# Local impacts on road networks and access to critical locations during
extreme floods

Simone Loreti  University of Bern, Institute of Geography, Bern, 3012,
Switzerland  University of Bern, Oeschger Centre for Climate Change Research,
Mobiliar Lab for Natural Risks, Bern, 3012, Switzerland  To whom
correspondence should be addressed. E-mail<EMAIL_ADDRESS>or
<EMAIL_ADDRESS>Enrico Ser-Giacomi  Massachusetts Institute of
Technology, Department of Earth, Atmospheric and Planetary Sciences, Cambridge
MA, 02139, United States  Andreas Zischg  University of Bern, Institute of
Geography, Bern, 3012, Switzerland  University of Bern, Oeschger Centre for
Climate Change Research, Mobiliar Lab for Natural Risks, Bern, 3012,
Switzerland  Margreth Keiler  University of Bern, Institute of Geography,
Bern, 3012, Switzerland  University of Bern, Oeschger Centre for Climate
Change Research, Mobiliar Lab for Natural Risks, Bern, 3012, Switzerland
University of Innsbruck, Department of Geography, Innsbruck, 6020, Austria
Austrian Academy of Sciences, Institute of Interdisciplinary Mountain
Research, Innsbruck, 6020, Austria  Marc Barthelemy  Institut de Physique
Théorique, CEA, CNRS-URA 2306, Gif-surYvette, F-91191, France  Centre
d’Analyse et de Mathématique Sociales (CNRS/EHESS), Paris, 75006, France  To
whom correspondence should be addressed. E-mail<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract

Floods affected more than 2 billion people worldwide from 1998 to 2017 and
their occurrence is expected to increase due to climate warming, population
growth and rapid urbanization. Recent approaches for understanding the
resilience of transportation networks when facing floods mostly use the
framework of percolation but we show here on a realistic high-resolution flood
simulation that it is inadequate. Indeed, the giant connected component is not
relevant and instead, we propose to partition the road network in terms of
accessibility of local towns and define new measures that characterize the
impact of the flooding event. Our analysis allows to identify cities that will
be pivotal during the flooding by providing to a large number of individuals
critical services such as hospitalization services, food supply, etc. This
approach is particularly relevant for practical risk management and will help
decision makers for allocating resources in space and time.

Date: 31 January 2022.
Disclaimer: This research article has been published the 28th of January 2022
on Scientific Reports.
DOI: https://doi.org/10.1038/s41598-022-04927-3.
Cite this article: Download citation from the Scientific Reports citation
tool.

## Introduction

Floods are the most frequent and life-threatening natural hazards related
disasters (1, 2, 3, 4) and are expected to increase in occurences and damages
due to a large variety of factors such as climate warming with intensification
of precipitation extremes, population growth, and rapid urbanization (5, 6, 7,
8, 9, 10, 11, 12). As other natural disasters, floods reduce accessibility and
serviceability of the road transportation network, which is one of our most
valuable infrastructure assests (13). These disruptive effects on the road
system are intimately related to the vulnerability and resilience of the
transportation network and are traditionally analysed with a number of
methodologies, such as systems dynamics models, stochastic and optimization
processes (14), network science (14, 15, 16, 17), demand and supply models
(17, 16), and approaches based on traffic data (15).

