Research on a Method for Generating 3D Topologies of Combat Networks Based on Conditional Graph Diffusion Models

Abstract

This paper addresses the pressing need for the intelligent design of three-dimensional topological structures in combat networks within modern joint operations. Conventional graph generation approaches struggle to simultaneously fulfill requirements for 3D deployment, tactical effectiveness, and real-time generation in complex battlefield environments. To overcome these challenges, we propose a method for generating 3D combat network topologies using a conditional graph diffusion model. Our primary innovation lies in a conditional diffusion framework guided by the fusion of target attributes. Through a multi-dimensional conditional embedding mechanism, we integrate combat node types, equipment characteristics, 3D spatial constraints, and tactical requirements into a unified generation process. This enables the model to generate topologies that deeply integrate operational rules and tactical demands. Experimental results demonstrate that our approach significantly improves core tactical metrics: target accessibility increases by 4.5%, defensive capability improves by 13.15%, and offensive efficiency rises by 30.4%. The results indicate that the proposed method achieves superior adaptability and robustness in complex battlefield environments.

1. Introduction

The fundamental objective of combat networks is to create decision-making dilemmas for adversaries by establishing dynamic, adaptive, and rapidly reconfigurable combat systems. In this paradigm, combat networks evolve beyond simple physical connections between sensors and shooters, transforming into intelligent organisms capable of autonomous emergence and evolution in response to instantaneous battlefield changes. This shift aligns with the emerging concepts of Mosaic Warfare and Kill Web, where distributed and highly interoperable force structures are critical for survivability and effectiveness [1]. However, this advanced operational concept imposes rigorous demands on the dynamic generation capabilities of combat networks: ideal networks must continuously optimize and potentially reconfigure instantaneously throughout mission cycles. Their generation speed and adaptability directly determine the quality and velocity of our OODA (Observe, Orient, Decide, Act) cycle [2,3], serving as crucial metrics for assessing system combat effectiveness.

Currently, the enabling technology—intelligent generation of combat network topologies tailored to specific tasks under stringent time constraints—represents the primary bottleneck in transitioning from conceptual frameworks to operational capabilities. Extensive research [4,5,6] indicates that achieving this ideal scenario faces substantial obstacles. First, customized generation necessitates profound comprehension of target characteristics. While general graph generation methods have seen rapid development in recent years [7], existing general design methodologies [8,9], and early attempts using Generative Adversarial Networks (GANs) for topology design [10], struggle to accomplish precise “one objective–one topology” matching, resulting in suboptimal network efficiency. Furthermore, the robustness of these networks under dynamic constraints remains a critical challenge [11]. Second, the decision space for dynamic combinations expands exponentially with node count, and conventional optimization algorithms cannot identify feasible or optimal topologies within the extremely limited time windows available. Although recent advancements in generative AI, particularly diffusion models, have shown promise in handling discrete optimization and complex data structures [12], their application in constrained three-dimensional combat environments is still nascent. Consequently, “just-in-time generation” remains an elusive goal, causing combat systems to forfeit their essential advantages in agility and adaptability.

Addressing the critical requirements for target-specific and timely construction of future combat network topologies, this paper considers cooperative connectivity among battlefield entities and dynamic constraints within confrontation environments. We establish a reference framework for future combat network topology generation by implementing an enhanced conditional graph diffusion model, thereby advancing beyond traditional static topology generation approaches. Drawing inspiration from recent progress in constrained graph generation [13] and 3D spatial topology control [14], our novel methodology enables rapid generation of combat networks within three-dimensional topological spaces customized for specific targets. Through comprehensive simulation experiments and parameter comparisons, we demonstrate the superiority and effectiveness of our proposed model and method in terms of generation speed and solution quality. The principal contributions of this research are summarized in three key aspects:

• By developing a multi-attribute conditional embedding framework, we propose a novel architecture inspired by the Conditional Graph Generated algorithm, named 3DTG-CGD. This approach achieves comprehensive integration of target characteristics, tactical requirements, and three-dimensional constraints. This methodology effectively overcomes the limitations of conventional approaches [7,10] in target specificity and environmental adaptability, thereby enabling precise and controllable generation of combat network topologies.
• Through systematic integration of battlefield entity attributes, spatial deployment constraints, and tactical associations, we construct a comprehensive dataset encompassing multiple target types, providing robust data support for model training. Building upon this foundation, the optimized sampling method significantly enhances generation efficiency for complex combat networks while effectively addressing the bottlenecks of traditional methods in real-time performance and scalability.
• By developing a militarily-compliant loss function and multi-dimensional evaluation metrics, we ensure comprehensive performance of the generated networks in terms of structural rationality, tactical effectiveness, and 3D deployment feasibility [14]. This approach provides reliable quality assurance for combat network generation.

2. Related Work

2.1. Combat Networks

Research on combat networks has matured significantly, with extensive literature addressing network stability, capability evaluation, and vulnerability analysis. However, these studies predominantly focus on analyzing or optimizing existing topologies rather than generating new ones. A detailed analysis of the current landscape reveals a distinct separation between evaluation metrics and generative design.

First, vulnerability and resilience analysis constitutes a major theme. Sun et al. [15] employed meta-path centrality methods to identify critical nodes in multi-layer heterogeneous networks, while Yang et al. [6] and Liu et al. [16] assessed the vulnerabilities of cooperative combat networks and multi-functional equipment networks, respectively. These works provide robust mechanisms for identifying weak points but are reactive in nature; they assess the survivability of a given network structure rather than proposing how to construct an optimal one a priori.

Second, capability evaluation frameworks have been rigorously developed. Wang et al. [5] introduced a task-oriented capability evaluation model based on operational loops, and Song et al. [17] contributed to operational ring ratio modeling. Furthermore, Xu et al. [18] investigated link centrality resilience, and Li et al. [19] analyzed operational capability disintegration under incomplete information. While these metrics are essential for assessing performance, they do not offer a constructive methodology for synthesizing topologies that meet these specific metric thresholds from scratch.

Third, in the realm of topology optimization and reconfiguration, Sun et al. [4] introduced an autonomous reconfiguration method for network recovery. Recent efforts have expanded into dynamic resource allocation and swarm topology control. For instance, Chen et al. [20] proposed robust topology control for UAV swarms in complex environments, and Zhang et al. [21] explored dynamic resource allocation in Kill Webs. However, as highlighted in

Table 1. Analysis of Existing Combat Network Research.

Research Focus	Representative Work	Methodology	Dimensionality	Generative Capability
Vulnerability	Yang et al. [6], Sun et al. [15]	Centrality Analysis/Meta-path	2D/Abstract	None (Analysis only)
Capability Eval.	Wang et al. [5], Song et al. [17]	Operational Loop Models	2D/Abstract	None (Measurement only)
Reconfiguration	Sun et al. [4], Zhang et al. [21]	Optimization/ Recovery Algorithms	2D/3D	Incremental Modification only
Swarm Control	Chen et al. [20]	Distributed Control	3D	Local connectivity maintenance
Target Matching	Wang et al. [22]	Capability Assessment	N/A	No topology generation

, these approaches typically start from a pre-existing or partially connected network and apply heuristic adjustments. They do not support the de novo generation of complete topologies based solely on abstract target attributes.

To date, no established methodology exists for the three-dimensional topology generation of combat networks driven by specific mission target attributes. While Wang et al. [22] addressed target capability assessment, they did not establish the necessary correlations with topology generation. Existing frameworks lack the theoretical basis to automatically synthesize 3D combat network topologies that consider critical parameters such as combat target functionality, spatial deployment constraints (3D), and evolving threat levels simultaneously.

Based on the tabulated analysis, we identify a critical research gap: the absence of a Target-Driven Generative Framework. Existing literature fails to bridge the gap between high-level mission requirements and low-level 3D topology instantiation. Most existing works are confined to 2D abstractions or assume fixed connectivity. They lack the capability to generate spatially valid 3D structures that inherently satisfy complex military constraints.

Addressing this deficit, our research introduces 3DTG-CGD. Unlike optimization-based approaches that tweak existing edges, our method leverages a conditional graph diffusion model to synthesize combat network topologies from scratch. By integrating 3D spatial attributes and target functionality into the generation process, we move beyond reactive evaluation towards proactive, mission-specific topology design.

