Entropy-regularized strategic dynamics on semantic complex networks: structured representation and equilibrium

1 Introduction

Complex human-centered systems are composed of humans, organizations, digital agents, infrastructures, and institutional rules [1]. Their decisions are interactive rather than isolated. A user’s route choice changes congestion. A platform’s recommendation changes group attention. A hospital’s resource allocation changes patient behavior [2]. A power-grid operator’s incentive changes residential demand [3]. Such systems appear in transportation, healthcare, social networks, public safety, energy management, online platforms, and human–AI collaboration [4, 5]. The main difficulty is that each agent acts with partial knowledge, bounded rationality, and context-dependent preferences [6]. Hence, the system is both a complex network and a strategic decision process. Classical equilibrium theory provides a rigorous language for interactions, but its standard form is often too abstract for knowledge-rich human systems [7].

Game theory, evolutionary dynamics, and distributed optimization have been widely used to model strategic interaction [8]. Potential games, originally formalized by [9], explain when unilateral incentives can be represented by a single global potential. Concave games and variational inequalities further provide existence, uniqueness, and sensitivity tools for continuous constrained strategy spaces [10, 11]. In parallel, entropy-regularized learning has a long connection with bounded rationality through logit or quantal response equilibria and smooth learning in games [12, 13, 14]. Recent network games and multi-agent decision systems extend these ideas to large-scale interactions [15, 16–18]. This body of work is mathematically mature, but it usually assumes that players, actions, payoff coefficients, and constraints are already available in a clean numerical form. In human-centered systems, these primitives are instead inferred from text, events, institutional rules, sensor records, and relational observations. A plain adjacency matrix cannot preserve the meanings of “trusts,” “competes with,” “violates,” “depends on,” or “shares a scarce resource” [19, 20].

Knowledge graphs offer a structured way to represent such meanings [21]. A knowledge graph stores entities and relations as triples, and its embeddings transform symbolic relations into continuous vectors [22]. Translational, rotational, bilinear, and hyperbolic embeddings have improved link prediction and relational reasoning [23]. Graph neural networks extend representation learning to irregular networks [24], and attention-based message passing assigns adaptive weights to heterogeneous neighbors [25]. Recent surveys and networked-knowledge studies show that knowledge graphs can connect semantic reasoning, complex networks, and decision intelligence [26, 27]. However, most knowledge graph methods focus on prediction, retrieval, or representation quality. They seldom impose the symmetry, monotonicity, or potential structure needed to obtain equilibrium guarantees.

The current literature therefore leaves a methodological gap. Game-theoretic models have deep equilibrium theory but weak semantic modeling, while knowledge graphs have strong semantic representation but weak strategic analysis [28]. Recent studies on graph entropy, game-theoretic clustering, hierarchical multi-agent learning, fairness-aware coordination, consensus games, cooperative game approximation, and hypergames indicate that this gap is becoming more important, especially when different agents may reason with different information sets [29, 30]. The novelty of this work is not to append embeddings as ordinary features. Instead, structured knowledge is projected into payoff tensors, feasible-set penalties, risk terms, and a logit-style entropy regularizer, after which a symmetrized potential layer is constructed so that equilibrium computation remains analyzable. This design directly links heterogeneous semantic representation with information-theoretic equilibrium learning, which is aligned with the theme of structured representation learning and information-theoretic principles.

In this article, we propose a knowledge graph-enhanced proximal strategy model, named KGE-GPS. The method first constructs a heterogeneous knowledge graph from actors, goals, resources, norms, risks, and contexts. It then learns an embedding vector for each entity and relation. The embeddings are projected into three strategic objects: a semantic payoff matrix, a constraint-weight vector, and a revision kernel. These objects define a semantic potential game over a dynamic human-centered network. We then design a proximal best-response update with entropy regularization; the entropy term acts as an information-theoretic stabilizer that discourages overly deterministic moves and captures bounded rationality. Recent Nash-seeking and game-based consensus studies show that event-triggered and hybrid mechanisms can improve multi-agent convergence. Our algorithm differs by using semantic knowledge and information-theoretic regularization to reshape the game itself.

The main contributions are summarized as follows.

First, we formulate an entropy-regularized, knowledge graph-enhanced semantic potential game that jointly models relational semantics, network coupling, and human-centered constraints. Related multi-agent reinforcement-learning studies have also exploited local-information attention and structured action graphs for complex decision-making [31, 32]. Second, we specify an explicit knowledge-to-game projection and symmetrization operator, so that directed and noisy semantic relations can be used while the potential-game condition is either exactly enforced or treated as a bounded perturbation. Third, we design a linearized entropy-proximal strategy algorithm with an interior-simplex safeguard and reproducible implementation details. Fourth, we prove existence, variational-inequality-based uniqueness, convergence, and robustness under explicit Lipschitz, monotonicity, entropy, and step-size assumptions. These results extend classical potential-game analysis to knowledge-perturbed complex networks and bridge structured representation learning and equilibrium computation.

The rest of this article is organized as follows. Section 2 introduces the problem description and preliminaries. Section 3 presents the knowledge graph-enhanced game model and its equilibrium structure. Section 4 designs the KGE-GPS algorithm. Section 5 analyzes convergence, equilibrium residuals, and robustness. Section 6 reports simulation results with tables and figures. Section 7 concludes the article and discusses future research directions. The main symbols and definitions used throughout the article are summarized in Table 1.