Network science is a multidisciplinary field (18, 19, 20, 21) which offers
robust and elegant tools and measures to investigate the topological features
and properties of a network (22, 23). In network science, a well-established
framework for assessing the robustness of any kind of disrupted network is
percolation theory (24, 25, 20), which was recently applied to road networks,
to investigate the disruptive effects of  earthquakes (26, 27, 28, 29) and
traffic congestion (30, 31, 32, 33), as well as of flooding events (34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44). Abdulla et al. (34) modelled the flood
propagation over one of the neighborhoods road network of Houston, U.S, as a
percolation mechanism. In the proposed process, they randomly removed
(flooded) the network’s nodes based on their elevation and their vicinity to
already flooded nodes (using Baye’s rule). Despite they mentioned to simulate
a percolation process, which would traditionally include the computation of
the giant connected component, they actually calculated the Latora &
Marchiori’s global efficiency (45) as a measure of network connectivity. Both
Abdulla & Birgisson (35) and Abdulla et al. (36) investigated the potential
failure of the Houston road network undergoing fluvial floods, by employing
different removal sequences of nodes in a flood-related percolation process.
For the entire set of nodes they first calculated four centrality measures,
i.e. degree, betweenness, closeness and eigenvector centrality, which then
served to rank their nodes. Following the nodes ranking from the highest to
the lowest rank, the authors performed the nodes’ removal sequence for each
centrality measure and measured the network connectivity through the giant
component size. Eventually, they found that the greatest damage to the network
was caused by the removal sequence associated to the betweenness centrality
measure, in agreement with results about the resilience of air-transportation
weighted networks (see for example Dall’Asta et al (46)). Abdulla et al. (36)
also presented a flood diffusion model and investigated the effects of the
initial conditions of the flood diffusion on the road network connectivity. To
mimic different initial conditions of the flood, the authors performed the
flood diffusion from different sets of nodes, correspondent to those nodes
which ranked the highest values of centrality measures, i.e. degree,
betweenness, closeness and eigenvector centrality. By measuring the size of
the giant component for each fraction of removed (flooded) nodes, they found
that the road networks connectivity was particularly vulnerable to the flood
propagation which started from nodes with high values of betweenness. However,
their framework did not include any hydrological or hydraulic model, nor was
validated. Abdulla & Birgisson (37) examined the impact of random and targeted
disruptions on the Houston road network. To represent the perturbative action
of floods on the road network they used the same mechanism of targeted
disruption as in Abdulla et al. (34). Through the calculation of the giant
component size, they compared the disruptive effects of targeted and random
failure on the road network. For low percentage of nodes removal (i.e. between
15% and 30%), they found that the network was more robust against targeted
failure than towards random one, while for a higher percentage of nodes
removal (i.e. between 30% and 70%), the network was instead more robust
against random failure than targeted disruptions. The also found similar
connectivity profiles (giant components size) by varying the road network
size. They concluded by emphasizing the suitability of the giant component in
assessing the robustness of road networks. Fan et al. (38) presented a
percolation-based diffusion model of flood propagation and recession, similar
to the susceptible-exposed-infected-recovered (SEIR) disease transmission
model. However, their model was not built up directly over the road network,
but over an overlapping network grid, composed of squared cells and each of
them containing a different number of road segments. Therefore, the flood
propagation occurred among grid cells, with a probability of flooding
proportional to the number of neighbouring flooded cells and to a constant
transmission rate. With a similar idea, they also simulated the recession
phase of the flood. Despite the mention to a percolation-based process, the
authors did not calculated the giant connected component or the equivalent
size of outbreak (20), but they characterised the entire process of flood
diffusion mainly by calculating the fraction of flooded cells. In addition,
all the model parameters, i.e. the transmission rate, the recovery rate, and
the exposed rate were estimated through a curve fitting with real data.
Farahmand et al. (39) proposed a probabilistic approach to the failure of road
networks (inversely proportional to the distance between roads and channels
network, and directly proportional to a channel’s vulnerability function),
which was assessed through the calculation of the giant component size of the
network. During the process of nodes removal, they found dramatic decreases
(leaps) in the connectivity profile represented by the giant component size,
suggesting the presence of critical roads. Dong et al. (28) proposed a
probabilistic links removal to mimic earthquakes-induced failures and
addressed the corresponding effects on the Portland (USA) road network through
a non-conventional percolation process. Instead of using the traditional giant
connected component, they introduced robust components defined as the union of
connected components in a network, containing at least one critical facility
(as a hospital), showing a two-phase transition. In their perspective, the
giant connected component doesn’t fairly represent the functionality of a
network undergoing such a disaster, claiming that measures of network’s
robustness should consider critical infrastructures. In another work,  Dong et
al. (40) employed the same idea of robust component for assessing the
robustness of a road network affected by a flood, resulting in an almost
gradual transition. They simulated the fluvial flood in the entire county by
assigning a probability of failure to each link, which was proportional to the
distance between the link and the floodway (but not considering any flood
depth). For each value of link removal probability, they calculated the robust
component and observed a complete shutter of the robust component at a
threshold of $p_{c}=0.8$. They also reported a sudden drop of the robust
component when $2\%$ of the roads were inundated ($p=0.02$), resulting in a
loss of accessibility to hospitals for a $22\%$ of the network. Even if the
papers related to the robust components might apparently represent a step
towards non-percolation approaches, these authors are still removing gradually
an increasing fraction of nodes from zero to one and monitor a cluster size
for each fraction of removed nodes. Instead of monitoring the size of the
largest connected cluster as in the traditional percolation, they monitor the
size of the disjoint union of all clusters containing at least a hospital and
consider it as if it were a single large component. The hospitals serve
uniquely in this work as markers for constructing the clusters that will
constitute the robust component.  Dong et al. (41) proposed to apply a
standard percolation process, i.e. with a random removal of the network nodes,
for assessing the robustness of twelve U.S. cities and three states’s road
networks, when subject to disaster-induced failures. For all the cities and
states networks, they showed a discrepancy between the results generated with
a simulation-based approach and an analytical one, i.e. the generating
function formalism (47, 48), due to the spatially embedded nature of the road
networks (indeed, by decreasing the node degree assortativity, the discrepancy
diminished). Therefore, they claimed about the inadequacy of the generating
functions formalism for assessing the robustness of road networks. However,
the Dong et al. (41) study and conclusions were limited to the random failure
mechanism of the network, which is far from real disruptive processes observed
in some natural disasters, as floods. Inspired by percolation theory, Ganin et
al. (42) analysed the increase of traffic delay in six U.S. cities’s road
networks with the increased severity of the road links disruptions, by
employing a mechanism of random links removal. A low, medium and high
percentage of disrupted road links represented, respectively, accidents (low),
power failures or sever flooding events (medium), and snow, ice, or dust
storms (high). The authors, therefore, used the traffic delay for assessing
the towns resilience, as alternative quantity to the traditional giant
component size. Wang et al. (43) compared three types of disruptions to both
USA and China’s road networks: random, flood-induced and localized failures.
They presented the flood-induced failure as a new typology of network
disruption, with intermediary features between a random and a localized
disruption, and resulting in an abrupt first-order phase transition.  Yadav et
al. (44) studied different failure scenarios on the London Rail Networks (LRN)
system, both as a supra-single network and a multi-layer interdependent
network (49, 50). The failure mechanisms included a random failure of the
overall network, a local random failure, a targeted failure based on
centrality measures and one scenario representing a flood event. The flood-
like failure consisted in dividing the nodes into three groups based on their
proximity to the river Thames and, within each group, randomly removing them
from the network.

In most of these previous studies related to the application of percolation-
based flood modeling on the road system (34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44), the network’s functionality was expressed through the size of the
largest connected component $P_{\infty}$ and its evolution when roads are
removed.  A summary of those studies and their key features related to the
percolation-based process is provided in LABEL:tab:table_literature_review.
However, $P_{\infty}$ is an aggregated quantity that does not capture the
entire network’s information and more importantly, does not reflect the local
reality. Therefore, more precise information is necessary, in particular at a
local scale, for a realistic evaluation of the disruption’s effects. Here, we
present a realistic and extreme flood scenario (51) based on physically
plausible rainfall scenarios and a high spatio-temporal resolution, and show
that it cannot be described adequately by a percolation transition and the
behavior of the giant component. Indeed, the giant component size $P_{\infty}$
never reaches zero during the flooding process and the percolation threshold
is never reached. This demonstrates the need for alternative useful measures
of the impact of flooding events on the road network. Here we propose such
measures and partition the network into _towns_ , through a Voronoi
tessellation, which allows us to extend the concept of _network’s
functionality_ to the entire network and at a local level. This approach led
us to define time-dependent measures alternative to percolation, both at a
local and at a global scale, which provide a realistic assessment of a flood-
induced disruptive event on a transportation network. In order to demonstrate
the relevance of these new measures, we use them on a realistic flood
simulation and also compare the results to a null random model.