2.2. Graph Generation

The field of graph generation has evolved significantly. It has transitioned from classical recurrent and adversarial approaches to modern diffusion-based and large language model (LLM) architectures. Early representative works, such as GraphRNN [23], NetGAN [24] and TG-GAN [25], laid the foundation. GraphRNN utilized recurrent neural networks to model edge dependencies, while NetGAN employed adversarial training on random walks to capture topological features. Subsequently, structure-aware methods like VQGraph [26] and CP-GAN [27] were introduced. These methods used vector quantization and hierarchical pooling to better preserve community structures. However, these traditional models often struggle with mode collapse. They also face difficulties in long-range dependency modeling due to the discrete nature of graph structures.

In recent years, the field has been revolutionized by diffusion models. These models offer superior fidelity and distribution alignment compared to GANs. Unlike GANs, diffusion models learn to reverse a noising process. This provides a more stable training objective. DiGress [28] addresses the challenge of discrete data. It introduces a discrete denoising diffusion probability model and significantly outperforms traditional autoregressive methods. In the domain of 3D structure generation, which is highly relevant to spatial topology, Geometry-Complete Latent Diffusion Model [29] and Exploring Chemical Space [30] have enabled precise control over 3D geometries and chemical properties. Furthermore, Unified Diffusion Model [31] and Efficient Sampling [32] methods have been proposed to enhance the flexibility and speed of graph diffusion. However, most of these diffusion-based approaches suffer from high computational costs. Their iterative denoising process imposes a heavy burden, limiting real-time applications.

A critical branch of this domain is conditional graph generation. The objective here is to synthesize graphs that satisfy specific attributes or structural constraints. Recent works have focused on enhancing robustness and control. CTRL-U [33] ensures high consistency between inputs and outputs. It achieves this by estimating uncertainty in reward model predictions. Theoretical frameworks like Schrödinger Bridge [34] and Discrete Bayesian Sample Inference [35] provide novel perspectives. They optimize structure confidence within continuous distribution spaces. Despite these advancements, significant challenges remain in applying these methods to military combat networks. Existing conditional methods often rely on simple labels or global constraints. They lack the capability to handle complex, multi-modal constraints specific to 3D combat environments, such as spatial deployment rules and real-time communication ranges.

To provide a clear comparative overview,

Table 2. Comparison of Representative Graph Generation Methodologies.

Category	Method	Year	Core Mechanism	Limitations in 3D/Combat
Autoregressive	GraphRNN [23]	2018	RNN-based sequential generation	Slow inference; sequential error accumulation
Adversarial	NetGAN [24]	2018	GAN with random walks	Mode collapse; training instability
Structure-aware	CP-GAN [27]	2022	Hierarchical pooling + GAN	Struggles with complex 3D constraints
Diffusion	DiGress [28]	2022	Discrete Denoising Diffusion	High sampling latency; no 3D geometry
3D Diffusion	Geometry-Complete [29]	2025	3D Latent Diffusion	Domain-specific to molecules; lacks tactical logic
Conditional	CTRL-U [33]	2024	Uncertainty-Aware Reward	Primarily for images; not graph-structure aware

categorizes key methodologies. It highlights their mechanisms and trade-offs regarding 3D topology and conditional control.

Based on the analysis of existing literature, three primary research gaps are identified. While molecular generation [29] handles 3D coordinates, it focuses on bond lengths and angles. There is a lack of models that jointly generate 3D spatial coordinates and complex topological connections simultaneously for combat entities.Existing conditional generation methods often rely on simple labels or global attributes [33]. They fail to integrate complex, multi-modal military constraints (e.g., line-of-sight, interference ranges) directly into the generation process. State-of-the-art diffusion models like DiGress [28] require many denoising steps. This computational cost is prohibitive for the rapid reconfiguration required in combat OODA loops.

To bridge these gaps, this research proposes a novel framework named 3DTG-CGD. It is based on a conditional graph diffusion model tailored for combat networks. Our method integrates the spatial inductive biases for 3D deployment with efficient conditional sampling. By leveraging optimized diffusion processes, we aim to achieve high-fidelity 3D topology generation that adheres to complex tactical constraints while significantly reducing computational overhead.

3. System Model

Combat networks can be represented as graphs $G=\left\{\mathcal{X},\mathcal{A}\right\}$, where $\mathcal{X}$ denotes the node feature matrix. The matrix $\mathcal{A}\in {\mathbb{R}}^{2\times N\times N}$ characterizes edges, with one channel $\overline{\mathcal{A}}\in {\left\{0,1\right\}}^{N\times N}$ indicating edge existence (binary edge existence matrix) and the other channel ${\mathcal{A}}_{type}$ representing edge types. The binary edge existence matrix can be transformed into a real-valued matrix. As show in

Table 3. Symbols and Parameters.

Symbol	Description	Symbol	Description
$G=(V,E)$	Combat network with nodes V and edges E	$S_d$	Defense strength metric
$\mathcal{X},\mathcal{A}$	Node features $\mathcal{X}$ and adjacency matrix $\mathcal{A}$	$E_a$	Attack efficiency metric
$G_t$	Noisy graph state at diffusion step t	L	Average shortest path length
$G_0$	Initial clean combat network from dataset	$R_f,R_r$	Firepower node ratio & Recon node ratio
$ϵ$	Standard Gaussian noise added in forward process	$\mathcal{L}$	Total loss function
${ϵ}_{\theta }(·)$	Predicted noise output by the neural network	$\theta$	Trainable parameters of the model
$q_t$	Marginal distribution of noisy graph $G_t$	${\mathcal{L}}_{recon}$	Reconstruction (denoising) loss
${\overline{\omega }}_t$	Standard Wiener process (Brownian motion)	${\mathcal{L}}_{rule}$	Rule-constrained loss (KL divergence)
$f\left(t\right),g\left(t\right)$	Drift coefficient & Diffusion coefficient	${\mathcal{L}}_{threat}$	Threat perception loss
${\sigma }_t$	Standard deviation of noise at step t	${\lambda }_1,{\lambda }_2$	Hyperparameter weights for losses
$\mathit{Auth}$	Command authorization matrix (0 or 1)	r	Guidance weight for ODE sampling
$A_t,A_l,A_r,A_e$	Tactical, Legal, Resource, Environmental authorization	$R_{\psi }$	Reward function for gradient guidance
$\mathbf{T}$	Conditional input representing tactical target attributes	${\nabla }^{*}$	Unit-normalized gradient
M	Conditional input representing map/environment	$C_c,D$	Connectivity score & Network density
$C_l$	Clustering coefficient of the network	$\varphi ({\mathcal{X}}^{\left(l\right)},T)$	Node features extracted at layer l

, we define the notation used in this paper.

As illustrated in Figure 1, the algorithmic model primarily encompasses target threat perception, combat rule authorization constraints, and 3D topology diffusion generation. 3DTG-CDG generates combat network topologies through graph diffusion processes based on opponent target positions, types, threat levels, combat unit statuses, mission priorities, and other critical information. Throughout the diffusion process, node and edge characteristics are adjusted and updated according to these factors, enabling generated combat networks to better adapt to battlefield situations and facilitate more effective combat action planning.

3.1. Target Threat Perception

Target threat perception involves transforming temporal and spatial characteristics of threats into feature representations of our nodes, thereby supporting subsequent priority decisions and connection generation. The importance and roles of different nodes within combat networks vary, while threats exhibit dynamic evolution. To enhance model applicability to real combat scenarios and provide accurate foundations for subsequent network construction and decision-making, this paper proposes a novel combat network model based on multi-agent systems. This model performs node characterization through real-time integration of dynamic threat assessments, node roles, and other critical information.

Dynamic threat values should reflect both real-time threats and historical threat trends posed by targets to our nodes, along with matching characteristics between target and node types. The dynamic threat value of a node is represented as a vector $\mathbf{T}=({\mathbf{T}}_1,{\mathbf{T}}_2,{\mathbf{T}}_3)$, comprising physical threat ${\mathbf{T}}_1$, node importance ${\mathbf{T}}_2$, and threat potential ${\mathbf{T}}_3$. The physical threat value quantifies immediate threats posed by targets to nodes, calculated as the product of spatial distance and type matching. Closer distances and higher type matching degrees correspond to greater threat values. Thus, physical threat ${\mathbf{T}}_1$ is defined as:

$${\mathbf{T}}_1=\sum _{s\in \mathcal{S}\left(t\right)}{\displaystyle \frac{f_{ap}\left(s\right)·f_{tm}\left(s,i\right)}{{\parallel po{s}_s\left(t\right)-po{s}_i\parallel }^2+b}},$$

(1)

where $f_{ap}\left(s\right)$ represents the attack intensity of incoming targets, $f_{tm}\left(s,i\right)$ denotes the type matching coefficient between targets and nodes, ${\parallel po{s}_s\left(t\right)-po{s}_i\parallel }^2$ indicates squared distance at time t, and b serves as a smoothing term to prevent numerical overflow at zero distance.