2 Problem description and preliminaries

2.1 Human-centered strategic network

Consider a human-centered system with n agents. The agents may be people, organizations, service units, vehicles, sensors with human supervision, or AI assistants acting for humans. Let N = { 1,2 , … , n }. Agent i chooses a mixed strategy x i ∈ Δ i, where Δ i = { x i ∈ R + m i : 1 ⊤ x i = 1 }. The joint strategy is x = ( x 1 , … , x n ) ∈ Δ = ∏ i Δ i. The interaction graph at time t is G t = ( N , E t , W t ). The element w i j , t ≥ 0 measures the intensity of influence from agent j to agent i.

A classical network game defines u i ( x i , x − i ; W t ) directly from W t. This is insufficient when the edge i ← j has a semantic type. For example, a relation may indicate trust, competition, dependence, regulation, resource sharing, social influence, or risk transmission. We therefore introduce a knowledge graph K = ( V K , R K , T K ), where T K ⊆ V K × R K × V K is the set of triples. The knowledge graph contains six entity groups: actors, goals, resources, norms, risks, and contexts. The aim is to use K to define a game whose equilibria are both strategically stable and semantically interpretable and to regularize the search via entropy.

TABLE 1. Main symbols and definitions.

2.2 Knowledge embedding and semantic coupling

For each entity e ∈ V K and relation r ∈ R K, let z e , z r ∈ R d be embeddings. These form the structured representations of the underlying relational data. A general triple score is written as Equation 1:where h is the head entity and q is the tail entity. The embedding loss is Equation 2:where T ̄ K is a negative triple set. The learned embeddings induce a semantic coupling (Equation 3).where r ( i , j ) is the dominant relation type between agents. The effective strategic weight is Equation 4:

ψ h , r , q = φ z h , z r , z q , (1)

L K = ∑ h , r , q ∈ T K ℓ ψ h , r , q , 1 + ∑ h ̄ , r ̄ , q ̄ ∈ T ̄ K ℓ ψ h ̄ , r ̄ , q ̄ , 0 , (2)

S i j = σ a ⊤ z i , z j , z r i , j + b , (3)

w ̃ i j , t = w i j , t 1 + α S i j , α ≥ 0 . (4)

This formulation preserves physical or social adjacency while allowing structured semantics to amplify or attenuate interaction. Information-theoretically, the entropy of the relational embeddings can be related to the uncertainty of the semantic coupling, but here we use the coupling directly as a deterministic weight modifier.

2.3 Equilibrium concepts

The utility of agent i is denoted by u i ( x i , x − i ; K , G t ). A strategy profile x ⋆ is a Nash equilibrium if Equation 5:For computation, we use the Nash residual (Equation 6).A profile is an ϵ-Nash equilibrium if R ( x ) ≤ ϵ.

u i x i ⋆ , x − i ⋆ ≥ u i y i , x − i ⋆ , ∀ y i ∈ Δ i , ∀ i . (5)

R x = max i ∈ N max y i ∈ Δ i u i y i , x − i − u i x i , x − i . (6)

Remark 1: This section defines the decision objects used throughout the article. The key modeling step is separating structural adjacency from semantic coupling. The adjacency matrix tells who interacts with whom, while the knowledge graph explains why the interaction matters. This separation allows later analysis to remain in a clear game-theoretic form while still representing human-centered semantics. Later, an entropy term will be added to stabilize the learning dynamics.

3 Model and analysis

3.1 Semantic potential game formulation

We define the utility of agent i as Equation 7:where Q i ≻ 0 is a self-cost matrix, B i j represents pairwise strategic complementarity or substitution, Ω i is a norm-induced regularizer, and C i is a risk-sensitive cost. The knowledge graph affects w ̃ i j, Ω i, and C i. A typical norm-induced term is Equation 8:where ν i k is high when strategy k violates a strong norm or consumes a scarce resource. A risk-sensitive term is Equation 9:where r i k ∈ [ 0,1 ) is the knowledge-inferred risk of strategy k and δ > 0 avoids singularity.

u i x = a i ⊤ x i − 1 2 x i ⊤ Q i x i + ∑ j ∈ N i w ̃ i j x i ⊤ B i j x j − λ i Ω i x i ; K − γ i C i x i ; K , (7)

Ω i x i ; K = ∑ k = 1 m i ν i k K x i k 2 , (8)

C i x i ; K = − log 1 − ∑ k = 1 m i r i k K x i k + δ , (9)

Figure 1 illustrates the overall framework. The left part collects logs, rules, records, and contextual observations from a human-centered system. These data are converted into a knowledge graph, whose embeddings generate payoff modifiers and semantic constraints. The dynamic interaction graph captures who can affect whom, while the knowledge graph clarifies the meaning of each influence. The equilibrium solver receives both sources and outputs a stable strategy profile. This design makes the strategic model more interpretable than a pure neural solver because each payoff component has a semantic source.