## Results

### Flooding is not a percolation transition

In order to test our approach and our new metrics, we will consider a portion
of the Swiss road network built up from the _swissTLM3D_ dataset (52) and
shown in Figure 1(a). This area comprises the major towns of Bern, Thun,
Interlaken and Brienz and two Swiss lakes, the Thun Lake and the Brienz Lake.
This portion of road network was selected in such a way to include the largest
extension of the disrupted zone, represented by the blue flooded roads, and a
surrounding area (limited by the green perimeter) where the effects of the
disruptions were expected to occur as well. We use a realistic flood model
(51) based on physically plausible rainfall scenarios, with a high spatio-
temporal resolution (a spatial resolution of $50$ m and temporal resolution of
a hour) and we model the impact of the rainfall in a deterministic way, based
on physical laws and ground morphology and properties. In addition, this model
takes into account both flood protection’s works and the elevation difference
between the river’s water surface and the bridge over it. In contrast to other
simplified models such as in  (34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44) ,
the model here takes into account all the relevant field features  (additional
information in the Section 0.4 $-$ Flood failure). The flooded roads (in blue
in Figure 1(a)) result from these simulations (51), while both the green and
red external borders were created from an alpha-shape perimeter of the flooded
roads (with shrinking factor $0.5$) by broadening the perimeter outwards over
a distance of $5000$ m and $1000$ m, respectively (see Figure 1(a)). We
compare our results to a random null model where we remove the roads at
random, i.e. by randomly reshuffling the water levels simulated by Zischg et
al. (51) over the entire set of roads. In this way, the null model has the
same number of flooded roads per hour as in the real-like floods, allowing the
comparison between the (time-dependent) random-like flood and the real-like
one. In the following, we will coin the simulation result as a _real-like
flood_ and the null model as a _random flood_.

For these flooding processes, we first monitor the quantity $P_{\infty}$ (see
Figure 1(b)), defined as the fraction of nodes that belongs to the giant
connected component. The simulations start at time $t_{0}=0$ hours where there
is no flood, the network is intact and $P_{\infty}$ is maximum and equal to
$1$. During the flooding, an increasing number of roads is covered by water
until the maximum flood extension (MFE) is reached at $t_{MFE}=75$ hours.
After this peak, the flood withdraws and the flooded roads eventually dry up,
becoming accessible again. We monitor this recovery phase until
$t=166\;h\approx$ 1 week. Note that not all flooded roads were removed from
the network, but only those ones covered by more than $0.3$ meters, which
corresponds to the height at which vehicles start to float (53). The results
presented in Figure 1(b) first show that the magnitude of the $P_{\infty}$’s
variations is affected by the choice of the border delimiting the flooded
area, which points to the problem of defining the area of study. Nevertheless,
the $P_{\infty}$’s minimum is reached at the same time $t_{MFE}$ for all
choices of borders and for both flood scenarios. A recovery phase then
follows, characterised by a slow growth of $P_{\infty}$, which yet doesn’t
reach the initial stage of $P_{\infty}=1$. The curves corresponding to the
random flood grow faster that those ones obtained for the real-like flood,
mostly due to the retention effect of lakes which slows down the flood
withdrawal.  Indeed, the flood duration for the roads adjacent to the lakes is
higher than for those ones which are only affected by river flooding. Although
we added additional time to the simulations to consider the slow process of
lake emptying, the full recovery would need more time. Focusing on the real-
like case, we observe jumps of $P_{\infty}$ during the flooding process
($t<t_{MFE}$), which are due to the presence of natural elements, such as
lakes or mountains, that constrain the road network. Flooding of those
constrained roads has then a dramatic impact on the giant component size and
creates these discontinuities in $P_{\infty}$. The existence of these
discontinuities depends obviously on the border definition: for the $5000\;m$
case, the behavior of $P_{\infty}$ is much smoother as there are multiple
paths connecting nodes in this larger network. Again, this rises the question
on the choice of the embedding region when studying a percolation related
quantity such as $P_{\infty}$. Also, we observe that these discontinuities are
washed out in the random case, leading to an artificially smooth evolution of
the giant component size. We don’t see jumps in the recovery phase since those
roads whose disruption would cause the jumps in the flood’s growing phase have
not been dried up yet (or completely) at the last time step of the
simulations. In previous studies where floods were considered in the framework
of percolation, as in (37, 40, 39, 43), we note that the authors reported a
complete (or nearly complete) disintegration of the giant connected component,
showing the decrease of $P_{\infty}$ (or $RC$ in (40)) from one to zero. This
process was carried out by removing a fraction of nodes from zero to one in
(37, 40, 39) and from zero to around $0.8$ in (43). This is in sharp contrast
with our observations for both real-like and random floods, where $P_{\infty}$
undergoes a reduction of less than $30\%$ (depending on the choice of
borders). This would suggest that percolation might not be the correct
framework, and $P_{\infty}$ not the most relevant quantity, for understanding
the impact of flooding on transportation networks.

