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

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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]

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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

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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

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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.

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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.

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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

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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

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Overall Pipeline

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