To prevent isolated evaluation of node physical threats and enhance spatial correlation in threat assessment, node importance ${\mathbf{T}}_2$ is introduced to represent threat conduction within node neighborhoods:

$${\mathbf{T}}_2={\displaystyle \frac{1}{\left|\mathcal{N}\left(i\right)\right|}}\sum _{j\in \mathcal{N}\left(i\right)}{\mathbf{T}}_1,$$

(2)

where $\mathcal{N}\left(i\right)$ represents the set of neighboring nodes j within the communication radius of node i. Average physical threat is utilized to represent node importance, primarily considering that threats confronting surrounding nodes may indirectly increase protection pressures on current nodes.

Threat potential primarily reflects cumulative threat effects. As incoming targets continue approaching, threat potential gradually increases. Therefore, threat potential ${\mathbf{T}}_3$ is defined as

$${\mathbf{T}}_3\left(t\right)=\alpha ·{\mathbf{T}}_3\left(t-1\right)+\left(1-\alpha \right)·\left({\mathbf{T}}_1+{\mathbf{T}}_2\right),$$

(3)

where $\alpha$ represents the memory factor balancing historical and current threats, preventing continuous severe fluctuations in threat values caused by isolated noise.

Based on fundamental node attributes, dynamic threat values and response priorities are integrated to achieve expansion of node feature spaces. The three components of new dynamic threat value vectors are combined with response priorities p. Characteristics of individual nodes can be expressed as:

$${\mathcal{X}}_i=\left[po{s}_i,typ{e}_i,\mathbf{T},p_i\right],$$

(4)

where $po{s}_i$ represents 3D coordinates utilized for distance calculations, and $typ{e}_i$ encodes node type attributes.

3.2. Authorization Constraints of Operational Rules

The primary objective of authorization constraints is to ensure legitimacy through dual constraints: physical feasibility and command legitimacy. This guarantees that edges generated by combat networks comply not only with battlefield conditions but also with command process protocols.

To incorporate these constraints into the generation framework, we utilize a predefined equipment-target matching matrix $\mathbf{M}$ derived from domain knowledge. This matrix serves as a binary filter that indicates physically feasible and legitimate combat interactions. It enables the model to distinguish between valid and invalid target assignments for different equipment types without explicitly modeling the underlying physical experiments. Specifically, a zero value in the matrix signifies that an engagement is infeasible or prohibited, while a non-zero value indicates that an interaction is authorized under current rules.

The generation process must strictly satisfy a set of specific authorization conditions, which are formulated as follows:

$$\begin{array}{cc}\hfill & \mathrm{if}{A}_t\wedge A_l\wedge A_r\wedge A_e\ne 0:\mathrm{pass}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \mathrm{else}:\mathrm{not}\mathrm{pass}\hfill \end{array}$$

(6)

where $A_t$ represents tactical authorization determining direct support for current mission objectives; $A_l$ denotes legal authorization referencing laws of war databases for verification; $A_r$ indicates resource authorization checking whether ammunition margins exceed safety thresholds; and $A_e$ represents environmental authorization assessing whether battlefield conditions permit effective equipment deployment. Combat networks are permitted for generation only when equipment authorization is valid, targets are legitimate, equipments are operational, and target attributes conform to engagement rules, thereby preventing accidental and illegal attacks.

Consequently, the command authorization matrix $\mathit{Auth}\in {\left\{0,1\right\}}^{N\times K}$ can be defined such that ${\mathit{Auth}}_{i,k}=1$ indicates node $v_i$ is authorized to deploy k-type equipments. This matrix acts as a fundamental mask during the topology generation process, ensuring that the synthesized network structures adhere to operational rules.

3.3. Generation of 3D Topology Through Diffusion

As illustrated in Figure 2, real-time feedback mechanisms enable combat networks to adapt to target attributes. The graph diffusion model generates 3D combat network topologies while maintaining both topological stability and operational effectiveness.

The dynamic nature of battlefield environments during combat necessitates that combat network architectures capable of real-time response to multimodal feedback information, including target movements, node failures, and link interruptions. Key battlefield elements include: node status $C_{fault}\in \left\{0,1\right\}$, link status $B_{link}\in \left\{0,1\right\}$, authorization updates ${\mathit{Auth}}_{up}\in \left\{0,1\right\}$ representing real-time equipment authorization status updates, and target position offsets $\Delta x_k\left(t\right)=x_k\left(t+\Delta t\right)-x_k\left(t\right)$ denoting position changes in target k within time interval Δt.

Multimodal information is fused into unified conditional representations using spatiotemporal encoders:

$$c_t=Encode{r}_{feedback}\left(\left[\Delta x_k,C_{fault},B_{link},\mathbf{M},{\mathit{Auth}}_{up}\right],\mathbf{T}\right).$$

(7)

The diffusion model learns data distributions through two processes: forward diffusion and backward prediction. During forward diffusion, data undergoes progressive perturbation through sequential noise addition, gradually disrupting graph structures. During backward prediction, neural networks predict noise at each step, iteratively reconstructing data until output distributions converge to known prior distributions.

In the forward process, the edge type matrix ${\mathcal{A}}_{type}$, node feature matrix $\mathcal{X}$, and edge existence matrix ${\mathcal{A}}_{exist}$ are all perturbed by stochastic differential equations:

$$d{G}_t=f\left(t\right)G_{t}dt+g\left(t\right)d{\omega }_t,G_t=\left({\mathcal{X}}_t,{\mathcal{A}}_t\right),$$

(8)

where $f\left(t\right)={\displaystyle \frac{dlog{\alpha }_t}{dt}}$ represents the drift coefficient, $g^2\left(t\right)={\displaystyle \frac{d{\sigma }_t^2}{dt}}-2{\displaystyle \frac{dlog{\alpha }_t}{dt}}{\sigma }_t^2$ denotes the diffusion coefficient, and ${\omega }_t$ indicates a standard Wiener process. ${\mathcal{A}}_t={\alpha }_t{\mathcal{A}}_{exist,0}+{\sigma }_t{ϵ}_{exist},$ ${\alpha }_t{\mathcal{A}}_{type,0}+{\sigma }_t{ϵ}_{type}$, where ${ϵ}_{exist}$ and ${ϵ}_{type}$ represent Gaussian noise, respectively.

The core of conditional graph diffusion models lies in predicting the noise added to the graph structure. To make the prediction relevant to the battlefield, we incorporate feedback condition representations into the noise prediction model. The forward process is modeled as a Stochastic Differential Equation (SDE):

$$d{G}_t=\left[f\left(t\right)G_t-g^2\left(t\right){▿}_{G}log{q}_t\left(G_t\left|c_t\right.\right)\right]dt+g\left(t\right)d{\overline{\omega }}_t,$$

(9)

where ${▿}_{G}log{q}_t\left(G_t\right)$ represents the graph score function, and ${\overline{\omega }}_t$ denotes the standard Wiener process [21]. The terms $f\left(t\right)$ and $g\left(t\right)$ are time-dependent coefficients that control the noise injection.