The proposed game is a semantic potential game if there exists Φ : Δ → R such that, for any i and any x i , y i ∈ Δ i (Equation 10),We now specify how this condition is enforced from a directed and heterogeneous knowledge graph. For an ordered pair ( i , j ), the raw semantic block is first computed as Equation 11:where T r maps relation r to a strategy-interaction matrix and c ijr is the confidence of relation r on edge ( i , j ). The directed block M i j keeps the semantic orientation learned from triples. The potential layer then uses the symmetric operator (Equation 12).and sets w ̃ i j B i j = A i j. The antisymmetric component E i j = M i j − A i j is not used in the exact-potential solver; if one keeps it for directional prediction, it is treated as a bounded perturbation in Theorem 3. Hence the implemented equilibrium algorithm runs on the symmetrized potential layer, while relation directionality still affects A i j through the confidence scores and relation-specific tensors before symmetrization.

u i x i , x − i − u i y i , x − i = Φ x i , x − i − Φ y i , x − i . (10)

M i j = w i j B 0 , i j + α ∑ r ∈ R c ijr T r z i , z j , z r , c ijr = exp ψ r z i , z j , z r ∑ q ∈ R ⁡ exp ψ q z i , z j , z q , (11)

A i j = S p o t M i j , M j i = 1 2 M i j + M j i ⊤ , A i j = A j i ⊤ , (12)

With this projection, a sufficient potential construction is Equation 13 because A i j = A j i ⊤ provides the exact-potential identity. The construction also provides an approximate guarantee: if max i , j ‖ E i j ‖ ≤ κ s y m, then the deviation from the exact-potential game is bounded by O ( κ s y m ) on compact strategy sets and is covered by the perturbation analysis.

Φ x = ∑ i a i ⊤ x i − 1 2 x i ⊤ Q i x i − λ i Ω i x i − γ i C i x i + 1 2 ∑ i ∑ j ≠ i x i ⊤ A i j x j , (13)

Figure 2 shows how relational knowledge is mapped into strategic objects. Actor-goal triples influence preference vectors. Actor-resource triples influence coupling strengths. Norm-actor triples generate feasible-set penalties. Risk-context triples produce risk-sensitive costs. This mapping is useful because human-centered interactions are rarely explained by one relation type. A platform user, for example, may pursue convenience, depend on limited attention, follow social norms, and react to perceived risks. The proposed graph representation keeps these sources explicit before equilibrium computation.

3.2 Existence of semantic equilibrium

We state the basic assumptions.

Assumption 1: Each Δ i is nonempty, compact, and convex. The functions Ω i and C i are continuously differentiable and convex on Δ i. The matrices Q i are positive definite. The semantic couplings S i j are bounded.

Assumption 2: The symmetrized interaction condition w ̃ i j B i j = ( w ̃ j i B j i ) ⊤ holds after semantic aggregation. Moreover, ∇ Φ is L-Lipschitz on Δ.

Theorem 1 (Existence and uniqueness of semantic Nash equilibrium): Under Assumptions 1 and 2, the knowledge graph-enhanced game admits at least one Nash equilibrium. If the pseudo-gradient F is μ-strongly monotone on Δ; equivalently, if F = − Φ is μ-strongly convex in the exact-potential case, the equilibrium is unique.

Proof: Because each Δ i is compact and convex, the joint feasible set Δ is compact and convex. By Assumption 1, u i ( x i , x − i ) is continuous in x and concave in x i. The Debreu–Glicksberg–Fan existence theorem therefore guarantees at least one Nash equilibrium. Equivalently, because the symmetrized potential Φ is continuous on compact Δ, it attains a maximizer x ⋆. For any unilateral deviation y i, the potential-game identity yields (Equation 14).so x ⋆ is a Nash equilibrium.

u i x i ⋆ , x − i ⋆ − u i y i , x − i ⋆ = Φ x i ⋆ , x − i ⋆ − Φ y i , x − i ⋆ ≥ 0 , (14)

For uniqueness, use the variational inequality form rather than a local-maximizer argument. Suppose x ⋆ and y ⋆ are two equilibria. Then Equation 15,Adding the two inequalities yields (Equation 16).However, strong monotonicity provides (Equation 17).The two inequalities imply ‖ x ⋆ − y ⋆ ‖ = 0; hence, x ⋆ = y ⋆. In the potential formulation, F = ∇ F and the strong convexity of F = − Φ provides the required strong monotonicity. Thus, the proof of Theorem 1 is completed.

〈 F x ⋆ , y ⋆ − x ⋆ 〉 ≥ 0 , 〈 F y ⋆ , x ⋆ − y ⋆ 〉 ≥ 0 . (15)

⟨ F x ⋆ − F y ⋆ , x ⋆ − y ⋆ ⟩ ≤ 0 . (16)

μ ‖ x ⋆ − y ⋆ ‖ 2 ≤ ⟨ F x ⋆ − F y ⋆ , x ⋆ − y ⋆ ⟩ . (17)

Figure 3 visualizes a two-dimensional projection of the simulated semantic potential, not a generic quadratic bowl. The first coordinate is the probability assigned by a representative agent to a resource-intensive strategy, and the second coordinate is the probability assigned by a neighboring agent to a coordination strategy after renormalizing the remaining strategy probabilities. The plotted surface is the potential value obtained from the same payoff tensors and norm-risk penalties used in the n = 100 experiment. The marked trajectory is one KGE-GPS run. The knowledge graph reshapes the landscape by lowering regions associated with norm violation and resource conflict and by lifting regions associated with compatible goals, which explains why the trajectory is guided toward a more stable high-potential region.

