# Evaluation of Railway Systems: A Network Approach

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## Abstract

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## 1. Introduction

- How effective is the proposed ERC measure, compared to classical topological metrics, in identifying critical stations and journeys within a railway network?
- What are the most important station and railway line connections that contribute to the overall efficiency and resilience of the Lombardy railway network?

## 2. Literature Review

## 3. Theoretical Background

#### 3.1. Some Preliminaries on Graph Theory

#### 3.2. Robustness Indicators

#### 3.2.1. Edge-Based Effective Resistance Centrality

#### 3.2.2. Vertex-Based Effective Resistance Centrality

## 4. Methodology

- Compute the weighted adjacency matrix $\mathbf{W}$ associated with the whole network;
- Calculate the weighted Laplacian matrix ${\mathbf{L}}^{W}$ as indicated in formula (1);
- Compute the weighted Laplacian eigenvalues ${\mu}_{i}^{W}$ of the matrix ${\mathbf{L}}^{W}$;
- Calculate the weighted normalized Kirchhoff index;
- Calculate the weighted normalized Kirchhoff index when a vertex or an edge is deactivated;
- Compute the weighted vertex-based $ERC$, ${R}_{K}^{W}({v}_{i},G)$, or edge-based $ERC$, ${R}_{K}^{W}(({e}_{i,j}),G)$;
- Detect the most strategic stations or journeys by ranking ${R}_{K}^{W}({v}_{i},G)$ and ${R}_{K}^{W}(({e}_{i,j}),G)$.

## 5. Research Materials

#### 5.1. Search Strategy and Data Collection

- The main lines (6131 km): They have high traffic and good infrastructure quality; they include all of the main lines between the main cities of the country.
- Complementary lines, which have less traffic and are responsible for connecting medium or small regional centers. Most of these lines are single tracks and some are not electrified.
- Junction lines (936 km): Connecting complementary and essential lines near metropolitan areas.

#### 5.2. Characteristics of the Network

#### 5.3. Railway Services Classification

- International transport: This is carried out by Trenitalia in collaboration with foreign railways, as well as by other companies, such as SNCF Voyages Italia, with the aim of connecting large international cities. EuroCity (of the Thello company) departs from Milano Centrale station and heads to Nice and Marseille; Eurocity (of SBB/FFS) departs from Milano and Venezia and heads to Geneve, Basel, and Zurich.
- Long-distance transport: This is carried out by Trenitalia with Intercity, Eurocity, Frecciabianca, Frecciargento, Frecciarossa, and Italo, in order to connect large cities.
- Regional transport: This is provided by various companies; it is based on service contracts that are stipulated by the region, partially subsidizing the costs and determining the tariffs.
- Suburban service: This is a subset of regional transport; it intends to guarantee regular journeys between Milan, the centers of its metropolitan area, and other important nearby cities, such as Lecco, Lodi, Mariano Comense, Novara, Saronno, Treviglio, and Varese.

- Dimensions of the station: The set of areas and surfaces accessible and frequented by the traveler/user.
- Attendance: The number of travelers and simple frequenters who engage IN the railway system on a daily basis (from a predetermined station).
- Interchange capacity: The attitude of a railway plant to connect, interact, and operate in an integrated manner with other public transport systems.
- Level of the commercial offer: The quality of the passenger service offered by the system to customers in terms of railway traffic.

#### 5.4. Railway Vertices Classification

- PLATINUM: This category includes large railway installations characterized by very high attendance (>25,000 average visitors/day) and high-quality passenger services for long, medium, and short distances.
- GOLD: This category includes medium-/large-sized railway facilities characterized by high attendance (>10,000 average visitors/day approximately) and high-quality passenger services for long, medium, and short distances.
- SILVER: This category includes systems characterized by medium/small dimensions, often without a ’travelers building’. It is only equipped with regional/metropolitan services characterized by high and consistent attendance (in some cases, >4000 average visitors/day), or stations and stops characterized by consistent attendance (>2500 average visitors/day approximately).
- BRONZE: This category includes small stations characterized by low attendance (generally <500 average visitors/day), without a travelers’ building; it is equipped with services for regional/local traffic.

## 6. Numerical Results and Discussion

## 7. Conclusions

## 8. Further Developments and Limitations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Milan network (rail and underground). Source: https://giromilano.atm.it (accessed on 9 May 2023).

**Figure 5.**Correlation between rankings obtained by $ERC$, betweenness centrality, eigenvector centrality, and residual closeness.

**Table 1.**Classical network indicators (see, among others, [41]).

Density | 0.03 |

Assortativity | 0.005 |

Average degree | 2.5 |

Average clustering | 0.1 |

**Table 2.**Normalized weighted Kirchhoff index for the Lombardy railway network and average normalized weighted Kirchhoff index for simulated networks.

Network | ${\mathit{K}}_{\mathit{N}}^{\mathit{W}}(\mathit{G})$ |
---|---|

Lombardy Railways | 3.53 |

ER | 2.16 |

WS | 1.63 |

BA | 1.97 |

STATION | ${\mathit{R}}_{\mathit{K}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ | Deg | STATION | ${\mathit{R}}_{\mathit{K}}^{\mathit{W}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ | Strength |
---|---|---|---|---|---|