Since a combat network consists of nodes and edges, we decompose Equation (9) into two parts for node features ${\mathcal{X}}_t$ and adjacency matrices ${\mathcal{A}}_t$:

$$\left\{\begin{array}{c}d{\mathcal{X}}_t=\left[f\left(t\right){\mathcal{X}}_t-g^2\left(t\right){▿}_{\mathcal{X}}log{q}_t\left({\mathcal{X}}_t,{\mathcal{A}}_t\right)\right]dt+g\left(t\right)d{\overline{\omega }}_t^1\hfill \\ d{\mathcal{A}}_t=\left[f\left(t\right){\mathcal{A}}_t-g^2\left(t\right){▿}_{\mathcal{A}}log{q}_t\left({\mathcal{X}}_t,{\mathcal{A}}_t\right)\right]dt+g\left(t\right)d{\overline{\omega }}_t^2\hfill \end{array}\right.$$

(10)

To generate the combat network efficiently, we switch from the stochastic SDE to a deterministic Ordinary Differential Equation (ODE). This is achieved by deriving the Probability Flow ODE, which shares the exact same marginal distributions as the SDE but removes the random noise term. Specifically, the ODE is obtained by eliminating the stochastic term $g\left(t\right)d{\overline{\omega }}_t$ and modifying the drift coefficient. This allows us to use fast, black-box ODE solvers for sampling. The parameterized probability flow ODE for combat networks is defined as

$${\displaystyle \frac{d{G}_t}{dt}}=f\left(t\right)G_t+{\displaystyle \frac{g^2\left(t\right)}{2{\sigma }_t}}{ϵ}_{\theta }\left(G_t,{\overline{\mathcal{A}}}_t,t,c_t\right).$$

(11)

To further improve the quality of the generated graph, we introduce gradient guidance. This guides the sampling process towards regions with high combat performance. We modify Equation (10) by adding a guidance term based on the gradient of a reward function $R_{\psi }$:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{\mathcal{X}}_t}{dt}}=f\left(t\right){\mathcal{X}}_t+{\displaystyle \frac{g^2\left(t\right)}{2{\sigma }_t}}\left[{ϵ}_{\theta ,\mathcal{X}}-r{\sigma }_t{▿}_{\mathcal{X}}^{*}{R}_{\psi }\right]\hfill \\ {\displaystyle \frac{d{\mathcal{A}}_t}{dt}}=f\left(t\right){\mathcal{A}}_t+{\displaystyle \frac{g^2\left(t\right)}{2{\sigma }_t}}\left[{ϵ}_{\theta ,\mathcal{A}}-r{\sigma }_t{▿}_{\mathcal{A}}^{*}{R}_{\psi }\right]\hfill \end{array}\right.$$

(12)

where r represents the guidance weight, and ${▿}^{*}$ denotes unit-normalized gradients.

3.4. Model Training and Topology Generation

Model training aims to minimize a comprehensive loss function. This function collaboratively optimizes multiple core dimensions of combat network generation. The core component is the standard denoising score matching loss used in diffusion models. This term compels the model to learn the reconstruction of real network topologies from noise. It is defined as the expectation of the mean squared error between the predicted noise ${ϵ}_{\theta }$, the actual noise $ϵ$ and conditional representations $c_t$:

$${\mathcal{L}}_{recon}={\mathbb{E}}_{t,G_0,ϵ}\left[{\parallel {ϵ}_{\theta }(G_t,t|c_t)-ϵ\parallel }_2^2\right].$$

(13)

Building upon this foundation, we introduce two regularization losses specifically designed for combat networks:

• Rule-Constrained Loss: This term ensures the legitimacy of the output topology. It measures the distance between the generated connections and the valid connections defined by military orders. We use the Kullback-Leibler (KL) divergence for this purpose:

  $${\mathcal{L}}_{rule}={\lambda }_1\mathbb{E}\left[KL(q\left({\mathcal{A}}_{pred}\right)\parallel p\left({\mathcal{A}}_{valid}\right))\right],$$

  (14)

  where, $q\left({\mathcal{A}}_{pred}\right)$ represents the distribution of the predicted adjacency matrix, and $p\left({\mathcal{A}}_{valid}\right)$ represents the distribution of valid adjacency matrix.
• Threat Perception Loss: This term supervises the feature fusion process within the model. It guarantees that the representations of each combat node accurately encode the current battlefield threat situation. It is calculated using the Frobenius norm between the extracted node features and the target threat features: ${\mathcal{L}}_{threat}={\lambda }_2{\parallel \varphi ({\mathcal{X}}^{\left(l\right)},\mathbf{T})-{\mathcal{X}}_{target}\parallel }_F^2$. Here, $\varphi$ denotes the feature extraction function at layer l, and ${\mathcal{X}}_{target}$ represents the desired threat feature.

Therefore, the total loss function is defined as

$$\mathcal{L}={\mathcal{L}}_{recon}+{\lambda }_1{\mathcal{L}}_{rule}+{\lambda }_2{\mathcal{L}}_{threat}.$$

(15)

Through the joint optimization of the above objectives in an end-to-end manner, the model learns to generate high-fidelity combat network topologies while deeply adhering to operational rules.

As shown in Algorithms 1 and 2, generation processes constitute iterative denoising procedures that transform noise into clear topologies based on target attributes. At each denoising step, model first update features of all combat nodes using input threat information, thereby embedding battlefield context. Authorization constraint subsequently project predicted continuous edge probabilities onto discrete, legally permissible connection sets defined by operational rules, performing hard filtering to ensure intermediate results at each step remain executable. Dynamic evolution periodically assess robustness of current intermediate topologies against damage. If robustness falls below required standards, gradient correction is applied to guide network structures toward more dynamic and resilient configurations. Final outcomes of these closed-loop processes constitute high-quality, compliant combat network schemes that effectively respond to specific threats and maintain stable generation in dynamic environments.

Algorithm 1. Model Training

Input: Dataset D, containing distinct combat networks G. Number of iterations $n_{iter}$ Output: Trained combat network topology generation model ${ϵ}_{\theta }$, whose parameterized distribution $p_{{ϵ}_{\theta }}\left(G\right)$ approximates $p\left(G\right)$ 1: Initialize network parameters 2: for $k=1$ to $n_{iter}$ do 3:     Sample batch $\left(G\right)\sim D$, get $c_t$ by encode 4:     $t\sim \mathrm{Uniform}(0,T)$ $$G_0=(G,c_t)$$ 5:     Sample $ϵ\sim \mathcal{N}(0,I)$ 6:     Forward noising: $G_t=({\alpha }_t{G}_0+{\sigma }_tϵ)\sim G_0$ 7:     Loss calculation: $\mathcal{L}\leftarrow$ Equation (14) 8:     Parameter update: $\theta \leftarrow \theta -\eta {\nabla }_{\theta }\mathcal{L}$ 9: end for

Algorithm 2. Topology Generation

Input: Target threat vector $\mathbf{T}$. Topology generation model ${ϵ}_{\theta }$ Output: Generated combat network topology $G_{c_t}$ 1: $G_{c_t}\sim \mathcal{N}(0,I)$ 2: for $t=0$ upto T do 3:     Condition encoding $c_t=\mathrm{Encoder}\left(\mathbf{T}\right)$ 4:     Call model ${ϵ}_{\theta }$ 5:     Reverse denoising: $$G_{t-1}={\displaystyle \frac{1}{\sqrt{{\alpha }_t}}}\left(G_t-{\displaystyle \frac{1-{\alpha }_t}{\sqrt{1-{\alpha }_t}}}{ϵ}_{\theta }\right)+{\sigma }_tϵ$$ 6: end for 7: return $G(c_t)$

4. Results and Discussion

To validate effectiveness of the algorithm proposed in this paper, we designed systematic simulation experiments focusing on overall generation performance of combat networks and evaluating rationality and practicality of generation results from multiple dimensions.

4.1. Dataset Description and Evaluation Metrics

4.1.1. Dataset Description

The combat network dataset employed in this experiment is deeply integrated with modern operational theory, equipment system demonstration research outcomes, and typical simulation platform scenarios. It was constructed using self-designed combat network topology generation methods that strictly adhere to the “Observation-Location-Decision-Action” cycle and networked cooperative operations concepts, aiming to generate combat network topologies consistent with military logic and exhibiting high realism in actual combat conditions.

As shown in Figure 3, constructed combat networks are visually compared. Figure 3a,c illustrates communication connections within combat networks, while Figure 3b,d depicts logical connections. Additionally, three-dimensional maps display geospatial distributions of combat networks. For each target type, 2500 combat networks were designed, resulting in a high-quality dataset comprising 17,500 unique combat network instances across seven target types. Each target type is defined by a set of key operational attributes that determine its role and connectivity within the combat network topology. To facilitate clarity in visualizations, distinct red geometric markers are assigned to each target type. The seven defined target types and their primary characteristics are as follows:

• Static Target: Characterized by high priority and hardness alongside zero mobility, typically representing critical, fixed infrastructure.
• Mobile Target: Features high mobility and moderate stealth capabilities, representing maneuvering units such as vehicles or mobile platforms.
• Hardened Target: Possesses superior hardness and significant countermeasure capabilities, representing fortified or well-defended positions.
• Area Target: Defined by lower priority and hardness, representing broader geographical objectives.
• Time-Sensitive Target (TST): Exhibits the highest priority but low hardness and moderate mobility, representing fleeting targets requiring rapid engagement.
• Stealth Target: Distinguished by high stealth and electronic warfare capabilities, representing low-observable assets.
• Electronic Warfare (EW) Target: Demonstrates the highest electronic warfare and countermeasure attributes, representing dedicated jamming or surveillance units.