3.3 Knowledge-to-game projection

The projection from the knowledge graph to the game is designed to be modular. Let P a, P c, and P r be three trainable projection matrices. For each agent i, the semantic preference vector, constraint vector, and risk vector are defined as Equation 18: where G i is the goal-neighbor set of agent i, C i is the context-neighbor set, and ‖ denotes concatenation. The vectors in these equations are inserted into a i, ν i, and r i. This mapping has two advantages. First, different semantic sources are separated before they are projected into utilities. Second, the resulting game remains differentiable so that equilibrium computation can use variational tools. The projection matrices can be learned from historical decisions or set by experts when data are limited.

a ̄ i = P a z i ‖ | G i | − 1 ∑ g ∈ G i z g ‖ | C i | − 1 ∑ c ∈ C i z c ,

ν ̄ i = s o f t p l u s P c z i ‖ ∑ r ∈ R i norm z r , r ̄ i = σ P r z i ‖ ∑ q ∈ R i risk z q , (18)

The semantic coupling matrix also admits a tensor form. Let B ∈ R m i × m j × d be a relation-specific interaction tensor. The payoff matrix between agents i and j is Equation 19:

B i j = B 0 , i j + ∑ q = 1 d θ ijq B : , : , q , θ i j = s o f t m a x P b z i ‖ z j ‖ z r i , j . (19)

Thus, knowledge does not only adjust the scalar edge weight. It can also change whether two strategies are complements or substitutes. For example, two agents sharing a resource may have negative cross-payoff for simultaneous high consumption, while two agents with compatible goals may have positive cross-payoff for coordinated action. This tensor view is the main modeling mechanism that provides the proposed framework more expressive power than ordinary weighted network games.

3.4 Semantic variational inequality

The equilibrium can be equivalently written as a semantic variational inequality. Define the pseudo-gradient (Equation 20).

F x = − ∇ x 1 u 1 x , … , − ∇ x n u n x ⊤ . (20)

A profile x ⋆ is a Nash equilibrium if and only if Equation 21:

⟨ F x ⋆ , y − x ⋆ ⟩ ≥ 0 , ∀ y ∈ Δ . (21)

The knowledge graph affects this variational inequality through F. This expression is important because it connects the proposed model to monotone operator theory. If F is strongly monotone, then the variational inequality has a unique solution. If F is merely monotone, the solution set is convex. If knowledge perturbation changes F smoothly, sensitivity can be bounded by the inverse monotonicity margin. This provides semantic information a precise mathematical role.

Corollary 1 (Monotone semantic game): If F is μ-strongly monotone and L-Lipschitz on Δ, then the semantic game has a unique Nash equilibrium. Moreover, the projected-gradient residual ‖ x − Π Δ ( x − F ( x ) ) ‖ is zero if and only if x is the equilibrium.

Proof: Strong monotonicity implies uniqueness of the solution of the variational inequality. Existence follows from compactness and continuity. The projected-gradient residual is the natural residual of the variational inequality. By the projection theorem, x = Π Δ ( x − F ( x ) ) is equivalent to ⟨ F ( x ) , y − x ⟩ ≥ 0 for every y ∈ Δ. Hence the residual vanishes exactly at the equilibrium. Thus, the proof of Corollary 1 is completed.

Remark 2: The model is not a simple feature-augmented game. The knowledge graph changes the geometry of the potential function, the feasible preference region, and the revision dynamics. This is the theoretical bridge between structured semantic representation and equilibrium analysis. It also explains why the same physical network may lead to different equilibria under different institutional norms or contextual risks.

4 Algorithm design

4.1 Design motivation

A direct best-response solver can be unstable in human-centered systems. First, agents may overreact to short-term payoff changes. Second, knowledge extracted from records can be noisy. Third, a purely deterministic update may converge to brittle equilibria. We therefore combine three principles. The first is proximal regularization, which prevents violent changes of strategies. The second is entropy stabilization, which serves as an information-theoretic regularizer that preserves exploration and models bounded rationality. The third is knowledge-guided payoff shaping, which injects semantic constraints into the update.

For agent i, define the interior simplex (Equation 22).The exact entropy-regularized proximal best response is Equation 23:where H ε f ( y i ) = ∑ l ( y i l + ε f ) log ( y i l + ε f ). This problem can be solved by an inner convex routine when an exact best response is needed. The algorithm implemented in this article uses its first-order linearized version,which yields the explicit projected update (Equation 25).Here, Π i ε f denotes Euclidean projection onto the floored simplex. Thus Equation 24 is not claimed to be algebraically identical to the exact proximal best response; it is the one-gradient-step linearized entropy-proximal rule analyzed below. The probability floor keeps all iterates in the interior, makes the logarithm well defined, and changes the equilibrium by at most an O ( ε f ) bias on compact feasible sets.

Δ i ε f = x i ∈ R m i : 1 ⊤ x i = 1 , x i l ≥ ε f , l = 1 , … , m i , 0 < ε f < 1 / m i . (22)

x ̄ i k + 1 = arg min y i ∈ Δ i ε f − u i y i , x − i k + ρ 2 ‖ y i − x i k ‖ 2 2 + τ H ε f y i , (23)

x i k + 1 = arg min y i ∈ Δ i ε f − ∇ x i u i x k + τ ∇ H ε f x i k , y i − x i k + 1 2 η ‖ y i − x i k ‖ 2 , (24)

x i k + 1 = Π i ε f x i k + η ∇ x i u i x k − η τ log x i k + ε f 1 + 1 . (25)

4.2 Algorithmic thought

The algorithmic idea is to convert heterogeneous knowledge into stable strategic gradients. Knowledge embeddings are not used as final predictions. They generate a semantic game whose potential can be optimized by decentralized updates. At each iteration, the algorithm estimates semantic couplings, computes local payoff gradients, applies entropy-proximal revision, and monitors the Nash residual. The update stops when the residual is below a tolerance.

