Evolutionary dynamics of adaptive multilayer economic networks: a stability-guarded potential game framework

1 Introduction

Complex social and economic systems emerge from repeated interactions among individuals, firms, platforms, public agencies, and institutions. Their macroscopic behavior depends not only on local decisions but also on the networks through which incentives, information, and influence propagate. Recent evolutionary-game studies have assessed low-carbon technology diffusion, opinion–epidemic coevolution, and group cooperation on heterogeneous higher-order networks [1–3]. These applications show that a fixed interaction graph can be too restrictive when trust, trading, communication, or institutional links change in response to observed behavior.

Adaptive-network models capture this feedback, but they often lack a common objective that remains valid when strategies and links are updated simultaneously. Weighted-network analysis, adaptive interbank-network simulation, and evolutionary risk-governance models demonstrate the importance of heterogeneous and state-dependent connections [4–6]. Studies on self-interaction learning, reputation-mediated game transitions, and other-regarding preferences show how bounded rationality reshapes cooperation [7–9]; uncertain information diffusion and platform-supervised purchasing behavior provide related social applications [10, 11]. However, aggressive co-adaptation may generate oscillations, transient clustering, or welfare loss unless the coupled dynamics are explicitly stabilized.

Distributed equilibrium-seeking and network optimization provide useful algorithmic foundations. Event-triggered, signed-network, and adaptive methods establish decentralized convergence under realistic communication conditions [12–14], while delayed, robust, and aggregative formulations extend these guarantees [15–17]. Potential-game offloading, compressed Nash seeking, and merely monotone game methods clarify how local updates can be coordinated at scale [18–20]. Finite-time network games, resilient task allocation, and fuzzy scheduling provide additional design principles [21–23], while resource-aware multi-agent decision-making highlights broader social objectives [24]. However, these methods generally treat the graph as fixed or externally specified rather than as an endogenous strategic variable.

This paper develops an adaptive multilayer evolutionary potential game, in which mixed strategies and market, information, and institutional links evolve under one dynamic potential. The proposed stability-guarded mirror-replicator (SGMR) dynamic combines entropy-regularized strategy revision, projected evidence-driven link rewiring, and spectral safeguard. Its algorithmic design is motivated by reinforcement learning-based network games and synchronous potential-game optimization [25, 26]; networked information-diffusion dynamics and high-fidelity economic simulation support its social system interpretation [27, 28].

The novelty lies in the closed analytical loop among incentives, endogenous topology, and stability control. Adaptive network evolutionary games usually prescribe rewiring through a behavioral heuristic without a common certificate, whereas distributed potential-game methods typically optimize over a fixed communication graph. In contrast, SGMR makes both strategy revision and evidence-guided rewiring improve one time-indexed potential, and it uses the current graph spectrum to restrict the two update steps before instability occurs. Thus, the target-link mechanism and spectral guard are not independent add-ons: the target specifies where the graph should move, while the guard specifies how rapidly the coupled strategy–graph state may move.

The main contributions of SGMR are as follows.

• The strategy and link blocks share an exact dynamic potential despite state-dependent multilayer weights.
• The mirror update is shown to recover replicator dynamics in the small-step limit.
• The spectral guard yields monotone potential improvement and a finite-horizon stationarity guarantee.
• A local input-to-state stability result quantifies the robustness to observation disturbances. Computational experiments compare the method with static mirror ascent, standard replicator dynamics, and fictitious play.

The remainder of this paper introduces the model, develops the co-evolutionary game and guarded dynamics, proves the theoretical properties, reports the experiments, and concludes with future directions.

2 Multilayer social–economic system and preliminaries

2.1 Agents, strategies, and multilayer social interactions

Consider n agents indexed by V = { 1 , … , n }. An agent can denote a firm, a consumer coalition, a digital platform, a local government, or another decision-making entity. The agents interact on L layers. The market interaction layer records exchange or competitive relationships. The information diffusion layer captures signals, forecasts, reputation, and imitation. The institutional coordination layer represents policies, contracts, compliance rules, or technological compatibility. At time t, layer ℓ is denoted by G ℓ , t = ( V , W ℓ , t ), where W ℓ , t = [ w i j ℓ , t ] is nonnegative, symmetric, and has zero diagonal.

Each agent selects one of m strategic alternatives. To permit bounded rationality and heterogeneous populations, we model the choice as a probability vector

Δ m = x ∈ R + m : 1 ⊤ x = 1 , x i t ∈ Δ m . (1)

For example, the coordinates can encode low-, moderate-, and high-cooperation investment actions. The mixed representation is not a randomization device alone. It can be read as the current behavioral propensity of an individual agent or, in a population interpretation, as the local share of subpopulations using each strategy. This dual interpretation makes the model compatible with both agent-based adaptation and evolutionary social dynamics under local payoff information.