In addition, we note that a flood event displays different dynamics. Indeed,
during the simulation we can also compute for each hour the number of flooded
roads, $F(t)$, which resulting curve is shown in Figure 2(a). The curve
displays a peak at the maximum flood extension $t_{MFE}$ and divides the flood
into two phases, the growing phase ranging from $t=0$ to $t_{MFE}$ and the
recovery phase (or flood’s withdrawal phase) for $t>t_{MFE}$. Interestingly
enough, we note that the fraction of disconnected nodes is smaller than $7\%$,
which is significantly smaller than what is usually assumed in percolation
type approaches. The beginning of the growing phase can be well fitted by an
exponential function with a time scale of order $14$ hours. The end of this
phase displays a saturation effect and is better fitted by a generalized
logistic function (see Supplementary Table 1 for details on the various fits
and Supplementary Text for details on the generalised logistic function).
These results suggests that the evolution of the number of flooded roads could
be described - at least in the first phase - by a generalized Verhulst’s
equation (54) of the form

$\begin{split}\frac{dF(t)}{dt}=\beta_{1}\Bigl{(}F-C_{1}\Bigr{)}&\left(1-\left(\frac{F-C_{1}}{D_{1}}\right)^{\gamma_{1}}\right)\\\
&\quad\text{for \; $t_{0}<t<t_{MFE}$}\end{split}$ (1)

where the intrinsic growth rate is here $\beta_{1}\approx 0.145$. The recovery
phase is well fitted by an exponential decay with a time scale of order $75$
hours, consistent with the observed slow decay. It would be interesting to
construct a simple model for deriving these equations and results, to better
understand the critical parameters governing these time scales.  Indeed, the
form of these equations might suggest a more complex flood propagation model
on road networks than those ones presented in (36, 38). In addition, the
Abdulla et al. (36) model does not include any hydrological or hydraulic
component, nor a validation, while the Fan et al. (38) model, which is applied
to grid cells instead of roads, does not consider the water depth, nor the
difference in height among the nodes (in their flooding probability). Another
related measure is the number of disconnected nodes in the whole network,
$N_{\infty}(t)$ (defined in Equation 6 in the Methods), which time evolution
is shown in Figure 2(b) for both flood scenarios. Despite the amount of
flooded roads per hour is the same in both flood scenarios, we observe a
faster increase of $N_{\infty}(t)$ during the growing phase and a slower
decrease during the recovery phase in the real-like flood, than in the random
flood. This difference is due to the retention effect of the lakes, which
implies a storage of flood water and a consequent time lag in releasing the
same water. Therefore, important roads located along the right and left
shorelines of both lakes (and connecting the upstream with downstream areas)
are inundated earlier in time than the rest of the basin and are drying up
much later because the lakes are going back to their normal level very slowly
(much more slowly than the river floodplains). These differences highlight
once again the danger of using simple random model for describing floods.

### Realistic measures of the local impact of flooding

We saw above that describing the flooding process by percolation could
actually be misleading and that the results could depend on a variety of
factors such as the choice of borders for example. Maybe more importantly, we
note that during floods, individuals will tend to reach the closest town
centers where they can find (at least the basic) supplies, services or
assistance. When the closest town cannot be reached anymore due to flooding,
individuals will have to reroute to the next closest town in order to fulfill
their needs. The rerouting process from a town to another can be viewed as a
varying attraction basin of the neighbouring town, as illustrated in Figure 3.
Some towns will see their attraction basin decrease, while others will gain
some nodes and attract additional individuals. In order to capture these
important effects that occur during disruptive events, we first identify the
nodes that correspond to towns and identify their attraction basin as their
Voronoi cell (for details, see Methods). The Voronoi cell of a town $c$ will
be denoted as $K_{c}(t)$ and is defined as
$K_{c}(t)\doteq\\{i\mid\min\limits_{c^{\prime}}(d_{ic^{\prime}}(t))=d_{ic}(t)\\}$
(its size is denoted by $|K_{c}(t)|$). The shortest distance $d_{ic}(t)$ from
a node $i$ to the town $c$ is computed using Dijkstra’s algorithm over the
graph of non-flooded roads at time $t$. We can then define the town’s loss
$L_{c}(t)=1-|K_{c}(t)|/|K_{c}(t_{0})|$ (see Methods for details on these
metrics). These metrics are shown in Figure 4(a,b,c) for each of the $67$
towns composing the network.

The evolution of $|K_{c}(t)|$ over the entire flood period (Figure 4(a))
displays both losses of nodes for certain towns and an increase of their
attractivity for others. In order to visualize in Figure 4(a) which towns
undergo a size reduction or increase, we represent them in different colors
according to their behavior. We find that about half of the towns ($54\%$) are
not affected by the flood, about $31\%$ undergo an overall size reduction, and
about $15\%$ display an overall size increase. Although we do not have any
spatial information from Figure 4(a), the distribution of non affected towns
indicates that the flood impact on the transportation network is limited
within the area defined in Figure 1 (i.e. within a perimeter of $5000\;m$),
and does not involve the entirety of the network here considered. A second
information we can infer from the affected towns is about their mechanism of
size variation. Indeed, Figure 4(a) shows clearly that this mechanism is a
mutual process among towns (at least two), as illustrated by two neighbouring
towns, Thun and Thierachern, shown in Figure 4(a, b). Although the maximum
size variation of Thun involves about twice as many nodes as for Thierachern,
the two $|K_{c}(t)|$ curves are partially mirrored, which means many nodes
lost by Thun are acquired by Thierachern. A third interesting point is related
to the types of dynamics of $|K_{c}(t)|$. We observe towns exhibiting large
size variations, reaching a sharp peak (maximum or minimum), and towns
displaying smaller jumps and plateaus (e.g. Bern). Also, the maximum size
changes can occur at times $t\neq t_{MFE}$, even during the phase of flood
withdrawal, as for the case of Thun and Thierachern. This is probably an
effect of the spatio-temporal variability, implying that local peaks of road
closures can occur earlier or later than the overall peak (whole river basin).
In the case of Thun and Thierachern, this is an effect of the time lag of the
lake Thun which needed to be filled first before being able to trigger
inundations along the shorelines.