Table 2 provides the computational procedure. The first two lines form a semantic game from the knowledge graph. The middle lines implement decentralized strategy revision with entropy regularization. The residual-estimation line is important because it connects optimization progress with equilibrium quality. In practical deployment, the embedding update can be slower than the strategy update. For example, knowledge can be refreshed daily, while strategies can be revised every few minutes. This separation reduces computational burden without removing semantic guidance.

TABLE 2. KGE-GPS algorithm pseudocode.

4.3 Computational complexity

Let m = max i m i, E G be the number of active interaction edges, and E K be the number of knowledge triples. The embedding update has complexity O ( E K d ) per negative-sampling pass. The semantic coupling update has complexity O ( E G d ). The strategy update has complexity O ( E G m 2 + n m ⁡ log ⁡ m ), where the first term comes from pairwise payoff gradients and the second term comes from simplex projection. If the interaction graph is sparse, E G = O ( n ), and the update is nearly linear in the number of agents for fixed m and d. A dense normal-form game would require O ( n 2 m 2 ) pairwise evaluation. Therefore, the proposed method gains scalability from two sources: graph sparsity and semantic filtering.

The algorithm also supports asynchronous implementation. Suppose only a subset S k ⊂ N updates at iteration k. The update becomes Equation 26:

x i k + 1 = Π i ε f x i k + η g i k − η τ log x i k + ε f 1 + 1 , i ∈ S k , x i k , i ∉ S k . (26)

If each agent is selected with positive probability and the step size is sufficiently small, the expected potential still decreases. This property is useful for human-centered systems because agents may not revise decisions at the same time. The asynchronous version also reduces communication pressure in large-scale platforms.

4.4 Interpretability of the update

Each term in the update has a behavioral meaning. The gradient g i k measures the immediate incentive of agent i. The proximal part keeps the new decision close to the previous one and represents behavioral inertia. The entropy part prevents premature collapse to one action and reflects uncertainty or bounded rationality from an information-theoretic viewpoint. The semantic coupling adjusts the gradient based on goals, norms, resources, and risks. Therefore, the update can be explained to domain experts. If a strategy is suppressed, one can trace whether the cause is high risk, norm conflict, resource competition, or weak coordination benefit. This traceability is essential in human-centered systems where opaque equilibrium recommendations may not be acceptable.

A concrete trace can be obtained by decomposing the gradient contribution of a candidate strategy l as Equation 27:If this component is negative, the largest terms can be mapped back to triples. For instance, a suppressed high-consumption action may be explained by an actor-resource competition triple that creates a negative A i j term and by an actor-norm violation triple that increases ∂ Ω i / ∂ x i l. Thus, the recommendation is not only a numerical move but also a semantic explanation.

g i k l = a i l − Q i x i k l + ∑ j ∈ N i e l ⊤ A i j x j k − λ i ∂ Ω i ∂ x i l − γ i ∂ C i ∂ x i l . (27)

Remark 3: The proposed algorithm has a clear division of labor. The knowledge module extracts stable semantic structure, while the game module computes strategic adaptation. The proximal term protects against abrupt behavioral shifts. The entropy term acts as an information-theoretic regularizer that represents bounded rationality and improves numerical stability. These design choices are useful in human-centered systems where decisions must be adaptive but should not oscillate sharply.

5 Algorithm analysis

5.1 Descent property

Let F ( x ) = − Φ ( x ) be the equilibrium-seeking objective. Because maximizing Φ is equivalent to minimizing F, the update can be analyzed as projected proximal descent on F with entropy regularization.

Assumption 3: On Δ ε f = ∏ i Δ i ε f, F = − Φ is L-smooth and bounded below. The floored entropy H ε f ( x ) = ∑ i , l ( x i l + ε f ) log ( x i l + ε f ) has a Lipschitz gradient with constant L H ≤ 1 / ε f. Let L τ = L + τ L H.

Lemma 1 (One-step descent): Suppose Assumptions 1–3 hold and 0 < η < 1 / L τ. Let F τ ( x ) = F ( x ) + τ H ε f ( x ). Then the update in Equation 24 satisfies (Equation 28).Equivalently, if one writes the result for F alone (Equation 29),where Equation 30:is the Bregman entropy variation.

F τ x k + 1 ≤ F τ x k − 1 2 η − L τ 2 ‖ x k + 1 − x k ‖ 2 . (28)

F x k + 1 ≤ F x k − 1 2 η − L 2 ‖ x k + 1 − x k ‖ 2 − τ D H x k + 1 , x k , (29)

D H x k + 1 , x k = H ε f x k + 1 − H ε f x k − ⟨ ∇ H ε f x k , x k + 1 − x k ⟩ ≥ 0 (30)

Proof: By L τ-smoothness of F τ, we have Equation 31:The projected linearized step is the minimizer of the first-order model of F τ plus ‖ y − x k ‖ 2 / ( 2 η ) over Δ ε f. Its variational optimality condition provides (Equation 32).Combining the two inequalities proves the descent inequality. The expression involving D H follows from the definition of the entropy Bregman divergence. Thus, the proof of Lemma 1 is completed.