The agent receives a private data-driven benefit b i t ∈ R m, which is obtained from rolling observations such as realized demand, local capacity, price spread, or policy intensity. We use the simple recursive estimatorwhere r i t is a vector of local reward estimates and r ̄ t is a normalization level. This step is intentionally lightweight. More elaborate econometric or neural estimators can replace it without changing the strategic layer of the framework.

b i t = 1 − ω b i t − 1 + ω r i t − r ̄ t 1 , 0 < ω ≤ 1 , (2)

2.2 Action compatibility and link evidence

A link should not be interpreted as a static similarity score. It is a modifiable economic channel. We construct an evidence-supported target matrix by combining action compatibility, observed co-movement, and layer-specific constraints:where a i j ℓ ∈ { 0,1 } describes an admissible relation, σ ( q ) = ( 1 + exp ( − q ) ) − 1, and s i j ℓ , t is a standardized local signal. The parameters ξ ℓ, ζ ℓ, and χ ℓ quantify baseline connectivity, strategic incompatibility, and observed evidence, respectively. Equation 3 does not provide a graph-prediction objective by itself. It provides a bounded reference that prevents link updates from amplifying temporary noise.

w ̂ i j ℓ , t = a i j ℓ σ ξ ℓ − ζ ℓ x i t − x j t 1 + χ ℓ s i j ℓ , t , i ≠ j , (3)

To couple the layers, we define a positive semidefinite Laplacianwhere D ℓ , t is the degree matrix. A large ν ℓ makes the corresponding layer more influential in maintaining diffusion consistency. This is useful when, for example, institutional compatibility is more reliable than a volatile social signal.

L c t = ∑ ℓ = 1 L ν ℓ D ℓ , t − W ℓ , t , ν ℓ ≥ 0 , ∑ ℓ = 1 L ν ℓ = 1 , (4)

2.3 Preliminary definitions

Let h ( x ) = ∑ k = 1 m x k ⁡ log x k be the negative-entropy mirror map. The associated Bregman divergence is

D h y , x = ∑ k = 1 m y k ⁡ log y k x k . (5)

Pinsker’s inequality yields D h ( y , x ) ≥ 1 2 y − x 1 2 and, thus, controls Euclidean changes in a mixed action. We use W for the closed convex set satisfying symmetry, zero diagonal, bounded weights 0 ≤ w i j ℓ ≤ w ̄ ℓ, and optional layer sparsity masks.

Assumption 1 (Regularity and bounded observations). For all t, the benefit vectors satisfy b i t 2 ≤ b ̄. The matrices B ℓ = ( B ℓ ) ⊤ have bounded spectral norm B ℓ 2 ≤ B ̄ ℓ. The feasible link set W is nonempty, compact, and convex. The signal process in Equation 3 is bounded, and the target link matrices are Lipschitz in X with constant L W ̂.

Remark 1 The multilayer representation does not require every economic system to have three literal networks. It is an analytical separation of interaction channels that may be estimated from different data sources. The value of the separation is that market externalities, information diffusion, and institutional constraints can receive different weights and update rates. This preserves the interpretability while still allowing a compact joint state for learning and control.

3 Co-evolutionary potential game and social–physics analysis

3.1 A co-evolving dynamic potential

We next construct a potential that links decentralized incentives with a welfare-relevant collective criterion. Let q i ∈ Δ m be a baseline strategic profile. For X = ( x 1 , … , x n ) and W = ( W 1 , … , W L ), definewhere W ̄ t = ∑ ℓ ν ℓ W ℓ , t and w ̄ i j t is its ( i , j ) entry. At iteration t, both W ̄ t and the target matrices W ̂ ℓ , t are computed from the current state and then held fixed during the two block updates. The coefficient λ / 4 compensates for the symmetric double counting of unordered pairs, so the action gradient of the diffusion term is − λ L c t X. The first line captures private benefits and resistance to implausibly abrupt action changes. The second line rewards strategic complementarity when B ℓ has positive off-diagonal effects but penalizes action disagreement across influential links. The last line makes links evidence-responsive and temporally smooth.

Φ t X , W = ∑ i = 1 n b i t ⊤ x i − α i 2 x i − q i 2 2 + 1 2 ∑ ℓ = 1 L ∑ i = 1 n ∑ j = 1 n w i j ℓ x i ⊤ B ℓ x j − λ 4 ∑ i = 1 n ∑ j = 1 n w ̄ i j t x i − x j 2 2 − κ 2 ∑ ℓ = 1 L W ℓ − W ̂ ℓ , t F 2 − τ 2 ∑ ℓ = 1 L W ℓ − W ℓ , t − 1 F 2 . (6)

The dynamic potential creates a transparent theoretical bridge. The common interaction term makes an agent internalize the effect of neighboring actions. The link term remains separable across entries after conditioning on actions, which allows localized projected updates. Unlike a static social-welfare objective, Equation 6 is recalibrated through the observed b i t and W ̂ ℓ , t. It can, therefore, absorb slowly changing economic conditions without redefining the game from scratch.

3.2 Local utility and exact-potential structure

For a fixed link state W, we give agent i the utility

u i t x i , X − i ; W = b i t ⊤ x i − α i 2 x i − q i 2 2 + ∑ ℓ = 1 L ∑ j = 1 n w i j ℓ x i ⊤ B ℓ x j − λ 2 ∑ j = 1 n w ̄ i j t x i − x j 2 2 . (7)

The term X − i denotes the actions of agents other than i. The link penalties in Equation 6 are handled by the network-update module rather than by a single agent because a symmetric link is a shared channel. Conditional on W, the agents play an exact potential game.

Theorem 1 (Exact-potential identity and equilibrium existence). Under Assumption 1, suppose W ℓ = ( W ℓ ) ⊤ and B ℓ = ( B ℓ ) ⊤ for every ℓ. For any i, any x i , x i ′ ∈ Δ m, and any fixed X − i and W, Consequently, every global maximizer of Φ t ( ⋅ , W ) over Δ m n is a Nash equilibrium of the action game. At least one such equilibrium exists.