Figure 4(b) shows $L_{c}(t)$ for all the $67$ towns. As seen above, this
measure can reach negative values, meaning that a town can gain nodes from
neighboring ones, due to the rerouting mechanism. As in the previous plot, we
group the $L_{c}(t)$ curves into three categories for better readability:
towns undergoing losses, incorporating new nodes, or constant. Both prior to
the flood onset and after the recovery phase, $L_{c}(t)$ is expected to be
zero or close to zero, while in the interval between these two extremes,
$L_{c}(t)$ is expected to take values $0<L_{c}(t)<1$ (for a town loss) or
$-1<L_{c}(t)<0$ (for a town gain). Although this behaviour is observed for
most of the towns in Figure 4(b), some of them (dotted lines) do not show the
same tendency for $L_{c}(t)$ to return to zero during the flood’s recovery
phase, i.e. $\lim_{t\to t_{end}}L_{c}\neq 0$. Indeed, the lake levels do not
return to the initial level during the timeframe of the simulations but need a
few days longer and some streets remain interrupted until the end of the
simulation period. Also, Figure 4(b) confirms the finding of Figure 4(a) that
the largest size variations of each town (the largest values of $L_{c}(t)$,
both positive and negative) do not always occur at the maximum flood
extension. For example, the largest positive value of $L_{c}(t)$, recorded
among all the towns, occurs around $t=60$ hours and belongs to the town of
Gelterfingen, around halfway between Bern and Thun. The largest negative
value, instead, was recorded for the town of Thierachern a few hours after the
maximum flood extension. This means that emergency services and assistance
should be dynamically allocated over the flood timespan. In this specific
example, Gelterfingen would have a priority over Thierachern, or Thun, which
both peak later. We also note the order of magnitude of the variations of
$L_{c}$ which for some cities can reach values as large as $60\%$. Such a
large increase potentially represents a very large logistic burden during the
flood, a crucial information that needs to be integrated in risk management.

The quantity $L_{c}(t)$ is a quantity which changes over time, and we define
the expansion/shrinking rate as $\zeta_{c}(t)=|L_{c}(t_{MFE})|$ (see Methods
for detail). This quantity can be used for each town as an overall indicator
of the size variation during the flood’s growing phase, and gives an
aggregated and more concise information about the impact of the flood on a
given city. We show in Figure 4(c) the value of this indicator and we thus
observe that some cities will indeed experience an important growth which is
of utmost importance for preparation planning as they will constitute a set of
resilient towns serving a large number of individuals. We also note that there
are some small discrepancies between $L_{c}(t)$ and $\zeta_{c}$ which are due
to the peaks (or in general to the largest values) of $|K_{c}|$ which can
occur at a different time than $t_{MFE}$. For example, the two towns which
exhibit the largest values of $L_{c}(t)$, both positive and negative, i.e.
Gelterfingen and Thierachern, do not score the largest values of $\zeta_{c}$
(see Figure 4(c)).

Rerouting entailed by a road disruption generally results in a longer path
than the initial one (see Figure 3). In other words, for a node $i$, the
distance to its closest town $c^{\prime}$ at time $t$, i.e. after rerouting,
is longer than the distance to its closest town at $t_{0}$:
$d_{ic^{\prime}}(t)\geq d_{ic}(t_{0})$. A simple way to characterise the
detour experienced by an individual living close to a city $c$ is then to
compute the quantity $\eta_{i}(t)=\frac{d_{ic^{\prime}}(t)}{d_{ic}(t_{0})}-1$,
in analogy to the detour index (55, 56). The larger $\eta_{i}(t)$, and the
larger the rerouting for connecting the node $i$ to its closest town’s center
$c^{\prime}$ (which is different from the initially assigned town’s center
$c$). At each time $t$ of the disruptive event, the average rerouting from an
initially assigned town center to a new closest one, can be captured by the
average detour defined as

$\overline{\eta}(t)=\frac{1}{m}\sum_{i=1}\bigg{(}\frac{d_{ic^{\prime}}(t)}{d_{ic}(t_{0})}-1\bigg{)}$
(2)

where $m$ is the number of nodes. This quantity, Equation 2, is a measure of
the global cost or (in)efficiency of a network undergoing a disruptive event,
since the rerouting would represent a time consuming and costly workaround to
roads blockage. The global metrics $\overline{\eta}(t)$ and the loss averaged
over all cities, $\overline{L}(t)=1/n\sum_{c=1}^{n}L_{c}(t)$, are shown in
Figure 4(d, top) and Figure 4(d, bottom), respectively. Both these metrics are
global indicators of the entire network performance, and, in contrast with
$P_{\infty}$, they encode the local information coming from the entire
network. Large values of $\overline{\eta}(t)$ and $\overline{L}(t)$ correspond
to a large rerouting and a large loss of nodes for most towns. Ideally, the
most efficient and robust network would therefore display a value
$|\overline{\eta}(t)-\overline{\eta}(t_{0})|=0$, and
$|\overline{L}(t)-\overline{L}(t_{0})|=0$, for all $t$. Despite sudden and
abrupt variations of $\overline{\eta}(t)$ during the real-like flood, the
network shows a significant higher efficiency than in a random flood scenario
which causes large and gradual variations of $\overline{\eta}(t)$ over the
entire flooding event. Surprisingly, the high efficiency of the network in the
real-like flood appears simultaneously with a lower robustness (represented by
$\overline{L}(t)$) than in the random flood which suggests that a low
rerouting can be achieved through a larger redistribution of the town’s
Voronoi cells. However, the difference in $\overline{L}(t)$ between the two
types of floods is not so pronounced as for $\overline{\eta}(t)$. A last
remarkable difference between the two flood scenarios can be observed in the
latest hours of the recovery phase: in the random flood, $\overline{L}(t)$ and
$\overline{\eta}(t)$ tend to zero (even though $\overline{\eta}(t)$ does not
arrive to zero), while in a real-like flood, they reach a non-zero positive
value (and a peak shifted in time). Basically, this is due to the retention
effect of lakes that causes flooding on roads later in time, and which lasts
for a few additional days up to return of the lakes’ water levels to their
initial ones.

