An analysis of the vulnerability of road networks in response to disruption events through accessibility indicators specification

Figures & data

Figure 1. Elements of vulnerability. Source: Husdal (2004).

Table 1. List of some vulnerability/criticality metrics.

Metric	Measure	Description	Reference
Accessibility index	$A_i=w_i^p\sum _{j=1}^{n-1}{w}_j^r{\left({\displaystyle \frac{f(t_{\mathrm{ij}})}{f^0(t_{\mathrm{ij}})}}\right)}^{-\alpha }$	$A_i:$accessibility index of the zone $i$; $w_i^p$: population weight of zone $i$which equals to the ratio of population in zone $i$ overall the population in the study area; $w_j^r$: weight of residence in zone $j$which equals to the ratio of residence in zone $j$overall the residence in the study area except for residence in zone $i$; $f(t_{\mathrm{ij}}):$ travel cost between region $i$ and $j$ after network degradation; $f^0(t_{\mathrm{ij}})$: travel cost between region $i$ and $j$ before network degradation; $\alpha$: travel cost decay parameter	Lu and Peng (2011)
Accessibility Index	$A{I}^w={\displaystyle \frac{q^w}{{\sum }_{k\in K^w}{f}_k^w{c}_k^w}}$	$A{I}^w$: accessibility index for each OD movement $w$; $q^w$: travel demand of OD movement $w$; $f_k^w$: flow on the path of OD movement $w$; $c_k^w$: mean travel cost of the path connecting OD movement $w$.	Luathep et al. (2011)
Regional accessibility	$A_s={\displaystyle \frac{{\sum }_i{\sum }_j{\displaystyle \frac{P_j}{T_{\mathrm{ij}}}{P}_i}}{{\sum }_i{P}_i}}$	$A_s$: regional accessibility for the $s$ scenario; $P_j$: resident population for each destination $j$; $T_{\mathrm{ij}}$: generalized travel time between origin $i$ to destination $j$; $P_i$: resident population for each origin $i$.	Martin et al. (2021)
Population weighted Hansen index	$A{1}_i={\displaystyle \frac{P_i}{P_{\mathrm{tot}}}{\displaystyle \frac{{\sum }_j{B}_{j}f(C_{\mathrm{ij}})}{{\sum }_j{B}_j}}}$	$A{1}_i$: Population weighted Hansen index; $P_i$: population of node $i$; $P_{\mathrm{tot}}$: total population; $B_j$: total number of opportunities in each traffic zone; $f(C_{\mathrm{ij}})$: impedance function.	Papilloud and Keiler (2021)
Network robustness index	$q_a=\sum _a{t}_a{x}_a\delta {}_a-\sum _a{t}_a{x}_a$	$q_a:\mathrm{value}\mathrm{of}\mathrm{NRI}\mathrm{for}\mathrm{a}\mathrm{link}a$ $x_a$: travel time on each link $a$ comprising the network; $t_a$: traffic flow on each link $a$ comprising the network; $\delta {}_a$ is equal to 1 if a link $a$ is not removed, 0 otherwise	Sullivan et al. (2010)
Network Vulnerability Index	$\mathrm{NVI}=\sum _{i=1}^A\left[\left({\displaystyle \frac{x_i^{\mathrm{before}}}{r_i^{\mathrm{before}}}{t}_i^{\mathrm{before}}}\right)\right]-\sum _{i=1}^A\left[\left({\displaystyle \frac{x_i^{\mathrm{after}}}{r_i^{\mathrm{after}}}{t}_i^{\mathrm{after}}}\right)\right]$	$\mathrm{NVI}$: network vulnerability index; $r_i$: total available capacity of link $i$/standard hourly link capacity per lane for the given type of road; $A$: number of links in a network; $x_i$: total flow on a link $i$; $t_i$: travel time on a link $i$.	Balijepalli and Oppong (2014)
Betweness centrality	$\mathrm{BC}(u)=\sum _{i\ne u\ne j\in G}{\displaystyle \frac{{\sigma }_{\mathrm{ij}}(u)}{{\sigma }_{\mathrm{ij}}}}$	$\mathrm{BC}(u)$: betweeness centrality of $u$ node; ${\sigma }_{\mathrm{ij}}(u)$: total number of shortest paths between nodes $i$ and $j$that pass through $u$; ${\sigma }_{\mathrm{ij}}$: total number of shortest paths between nodes $i$ and $j$.