Proof: Fix an agent i , and write Δ x i = x i − x i ′. The private term of the potential changes bywhich equals the corresponding utility difference. Next, consider a layer ℓ. Since both W ℓ and B ℓ are symmetric,where x p ′ = x p for p ≠ i. The equality follows because the terms with p = i and j = i are equal after transposition, while the diagonal term is included consistently in both expressions. This is exactly the difference in the interaction component in Equation 7.

b i t ⊤ Δ x i − α i 2 x i − q i 2 2 − x i ′ − q i 2 2 ,

1 2 ∑ p , j w p j ℓ x p ⊤ B ℓ x j − 1 2 ∑ p , j w p j ℓ x p ′ ⊤ B ℓ x j ′ = ∑ j = 1 n w i j ℓ Δ x i ⊤ B ℓ x j ,

For the diffusion term, only pairs containing i change. The symmetry of W ̄ t yieldsafter the two identical half-sums are combined. This coincides with the last term of the utility difference. The terms involving W ̂ ℓ , t and W ℓ , t − 1 do not depend on X when W is fixed and hence cancel. Adding the three components proves Equation 8.

− λ 2 ∑ j w ̄ i j t x i − x j 2 2 − x i ′ − x j 2 2

The set Δ m n is compact and nonempty, and Φ t ( ⋅ , W ) is continuous. Thus, a global maximizer X ⋆ exists. If an agent could profitably deviate from x i ⋆, Equation 8 would imply a strict increase in Φ t, contradicting global maximality. Therefore, X ⋆ is a Nash equilibrium.

3.3 Equilibrium selection and network externalities

The potential formulation also provides a direct social–physics interpretation of equilibrium selection and collective strategic alignment. Consider two actions k and r of agent i. At an interior action equilibrium, the simplex Karush–Kuhn–Tucker (KKT) conditions imply that every action with positive probability has the same marginal potential. Hence,

0 = g i k ⋆ − g i r ⋆ = b i k ⋆ − b i r ⋆ − α i x i k ⋆ − q i k − x i r ⋆ − q i r + ∑ ℓ = 1 L ∑ j = 1 n w i j ℓ , ⋆ B ℓ x j ⋆ k − B ℓ x j ⋆ r − λ ∑ j = 1 n w ̄ i j ⋆ x i k ⋆ − x j k ⋆ − x i r ⋆ − x j r ⋆ . (9)

Equation 9 distinguishes a private marginal advantage from two externalities. The first externality is strategic complementarity. A cooperative option becomes more attractive when well-connected neighbors assign mass to actions that interact favorably through B ℓ. The second is diffusion consistency. It discourages a local strategic move that departs sharply from the weighted neighborhood. These two channels need not agree. A new policy may privately reward a high-investment action, while the information network still supports a conservative action. The adaptive link term in Equation 6 allows the evidence to slowly resolve such tension without treating every short-lived discrepancy as a reason for full rewiring.

A useful question in this context is whether a few high-degree agents can create unbounded influence. The next proposition gives a simple answer. It bounds the marginal strategic externality by the local weighted degree and makes the role of the spectral guard explicit.

Proposition 1 (Bounded marginal externality). Let d i ℓ = ∑ j w i j ℓ and assume B ℓ 2 ≤ B ̄ ℓ. For any two action components k , r and any profile X ∈ Δ m n, the complementarity component of the marginal difference satisfies

Proof. Let e k and e r be canonical unit vectors. For each layer and neighbor,

B ℓ x j k − B ℓ x j r = e k − e r ⊤ B ℓ x j .

By Cauchy–Schwarz inequality,

e k − e r ⊤ B ℓ x j ≤ e k − e r 2 B ℓ 2 x j 2 .

Because x j ∈ Δ m, x j 2 ≤ 1, and e k − e r 2 = 2. Multiplication by the non-negative weight w i j ℓ and summation over j and ℓ prove Equation 10. For the diffusion difference, each coordinate of a simplex vector lies in [0, 1]. Thus,

x i k − x j k − x i r − x j r ≤ 2 .

Multiplication by λ w ̄ i j and summation complete the second bound.

The proposition is not only a technical estimate. It explains the designer why step sizes must react to local degree and complementarity. A dense market layer with a large B ̄ ℓ can magnify a small incentive shock. The spectral guard is a network-level counterpart of the component-wise bound: it controls the collective amplification that arises when many such local effects act at once.

3.4 Curvature and stability interpretation

The game can have several equilibria when complementarity is strong. We, therefore, identify a tractable region where the potential has adequate action curvature. Letwhere α min = min i α i. When μ x t > 0, the negative private-resistance curvature dominates the maximum positive curvature contributed by strategic complementarity. The diffusion Hessian is − λ L c t ⊗ I m ⪯ 0 and, therefore, cannot weaken concavity; it can only strengthen the curvature away from the consensus subspace. This condition prevents the coordination term from turning every update into a runaway imitation process.

μ x t = α min − ∑ ℓ = 1 L W ℓ , t 2 B ℓ 2 , (11)

The link block has curvature μ W = κ + τ. The parameter κ expresses confidence in the evidence-supported target, while τ suppresses excessive day-to-day reconfiguration. Their sum is strictly positive. The two blocks remain coupled through w i j ℓ x i ⊤ B ℓ x j, which is the source of both useful adaptation and possible instability. We quantify it by applying the cross-sensitivity constant

C x W = sup X , W ∈ Δ m n × W ∇ X W 2 Φ t X , W 2 . (12)

The SGMR guard will use the estimates of Equations 11, 12 to prevent simultaneous large steps.

