# Methodology

## 3.1 Three-Layer Capacitated Network Modeling

We model the system as a directed capacitated network:

G = (V, E)

Where: - V: set of nodes - E: set of edges

The network contains three layers:

-   Layer 1: Source layer
-   Layer 2: Intermediate layer
-   Layer 3: Sink layer

Node set:

V = V1 ∪ V2 ∪ V3

Edge set:

E = E12 ∪ E13 ∪ E23 ∪ E1 ∪ E2 ∪ E3

Where:

  Edge Type   Description
  ----------- -----------------
  E12         Layer1 → Layer2
  E13         Layer1 → Layer3
  E23         Layer2 → Layer3
  E1          within Layer1
  E2          within Layer2
  E3          within Layer3

### Random Capacity Modeling

Each edge e has a stochastic capacity:

Ce

Capacity is a discrete random variable:

Ce ∈ {c1, c2, ..., ck}

Probability:

P(Ce = ci) = pi

with:

Σ pi = 1

#### Intra-layer edges

Randomly select 1--5 capacity levels and assign random probabilities.

Example:

capacity: \[100,200,300\] probability: \[0.2,0.5,0.3\]

#### Inter-layer edges

Only one capacity value is selected with probability 1.

Example:

capacity: \[400\] probability: \[1\]

------------------------------------------------------------------------

## 3.2 Edge-Based Graph Construction

Traditional graph learning uses nodes as entities. However, in
capacitated flow networks the key variable is **edge capacity**.
Therefore, we transform edges into nodes.

We construct a **Line Graph**:

Ge = (Ve, Ee)

Where:

Ve = original network edges

Two edges are connected if they share a node.

Example:

(A,B) and (A,C) share node A → connected in line graph.

This produces adjacency matrix:

Ae

------------------------------------------------------------------------

## 3.3 Edge Feature Representation

Each edge (treated as a node in the line graph) is associated with a
time series feature vector.

For edge e:

xe(t)

Basic feature:

xe(t) = Ce(t)

Optional structural features:

xe(t) = \[Ce(t), layer_type, centrality\]

Input tensor:

X ∈ R\^(Ne × T × d)

Where:

-   Ne = number of edges
-   T = number of time steps
-   d = feature dimension

------------------------------------------------------------------------

## 3.4 Spatial Attention Module

Spatial attention captures dependencies between adjacent edges.

Attention score:

a(i,j) = (Qhi)(Khj)\^T / √d + φ(i,j)

Where:

Q,K = learnable parameters

Normalize:

α(i,j) = softmax_j(a(i,j))

Update representation:

hi = Σ α(i,j) hj

This mechanism allows the model to focus on important neighboring edges.

------------------------------------------------------------------------

## 3.5 Temporal Attention Module

Temporal attention models temporal dependencies in edge states.

For time steps ts and tr:

b(ts,tr) = (Qh_ts)(Kh_tr)\^T / √d

Normalize:

β(ts,tr)

Update representation:

h(ts) = Σ β(ts,tr) h(tr)

This allows the model to identify important historical states.

------------------------------------------------------------------------

## 3.6 Feature Fusion

Spatial and temporal features are fused using a gating mechanism.

H = r ⊙ H_prev + (1-r) ⊙ (gs Hs + gt Ht)

Where:

-   r: gating parameter
-   gs: spatial weight
-   gt: temporal weight

------------------------------------------------------------------------

## 3.7 Reliability Prediction

Finally, a prediction layer estimates network reliability.

Input:

H

Prediction:

R̂ = f(H)

Where:

R represents:

R = P(all demands satisfied)

Loss function:

L = \|\|R - R̂\|\|²

------------------------------------------------------------------------

## Overall Pipeline

Three-layer network → Random capacity generation → Edge-to-node
transformation → Edge time-series features → Spatial attention →
Temporal attention → Feature fusion → Reliability prediction


methodology_section

# 三层多商品随机容量网络可靠性
# Three-Layer Multicommodity Stochastic Capacity Networks

## 1.问题定义\| Problem Definition

构建一个三层有向网络。\|Construct a three-layer directed network.