F τ x k + 1 ≤ F τ x k + ⟨ ∇ F τ x k , x k + 1 − x k ⟩ + L τ 2 ‖ x k + 1 − x k ‖ 2 . (31)

⟨ ∇ F τ x k , x k + 1 − x k ⟩ ≤ − 1 η ‖ x k + 1 − x k ‖ 2 . (32)

5.2 Convergence to approximate equilibrium

Theorem 2 (Convergence): Under Assumptions 1–3, if 0 < η < 1 / L τ, every limit point of { x k } generated by KGE-GPS solves the variational inequality (Equation 33).

⟨ ∇ F x ̄ + τ ∇ H ε f x ̄ , y − x ̄ ⟩ ≥ 0 , ∀ y ∈ Δ ε f . (33)

If F is μ-strongly convex and 0 < η < 2 μ / L τ 2, then the iterates converge linearly to the unique solution x τ ⋆ of the entropy-regularized problem, and Equation 34:

‖ x k − x τ ⋆ ‖ 2 ≤ 1 − 2 η μ + η 2 L τ 2 k ‖ x 0 − x τ ⋆ ‖ 2 . (34)

Moreover, if the unregularized equilibrium x ⋆ lies in the relative interior and the sensitivity constant is finite, then ‖ x τ ⋆ − x ⋆ ‖ = O ( τ + ε f ) (Equation 35).

Proof: Lemma 1 shows that F τ ( x k ) decreases and is bounded below on compact Δ ε f. Therefore,Taking limits in the projected optimality condition yields the stated variational inequality. If F is μ-strongly convex, then F τ is also μ-strongly convex and has a unique minimizer x τ ⋆. The projected-gradient mapping of a μ-strongly monotone and L τ-Lipschitz operator is contractive for 0 < η < 2 μ / L τ 2, which provides the displayed linear bound. Finally, the difference between x τ ⋆ and x ⋆ follows from Section 6 sensitivity: the perturbing operator is τ ∇ H ε f plus the feasible-set flooring error, whose norms are bounded on Δ ε f. Thus, the proof of Theorem 2 is completed.

∑ k = 0 ∞ ‖ x k + 1 − x k ‖ 2 < ∞ , ‖ x k + 1 − x k ‖ → 0 . (35)

5.3 Robustness to knowledge perturbation

Let K ̂ be a perturbed knowledge graph, and let Φ ̂ be the corresponding potential. The perturbation may come from three observable sources: missing or spurious triples, relation-confidence uncertainty, and embedding drift. If a triple perturbation changes confidence scores by at most δ c, relation embeddings by at most δ z, and the symmetrization residual by at most κ s y m, then the induced gradient perturbation can be bounded as Equation 36:where C c , C z , C s depend on the bounded strategy sets and projection matrices. In experiments, δ c is estimated by randomly deleting or corrupting a fraction of triples, δ z by the embedding validation loss, and κ s y m by max i j ‖ M i j − M j i ⊤ ‖ before projection.

sup x ∈ Δ ε f ‖ ∇ Φ ̂ x − ∇ Φ x ‖ ≤ ξ K ≔ C c δ c + C z δ z + C s κ s y m , (36)

Theorem 3 (Robustness): If − Φ is μ-strongly convex and the knowledge perturbation satisfies (Equation 36), then the equilibrium x ̂ ⋆ of the perturbed game satisfies (Equation 37).Moreover, the exact Nash residual of x ̂ ⋆ in the original game is bounded by Equation 38:where L D is the Lipschitz constant of unilateral payoff improvement.

‖ x ̂ ⋆ − x ⋆ ‖ ≤ ξ K μ . (37)

R e x x ̂ ⋆ ≤ L D ξ K μ , (38)

Proof: Let F = − Φ and F ̂ = − Φ ̂. The optimality conditions for x ⋆ and x ̂ ⋆ are variational inequalities (Equation 39).

〈 ∇ F x ⋆ , y − x ⋆ 〉 ≥ 0 , 〈 ∇ F ̂ x ̂ ⋆ , y − x ̂ ⋆ 〉 ≥ 0 . (39)

Set y = x ̂ ⋆ in the first inequality and y = x ⋆ in the second. Adding them yields (Equation 40).

⟨ ∇ F x ⋆ − ∇ F ̂ x ̂ ⋆ , x ̂ ⋆ − x ⋆ ⟩ ≥ 0 . (40)

Add and subtract ∇ F ( x ̂ ⋆ ) and use strong monotonicity of ∇ F (Equation 41).

μ ‖ x ̂ ⋆ − x ⋆ ‖ 2 ≤ ‖ ∇ F x ̂ ⋆ − ∇ F ̂ x ̂ ⋆ ‖ ⋅ ‖ x ̂ ⋆ − x ⋆ ‖ . (41)

The perturbation bound gives ‖ x ̂ ⋆ − x ⋆ ‖ ≤ ξ / μ. Because unilateral payoff improvement is Lipschitz on compact Δ, the residual bound follows from R ( x ⋆ ) = 0 and Equation 42:

R m e x x ̂ ⋆ − R m e x x ⋆ ≤ L D ‖ x ̂ ⋆ − x ⋆ ‖ . (42)

Thus, the proof of Theorem 3 is completed.