Figure 1 shows the modeling architecture. The right side of the architecture summarizes local payoff and action information, signals and beliefs, and policy and constraint information, while the lower module jointly updates actions and links. The construction avoids a centralized welfare oracle. Each node needs local payoff estimates, neighboring strategies, and incident links. The framework is consequently compatible with privacy-sensitive market settings in which global data pooling is impractical.

Remark 2 The potential is deliberately dynamic rather than time-invariant. A static potential can certify equilibrium under a fixed environment, but it cannot distinguish genuine structural adaptation from a response to temporary evidence. The target-link and temporal-smoothing terms separate these effects. They also create explicit parameters for policy interpretation: one can examine how confidence in data, inertia of relationships, and diffusion pressure may affect equilibrium selection.

4 Stability-guarded evolutionary dynamics

4.1 Design motivation and main idea

A decentralized update must solve three problems at once. It should improve a local action using only neighborhood information. It should adapt links toward economically supported relations instead of fitting noise. Finally, it should avoid an aggressive action update when the evolving network increases the effective curvature of the game. A direct replicator rule addresses the first but not the second or third problem. A direct graph-learning rule addresses the second issue but can destabilize the strategic state.

The proposed SGMR algorithm solves this by alternating an entropic action step and a projected link step. The action step is a mirror-replicator update. Its logarithmic geometry respects the simplex and retains exploratory probability mass. The link step is a projected gradient update toward the potential-improving direction. A spectral guard computes a conservative local Lipschitz estimate and clips both rates. The action and link blocks are, therefore, not merely concatenated; they are jointly conditioned on the current graph spectrum.

For agent i, we define the local gradient

g i t = ∇ x i u i t = b i t − α i x i t − q i + ∑ ℓ = 1 L ∑ j = 1 n w i j ℓ , t B ℓ x j t − λ ∑ j = 1 n w ̄ i j t x i t − x j t . (13)

The action update iswhere ⊙ denotes element-wise multiplication. Equation 14 follows from maximizing a linearized payoff minus the entropy divergence. It can be executed with local actions and weighted neighbor aggregates.

x i t + 1 = x i t ⊙ exp η t g i t 1 ⊤ x i t ⊙ exp η t g i t , (14)

Proposition 2 (Replicator limit of the mirror strategy revision). For an interior strategy x i t and a bounded payoff gradient g i t, the update in Equation 14 satisfies, component-wise,Consequently, as η t → 0, the strategy block approaches the replicator dynamic

Proof. Using exp ( η t g i k t ) = 1 + η t g i k t + O ( η t 2 ) and ∑ r x i r t = 1, the denominator in Equation 14 is

1 + η t x i t , g i t + O η t 2 .

Applying ( 1 + a ) − 1 = 1 − a + O ( a 2 ) and collecting first-order terms yieldwhich proves Equation 15. The continuous-time Equation 16 follows by dividing by η t and taking the small-step limit.

x i k t + 1 = x i k t + η t x i k t g i k t − x i t , g i t + O η t 2 ,

This result makes the evolutionary content of SGMR explicit. Strategies with above-average marginal payoff increase, whereas strategies with below-average payoff decline. Unlike a standard replicator process on a fixed graph, however, the payoff landscape here changes endogenously because the multilayer links are updated in the second block.

After the action update, the link gradient on layer ℓ isfollowed by symmetrization and zeroing the diagonal. The updatepreserves feasibility. Local implementation is possible because each entry h i j ℓ , t depends only on two endpoints, their current actions, and the local target-link estimate.

H ℓ , t = ∇ W ℓ Φ t X t + 1 , W t = 1 2 X t + 1 B ℓ X t + 1 ⊤ − κ W ℓ , t − W ̂ ℓ , t − τ W ℓ , t − W ℓ , t − 1 , (17)

W ℓ , t + 1 = Π W ℓ W ℓ , t + β t H ℓ , t , (18)

4.2 Spectral stability guard

Let L ̂ x t be a computable upper bound for the action-block Lipschitz constant, and let L ̂ W t denote the exact link-block constant:

L ̂ x t = α max + ∑ ℓ = 1 L W ℓ , t 2 B ℓ 2 + λ λ max L c t , (19)

L ̂ W t = κ + τ . (20)

For fixed X, the interaction term is linear in each W ℓ, so the link-block gradient curvature is generated only by the two quadratic regularizers. The required spectral norms of the sparse adjacency and coupling-Laplacian matrices can be estimated by a few local power iterations. The guard setswhere ϑ x , ϑ W ∈ ( 0,1 ) and ϵ > 0. The rate ρ t is used when the target-link estimator applies a moving-average correction. This rate decreases when the cross-sensitivity is high. Hence, the system slows link adaptation precisely when an action perturbation has an unusually large effect on the link gradient.

η t = min η max , ϑ x L ̂ x t + ϵ , β t = min β max , ϑ W L ̂ W t + ϵ , ρ t = min ρ max , ϑ ρ 1 + C x W t , (21)

Table 1 positions SGMR relative to neighboring methodological families, while Table 2 summarizes the notation and definitions used throughout the study. Table 3 summarizes the algorithm. The method separates observation calibration from strategic optimization, which prevents raw data fluctuations from entering the payoff gradient without filtering. The action stage is fully parallel. The link stage is edge-local and can also be parallelized after endpoint reconciliation. The only network-wide quantity is the conservative spectral estimate; it can be approximated through distributed power iterations. Thus, SGMR retains the distributed character required for large economic systems.