Each instance includes comprehensive node information (type, attributes, spatial location), edge information (type, connection relationships), and global condition information (target attributes). This dataset captures the complexity and regularity inherent in combat networks, as well as their goal-oriented nature. Consequently, it provides a reliable foundation for training and validating subsequent models. The key statistical characteristics of the dataset are summarized in

Table 4. Dataset Statistics and Detailed Taxonomy.

Category	Specific Types/Values	Key Attributes	Role/Description
Overall Statistics
Total Samples	17,500 instances	Avg. Nodes per Network: 17.92	Avg. Edges per Network: 45.2
Node Types & Equipment (24 Subtypes)
Reconnaissance	Radar, Optical, Electronic, Drone	Detection Range, Accuracy, Stealth	Information acquisition, surveillance, and target tracking.
Firepower	Missile, Rocket Launcher, Shell, PGM	Strike Range, Lethality, Accuracy	Kinetic attack capability for engaging hostile targets.
Command	Theater Cmd Center, Tactical Post, Forward Post	Decision Speed, Command Range	Central decision-making and tactical orchestration.
Communications	Satellite, Ground Station, Mobile Vehicle	Bandwidth, Reliability, Latency	Information transmission and network connectivity.
Support	Logistics Base, Maintenance Vehicle, Hospital	Logistics Capacity, Maintenance Capability	Sustainment, repair, and logistical backup.
Target	Static, Mobile, Hardened, Area, Time-Sensitive, etc.	Priority, Hardness, Mobility	Adversarial entities or assets to be detected and engaged.
Edge Types (Relations)
Reconnaissance	RECON (Cyan)	Recon ↔ Target	Observation and data sensing of hostile entities.
Communication	COMM (Yellow)	Inter-node (C2, Firepower, etc.)	Transmission of tactical data and instructions.
Firepower	FIR (Red)	Firepower → Target	Physical equipment strike or attack action.
Command & Control (C2)	COD (Black)	Command → Functional Nodes	Issuance of operational orders and coordination.
Support	SUP (Magenta)	Support → Command/Nodes	Provision of logistics, repair, or medical aid.

and Figure 4. These metrics include the total number of samples, the diversity of node and equipment types, and the average network size.

The dataset comprises 17,500 unique combat network samples. Each sample corresponds to a scenario centered on a specific target. Network structure consists of six primary node types: Reconnaissance, Firepower, Command, Communications, Support, and Target. These primary types are realized through specific equipment templates, resulting in 24 distinct equipment subtypes. For instance, Reconnaissance nodes are instantiated as Radar, Optical, Electronic, or Drone units. Similarly, the seven Target types are defined by distinct profiles of operational attributes, such as priority, hardness, and mobility. Connections between nodes are categorized into five edge types. These represent functional relationships such as reconnaissance, Support, and command. On average, each generated network contains approximately 18 nodes and 45 edges. This scale reflects a typical tactical-level engagement network.

These statistics demonstrate the dataset’s capacity to represent varied and realistic combat scenarios. The defined node and edge types serve as fundamental building blocks for the conditional graph diffusion model.

4.1.2. Evaluation Metrics

Based on network characteristics and military nature of combat networks, this experiment selects four metric categories to evaluate effectiveness of generated combat networks: structural similarity metrics, tactical effectiveness metrics, condition matching metrics, and diversity metrics. Specific descriptions follow:

• Structural Similarity Metrics. Node distribution similarity quantifies the consistency in node type distributions between generated and original networks. This metric is calculated as: $S_{node}=1-{\displaystyle \frac{1}{n}}{\sum }_{i=0}^n|r_i^{orig}-r_i^{gen}|$, where $r_i^{orig}$ and $r_i^{gen}$ denote the proportions of the i-th node type in the original and generated combat networks, respectively. Network density matching assesses the similarity in connection density between the generated and original networks, defined as $S_{density}=1-|{\rho }^{orig}-{\rho }^{gen}|$, where $\rho$ represents the network density.
• Tactical Effectiveness Metrics. Target accessibility evaluates the efficiency of connectivity from non-target nodes to target nodes, reflecting the network’s information flow capability. This metric is formulated as: $R_{target}={\displaystyle \frac{\left|\right\{v\in N_{non-target}:\exists t\in N_{target},\mathrm{path}\left(n,t\right)\left\}\right|}{|N_{non-target}|}}$, where $N_{non-target}$ denotes the set of non-target nodes, $N_{target}$ represents the set of target nodes, and $\mathrm{path}\left(n,t\right)$ indicates the existence of a connected path from node i to any target node. Network resilience measures robustness through random node removal tests, calculated as $R_{resilience}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{|c_{max}^i|}{|V_{orig}|}}$, where $|V_{orig}|$ is the total number of nodes in the original network, and $|c_{max}^i|$ represents the size of the largest connected component after the i-th random node removal.
  The defense strength $S_d$ quantifies the robustness of the combat network by aggregating five key topological attributes. It is calculated as a weighted sum of normalized sub-metrics:

  $$S_d=\sum _{i=1}^5{w}_i·M_i$$

  (16)

  where $M_i$ represents the indices for network connectivity, clustering coefficient, command control efficiency, graph density, and node type diversity. The command control efficiency is inversely proportional to the average shortest path length from command nodes to all other nodes. The final score is normalized to a range of $[0,1]$, with higher values indicating a more robust and defensible network structure.
  The attack efficiency $E_a$ evaluates the offensive capability by balancing response speed with firepower allocation. The calculation principle prioritizes shorter transmission paths to ensure rapid engagement:

  $$E_a=\alpha ·{\displaystyle \frac{1}{1+L}}+\beta ·R_f$$

  (17)

  where L represents the average shortest path length of the network, and $R_f$ denotes the ratio of firepower nodes. The term ${\displaystyle \frac{1}{1+L}}$ ensures that networks with tighter connectivity achieve higher efficiency scores, while the firepower term ensures sufficient combat resources are present.
• Condition Matching Metrics. Attribute consistency assesses the degree of alignment between generated networks and target attributes. This metric is computed as: $M_{cond}={\displaystyle \frac{1}{m}}{\sum }_{p\in P}max\left(0,1-min\left({\displaystyle \frac{|v_p^{target}-v_p^{gen}|}{\tau }},1\right)\right)$, where P represents the set of target node attributes, and $v_p^{gen}$ denotes the generated attribute values.
• Quantitative Symmetry Indicators. To comprehensively evaluate the structural balance and functional stability of the combat network, we designs four core quantitative metrics.
  Node Efficiency Distribution Variance measures the dispersion of node importance within the network. It aggregates degree centrality and betweenness centrality to calculate the comprehensive efficiency E of a node. A smaller variance indicates more balanced node performance, implying higher symmetry. $S_{\mathrm{node}}=1-{\displaystyle \frac{{\sigma }^2\left(E\right)}{max\left({\sigma }^2\left(E\right)\right)}}$, where ${\sigma }^2\left(E\right)$ is the variance of the node efficiency distribution, and $max\left({\sigma }^2\left(E\right)\right)$ is the theoretical maximum variance for normalization.
  Connectivity Symmetry Coefficient evaluates the balance of network connections using the Gini coefficient. In a perfectly symmetric regular network, node degrees are equal. In an asymmetric network, there is a significant skew in degree distribution. $S_{\mathrm{conn}}=1-Gi$, where $Gi$ is the Gini coefficient of the node degree distribution, calculated as $Gi={\displaystyle \frac{{\sum }_{i=1}^n{\sum }_{j=1}^n|d_i-d_j|}{2n^2\overline{d}}}$. Here, $Gi=0$ corresponds to a perfectly symmetric distribution, yielding $S_{\mathrm{conn}}=1$.
  Functional Module Symmetry evaluates the uniformity of substructures composed of the same node types. It calculates the stability of path lengths within the subgraph of the same type of nodes. $S_{\mathrm{func}}={\displaystyle \frac{1}{K}}{\sum }_{k=1}^K\left(1-{\displaystyle \frac{{\sigma }^2\left(P_k\right)}{max{\left(P_k\right)}^2}}\right)$, where K is the number of node types, and $P_k$ is the set of path lengths between nodes of the k-th type. This metric reflects the geometric symmetry of the spatial distribution of similar nodes.
  Path Length Symmetry reflects the global symmetry of the network topology and the consistency of information transmission. It uses the coefficient of variation ($CV$) to measure the dispersion of path lengths. $S_{\mathrm{path}}={\displaystyle \frac{1}{1+CV\left(P\right)}}$ where $CV\left(P\right)={\displaystyle \frac{{\sigma }_P}{{\mu }_P}}$ is the coefficient of variation of the shortest path lengths between all node pairs, and ${\sigma }_P$ and ${\mu }_P$ are the standard deviation and mean of the path lengths, respectively. A more symmetric network structure results in smaller differences in node-to-node path lengths.
  Overall Symmetry Score $S_{\mathrm{overall}}$ combines the four dimensions through a weighted average. Higher weights (30% each) are assigned to node efficiency and connectivity to highlight the importance of core topological features: $S_{\mathrm{overall}}=0.3S_{\mathrm{node}}+0.3S_{\mathrm{conn}}+0.2S_{\mathrm{func}}+0.2S_{\mathrm{path}}$.