Network efficiency	$\epsilon (G,d)={\displaystyle \frac{{\sum }_{w\in W}{\displaystyle \frac{d_w}{{\lambda }_w}}}{n_w}}$	$\epsilon (G,d)$: network efficiency for a given network topology $G$ and travel demand $d$; $d_w$: travel demand between O-D pair $w$; ${\lambda }_w$: travel cost between O-D pair $w$; $n_w$: number of O-D pairs.	Nagurney and Qiang (2007)
Closeness index	$\mathrm{VI}={\displaystyle \frac{1}{|N|(|N|-1)}{\displaystyle \frac{{\sum }_{i\ne j\in N}(d_{\mathrm{ij}}^{\mathrm{\prime }}-d_{\mathrm{ij}})}{{\sum }_{i\ne j\in N}{d}_{\mathrm{ij}}}}}$	$\mathrm{VI}$: Closeness index; $N$: number of nodes of a network; $d_{\mathrm{ij}}$: shortest path between $i$ and$j$ before disruption; $d_{\mathrm{ij}}^{\mathrm{\prime }}$: shortest path between $i$ and $j$ after disruption.	Gu et al. (2020)
Efficiency of alternative route	$\eta (o,d)={\displaystyle \frac{d_0(o,d)}{d_1(o,d)}}$	η ($o,d)$: efficiency of alternative route $d_o(o,d)$: length of the shortest route between nodes before the disruption $d_1(o,d)$: length of the shortest route between nodes after the disruption.	Morelli and Cunha (2021)
Traffic volume centrality	$\mathrm{UE}-\mathrm{Traffic}-\mathrm{Volume}-C={\displaystyle \frac{x_g}{{\sum }_{i\ne j\in V_d}{D}_{\mathrm{ij}}}}$	$x_g$: equilibrium traffic volume on link $g$; $D_{\mathrm{ij}}$: demand corresponding to OD pair $\mathrm{ij}$	Mylonas, Mitsakis, and Kepaptsoglou (2023)
Betweness-accessibility	$C_{\mathrm{pop}}(l_i)=\sum _{j,kj\ne k}{\displaystyle \frac{\sigma (p_j,p_k|l_i)}{\sigma (p_j,p_k)}\mathrm{po}{p}_j{\displaystyle \frac{\mathrm{emp}{l}_{k}f(t{t}_{\mathrm{jk}})}{\mathrm{Ac}{c}_j^{\mathrm{empl}}}}}$	$\sigma (p_j,p_k|l_i)$: shortest path between $p_j$ and $p_k$ that pass through link $l_i$; $\sigma (p_j,p_k)$: total number of shortest paths between $p_j$ and $p_k$; $\mathrm{po}{p}_j$: population of node $j$; $\mathrm{emp}{l}_k$: employments of node $k$; $f(t{t}_{\mathrm{jk}})$: cost function of trip production; $\mathrm{Ac}{c}_j^{\mathrm{empl}}$: accessibility of node $j$.	Sarlas, Páez, and Axhausen (2020)
Importance of a link	$\mathrm{Importanc}{e}_k(k)={\displaystyle \frac{{\sum }_i{\sum }_{i\ne j}{w}_{\mathrm{ij}}(c_{\mathrm{ij}}^k-c_{\mathrm{ij}}^0)}{{\sum }_i{\sum }_{i\ne j}{w}_{\mathrm{ij}}}}$	$\mathrm{Importanc}{e}_k(k)$: importance of a link $k$; $w_{\mathrm{ij}}$: weight assigned to each OD pair in a network; ${\mathrm{c}}_{\mathrm{ij}}^k$: travel cost of the damaged network; $c_{\mathrm{ij}}^1$: travel cost af the undamaged network.	Jenelius, Petersen, and Mattsson (2006)
Vulnerability Index	$I_{V_j}=\sum _{i=1}^n{w}_i{P}_i$	$I_{V_j}$: vulnerability index; $P_i$: property of the i-th road; $w_i$: weight of the i-th properties; $n$: number of properties.	Francini et al. (2020)
Link Importance Index	$L{I}_j=\beta \cdot F(\mathrm{AD}{T}_j)+(1-\beta )\cdot G(\mathrm{\Delta }{C}_j)$	$L{I}_j$: Link Importance Index of the link $j$; $F(\mathrm{AD}{T}_j)$: function of the average daily volume traffic along link $j$; $G(\mathrm{\Delta }{C}_j)$: function of increase in the network users’ total cost due to the interruption of link $j$; $\beta$: sensitivity parameter.	Rupi et al. (2015)
Weighted network travel time increase	$\mathrm{\Delta }t(e)=\sum _{\mathrm{ij}}{w}_{\mathrm{ij}}\mathrm{\Delta }{t}_{\mathrm{ij}}^e$	$\mathrm{\Delta }t(e)$: weighted network travel time increase; $w_{\mathrm{ij}}$: normalized weight for travel between $i$ and $j$; $\mathrm{\Delta }{t}_{\mathrm{ij}}^e$: travel time increase between i and j when link e is disrupted.	Taylor, Sekhar, and D'Este (2006)

Table 2. The statistical tests for the gravity model.