Remark 4: The analysis shows that semantic knowledge does not destroy equilibrium theory if it enters through bounded and smooth payoff modifiers. The convergence proof relies on the symmetrized potential layer and the floored entropy regularizer, while the robustness proof quantifies the cost of triple noise, relation uncertainty, embedding drift, and residual directionality. Thus, the framework is suitable for data-driven settings where extracted triples may be incomplete or noisy.

6 Simulation and analysis

6.1 Experimental setting

We simulate a human-centered decision network with n ∈ { 50,100,200,400 } agents and m i = 5 strategies per agent. Unless otherwise stated, agents are connected by a Watts–Strogatz small-world graph with mean degree of 6 and a rewiring probability of 0.1. The knowledge graph contains six entity categories, namely, actors, goals, resources, norms, risks, and contexts, and eight relation types: pursues, uses, competes, cooperates, violates, exposed-to, located-in, and observes. For the default n = 100 case, we generate 12 n triples, including actor–goal, actor–resource, actor–norm, actor–risk, context–risk, and actor–actor triples. Entity and relation embeddings are trained with a TransE-style margin loss, embedding dimension d = 32, five negative samples per positive triple, margin 1.0, Adam learning rate 1 0 − 3, and 200 epochs. Payoff matrices use Q i = q i I + D i with q i ∈ [ 0.8 , 1.2 ], bounded random base matrices B 0 , i j, and semantic tensors normalized to spectral norm at most one. The default parameters are α = 0.6, η = 0.04, ρ = 1, τ = 0.03, λ i = 0.2, γ i = 0.1, ε f = 1 0 − 5, and K max = 500. We use random seeds 1–20 and report mean values with standard deviations.

The baselines are implemented with the same graph, payoff initialization, stopping rule, and random seeds. The projected best response uses x i k + 1 = Π i ε f ( x i k + η ∇ x i u i ( x k ) ) [33]. Replicator dynamics uses the exponentiated update x i l k + 1 = x i l k ⁡ exp ( η π i l k ) / ∑ s x i s k ⁡ exp ( η π i s k ), where π i l k is the payoff of pure action l [34, 35]. The no-knowledge graph (KG) variant sets α = 0 and removes Ω i and C i. The static-graph variant freezes w ̃ i j at its initial value and does not refresh semantic couplings. The exact Nash residual is Equation 43:where the inner problem is solved as a constrained simplex subproblem rather than by a one-step approximation. Social welfare is W ( x ) = n − 1 ∑ i u i ( x ), oscillation is T − 1 ∑ k = 1 T ‖ x k − x k − 1 ‖, the robustness score is 1 / ( 1 + R ̄ n o i s e / R ̄ 0 ), and time and memory indices are normalized by KGE-GPS at n = 100.

R e x x = max i max y i ∈ Δ i ε f u i y i , x − i − u i x i , x − i , (43)

Table 3 compares the proposed method with four baselines. KGE-GPS achieves the lowest exact equilibrium residual and the highest average welfare over 20 seeds. The iteration count is lower than those of projected best response and replicator dynamics, while the runtime index clarifies that fewer iterations do not imply zero computational overhead. Although KGE-GPS has an additional knowledge-processing module, fewer strategy-revision iterations keep total runtime competitive. The result supports the claim that semantic payoff shaping and entropy regularization improve equilibrium quality and iteration efficiency.

TABLE 3. Main simulation results.

Bold values indicate the best performance in the principal equilibrium-quality metrics; the time column is a normalized runtime index and is not part of the bolding criterion.

Table 4 adds validation beyond a single synthetic topology. Performance degrades mildly when the topology becomes more random or more heterogeneous, but the residual remains below 0.05 in all tested regimes. This indicates that the observed advantage is not tied to a single small-world generator.

TABLE 4. Additional validation under different graph regimes.

Bold values indicate the best result in each column among the tested graph regimes.

Figure 4 presents the residual curves. All methods reduce the residual, but their rates clearly differ. KGE-GPS decreases rapidly in the first 40 iterations and then approaches a stable low-residual region. The projected best response is slower because it lacks entropy-stabilized smoothing. Replicator dynamics has the largest residual tail because the strategy probability of weak actions changes slowly. The no-KG variant converges, but its final residual is higher. This indicates that knowledge guidance and entropy regularization affect not only the transient speed but also the final equilibrium quality.

6.2 Ablation study

Table 5 reports the contribution of each component. Removing the knowledge graph causes the largest degradation, showing that semantic coupling is central to the method. Removing entropy increases oscillation because agents become overly deterministic, highlighting the information-theoretic benefit of entropy regularization. Removing inertia also increases oscillation, which confirms the stabilizing role of the proximal term. The static-graph variant performs worse than the full model because it cannot adjust interaction weights according to semantic context. Overall, the components are complementary rather than redundant.

TABLE 5. Ablation study of model components.

Bold values indicate the best result in each metric among the ablation variants.

Figure 5 visualizes the final residuals of the ablation variants. The full model has the smallest residual. The variant without the knowledge graph has the largest residual, which suggests that relational semantics are not merely auxiliary features. The entropy-free and static-graph variants fall in between. This pattern is consistent with the theoretical design. The knowledge graph reshapes the potential landscape, entropy stabilizes bounded-rational revisions, and dynamic coupling adapts the game to contextual change. Removing any part weakens the equilibrium computation.