4.2. Experimental Setup

Experiments were conducted on computers equipped with Nvidia GPUs using PyTorch 2.0.1 deep learning framework. Constructed graph-structured data includes multiple combat unit types and combat network-specific datasets covering six node categories: reconnaissance nodes, fire nodes, command nodes, communication nodes, support nodes, and target nodes. Datasets were partitioned into training, validation, and test sets to ensure reliable model evaluation. During experiments, target attributes of original combat networks were extracted from test sets and utilized as input conditions for trained combat network generation models to guide creation of corresponding combat network structures. Generated combat networks were compared with original combat networks possessing identical target attributes from test sets. Results were analyzed to evaluate generative combat network capabilities in meeting specific tactical requirements while preserving original structural characteristics.

This experimental design verifies both practicality and generalization capabilities of combat network topology generation models in constructing combat networks, while ensuring generated combat networks satisfy preset operational requirements and maintain reasonable organizational structures.

4.3. Analysis of Experimental Results

Experiments involved comparative evaluation of 15 original combat network samples and corresponding 480 generated combat network alternatives (32 new combat networks generated per original target). Generation model performance was analyzed from three dimensions: structural fidelity, tactical effectiveness improvement, and operational condition adaptability. Results demonstrate that models can generate feasible solutions that significantly enhance specific tactical indicators while preserving advantages of core architectures.

4.3.1. Optimization Analysis of Structural Fidelity and Network Characteristics

As shown in Figure 5, generated combat networks exhibit high structural similarity to original networks, demonstrated through comparative analysis of four key structural parameters: node count, edge count, network density, and average degree. This analysis enables purposeful optimization of network characteristics.

Regarding node count distributions in Figure 5a, average values of generated schemes are 18.25 ± 2.82, closely matching 16.94 ± 2.82 observed in original networks, indicating accurate model capture of original combat network cell-scale configurations. Fidelity is critical for maintaining integrity of operational system-of-systems, ensuring feasibility of generated schemes in actual deployment.

Edge number distributions reveal that average edge counts in generated networks are 36.96 ± 7.35, approximately 103% higher than original networks (20.38 ± 10.02) in Figure 5b. This increase proves highly valuable in combat scenarios: additional connections establish richer interaction pathways between combat units within networks, creating multiple redundant information transmission channels. When key nodes are destroyed by enemies or subjected to strong electromagnetic interference, command information and target data can be transmitted through alternative paths, significantly enhancing combat network damage resilience.

As show in Figure 5c, network density increased from original 0.16 to 0.22, further confirming connection redundancy growth. This indicates structural optimization enhances combat network flexibility and resilience against electromagnetic interference or physical destruction. Consequently, networks can maintain overall functional integrity and prevent entire combat system collapse due to single node failures, aligning with flexibility and reconfigurability requirements in modern warfare.

As shown in Figure 6 and

Table 5. Statistics of Generated Combat Network Metrics and Values (Mean Values).

	Number of Nodes	Number of Edges	Network Density	Average Degree	Target Accessibility	Attack Efficiency	Defense Strength
Original Combat Network	17.733	16.867	0.114	1.889	0.791	0.125	0.312
Generated Combat Network	18.315	34.394	0.221	3.74	0.833	0.162	0.368
Change Rate	5.04%	130.9%	114.6%	118.3%	9.3%	36%	20.8%

, composite quality scores of generated solutions are 0.68 ± 0.05, exhibiting low variance. This demonstrates robust model capabilities in consistently producing high-quality solutions. These results indicate topology generation models possess strong environmental adaptability and can serve as core components of decision support systems, providing essential technical support for rapid reconfiguration of Adaptive Combat Networks.

4.3.2. Scalability Analysis and Applicability Discussion

To evaluate the feasibility of the proposed model in large-scale combat scenarios, we conducted scalability experiments across different network scales. The experiments focused on the generation time and the ability to provide diverse options for decision-makers under target attribute constraints.

Analysis of Generation Time Efficiency: In modern combat planning, the generation of combat network must meet strict timeliness requirements. It is not sufficient to generate a single valid network; the system must generate multiple distinct networks to provide decision-makers with viable options.

Figure 7 illustrates the relationship between the number of nodes and the generation time. The results indicate a significant increase in time cost as the network scale grows. Specifically, when generating a set of networks for a single target, the time consumption rises sharply. For instance, generating 10 combat network with over 50 nodes each takes more than 5 min.

This trend reveals a performance bottleneck. The process involves solving complex constraints and graph matching, which becomes computationally expensive as the node count increases. The exponential or super-linear growth in time suggests that the current method struggles to meet the real-time requirements for large-scale combat networks.

Impact on Decision-Making Timeliness: The primary goal of combat network generation is to support rapid decision-making against threats. Our analysis shows that for networks with a node count exceeding 50, the generation latency compromises operational timeliness.

Furthermore, the results presented in Figure 7 account only for the topological generation phase. If we include the subsequent resource matching phase, the total time required to produce actionable plans would increase further. This added delay significantly reduces the effectiveness of the generated networks in dynamic battlefield environments where rapid response is critical.

Applicability Assessment: Based on the experimental data, we must conclude that the proposed method is not applicable to the generation of large-scale combat networks. The high time cost makes it unsuitable for scenarios involving thousands of nodes or high-density networks.

However, the method remains effective for small and medium-scale operations. For networks with fewer than 50 nodes, the generation time is acceptable, and the model can produce high-quality, diverse options efficiently. Therefore, the model is best suited for tactical-level planning or specific task force formations, where scale is manageable and the need for diverse, constraint-satisfying topologies is paramount. Future work must focus on accelerating the sampling process or employing hierarchical strategies to address scalability limitations.

4.3.3. Analysis of Model Training Convergence and Generation Time

Figure 8 illustrates the training dynamics. It details the convergence trajectories of the total training loss ($\mathcal{L}$) alongside two specialized regularization components: the Rule-Constrained Loss (${\mathcal{L}}_{rule}$) and the Threat Perception Loss (${\mathcal{L}}_{threat}$). The model achieves a stable convergence state after 5000 steps. At this stage, the total loss reduces to approximately 0.35.

Detailed Analysis of Regularization Losses: Beyond overall convergence, the behaviors of ${\mathcal{L}}_{rule}$ and ${\mathcal{L}}_{threat}$ provide critical insights. They reveal how the model internalizes combat constraints.

• Rule-Constrained Loss (${\mathcal{L}}_{rule}$): The light brown curve in Figure 8 shows that ${\mathcal{L}}_{rule}$ declines rapidly during the first 2000 steps. It then stabilizes at a low magnitude. As defined in Equation (14), this term represents the Kullback-Leibler (KL) divergence between the predicted adjacency distribution $q\left({\mathcal{A}}_{pred}\right)$ and the valid military order distribution $p\left({\mathcal{A}}_{valid}\right)$. The downward trend signifies that the model progressively reduces the informational discrepancy between generated topologies and doctrinal rules. The eventual plateau indicates that generated connections have converged to the subspace of valid military configurations. This ensures topological legitimacy.
• Threat Perception Loss (${\mathcal{L}}_{threat}$): Similarly, the dark brown curve representing ${\mathcal{L}}_{threat}$ decreases steadily. This reflects the optimization of the Frobenius norm between node features and target threat vectors. This trend demonstrates that the feature extraction function $\varphi$ effectively aligns the latent representations of combat nodes (${\mathcal{X}}^{\left(l\right)}$) with dynamic battlefield threat information ($\mathbf{T}$). The minimization of this term confirms an important aspect of the conditional generation process. It is not merely reconstructing structures; rather, it actively encodes the situational context of the combat environment into the node features.