Attributes			Passenger cars
${\mathit{\beta }}_1$	Robust Std err	Robust t-test	1.17	+2.07
${\mathit{\beta }}_2$	Robust Std err	Robust t-test	−1.14	−7.800

Parameters	Passenger cars
Adjusted rho-square	0.659
Final log-likelihood	−34.142
Null log-likelihood	−106.52

Figure 2. Active population distribution of the metropolitan area of the city of Naples. Source: Author’s elaboration.

Figure 3. Jobs distribution of the metropolitan area of the city of Naples. Source: Author’s elaboration.

Figure 4. Main links in the study area. Source: Author’s elaborations.

Figure 5. Districts of the metropolitan area of the city of Naples. Source: Author’s elaboration.

Figure 6. Distribution of the purpose of movement during the morning on a typical weekday. Source: Author’s elaboration based on Urban Plan for Sustainable Mobility.

Figure 7. Modal distribution for internal movements within the metropolitan area of the city of Naples during the morning peak hours in a typical weekday. Source: Author’s elaboration based on Urban Plan for Sustainable Mobility.

Figure 8. Links investigated. Source: Author’s elaboration.

Table 3. Main characteristic of the case study.

Region	Municipalities	Population	Active population	Jobs	Road length (km)	Road types considered for the elimination of sections and length (km)
Metropolitan area of city of Naples	112	3,055,196	2,008,788	783,490	20,623.92	Motorway (20.2) Trunk road (103.1) Primary link (1.5)

Figure 9. Damaged districts by removing the road ‘Tangenziale di Napoli’ (scenario 1) from the network. Source: Author’s elaboration.

Figure 10. Damaged districts by removing the ‘Galleria della Vittoria and Via Nuova Marina’ (scenario 2) from the network. Source: Author’s elaboration.

Figure 11. Damaged districts by removing the road ‘Strada Statale 7 quater Via Domitiana’ (scenario 3) from the network. Source: Author’s elaboration.

Figure 12. Damaged districts by removing the road ‘Strada Statale 87 Sannitica’ (scenario 4) from the network. Source: Author’s elaboration.

Figure 13. Damaged districts by removing the road ‘Strada Statale 145 Sorrentina’ (scenario 5) from the network. Source: Author’s elaboration.

Figure 14. Damaged districts by removing the road ‘Strada Statale 162 NC Asse Mediano’ (scenario 6) from the network. Source: Author’s elaboration.

Figure 15. Damaged districts by removing the road ‘Strada Statale 268 del Vesuvio’ (scenario 7) from the network. Source: Author’s elaboration.

Table 4. Results.

Scenario	Links removed from the network	Average Accessibility variation for the metropolitan area	Average Accessibility variation for city of Naples	Number of districts with accessibility variation greater than 10%
1	Tangenziale di Napoli (A56)	−16.83%	−43.08%	39
2	Galleria della Vittoria and Via Nuova Marina	−2.22%	−7.69%	5
3	Strada Statale 7 quater Domitiana	−2.55%	−1.21%	4
4	Strada Statale 87 Sannitica	−1.96%	−1.06%	3
5	Strada Statale 145 Sorrentina	−9.27%	−0.89%	14
6	Strada Statale 162 NC Asse Mediano	−8.44%	−10.17%	24
7	Strada Statale 268 del Vesuvio	−8.33%	−3.56%	18

Source: Author’s elaborations.

Table 5. Link Importance Index.

Scenario	Road	Link Importance Index	Ranking based on LII
1	Tangenziale di Napoli	0.275	1
2	Galleria della Vittoria-Via Nuova Marina	0.061	6
3	Strada Statale 7 quater Domitiana	0.013	7
4	Strada Statale 87 Sannitica	0.166	3
5	Strada Statale 145 Sorrentina	0.147	4
6	Strada Statale 162 NC Asse Mediano	0.086	5
7	Strada Statale 268 del Vesuvio	0.267	2

Source: Author’s elaborations.

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