TABLE 1. Positioning of SGMR relative to neighboring methodological families.

TABLE 2. Notation and definitions used in this study.

TABLE 3. Pseudocode of the stability-guarded mirror-replicator algorithm.

Figure 2 shows a representative guard landscape. The white boundary separates combinations of the action-update step size η and link-update step size β with estimated contraction margin above or below zero. The selected operating point is intentionally interior to the stable region. This margin provides more information than a fixed learning rate because it reveals how a denser or more complementary network reduces the permissible adaptation speed. The same calculation can, therefore, act as a diagnostic for economic policy interventions that abruptly alter network connectivity.

Remark 3 The main algorithmic innovation is not simply an additional regularizer. It is the coupling of an exact-potential action update with a link update, whose rate is governed by the estimated network curvature. This makes the algorithm interpretable: rapid adaptation is allowed in weakly coupled regions, whereas strongly coupled regions activate a conservative regime. The guard, thereby, turns spectral information into an operational stability decision rather than a post hoc diagnostic.

5 Analysis of the co-evolutionary dynamics

5.1 One-step potential improvement

We first establish a descent-style property for the negative of the dynamic potential. The result permits nonconvexity of the full joint objective because it treats action and link blocks separately. Let Φ t ⋆ = max X , W Φ t ( X , W ), and let the environment be fixed within a single update. The time-varying case is addressed after the theorem.

Assumption 2 (Block smoothness). For fixed W , Φ t ( ⋅ , W ) has an L x t-Lipschitz gradient on Δ m n. For fixed X , Φ t ( X , ⋅ ) has an L W t-Lipschitz gradient on W. The guard estimates satisfy L ̂ x t ≥ L x t and L ̂ W t ≥ L W t.

Theorem 2 (Monotone potential improvement). Under Assumptions 1 and 2, suppose η t < 1 / L x t and β t < 1 / L W t. Then, one SGMR update satisfiesIn particular, the potential is nondecreasing.

Proof. For the action block, use the standard smoothness inequality with W = W t:

Φ t X t + 1 , W t ≥ Φ t X t , W t + ∑ i = 1 n g i t , x i t + 1 − x i t − L x t 2 ∑ i = 1 n x i t + 1 − x i t 2 2 . (23)

The mirror update Equation 14 is the solution of

arg max x i ∈ Δ m g i t , x i − 1 η t D h x i , x i t .

Comparing its objective value with the feasible point x i t yields

g i t , x i t + 1 − x i t ≥ 1 η t D h x i t + 1 , x i t .

Pinsker’s inequality yields

1 2 x i t + 1 − x i t 2 2 ≤ 1 2 x i t + 1 − x i t 1 2 ≤ D h x i t + 1 , x i t .

Substitution into Equation 23 proves

Φ t X t + 1 , W t − Φ t X t , W t ≥ 1 η t − L x t ∑ i D h x i t + 1 , x i t . (24)

For the link block, projected-gradient optimality yields

∇ W Φ t X t + 1 , W t , W t + 1 − W t ≥ 1 β t W t + 1 − W t F 2 .

Apply block smoothness to Φ t ( X t + 1 , ⋅ ):

Φ t X t + 1 , W t + 1 ≥ Φ t X t + 1 , W t + ∇ W Φ t , W t + 1 − W t − L W t 2 W t + 1 − W t F 2 ≥ Φ t X t + 1 , W t + 1 β t − L W t 2 W t + 1 − W t F 2 .

Combining this inequality with Equation 24 yields Equation 22. Both coefficients are positive under the stated rates, so the potential is nondecreasing.

5.2 Stationarity and finite-horizon complexity

The potential is bounded above because the action and link sets are compact. Theorem 2, therefore, implies that the cumulative action divergence and link motion are finite. To translate this into a familiar complexity measure, define

G t 2 = ∑ i = 1 n 2 η t 2 D h x i t + 1 , x i t + 1 β t 2 W t + 1 − W t F 2 . (25)

The first term is a mirror-projected residual. The second is the projected link-gradient residual. Both vanish at a fixed point of the update map.

Theorem 3 (Sublinear stationarity). Let η ̲ ≤ η t ≤ η ̄ < 1 / L x t and β ̲ ≤ β t ≤ β ̄ < 1 / L W t. DefineThen, for any T ≥ 1,where Φ T ⋆ = max X , W Φ T ( X , W ) and V T = ∑ t = 0 T − 1 sup X , W | Φ t + 1 ( X , W ) − Φ t ( X , W ) | quantifies the environmental variation. If the environment is stationary, V T = 0, and the best residual is O ( T − 1 / 2 ).

Proof. By Equation 25, ∑ i D h ( x i t + 1 , x i t ) ≥ η t 2 2 G x , t 2 and W t + 1 − W t F 2 = β t 2 G W , t 2. Using these relations in Equation 22, together with the uniform step bounds, yields

Φ t X t + 1 , W t + 1 − Φ t X t , W t ≥ c G t 2 .