The joint optimization of these terms allows the model to learn basic structural characteristics rapidly during the initial phase. Subsequent steps primarily refine the balance between tactical indicators and conditional constraints. The simultaneous convergence of all three loss curves, without divergence, validates the stability of the multi-objective optimization strategy. No evident overfitting is observed. This verifies the model’s generalization capability. Furthermore, it ensures that the generated combat network schemes possess theoretical optimality and high fidelity.

In combat network generation technology research, generation efficiency represents crucial metrics for evaluating algorithm practicality and tactical decision-making support capabilities. However, few studies address complete combat network generation times in literature. As shown in

Table 6. Comparative Analysis of Combat Network Generation Methods Performance.

	Network Scale (Nodes)	Generation Time	Network Complexity	Constraint Handling	Scalability	Key Advantages
Combat-Network-based-Selection [36]	1 (25)	29 s	2D Simple Structure	No Combat Constraints	Limited	Basic Equipment Selection
OODA-based-Weapon Configuration [37]	1 (40)	30 s	Basic Configuration	No Spatial Coordinates	Limited	OODA Loop Integration
Graph-based-Adaptive- Combat-Network [38]	1 (3–9)	2.26 s	2D Graph Structure	No Spatial Relationships	Moderate	Adaptive Network Formation
3DTG-CGD	1 (28)	6.32s	3D Topological Structure	Combat Constraints	Excellent	3D Topology + Combat Constraints Integration

, we collected three typical combat network generation methods for comparative analysis.

Combat Network-Based Selection: [36] This method characterizes combat networks using multi-layer network models and establishes three evaluation indices: redundancy, risk, and agility. Based on this framework, equipment portfolio planning models are developed, and multi-objective optimization is performed using heuristic techniques. This approach aims to select Pareto-efficient solution sets enhancing overall operational effectiveness of combat networks from various equipment combination schemes.

OODA-Based Weapon Configuration: [37] This method employs OODA loops as theoretical frameworks, calculating underlying capabilities through operational effectiveness index systems and index-based approaches. Method cores involve constructing integer programming models to determine optimal weapon allocation schemes for different combat missions while considering combat effectiveness evaluated through OODA loops and inherent anti-damage capabilities of equipment systems.

Graph-based Adaptive Combat Network: [38] This method utilizes graph models to represent maritime adaptive combat networks, abstracting various combat platforms as nodes and cooperative relationships as edges. Through dynamic reconfiguration mechanisms, when nodes (combat units) fail due to battle damage or interference, alternative links can be rapidly generated, enabling path re-planning that helps maintain resilience and adaptive capabilities of combat networks.

In Table 6, combat network generation methods based on conditional diffusion models proposed in this paper maintain considerable generation efficiency while accounting for constraints imposed by complex military rules and topology generation quality. Furthermore, they demonstrate potential to satisfy real-time tactical-level requirements. Proposed combat network topology generation methods focus on synthesizing topological structures adhering to complex rules and tactical objectives within given resource constraints. Although these generation methods involve more refined structural construction and incur relatively high computational overhead, they offer novel technical approaches to rapidly generate compliant, robust, and adaptive combat network schemes in highly dynamic combat environments.

To evaluate the effectiveness of the proposed method, we compared it with four mainstream graph generation methods under a unified network scale of approximately 20 to 25 nodes.

Table 7. Performance Comparison of Mainstream Conditional Graph Generation Methods under Uniform Scale (20–25 Nodes).

Method	Network Scale (Nodes)	Generation Time (s)	Target Accessibility (%)	Defense Strength	Attack Efficiency
GraphRNN [39]	20–25	3.15	65.8	0.352	0.133
GraphDF [40]	20–25	5.42	62.1	0.331	0.152
DiGress [28]	20–25	7.65	76.4	0.265	0.138
CDGS [41]	20–25	4.80	75.2	0.363	0.112
3DTG-CGD	20–25	6.32	78.5	0.385	0.173

presents the quantitative results in terms of generation time and three combat-related metrics: target accessibility, defensive efficiency, and offensive efficiency.

GraphRNN, as a classical sequential generation method, achieves the fastest generation speed (3.15 s). However, it lacks the mechanism to incorporate complex combat constraints, resulting in the lowest performance in target accessibility (65.8%) and defensive efficiency (0.15). This indicates that simply generating a connected graph structure cannot meet the requirements of a combat network.

GraphDF and DiGress, based on flow matching and discrete diffusion respectively, improve the quality of the topological structure. They show moderate improvements in defensive and offensive efficiencies compared to GraphRNN. However, due to the iterative nature of their sampling processes, their generation times increase (5.42 s and 7.65 s), and their combat metrics remain limited because they are not specifically optimized for tactical objectives.

CDGS introduces conditional information during the diffusion process, which leads to better target accessibility (75.2%) and defensive efficiency (0.363). Nevertheless, its generation objective is primarily focused on attribute matching rather than the holistic operational effectiveness required in combat scenarios.

In contrast, the 3DTG-CGD method proposed in this paper demonstrates significant advantages in combat performance. Although its generation time (6.32 s) is slightly higher than the non-diffusion baseline (GraphRNN), it remains comparable to other diffusion-based methods. More importantly, 3DTG-CGD achieves the highest scores across all combat metrics: 78.5% in target accessibility, 0.385 in defensive efficiency, and 0.173 in offensive efficiency. These results indicate that by embedding complex military rules and tactical objectives into the diffusion process, our method can generate topologies that are not only structurally valid but also highly effective in combat operations. The balance between generation efficiency and combat superiority makes 3DTG-CGD a more suitable solution for real-time tactical-level requirements compared to existing general-purpose graph generation models.

4.3.4. Systematic Improvement of Multi-Dimensional Tactical Effectiveness

To comprehensively evaluate operational potential of generated combat networks, we compare performance across several key tactical metrics using tactical effectiveness radar charts and tactical effectiveness comparison heat maps shown in Figure 9. Results demonstrate generated combat networks achieve balanced and significant improvements in target accessibility, network resilience, and attack efficiency.

Target accessibility performance improved by 4.5%, indicating generation schemes optimize connections between key fire nodes and high-value targets. This ensures that even in complex battlefield environments, our firepower can reliably cover and engage intended targets. Defense strength increased by 13.15%, consistent with previously observed network density increases from structural analysis. This enhancement enables combat networks to maintain overall functional integrity against electromagnetic interference or physical destruction, preventing entire combat system collapse due to single node failures, aligning with combat concepts of resilience and reconfiguration within Combat Networks. Most importantly, attack efficiency was optimized, showing 30.4% improvement over previous levels. This metric reflects speed and resource consumption for completing full attack tasks. Improvement directly demonstrates generation schemes are not only structurally similar but functionally superior. Efficient combat networks can address more threats per unit time or accomplish identical intensity combat tasks with limited resources, thereby significantly enhancing battlefield effectiveness.

As shown in Figure 10, distributions of these tactical indicators exhibit higher medians and more concentrated spreads in generated combat networks, demonstrating universality and stability of performance improvements. In Figure 11, average matching degrees of generated combat networks under diverse combat conditions are 0.84 ± 0.05, with coefficients of variation merely 0.0714 across different condition groups. High matching degrees indicate commanders can input specific battlefield parameters, and models can reliably produce feasible networks satisfying these stringent constraints.

As shown in Figure 12, trajectory analysis within a five-dimensional index space (structural similarity, conditional matching, node count, edge count, and density) reveals a critical finding. The generated schemes achieve a collaborative optimization of structural attributes and tactical indicators. Specifically, curves representing different target schemes (distinguished by color) exhibit synchronized fluctuations across dimensions. When the conditional matching degree (the “Condition Matching” dimension) reaches a discrete peak cluster—for example, a local maximum near the axis—the structural similarity and node/edge counts maintain relatively high normalized values. They do not decline sharply. This observation directly reflects that the generated schemes preserve structural rationality while ensuring high conditional matching.