- 每层包含若干节点（nodes数量每层随机）
- 每层内有随机连接边（intra_edges数量层内随机）
- 层内边的容量随机（capacity随机）
$$
  C_{i} = \left \{100,200,300,400,500\right \} \quad (i=0,1,2,3,4)
  \\ k=100,200,300,400,500
  \\C_{i} \quad  the \quad maximal \quad capacity\quad  of \quad route\quad and\quad it\quad is\quad integer
  \\L_{i}N_{j} \quad取整数，边最大容量
$$
- 层内边的随机容量对应的概率随机（Probability随机）
 $$
P(C_{i} = k) = p_{i}
$$
$$
\sum_{i}^{len(i)} P(C_{i})=1 
$$
- 层间有跨层连接边（1层可与2或3层连接、2层只能与3层连接，inter_edges数量层间随机）

- 层间节点假设绝对可靠（inter_edges层间Probability=1）
$$
P = 1
$$
- 总需求量（demands随机）



data_generator_update

# Edge-Based Spatio-Temporal Attention Model for Multilayer Network Reliability Estimation

------------------------------------------------------------------------

## Abstract

Network reliability evaluation in multilayer stochastic systems is
computationally expensive due to combinatorial explosion in capacity
states. This paper proposes an edge-based spatio-temporal attention
learning framework to approximate multilayer network reliability. Monte
Carlo simulation is first used to generate reliability labels. A
temporal-spatial attention model is then trained to learn the nonlinear
mapping from stochastic edge features to network reliability. The
proposed approach significantly reduces computational cost while
maintaining high estimation accuracy.

------------------------------------------------------------------------

## Keywords

Network Reliability; Monte Carlo Simulation; Spatio-Temporal Attention;
Edge-Based Learning; Deep Learning Approximation

------------------------------------------------------------------------

# 1. Introduction

Network reliability plays a critical role in transportation systems,
communication infrastructures, and energy grids. Traditional reliability
computation relies on exhaustive state enumeration or Monte Carlo
simulation, both of which suffer from high computational complexity.

To address this limitation, we reformulate reliability estimation as a
supervised learning problem and propose an edge-based spatio-temporal
attention model.

------------------------------------------------------------------------

# 2. Problem Formulation

We consider a multilayer directed network:

G = (V, E)

Nodes are assumed perfectly reliable.

Each edge e ∈ E has stochastic capacity:

P(C_e = k) = p_k

Given demand D, network reliability is defined as:

R = P(max_flow(G) ≥ D)

------------------------------------------------------------------------

# 3. Monte Carlo Data Generation

To estimate ground-truth reliability:

1.  Generate random multilayer topology
2.  Sample edge capacities
3.  Compute maximum flow
4.  Record success indicator

Reliability estimator:

R ≈ (1/N) Σ I(flow_i ≥ D)

Computational complexity:

O(N · \|E\| · FlowComplexity)

------------------------------------------------------------------------

# 4. Edge-Based Spatio-Temporal Attention Model

## 4.1 Input Representation

Edge feature tensor:

X ∈ R\^(T × E × F)

Where:

T = time steps\
E = number of edges\
F = feature dimension

------------------------------------------------------------------------

## 4.2 Temporal Attention

Temporal attention weights:

α_t = softmax(W_t X_t)

Temporal aggregation:

Z_e = Σ α_t X_t

------------------------------------------------------------------------

## 4.3 Spatial Attention

Spatial attention weights:

β_e = softmax(W_s Z_e)

Spatial aggregation:

Z = Σ β_e Z_e

------------------------------------------------------------------------

## 4.4 Reliability Prediction

Final prediction:

R_hat = sigmoid(WZ + b)

------------------------------------------------------------------------

# 5. Training Objective

Loss function:

L = (R_hat - R_true)\^2

Optimization performed using Adam optimizer.