CODOGNO | 30.39% | 4 | PONTE SP | 22.2% | 158 |

PONTE SP | 24.75% | 3 | BERGAMO | 21.9% | 183 |

BERGAMO | 24.74% | 3 | CODOGNO | 20.4% | 172 |

BUSTO | 20.63% | 3 | MILANO ROGOREDO | 17.9% | 915 |

RHO FIERA | 12.73% | 3 | RHO FIERA | 16.4% | 648 |

BRESCIA | 12.64% | 5 | BUSTO | 15.8% | 401 |

ALBATE | 11.79% | 3 | MONZA | 14.6% | 761 |

MILANO ROGOREDO | 10.14% | 3 | BRESCIA | 11.0% | 295 |

CAVA CARBONARA | 9.76% | 3 | DESENZANO | 10.6% | 157 |

CARNATE | 7.29% | 4 | CARNATE | 9.7% | 266 |

Cut-Vertices | Cut-Vertices | ||
---|---|---|---|

Station | ${\mathit{R}}_{\mathit{K}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ | Station | ${\mathit{R}}_{\mathit{K}}^{\mathit{W}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ |

CREMONA | 26.38% | MILANO | 20.26% |

PAVIA | 16.5% | TREVIGLIO | 20.14% |

PIADENA | 16.30% | CREMONA | 19.73% |

MILANO | 8.14% | PIADENA | 14.47% |

COMO | 4.81% | PAVIA | 13.59% |

VARESE | 3.21% | PALAZZOLO | 2.88% |

MORTARA | 2.48% | SARONNO | 1.62% |

SARONNO | 1.95% | VARESE | 1.01% |

PALAZZOLO | 1.94% | COMO | 0.80% |

TREVIGLIO | 1.76% | VOGHERA | 0.79% |

JOURNEY | ${\mathit{R}}_{\mathit{K}}(({\mathit{e}}_{\mathit{i},\mathit{j}}),\mathit{G})$ | JOURNEY | ${\mathit{R}}_{\mathit{K}}^{\mathit{W}}(({\mathit{e}}_{\mathit{i},\mathit{j}}),\mathit{G})$ | Numtrain |
---|---|---|---|---|

CODOGNO–CREMONA | 26.6% | CASSANO-TREVIGLIO | 34.9% | 100 |

BERGAMO–PONTE SP | 24.2% | TREVIGLIO-TREVIGLIO OVEST | 34.3% | 56 |

BUSTO–GALLARATE | 17.9% | BERGAMO–PONTE SP | 21.5% | 88 |

MANTOVA–PIADENA | 17.2% | MANTOVA–PIADENA | 18.1% | 28 |

ALBATE-COMO | 10.7% | CODOGNO–CREMONA | 16.1% | 35 |

CAVA CARBONARA-PAVIA | 10.2% | MILANO-MILANO ROGOREDO | 15.8% | 584 |

MILANO-MILANO ROGOREDO | 9.2% | BUSTO–GALLARATE | 14.6% | 203 |

NOVARA–RHO FIERA | 8.2% | BRESCIA–DESENZANO | 14.1% | 78 |

GALLARATE-VARESE | 8.1% | NOVARA–RHO FIERA | 10.8% | 120 |

MORTARA–NOVARA | 7.3% | DESENZANO-VERONA | 10.5% | 79 |

**Table 6.**The top 15 stations in terms of ${R}_{K}^{W}({v}_{i},G)$. Additionally, the ranking provided by betweenness centrality, eigenvector centrality, and residual closeness is provided together with the RFI and FN classifications.

Station | Rank ${\mathit{R}}_{\mathit{K}}^{\mathit{W}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ | Rank Betweenness | Rank Eigenvector Centrality | Rank Residual Closeness | Classification |
---|---|---|---|---|---|

PONTE SP | 1 | 9 | 27 | 16 | Silver |

BERGAMO | 2 | 8 | 39 | 8 | Platinum |

CODOGNO | 3 | 2 | 20 | 11 | Gold |

MILANO | 4 | 24 | 1 | 71 | Platinum |

TREVIGLIO | 5 | 31 | 51 | 10 | Gold |

CREMONA | 6 | 4 | 38 | 6 | Gold |

MILANO ROGOREDO | 7 | 53 | 2 | 62 | Gold |

RHO FIERA | 8 | 16 | 4 | 49 | Gold |

BUSTO | 9 | 19 | 10 | 55 | Gold |

MONZA | 10 | 36 | 3 | 52 | Gold |

PIADENA | 11 | 1 | 53 | 1 | Silver |

PAVIA | 12 | 3 | 6 | 15 | Gold |

BRESCIA | 13 | 5 | 61 | 3 | Platinum |

DESENZANO | 14 | 53 | 65 | 37 | Gold |

CARNATE | 15 | 49 | 11 | 17 | Silver |

RFI Classification | N${}^{\circ}$ Stations | Av. ${\mathit{R}}_{\mathit{K}}^{\mathit{W}}({\mathit{v}}_{\mathit{i}},\mathit{G})$ | Av. Betweenness | Av. Eigenvector Centrality | Av. Strength | Av. Residual Closeness |
---|---|---|---|---|---|---|

Platinum | 5 | 11.04% | 399.20 | 0.20 | 470.20 | 53.67 |

Gold | 26 | 4.83% | 306.50 | 0.09 | 256.62 | 52.12 |

Silver | 37 | 0.31% | 178.41 | 0.02 | 107.08 | 50.69 |

Bronze | 13 | −0.37% | 224 | 0.00 | 76.385 | 55.83 |

Total | 81 | 2.31% | 240.47 | 0.050 | 172.57 | 52.15 |

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**MDPI and ACS Style**

Cornaro, A.; Grechi, D.
Evaluation of Railway Systems: A Network Approach. *Sustainability* **2023**, *15*, 8056.
https://doi.org/10.3390/su15108056

**AMA Style**

Cornaro A, Grechi D.
Evaluation of Railway Systems: A Network Approach. *Sustainability*. 2023; 15(10):8056.
https://doi.org/10.3390/su15108056

**Chicago/Turabian Style**

Cornaro, Alessandra, and Daniele Grechi.
2023. "Evaluation of Railway Systems: A Network Approach" *Sustainability* 15, no. 10: 8056.
https://doi.org/10.3390/su15108056