## Discussion

The discussion on a realistic flood simulation presented here allowed us to
show the inadequacy of percolation theory for evaluating the robustness of a
transportation network, when subject to a realistic flooding event.  The
standard percolation process is characterised by a simultaneous and
independent potential removal, usually using a probability $1-p$, of the
entire set of edges. This means that all the edges undergo the removal process
given by a probability $1-p$, and not just a subset. In this framework,  the
size of the giant component, $P_{\infty}$ is a decreasing function of the
growing fraction of removed nodes (or links) up to a certain threshold that
depends on the specific network and where $P_{\infty}=0$. In a real-like
flooding,  instead,  the process of water propagation through the network can
be thought as a localized attack both in time and space,  where only a subset
of the network’s edges can be affected. Therefore, the network cannot be
completely disrupted if it is larger than the flooded area.  The realistic
simulation of the flooding shows that, in general, we don’t observe
$P_{\infty}=0$, even in an extreme event and at the peak of the flooding.
Possible uncertainties in a flood propagation would not affect our
observation, since a realistic mechanism of edges removal would not lead to
$P_{\infty}=0$, if the considered network is large enough. Therefore,  the
absence of an observed transition limits severely the relevance of percolation
for studying this type of catastrophic event. In addition, we showed that the
relevance of the giant component is debatable and we introduced metrics based
on the idea that the crucial point for individuals during a disaster is the
possibility to reach a town, even if it is not the closest one in the normal
regime (see Supplementary Figure 1 for a representative visual comparison
between the percolation approach and one of the proposed metrics).

The approach presented here answers questions relevant to flood risk
management such as the spatio-temporal aspects of the flood’s impacts and
where and when resources should be allocated during a flooding event. Our
study displays the whole dynamics of the flooding and the sequence of
important events, the location and time of the first relevant impact, how the
road closures spread, which towns become less accessible, etc. In addition,
our metrics about the dynamics of rerouting and rising/shrinking towns
contains the relevant information for risk management decision makers, in
order to allocate resources in space and time (e.g. hospitalization services,
food supply, etc.). Already at the earliest moments of the flood, it could be
beneficial to transfer resources to towns with a low potential of inundation
and a large expansion/shrinking ratio $\zeta_{c}$.

Our study showed that the standard percolation approach might have limits of
applicability to real-world systems, such as in this case of flood-induced
disruption of a single road network. Future studies might investigate whether
such limits could occur in other real-world scenarios and potentially provide
a more general theory of percolation or a more formal description of its
limits of applicability. For example, still in transportation networks, what
happens if we apply the percolation framework to disrupted multi-layer
networks (49, 50) which include road networks? Also, how can we address the
issue related to the border size selection? Indeed, the size of the border
determines the number of opportunities for deviating routes: a tiny border
would eliminate many alternative routes, whereas a large border would enable a
large number of possible alternatives. Therefore, future studies could
investigated how far or how close the system can be delimited without losing
potential routing alternatives. Also, the influence of the network topography
on the flood’s impact could be further investigated in the future. Indeed, in
our case, we studied the flood impacts on a road network situated in
mountainous and hilly regions, while other authors considered quite flat
areas, as Abdulla et al. (34, 35, 36, 37) with the Houston road network.
Another direction for future works is related to a simple model for deriving
Equation 1 and understanding its parameters. Such a simple mechanistic model
might include basic principles of hydrology and hydrodynamics and, at least,
should account for the difference in height among each pair of nodes, in order
to provide the water flows directions. This simple model might also fulfil the
criteria of universality, i.e. being applicable to a vast variety of scenarios
and road networks. Indeed, so far, the Abdulla et al. (36) and Fan et al. (38)
models of flood propagation (in a road network) seem to be case-dependent, and
their parameters have to be estimated for any network and scenario. Another
limitation that could be addressed in future studies is related to the use of
realistic time-varying travel times. Indeed, travel times and the distribution
of traffic congestion change during a flood and the corresponding network
weights (represented by travel times) should be updated at each time step, in
order to get more reliable results.

## Methods

###  0.1 Definition of town center on a transportation network

The definitions of the new proposed measurements (alternative to percolation)
are built up around the concept of town center (or city center), which could
be intuitively thought as the core area of a town, for social, business, and
cultural activities. Despite the common idea of an extended area, the town
center was here redefined as a node of a road network, as representative spot
for the town center. For each Swiss municipality, two entities were employed
for selecting the proxy node of a town center, i.e. the road network and
polygons called (town or city) _central zones_ by the Swiss Federal Office
(57). Since several Swiss towns included a multitude of those polygons, which
were all named as central zones, the largest polygon was selected as _town
center_ , often corresponding to the historical town center, while the other
smaller polygons were neglected. The largest polygon was chosen by assuming it
could include the largest quantity of services, useful for a population. Then,
the roads overlapping and intersecting the largest polygon were selected, and
the centroid of their endpoints was employed to infer the proxy node. The node
representing the town center, was defined as the closest node to the roads
endpoints’ centroid, calculated from the roads which overlapped the largest
polygon. Obviously, other definitions of town center’s node are possible and
can be used. For example, the small central zones and their overlapping roads
could be also included in calculating the roads endpoints’ centroid, but with
the risk of obtaining a centroid far from the largest polygon.

###  0.2 Definition of town and temporal evolution of a flood on a
transportation network

The town centers’ nodes previously described in Section 0.1 (Definition of
town center on a transportation network) represent an essential ingredient for
the new metrics definitions. All those metrics are indeed based on the
shortest distances $d_{ic}$ and on the quickest paths $\tau_{ic}$ among all
the pairs of $i-$nodes of the road network and $c-$nodes of the town centers.
At each time, we thus have matrices $d_{ic}$ and $\tau_{ic}$ where
$i=1,\dots,m$, $c=1,\dots,n$. The quantities $m$ and $n$ represent the total
numbers of road network’s nodes and town center’s nodes, respectively. The
shortest distances and the quickest travel times were calculated with the
Dijkstra algorithm (58), by employing single-road lengths $d_{road}$ $[m]$ and
single-road travel times $\tau_{road}$ $[s]$ as network weights (see Figure
3(a)). The single-road lengths were collected from the Swisstopo database
(52), while the single-road travel times were approximated by the following
formula:

$\tau_{road}=\frac{d_{road}}{f\frac{v_{lim}}{3.6}}$ (3)

where, $f=0.8$ is the reduction speed factor which considers the relationships
between speed limits and traffic flows (59), $v_{lim}$ is the speed limit
$[km/h]$ from Supplementary Table 2, and the number $3.6$ was used to convert
$v_{lim}$ from $[km/h]$ into $[m/s]$. Once the elements $d_{ic}$ and
$\tau_{ic}$ are computed, the minimum value in each row of the corresponding
matrices $D$ and $\Tau$ is selected, i.e. the town center’s node with minimum
distance to the $i^{th}-$node. This corresponds to assign an $i^{th}-$node to
the closest town center’s node, at $t_{0}$. As a consequence, at time $t_{0}$,
all nodes in the network are associated to their closest town centers’ nodes
(Figure 3(b, top)), creating $n$ sets of nodes
$K_{c}(t_{0})=\\{i\mid\min\limits_{c^{\prime}}(d_{ic^{\prime}}(t_{0}))=d_{ic}(t_{0})\\}$.
These sets, $K_{1},K_{2},\dots,K_{n}$, correspond to Voronoi cells on the
network and could be written also for travel times $\tau_{ic}$. We defined the
generic $c-$th _town_ as:

$K_{c}\equiv
K_{c}(t_{0})\doteq\\{i\mid\min\limits_{c^{\prime}}(d_{ic^{\prime}}(t_{0}))=d_{ic}(t_{0})\\}$
(4)

such that $|\bigcup_{c=1}^{n}K_{c}|=m$. During the flood, the elements of $D$
and $\Tau$ could change due to the disruptions of the road network and we
recompute them at each time step $t_{0},t_{1},t_{2},\dots$, where $t_{0}$ is
the time just before the flood start, with no disruptions (Figure 3(b, top)),
and $t_{1}$ is the first time with disruptions (Figure 3(b, middle)). During
the flood, some roads are disconnected and some nodes have to be rerouted,
resulting in a reorganization of the Voronoi cells of each town. We indicate
the size of a town $c$ which can be variable by

$K_{c}(t)\doteq\\{i\mid\min\limits_{c^{\prime}}(d_{ic^{\prime}}(t))=d_{ic}(t)\\}$
(5)

###  0.3 New metrics

#### Detour index

Rerouting entailed by a road disruption generally results in a longer path
than the initial one, i.e. $d_{ic^{\prime}}(t)\geq d_{ic}(t_{0})$, where
$c^{\prime}\neq c$ is the closest town’s center at time $t$, to which the
$i-$node would be reassigned. Then, one of the simplest ways to measure the
deviation of a path’s distance at time $t$ from its original value at $t_{0}$,
would be straightforwardly the ratio between the two distances, or the
_detour_ , $\eta_{i}(t)=\frac{d_{ic^{\prime}}(t)}{d_{ic}(t_{0})}-1$, in
analogy to the detour index (55, 56). Since $d_{ic^{\prime}}(t)\geq
d_{ic}(t_{0})$, we would measure $\eta_{i}(t)\geq 0$ $\forall i$, with
$\eta_{i}(t)=0$ indicating the absence of rerouting to the closest
neighbouring town. Therefore, the larger the $\eta_{i}(t)$, the larger the
rerouting for connecting the $i-$node to the closest town’s center
$c^{\prime}$ (which is different from the initially assigned town’s center
$c$). At each time $t$ of the disruptive event, the average rerouting from an
initially assigned town center to a new closest one, can be captured by the
_average detour_ , defined as in Equation 2. We would consider Equation 2 as a
measure of global (in)efficiency of a network undergoing a disruptive event,
since the rerouting would represent a time consuming and costly - i.e.
inefficient - workaround to roads blockage. However, $\overline{\eta}(t)$
would differ from the well-established Latora & Marchiori’s (45) or from the
Vragović’s (60) global network efficiencies, which are basically averages of
$\frac{1}{d_{ij}}$ and $\frac{d_{ij}^{Euclid.}}{d_{ij}}$, respectively, over
all pairs of nodes. Indeed, the average in Equation 2 would be only over those
pairs of nodes which include the towns’ center nodes, and not over all pairs
of nodes.

#### Disconnected nodes

The distances and their variations over time, as well as the relative travel
times, can also be employed to quantify the _total number of disconnected
nodes_ in the entire network:

$\displaystyle N_{\infty}(t)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}i=1\end{subarray}}^{m}b_{ic}$
(6) $\displaystyle\text{where}\quad b_{ic}$
$\displaystyle=\begin{cases}1,&\text{if
}d_{ic}(t)=\infty\quad(\text{or}\;\tau_{ic}(t)=\infty)\\\
0,&\text{otherwise}\end{cases}$

#### Town loss

Our understanding of the flood’s effects can be further enriched by metrics
related to the size’s variations of each town, $|K_{c}(t)|$. The size
reduction of a town during a flood is intuitively expected and can be
expressed, analogously to $\eta_{i}(t)$ for the distances, through a ratio
between the size of the damaged (and diminished) town at time $t$, and the
original size at time $t_{0}$:

$L_{c}(t)=1-\frac{|K_{c}(t)|}{|K_{c}|}$ (7)

For example, we can have a reduction of the town size from an initial
$|K_{c}|=2500$ nodes to $|K_{c}(t)|=2000$ nodes at time $t$, or rather a size
loss equal to $L_{c}(t)=0.2$, i.e. a $20\%$ less than the initial size at time
$t_{0}$. In parallel to $\eta_{i}(t)$, we can also define a global measure for
$L_{c}(t)$, the _average town loss_ , simply by averaging $L_{c}(t)$ over all
towns:

$\overline{L_{c}}(t)=\frac{1}{n}\sum_{c=1}\bigg{(}1-\frac{|K_{c}(t)|}{|K_{c}|}\bigg{)}$
(8)

#### Expansion/shrinking ratio

While Equation 7 provides information at each time $t$ during the entire flood
event, the size variations of every town can be even characterized by a single
representative value, which relates the town size at the time of the maximum
flood extension, $t_{MFE}$, to the town size at $t_{0}$. We call it the
_expansion/shrinking rate_ :

$\zeta_{c}=\frac{\displaystyle\left\lvert|K_{c}(t_{MFE})|-|K_{c}|\right\rvert}{|K_{c}|}$
(9)

Although we would expect the maximum network’s disruption at $t_{MFE}$, where
the flood reaches the maximum extension, it might be that some towns could
reach their maximum disruption at another time $t\neq t_{MFE}$. Therefore,
$\max\limits_{t}\big{(}|K_{c}(t)|\big{)}$ might be different from
$|K_{c}(t_{MFE})|$, and Equation 7 and Equation 9 could eventually indicate
different size variations for the same town.