The potential used by the next iteration may change because rewards and target links are refreshed. Thus,where υ t = sup X , W | Φ t + 1 ( X , W ) − Φ t ( X , W ) |. Adding this inequality to the one-step improvement bound and summing from t = 0 to T − 1 yield

Φ t + 1 X t + 1 , W t + 1 ≥ Φ t X t + 1 , W t + 1 − υ t ,

c ∑ t = 0 T − 1 G t 2 ≤ Φ T ⁢ X T , W T − Φ 0 ⁢ X 0 , W 0 + ∑ t = 0 T − 1 υ t ≤ Φ T ⋆ − Φ 0 ⁢ X 0 , W 0 + V T .

The smallest residual cannot exceed the average residual, which proves Equation 26. Under stationarity, V T = 0. Taking square roots yields the stated O ( T − 1 / 2 ) residual behavior.

The theorem provides a meaningful guarantee for a nonstationary economic system. The variation term is explicit. Slow changes in rewards or policy signals mildly enlarge V T, while abrupt institutional shocks create a larger tracking burden. The guard cannot remove this fundamental limit, but it prevents the update rule from adding avoidable instability on top of it.

5.3 Communication and computational complexity

The SGMR update has a different cost profile from centralized welfare optimization. Let E ℓ be the active edge set of layer ℓ , and let | E | = ∑ ℓ | E ℓ |. Computing all local gradients in Equation 13 requires O ( | E | m 2 ) arithmetic operations when the interaction matrices are dense. In the common case where B ℓ is diagonal or low rank, this reduces to O ( | E | m ). The edge-wise link update requires O ( | E | m 2 ) operations and O ( | E | ) messages. A K-round power iteration for the largest spectral value costs O ( K | E | ) messages. No global payoff matrix, full agent-action history, or centralized graph storage is needed.

Proposition 3 (Local implementation complexity). Assume that each agent sends its current m-dimensional action to every neighbor once per iteration and each active edge exchanges one scalar link update per layer. Then, one iteration of SGMR has message complexityand memory complexity O ( d i m + L d i ) at agent i, where d i = ∑ ℓ | N i ℓ |. If every B ℓ is diagonal, the arithmetic complexity of an iteration is O ( | E | m + K | E | ).

Proof. For the action stage, agent i needs the m-vector x j from each neighboring agent j. Across all directed communication events, this is O ( | E | m ). The link stage exchanges a scalar value for each active undirected edge and layer; since | E | already counts layer-specific active edges, this cost is O ( | E | ). A power iteration consists of one scalar aggregation per edge; K iterations consequently add O ( K | E | ). Summing the three terms yields Equation 27.

Agent i stores its own action, its received neighbor actions, and incident link weights. The action cache contains O ( d i m ) values, while link information contains O ( L d i ) values. For a diagonal B ℓ, the product B ℓ x j uses O ( m ) operations rather than O ( m 2 ). Summation over active edges yields O ( | E | m ) arithmetic operations, and the power estimator contributes O ( K | E | ).

This result clarifies that the stability guard is not a prohibitive centralized calculation. It adds a tunable communication overhead K | E |, with small K typically sufficient because the guard only needs a conservative spectral upper estimate. The overhead is especially modest when the graph evolves slowly, in which case the spectral vector from the previous iteration provides an effective warm start.

5.4 Local robust stability under disturbances

We now examine the behavior near an isolated equilibrium Z ⋆ = ( X ⋆ , W ⋆ ). Let the actual update contain an additive perturbation δ t that aggregates reward noise, imperfect spectral estimates, and delayed neighbor messages:

Z t + 1 = T t Z t + δ t . (28)

The following result provides a precise mathematical characterization of the spectral guard’s role.

Theorem 4 (Input-to-state stability near a regular equilibrium). Suppose that Z ⋆ = ( X ⋆ , W ⋆ ) is a fixed point shared by the local update maps in a neighborhood N. Assume that, for all Z ∈ N, the unperturbed block updates satisfywhere q x , q W ∈ [ 0,1 ) are uniform local block-contraction factors. Defineand the weighted product normThen, every perturbed trajectory that remains in N satisfiesHence, bounded disturbances produce bounded equilibrium deviations, and the zero-disturbance dynamics converge locally at a linear rate in the weighted norm.

Proof. Write Δ X = X − X ⋆ and Δ W = W − W ⋆. Multiplying the second assumed block bound by η / β and adding it to the first yield

T t Z − Z ⋆ η , β ≤ q x + η β C x W Δ X 2 + q W + η β C x W η β Δ W F ≤ q Z − Z ⋆ η , β .

Using Equation 28 and the triangle inequality, therefore, yieldsIteration of this recursion yieldsBounding the finite geometric sum proves Equation 30.

Z t + 1 − Z ⋆ η , β ≤ q Z t − Z ⋆ η , β + δ t η , β .

Z t − Z ⋆ η , β ≤ q t Z 0 − Z ⋆ η , β + ∑ s = 0 t − 1 q t − 1 − s δ s η , β .

Remark 4 The analysis distinguishes two forms of convergence. The global result in Theorem 3 yields stationarity of the alternating method with an explicit cost for environmental variation. The local result in Theorem 4 yields a sharper linear bound near a regular equilibrium. This distinction is important in economic networks because a model may need broad exploratory adaptation before it enters a neighborhood where policy-sensitive stability is most important.