------------------------------------------------------------------------

# 6. Computational Advantage

Once trained:

Inference complexity:

O(EF)

Compared to Monte Carlo:

O(N · \|E\| · FlowComplexity)

This provides significant computational acceleration.

------------------------------------------------------------------------

# 7. Experimental Design

Experiments include:

-   Reliability vs Demand curves
-   Monte Carlo vs Model prediction comparison
-   Error distribution analysis
-   Sensitivity to edge density

Evaluation metrics:

-   MSE
-   MAE
-   R²

------------------------------------------------------------------------

# 8. Discussion

The model approximates a high-dimensional stochastic flow probability
function. Attention mechanisms enable automatic identification of
critical edges and critical time steps.

The framework is extendable to:

-   Node reliability modeling
-   Dynamic capacity processes
-   Large-scale infrastructure networks

------------------------------------------------------------------------

# 9. Conclusion

This paper presents an edge-based spatio-temporal attention framework
for multilayer network reliability estimation. By integrating Monte
Carlo simulation with deep learning approximation, the approach
significantly reduces computational burden while preserving estimation
accuracy.

Future work includes theoretical error bounds and real-world
infrastructure validation.

------------------------------------------------------------------------

# Appendix: Reference Implementation (PyTorch)

``` python
import torch
import torch.nn as nn

class TemporalAttention(nn.Module):
    def __init__(self, feature_dim):
        super().__init__()
        self.attn = nn.Linear(feature_dim, 1)

    def forward(self, x):
        weights = torch.softmax(self.attn(x), dim=0)
        return torch.sum(weights * x, dim=0)

class SpatialAttention(nn.Module):
    def __init__(self, feature_dim):
        super().__init__()
        self.attn = nn.Linear(feature_dim, 1)

    def forward(self, x):
        weights = torch.softmax(self.attn(x), dim=0)
        return torch.sum(weights * x, dim=0)

class ReliabilityModel(nn.Module):
    def __init__(self, feature_dim):
        super().__init__()
        self.temporal_attn = TemporalAttention(feature_dim)
        self.spatial_attn = SpatialAttention(feature_dim)
        self.fc = nn.Linear(feature_dim, 1)
        self.sigmoid = nn.Sigmoid()

    def forward(self, x):
        x = self.temporal_attn(x)
        x = self.spatial_attn(x)
        return self.sigmoid(self.fc(x))
```


Full_IEEE_Style_Paper

# Methodology 方法论

## 1. Problem Formulation 问题建模

We consider a multilayer network G = (V,E).\
我们考虑一个多层网络 G = (V,E)。

Edge capacity:

P(C_e = k) = p_k

Reliability:

R = P(max_flow ≥ D)

------------------------------------------------------------------------

## 2. Monte Carlo Estimation 蒙特卡洛估计

R ≈ (1/N) Σ I(flow_i ≥ D)

------------------------------------------------------------------------

## 3. Attention-Based Learning 基于注意力的学习

The model learns mapping:

f: Edge Features → Reliability

模型学习从边特征到可靠性的映射。


04_Methodology_Bilingual

# 整体逻辑框架

# Overall Logical Framework

## 流程 \| Pipeline

随机网络生成\
Random network generation

→ Monte Carlo 计算真实可靠性\
→ Monte Carlo computes true reliability

→ 构建数据集\
→ Dataset construction

→ 训练时空注意力模型\
→ Train spatio-temporal attention model

------------------------------------------------------------------------

## 核心理解 \| Key Insight

可靠性本质是：\
Reliability fundamentally measures:

随机容量是否满足需求的概率\
Probability that stochastic capacities satisfy demand


Methodology

3.1 Three-Layer Capacitated Netw

三层多商品随机容量网络可靠性

Three-Layer Multicommodity Stoc

Edge-Based Spatio-Temporal Attention Model for M

Methodology 方法论

1. Problem Formulation 问题建模

整体逻辑框架

Overall Logical Framework

流程 | Pi