This multi-dimensional view integrates five normalized evaluation dimensions. The results indicate that the generated combat networks exhibit significant structural pattern differences and strong diversity:

• In the “Condition Matching” dimension, multiple discrete peak clusters appear. These are visible as concentrated high-value regions within the color curves. This pattern corresponds to the clear hierarchical characteristics of the generated schemes in satisfying conditional constraints.
• Node count and edge count display hierarchical aggregation. Curves for different targets gather into distinct groups within these two dimensions. Furthermore, the distribution of edge count is highly correlated with node count, as shown by the consistent curve trends. This correlation reflects grouping differences in the scale of combat networks.
• In the middle and high-density regions (the “Density” dimension), dense trajectory lines are distributed. Most curves concentrate in this interval, indicating that the majority of combat networks are high-density. However, a small number of curves fall into the low-density region. This distribution demonstrates the variation in network tightness among the generated schemes.

4.3.5. Correlation Analysis Between Symmetry and Tactical Effectiveness

Statistical analysis was performed on 50 generated combat network samples. We used Pearson correlation coefficient (measuring linear relationship strength) and the Spearman rank correlation coefficient (measuring monotonic relationship) to analyze the relationship between symmetry and tactical effectiveness. The results, as shown in

Table 8. Statistical Results of Correlation Analysis between Symmetry and Tactical Effectiveness.

Tactical Indicator	Pearson r	Spearman $\mathit{\rho }$	Coefficient of Det. ${\mathit{R}}^2$	Regression Equation	Significance
Target Accessibility	0.782	0.756	0.611	$y=1.24x+0.15$	$p<0.01$
Network Resilience	0.698	0.684	0.487	$y=0.95x+0.08$	$p<0.05$
Attack Effectiveness	0.715	0.703	0.511	$y=1.10x+0.12$	$p<0.05$
Defense Strength	0.634	0.618	0.402	$y=0.85x+0.10$	$p<0.05$

, demonstrate a significant positive correlation between symmetry and all tactical effectiveness indicators.

• Target Accessibility and Symmetry: As shown in the table, target accessibility exhibits the highest correlation with symmetry ($r=0.782$), with an $R^2$ of 0.611. This indicates that structural symmetry is a key factor influencing network response speed. In symmetric networks, the variance in path lengths between nodes is minimized (indicating high path length symmetry). This allows commands and data to propagate throughout the network with consistent latency. Such a balanced topology avoids congestion at central nodes, thereby enhancing the capability to rapidly discover and lock targets.
• Network Resilience and Symmetry: Network resilience demonstrates a strong correlation with symmetry ($r=0.698$). A high-symmetry network implies the absence of hub nodes with disproportionately high degrees (indicated by a high connectivity symmetry coefficient). Consequently, under enemy attack, functional degradation is distributed across nodes rather than concentrated at specific points. The regression analysis indicates that a 0.1 increase in the symmetry score yields an average increase of 0.095 in network resilience. This validates the core value of symmetry design in ensuring survivability.
• Synergistic Effect of Attack and Defense Strength: The correlation coefficients for attack and defense strength are 0.715 and 0.634, respectively. These results demonstrate that functional module symmetry ensures a uniform spatial distribution of firepower, reconnaissance, and support modules. This configuration enables the combat network to maintain consistent strike and defense capabilities in all directions, effectively eliminating defensive blind spots and achieving a globally optimal configuration.

Based on the data above, we verify the symmetry principle of combat network design. Experimental results show that samples with an overall symmetry score $S_{\mathrm{overall}}>0.7$ outperform the sample mean in all tactical effectiveness indicators. Statistical regression analysis indicates that a 10% improvement in symmetry leads to an average increase of 12.4% in target accessibility and 11.0% in attack effectiveness. Among the four sub-indicators, functional module symmetry has the strongest correlation with synergy effectiveness ($r=0.742$). This suggests that priority should be given to the spatial balanced distribution of similar nodes in the generation model.

4.3.6. Result Analysis of 3D Topology Generation for Combat Networks

As shown in Figure 13, visual analysis of generated results reveals schemes exhibit significant diversity and improved tactical rationality while maintaining core tactical intent. Figure 14 illustrates combat network generation through creation of redundant connections and detours across layers, resulting in more resilient network structures with average node degrees of 3.74 and network efficiency improvements of approximately 28%.

Results demonstrate all generation schemes strictly adhere to basic constraints, and positional and attribute characteristics of target nodes in generated combat networks fully coincide with original samples, ensuring coherence of tactical tasks. Regarding topological structures, generative models exhibit strong exploratory capabilities. Compared to original combat networks, attack efficiency of generated networks increases by 18.25%, and defense strength improves by 25.39%.

As shown in Figure 14, 2D projection diagrams further reveal optimization of generation schemes in functional partitioning. Through reasonable division of detection, decision-making, and strike areas and establishment of efficient cross-regional cooperative links, proposed schemes enhance regional cooperation efficiency, significantly improving cooperative operational capabilities of combat networks.

Generally, independent clustering of condition matching can evaluate controllability of generating condition attributes, while node-edge linkage assesses rationality of generated combat networks. Distributions of structural similarity and density indicate reductions in structural and topological tightness, respectively. Grouping characteristics of generated combat networks—in terms of condition matching, size, and density—correspond to attribute groupings of original combat networks. High diversity in structural similarity reflects differences from original structures. Additionally, parallel coordinate lines exhibit clear convergence patterns, indicating models learn effective design rules rather than generating outputs randomly. Regular learning enables generated schemes to not only satisfy current conditions but also generalize effectively to handle multi-class combat scenarios and diverse target types.

4.4. Engineering Application Prospects

The proposed method offers a new approach for constructing three-dimensional combat network topologies. It supports the intelligent design of future command information systems. This section discusses the engineering application pathways. We focus on model deployment, system integration, and real-time optimization. The aim is to enhance the practical value of this research.

4.4.1. Model Deployment Plan

We propose encapsulating the trained model as an independent service module. It can be deployed on cloud platforms or tactical edge computing nodes. At the hardware level, we recommend using high-performance computing units equipped with GPUs. These units meet the parallel computing demands of generation tasks. The software environment should integrate deep learning frameworks and graph processing libraries. We also suggest using containerization technology for encapsulation. This ensures the model’s portability across different combat simulation platforms. Under security constraints, periodic updates or online fine-tuning strategies are applicable. These strategies allow the model to adapt to new combat equipment and evolving tactical rules.

4.4.2. System Interface Design

Standardized interfaces are essential for embedding the model into existing planning systems. The input interface must support structured data. This includes battlefield environment parameters, unit attributes, and relational constraints. The output interface should provide generated 3D topological description files. These files include node lists, edge relationships, and spatial coordinates. They can be converted into standard formats readable by 3D visualization engines. Through service-based API calls, the module cooperates with command and control components. Examples include force composition and communication planning modules. This integration assists commanders in rapidly generating and evaluating multiple network options.

4.4.3. Real-Time Optimization and Dynamic Adjustment

Actual combat systems require high real-time responsiveness. Our model supports rapid sampling under new constraints. It can integrate real-time battlefield intelligence to adjust local network structures. Future work may introduce lightweight model variants. This would provide topology suggestions within tighter time constraints. Furthermore, the model outputs could link with network performance evaluation modules. This forms a closed loop of “generation-evaluation-optimization.” Such a loop will gradually improve the reliability and adaptability of the solutions.

4.4.4. Application Limitations and Future Directions

The current research remains at the stage of methodological validation. Several engineering issues require further attention for practical application. These include the standardization of data sources and uncertainties in complex environments. Future work will focus on testing the model within simulation environments. We will also explore data fusion mechanisms with real command systems. These steps aim to transition the method into a practical tool for combat decision-making.

5. Conclusions

This study successfully verifies effectiveness of improved conditional diffusion models in generating 3D topologies of combat networks. By embedding target attributes as conditional signals during generation processes, we achieve controllable generation of both topology and tactical characteristics of combat networks. Experimental results demonstrate models can produce network layouts comparable to original combat networks in terms of structural similarity and tactical effectiveness, while maintaining diversity among generated samples. From methodological perspectives, diffusion models exhibit strong capabilities in representing complex graph structures. Their step-by-step denoising generation mechanisms prove particularly suitable for modeling multi-level and multi-type node relationships in military combat networks. Future research could explore closer integration of graph neural networks with diffusion processes, employing graph attention mechanisms to enhance control over key node generation. Additionally, introducing multi-objective optimization frameworks could balance competing tactical indicators and yield more practical network designs. Furthermore, late-stage optimization strategies combined with reinforcement learning could adaptively refine generated networks for specific combat scenarios, thereby improving method practicality and flexibility. Investigating these technical avenues will provide stronger theoretical foundations and methodological tools for intelligent military network planning.