Remark 5 (Local stationarity and a common fixed point). Local stationarity means that the reward estimator, admissible-link mask, and policy regime change on a time scale longer than the inner SGMR settling horizon. A shared local fixed point is, therefore, a reasonable approximation during a rolling window in which the same market regime and institutional rules remain active or when successive target-link and reward estimates vary only by small sampling errors. After an abrupt policy, demand, or information shock, this approximation need not hold; the approximation is thus the finite-horizon tracking result in Theorem 3, where the variation budget V T explicitly records the loss of stationarity.

6 Computational experiments and social-system analysis

6.1 Experimental protocol

We use reproducible synthetic economic networks to isolate the mechanisms of strategy–network coevolution. Each experiment uses three behavioral categories: conservative, balanced, and cooperative investment. The resulting strategy shares can be interpreted either as mixed choices of heterogeneous agents or as local population fractions, consistent with Proposition 2. The topology study uses a Watts–Strogatz graph with a mean degree of 6 and a rewiring probability of 0.15 for the small-world case, a Barabási–Albert graph with three links per arriving node for the scale-free network, and three approximately equal communities with within-/between-community probabilities 0.09 / 0.018 for the modular case. The institutional mask retains only admissible within- and cross-community relations, while initial nonzero weights equal the evidence target plus zero-mean Gaussian perturbations of standard deviation 0.06, clipped to [0, 0.82]. Local reward vectors follow a bounded autoregressive process. Each reported number is an average over 50 independent seeds, with common initialization across methods. All “±” values denote one standard deviation across these seeds, not a standard error or confidence interval.

We compare SGMR with static mirror ascent (SMA), standard replicator dynamics (RD), and damped fictitious play (DFP). SMA uses the same payoff model but freezes links at their initial state. RD updates actions through normalized payoff differences and uses a fixed link correction. DFP maintains empirical opponent actions but does not use a spectral guard. Initial mixed strategies are drawn from Dirichlet ( 2,2,2 ); the baseline profiles use Dirichlet ( 2.2 , 3.2 , 2.4 ). Hyperparameters are selected on 10 held-out seeds by maximizing final welfare while requiring a residual below 10⁻⁴ and a nonnegative estimated contraction margin; the selected hyperparameters are then frozen for all reported topologies. The added diagnostics were executed using Python 3.13 with NumPy 2.3.5, NetworkX 3.6.1, and Matplotlib 3.10.8. The archive contains reproducibility_sgmr.py, fixed master seeds of 20, 260, 723–202, 60, 725, and the generated metric file.

Table 4 specifies the reproducible setting rather than a hidden empirical benchmark. The chosen private curvature exceeds the strongest combined complementarity under the typical graph spectrum, which places the main operating regime inside the stable region predicted by Section 5. The three layers have different coupling strengths to represent a market interaction layer with stronger payoff effects, a weaker information diffusion layer, and a constrained institutional coordination layer. Varying network size and observation noise then evaluates whether the spectral guard has a meaningful effect beyond one favorable configuration.

TABLE 4. Computational settings for adaptive multilayer economic networks.

A concrete empirical mapping would construct the market interaction layer from transaction, supplier–buyer, credit-exposure, or co-investment records; the information diffusion layer from communication, news co-mention, repost, search, or forecast-sharing data; and the institutional coordination layer from shared contracts, regulatory jurisdictions, standards, or eligibility constraints. The private reward vector could be estimated from realized profit, adoption return, service quality, or policy compliance, while s i j ℓ , t could be a rolling co-movement, interaction frequency, or compatibility score. These measurements would replace the synthetic generators without altering the SGMR update equations.

6.2 Convergence and welfare

Figure 3 plots the normalized potential against iteration for n = 120. SGMR increases sharply during the first 45 iterations and approaches a stable plateau near 0.915. SMA improves more slowly because it cannot reinforce strategically compatible links. RD is competitive early but develops small oscillations caused by its fixed adjustment rate. DFP has the lowest plateau because empirical-belief averaging delays its response to a changing graph. The separation after iteration 60 indicates that the gain is not merely an initialization effect; it is sustained by coordinated action and link adaptation.

To examine Theorem 3 directly, we ran an additional stationary diagnostic for 700 iterations and recorded the residual in Equation 25. A log–log regression over iterations 60–600 yields a slope of − 3.108 with R 2 = 0.998, which is substantially steeper than the worst-case − 1 / 2 exponent. The theorem is, therefore, not tight for this regular instance; it remains a uniform upper bound intended to cover unfavorable curvature and environmental variation rather than predict an exact empirical exponent. Figure 4 shows the best-so-far SGMR residual together with the T−1/2 worst-case reference rate.

Table 5 reports normalized welfare and iterations to the residual criterion. SGMR is the best in every topology. The largest advantage is observed for the modular graph, where disconnected communities create the most scope for incorrect local reinforcement. There, adaptive links identify useful cross-community relations, while the spectral guard limits overreaction. The gap between SGMR and SMA shows the value of link adaptation. The additional gap over RD and DFP shows that adaptation without curvature control is insufficient when the local network structure is heterogeneous.

TABLE 5. Collective outcomes across representative complex-network topologies.

Because identical initialization seeds were used across methods, a paired test would normally be preferred; the archived summary contains only marginal means and standard deviations, so Table 6 reports the more conservative independent-sample Welch calculation. All three comparisons remain significant after Bonferroni correction, with large standardized effects.

TABLE 6. Conservative two-sided Welch tests for SGMR versus the strongest welfare baseline (SMA).

