# Transportation Geography and Network Science/Robustness

For road networks, the measure of robustness is used to evaluate its ability to cope with disturbances. Unlike reliability, robustness is the property for the network itself rather than for the users. But a road network with higher robustness could provide users a higher reliability [1].

Many measures of the network affect the network robustness, such as redundancy [2]. Redundancy could be defined as the existence of alternatives for certain travel demand, such as alternative traffic modes or alternative paths. When disruptions occur, an efficient corresponding alternative could guarantee an acceptable condition of the network. The effect of interdependency on road network robustness could be shown by the cascading disruptions that may occur over a large part of the road network because of the failure of a critical node or link. Hence, maintaining a reasonable hierarchy to minimize the interdependency is important to improve the robustness for a road network.

Nagurney and Qiang used unified network performance to measure the robustness of road network [3]. And for a given road network G and a vector of the equilibrium demand, the network performance measure could be expressed as [4],

${\displaystyle \epsilon =\epsilon (G,d)={\frac {\sum _{w\in W}{\frac {d_{w}}{\lambda _{w}}}}{n_{W}}}}$



Where ${\displaystyle d_{w}}$ is the demand in the equilibrium condition, while ${\displaystyle n_{w}}$ is the total number of OD pairs in the network. ${\displaystyle \lambda _{w}}$is defined as the equilibrium travel disutility, and it equals to the minimal cost of the path if the equilibrium traffic flow is not 0 [5].

The cost of a path is the function of its traffic flow. Considering the road connectivity, the cost function of path P could be expressed as [6],

${\displaystyle C_{p}=\sum _{a\in L}c_{a}\delta _{ap}}$


Where ${\displaystyle c_{a}}$ is the cost of link a, while ${\displaystyle \delta _{ap}}$is a binary function and it equals to 1 is link a is contained in the path. The importance of a node or a link could be evaluated by the following expression, G-g stands for the new network that ${\displaystyle g}$ has been removed.

${\displaystyle I(g)={\frac {\delta _{\epsilon }}{\epsilon }}={\frac {\epsilon (G,d)-\epsilon (G-g,d)}{\epsilon (G,d)}}}$


Considering the capacity degradation rather than the connections only, the capacity could be reflected in the cost function, which is shown as ${\displaystyle \mu _{a}}$ in the following equation:

${\displaystyle c_{a}f(a)=t_{a}^{0}[1+k({\frac {f_{a}}{\mu _{a}}})^{\beta }],\forall a\in L}$

And the network robustness measure is

${\displaystyle R^{\gamma }=R(G,c,\gamma ,\mu )={\frac {\epsilon ^{\gamma }}{\epsilon }}*100}$


Where ${\displaystyle \gamma }$ is the degradation ratio of the capacity.

## References

1. Immers, Ben, et al. “Robustness And Resilience Of Road Network Structures.”NECTAR Cluster Meeting on Reliability of Networks. 2004.
2. Immers, Ben, et al. “Robustness And Resilience Of Road Network Structures.”NECTAR Cluster Meeting on Reliability of Networks. 2004.
3. Nagurney, Anna, and Qiang Qiang. “Robustness of transportation networks subject to degradable links.” EPL (Europhysics Letters) 80.6 (2007): 68001.
4. Nagurney, Anna, and Qiang Qiang. “Robustness of transportation networks subject to degradable links.” EPL (Europhysics Letters) 80.6 (2007): 68001.
5. Nagurney, Anna, and Qiang Qiang. “Robustness of transportation networks subject to degradable links.” EPL (Europhysics Letters) 80.6 (2007): 68001.
6. Nagurney, Anna, and Qiang Qiang. “Robustness of transportation networks subject to degradable links.” EPL (Europhysics Letters) 80.6 (2007): 68001.