6.3 Robustness and scalability

Table 6 evaluates residuals under increasing knowledge perturbation. The no-KG method is nearly insensitive to knowledge noise because it does not use semantic information, but its baseline residual is high. KGE-GPS is affected by perturbation, yet it remains better than all baselines across the tested range. This agrees with Theorem 3: the equilibrium error grows with the perturbation magnitude but is controlled by the strong monotonicity constant. The result suggests that the proposed method is useful even when extracted knowledge is imperfect.

TABLE 6. Robustness under knowledge perturbation.

Figure 6 shows a heatmap of residuals under two uncertainty sources. The horizontal axis represents knowledge uncertainty, and the vertical axis represents observation noise. Residuals increase as either uncertainty source grows. The increase is gradual rather than abrupt, which is consistent with the perturbation bound. The lower-left region performs best because both relational knowledge and observations are reliable. The upper-right region is the most difficult, but the residual remains within a controllable range in the simulation.

Table 7 reports scalability results. As the number of agents increases from 50 to 400, the residual grows only moderately. Iterations also increase, but the growth is sublinear relative to network size. The time and memory indices increase because more interactions and strategy variables must be processed. However, the sparse semantic coupling avoids the quadratic explosion of dense solvers. This is important for complex human-centered systems, where many agents interact through sparse but meaningful relations.

TABLE 7. Scalability with increasing network size.

Figure 7 compares the running-time index of KGE-GPS with a dense solver. The dense solver grows rapidly because it treats all pairwise interactions as active. KGE-GPS uses the sparse interaction graph and semantic filtering, so its growth is much slower. This does not mean that knowledge processing is free. Instead, the benefit comes from avoiding irrelevant interactions after semantic screening. The results support using knowledge graphs as a structural prior for scalable game analysis.

6.4 Sensitivity to entropy temperature

The entropy temperature τ controls the information-theoretic tradeoff between exploitation and stability, and the semantic strength α controls how strongly knowledge modifies the physical interaction graph. Table 8 reports a two-parameter sensitivity check on the default n = 100 regime.

TABLE 8. Sensitivity to entropy temperature and semantic strength.

Bold values indicate the default operating point (alpha = 0.6, tau = 0.03), selected for the overall residual-oscillation trade-off.

A very small τ makes the update close to a deterministic best response and increases oscillation. A very large τ over-smooths strategies and raises the final residual. Moderate entropy yields the best welfare-residual tradeoff. Increasing α improves performance at first because useful semantic relations are amplified, but overly large α can overfit noisy triples and distort physical interactions. Therefore, τ and α should be selected by validation, and relation confidence can be used to adapt α edge by edge.

6.5 Discussion of practical deployment

In practice, the framework can be deployed in three layers. The data layer extracts candidate triples from structured records, domain rules, and text. The semantic layer validates triples, updates embeddings, and estimates confidence. The game layer computes strategy profiles and reports residuals. Human operators can inspect high-impact triples and adjust relation confidence before the game is solved. This human-in-the-loop design is consistent with the nature of human-centered systems. It also reduces the risk that an erroneous knowledge relation directly changes a policy recommendation.

The proposed framework is general, but domain adaptation is still necessary. In transportation, strategies may be route recommendations or pricing actions. In energy systems, they may be demand–response choices. In public health, they may be resource allocation or appointment guidance. In online platforms, they may be content exposure or moderation actions. The same mathematical structure can be retained, but domain experts should specify the entity types, relation types, and payoff terms.

Remark 5: The simulations support three conclusions. First, knowledge guidance and entropy regularization together improve equilibrium quality. Second, entropy and proximal inertia stabilize strategy revision. Third, the method remains robust when knowledge is noisy and scalable when the network grows. These findings are consistent with the theoretical results, although real deployment would require domain-specific knowledge extraction and validation.

7 Conclusion and future research work

This article proposed an entropy-regularized, knowledge graph-enhanced game-theoretic model for strategy interaction and equilibrium analysis in complex human-centered systems. The model uses a heterogeneous knowledge graph to provide structured representations of actors, goals, resources, norms, risks, and contexts. Knowledge embeddings are mapped into payoff modifiers, semantic constraints, and revision kernels. Based on this structure, we formulated a semantic potential game and designed the KGE-GPS algorithm, in which entropy serves as an information-theoretic stabilizer. Theoretical analysis proved equilibrium existence, convergence to an approximate Nash equilibrium, and robustness under bounded knowledge perturbations. Simulations showed that the proposed method improves residual, welfare, robustness, and scalability compared with several baselines, confirming the value of coupling structured representations with entropy regularization.

Future work will proceed in three directions. First, we will study online knowledge graph updating, where triples and relations change continuously with streaming events. Second, we will extend the model to hypergames with misaligned perceptions, so that different agents may hold different knowledge graphs. Third, we will validate the framework in real human-centered applications such as transportation guidance, energy-demand response, and human–AI collaborative decision-making while further exploring information-theoretic criteria for adaptive entropy tuning and structured representation refinement.

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Footnotes

Edited by: Asokan K, College of Engineering, India

Reviewed by: Zeling Xu, The Hong Kong University of Science and Technology Library, Hong Kong SAR, China

Vince Trencsenyi, University of London, United Kingdom