Figure 5 depicts the evaluation of scalability. Welfare increases slightly with system size for all methods because larger graphs create more opportunities for beneficial matching. However, the slope differs. SGMR continues to gain value as the number of agents increase, while SMA and RD flatten after approximately 160 agents. The flattening indicates that a static or unguided update fails to exploit additional network diversity. The distributed implementation remains practical because each action step uses neighborhood aggregates, and the extra guard requires only a lightweight spectral approximation.

6.3 Ablation and network adaptation

We next remove individual components of SGMR. “No target” replaces the evidence-supported link target by a constant prior. “No guard” fixes η and β at their average values. “No inertia” sets τ = 0. The full model remains best across all criteria.

Table 7 isolates the algorithmic mechanisms. Removing the target link degrades welfare and increases turnover, which indicates that a graph should respond to economically grounded evidence rather than only to current actions. Removing the guard has the strongest effect on the potential variance and final residual. This matches Theorem 4: simultaneous large action and link movements amplify cross-sensitivity. Removing inertia produces lesser welfare loss but the highest link turnover, confirming that the temporal term improves interpretability by discouraging unstable relationship switching.

TABLE 7. Ablation results for the proposed method.

Figure 6 shows an illustrative scale-free market interaction layer before and after learning. Node shading represents the cooperative-action share, while edge thickness reflects the learned interaction strength. The initial graph contains several strong edges between agents with divergent strategic propensities. After learning, weights concentrate around compatible action neighborhoods, yet the graph does not collapse into isolated clusters. This pattern is economically desirable. It retains bridges that support useful diffusion while reducing channels that repeatedly transmit incompatible incentives. The temporal penalty prevents a visually implausible full rewiring.

6.4 Robustness to observation noise

The final experiment adds zero-mean noise to local reward estimates and relation signals. The disturbance enters both Equations 2, 3. We report equilibrium welfare retention, defined as the welfare relative to the corresponding noise-free run. This directly evaluates the bounded-deviation statement in Theorem 4.

Figure 7 confirms that SGMR degrades gradually as the observation noise increases. At a standard deviation of 0.18, it retains approximately 0.846 of normalized welfare, compared with 0.715 for SMA and 0.616 for RD. The shaded band denotes the observed seed variation for SGMR. The improvement is consistent with the theory: the target-link term filters transient evidence, temporal inertia limits rapid rewiring, and the guard reduces step sizes when the noisy state creates a larger estimated curvature. No method can eliminate information loss from severe noise, but SGMR delays the resulting welfare collapse.

For this diagnostic, the clean fixed point is first computed to numerical tolerance, and the weighted norm in Theorem 4 is then evaluated directly. The 90th-percentile local contraction estimate is q ̂ = 0.905. For every tested noise level, the measured steady-state state deviation lies below max t ‖ δ t ‖ η , β / ( 1 − q ̂ ), and the empirical disturbance-to-state gain remains between 3.41 and 3.58. This provides a direct state-space verification rather than relying only on welfare preservation. The corresponding numerical state-deviation results and empirical ISS envelope are summarized in Table 8 and Figure 8, respectively.

TABLE 8. Direct state-deviation verification of Theorem 4.

Remark 6 The simulations are designed to evaluate co-evolutionary mechanisms rather than to claim a universal empirical effect for every economy or social system. They demonstrate that the theoretical quantities have observable consequences: the spectral guard reduces variance, the target-link term improves adaptive matching, and temporal inertia reduces unnecessary turnover. A domain deployment should replace the synthetic reward process with measured variables and validate the assumed network layers using domain-specific economic evidence.

7 Conclusions and future work

We developed a social–physics framework for the evolutionary dynamics of adaptive multilayer economic networks. The model unifies behavioral propensities, strategic complementarities, information diffusion, institutional constraints, and data-calibrated link rewiring in a single dynamic potential. The proposed SGMR process combines entropy-regularized strategy revision with projected network adaptation and a spectral stability guard. Its small-step limit recovers a replicator equation, while the discrete-time analysis establishes the exact-potential property, monotone potential improvement, sublinear stationarity, and local input-to-state stability under disturbances. Computational experiments show that coordinating strategy revision with network adaptation improves collective welfare, convergence speed, and resilience relative to fixed-network and unguarded evolutionary baselines.

The results highlight a general mechanism for complex social systems: cooperation and collective performance depend not only on payoff differences but also on how interaction channels are reinforced, weakened, and filtered over time. The present evidence is, nevertheless, synthetic and mechanism-oriented; it does not establish causal validity, forecasting accuracy, or policy effectiveness for a particular economy. A real-data study should estimate transaction/exposure links, information-sharing links, and institutional compatibility from time-stamped records, infer rewards from measurable outcomes, and evaluate both out-of-sample fit and policy counterfactuals. Three extensions are especially important. First, the multilayer structure should be estimated from real transaction, policy, communication, and platform data while accounting for missing links and strategic privacy. Second, directed, signed, higher-order, and coalition interactions should be incorporated to represent the asymmetric influence and group-level decision-making. Third, mechanism-design variants could allow regulators or platforms to learn incentives that improve welfare, fairness, resilience, and sustainability under explicit budget constraints. These directions would connect the present analytical model more closely with empirical evolutionary dynamics and policy-oriented applications in complex social systems.

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Footnotes

Edited by: Kaipeng Hu, Yunnan University of Finance and Economics, China

Reviewed by: Shitao Quan, Yunnan University of Finance and Economics, China

Yong Ding, Nanchang University, China