium $x^{-}$ that attracts every nonzero feasible initial state. Consequently, $limin{f}_{\left\{t\to \infty \right\}}{x}_i\left(t\right)\ge x_i^{-}>0$ for every component. The comparison requires no convergence assumption for y(t); it uses only its invariant bounds. The conditions are sufficient rather than necessary, leaving the intermediate region state-dependent.

Appendix A.7. Detailed Proof of Proposition 2

Define the fixed point map $\Phi =\left({\Phi }_x,{\Phi }_y\right)$ on $\Omega$ by

$${\Phi }_{x,i}\left(x,y\right)={\displaystyle \frac{h_i\left(y\right)+{\beta }_i^{\mathrm{eff}}\left(y\right)R_i\left(x\right)}{h_i\left(y\right)+{\beta }_i^{\mathrm{eff}}\left(y\right)R_i\left(x\right)+d_i\left(y\right)}},$$

(A24)

$${\Phi }_{y,i}\left(x,y\right)={\displaystyle \frac{{\ell }_i{u}_i+{\tau }_i{K}_i\left(y\right)}{{\ell }_i{u}_i+{\tau }_i{K}_i\left(y\right)+{\delta }_i+{\zeta }_i{x}_i}}.$$

(A25)

All components are continuous because the logistic functions, matrix products, sums, products, and quotients involved are continuous. The denominators are strictly positive: $d_i\left(y\right)\ge {\gamma }_i>0$ and ${\delta }_i>0$. The numerators are non-negative and do not exceed their corresponding denominators. Hence,

$$0\le {\Phi }_{x,i}\left(x,y\right)\le 1,\hspace{1em}\hspace{1em}0\le {\Phi }_{y,i}\left(x,y\right)\le 1.$$

(A26)

Thus, $\Phi$ maps the compact convex set $\Omega ={\left[0,1\right]}^N\times {\left[0,1\right]}^N$ continuously into itself. Brouwer’s fixed point theorem guarantees at least one fixed point $\left(x^{\mathrm{eq}},y^{\mathrm{eq}}\right)\in \Omega$. By construction, this fixed point satisfies the equilibrium equations of the forced system. This proves Proposition 2.

Appendix A.8. Detailed Proof of Proposition 3

Fix the current state $x_i$, $y_i$, the local exposure $R_i$, and the associated success probability $p_i$. The dependence of the distress drift on innovation intensity $u_i$ enters through

$$h_i={\alpha }_i+{\xi }_i{u}_i\left(1-p_i\right),$$

(A27)

$${\beta }_i^{\mathrm{eff}}={\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right)\left(1-{\varphi }_p{u}_i{p}_i\right),$$

(A28)

$$d_i={\gamma }_i+{\eta }_i{y}_i+{\omega }_i{u}_i{p}_i.$$

(A29)

Differentiating with respect to $u_i$ while holding the current state fixed gives

$${\displaystyle \frac{\partial h_i}{\partial u_i}}={\xi }_i\left(1-p_i\right),$$

(A30)

$${\displaystyle \frac{\partial {\beta }_i^{\mathrm{eff}}}{\partial u_i}}=-{\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right){\varphi }_p{p}_i,$$

(A31)

$${\displaystyle \frac{\partial d_i}{\partial u_i}}={\omega }_i{p}_i.$$

(A32)

For $0<x_i<1$, define

$${\Psi }_i={\beta }_0\left(1+{\kappa }_C{C}_i^R\right)\left(1-{\varphi }_y{y}_i\right){\varphi }_p{R}_i+{\displaystyle \frac{x_i}{1-x_i}}{\omega }_i\ge 0.$$

(A33)

Substitution into the distress equation yields

$${\displaystyle \frac{\partial {\dot{x}}_i}{\partial u_i}}=\left(1-x_i\right)\left[{\xi }_i\left(1-p_i\right)-p_i{\Psi }_i\right].$$

(A34)

If ${\xi }_i>0$, the derivative is positive if and only if

$${\xi }_i\left(1-p_i\right)>p_i{\Psi }_i\hspace{1em}\iff \hspace{1em}{p}_i<{\displaystyle \frac{{\xi }_i}{{\xi }_i+{\Psi }_i}}=p_i^{\mathrm{crit}}.$$

(A35)

The denominator is strictly positive because ${\xi }_i>0$. If ${\xi }_i=0$, Equation (A34) reduces to $-\left(1-x_i\right)p_i{\Psi }_i\le 0$, so no local innovation–failure region exists. This proves Proposition 3.

Appendix A.9. Scope of the Appendix Results

Theorems 3 and 4 apply to deliberately isolated source-free and conditional unforced subsystems. Theorem 6 extends the analysis to the fully coupled unforced system through sufficient comparison bounds, but it does not provide a sharp threshold in the intermediate region and does not apply to the externally forced benchmark. Proposition 2 guarantees the existence, not uniqueness or global attraction, of a forced equilibrium. Proposition 3 is local and instantaneous. These scope restrictions are maintained in the abstract, highlights, discussion, and conclusions.