###  0.4 Flood failure

A realistic and extreme flood scenario (51) based on physically plausible
rainfall scenarios and a high spatio-temporal resolution was used to generate
the road network disruptions. The spatio-temporal patterns of rainfall of a
three-day probable maximum precipitation event were generated with a Monte
Carlo procedure that considers only physically plausible patterns. The
hydrological model simulated the outflows from the subcatchments. A 1D
hydrodynamic flood model simulated the water fluxes from the tributaries
through the main river network and the lakes. The 1D hydrodynamic model
provided the boundary conditions for the 2D flood simulation models that had
been set up for each floodplain. The 2D hydrodynamic model had a spatial
resolution of 50 m and computed the flood depths for each timestep (hourly)
and thus provided the necessary information for the impact model. The impact
model simulated the water depths at the road edges. The model considers the
location of the bridges in the third dimension, i.e., road bridges located
above the simulated water surface elevation are considered as not affected by
the flood. In addition, the model has been validated with historic events.

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## Acknowledgements

S.L. thanks Alexander Krok, Maria Arduca and Riccardo Turin for useful
discussions. S.L. thanks the Mobiliar Lab for Natural Risks for funding and
supporting this research project.

## Author contributions statement

Author contributions: S.L., A.Z., M.K. and M.B. designed research; S.L.
performed research; S.L., E.S.-G., A.Z., M.K. and M.B. analyzed data; S.L. and
M.B. wrote the paper.

## Additional information

Competing interests: The authors declare no competing interest.

Figure 1: (a) Definition of borders. Given the set of endpoints (nodes) of the
flooded roads at the maximum flood extension, a tight $2-$D boundary (61)
surrounding those points was drawn. The boundary was created with a shrink
factor (61) equal to $s_{f}=0.5$, which is between a convex hull ($s_{f}=0$)
and the tightest single-region boundary ($s_{f}=1$). That boundary corresponds
to the dash-dot line called border: $0$ meters. Larger boundaries,
corresponding to border: $1000$ meters and border: $5000$ meters, were created
by radially expanding (62) the border: $0$ meters by a distance of $1000$ and
$5000$ meters, respectively. (b) Evolution during the flood of the fraction
$P_{\infty}$ of nodes belonging to the giant connected component computed for
both real-like floods (51) and randomly generated floods and for both the
$1000$ and $5000$ meter-borders shown in (a). The curves representing the
random floods were computed by averaging the values of $20$ realizations.
Here, they are shown with $1-$standard deviation error bars. More precisely,
those average values resulted from the arithmetic mean performed over the $20$
values of $P_{\infty}$ (related to the $20$ realizations of random floods), at
each time-step, while the vertical bars around the average values represented
the standard deviation performed over the $20$ values of $P_{\infty}$, still
at each time step. Therefore, both mean and standard deviation (1-std) of the
$20$ values of $P_{\infty}$ are here shown, for each time step (obviously,
those $20$ values are different at each time step). This figure was produced
with Matlab (63). Figure 2: (a) Time evolution of the number of flooded roads,
$F(t)$. During the Zischg et al. (51) simulations, roads were marked as
flooded if the water depth was greater than $0.3$ meters, which is considered
as the limit of water’s depth for driving a car (with average ground
clearance) (53). In terms of number of flooded roads, the earliest times of
the flooding event are well characterised by an exponential growth. However,
this function does not describe the saturation phase here observed, which is
instead typical of logistic-like functions. A fair approximation of the entire
growing phase is given by a generalised logistic function, which fits
particularly well the saturation phase, up to the maximum flood extension. The
recovery phase, that is the phase of flood withdraw, follows instead a slow
exponential decay. (b) Time evolution of the number of disconnected nodes,
$N_{\infty}(t)$. This metric is here illustrated for both the real-like flood
simulated by Zischg et al. (51) and the randomly generated flood. For the
real-like flood, the dynamics of $N_{\infty}(t)$ is well approximated by an
exponential growth and decay, respectively for the increasing and recovery
phases of the flood. Although both $N_{\infty}(t)$ \- calculated for the two
types of floods -, reach a similar peak at $t_{MFE}$, the number of nodes
which disconnect from the rest of the network is generally larger for real-
like floods than in random floods, in both phases of flood’s increase and
withdraw. This occurs since real-like floods are more spatially concentrated
(i.e. less dispersed) than random floods, with a higher probability of
surrounding and isolating nodes. Therefore, modelling and simulating a flood
event with a random blockage of roads would be misleading, generally resulting
in an underestimation of $N_{\infty}(t)$, in several moments of a flooding
event. This figure was produced with Matlab (63). Figure 3: Temporal evolution
of towns during a flood event. (a): Part of a weighted road network,
illustrating a typical network’s $i-$node and two $c-$nodes, which represent
two town centers. Single-road lengths $d_{roads}$ and single-road travel times
$\tau_{roads}$ were used as network weights. (b, top): Definition of towns,
i.e. assignment of the road network’s nodes to the nearest town centers at
$t_{0}$. In (b, middle), all nodes tend to reach the closest town centers,
even if different from their initially assigned ones at time $t_{0}$. That
means that the two nodes at the top left corner, which were yellow at $t_{0}$,
become green at $t_{1}$, since now closer to the town center $c_{1}$. As
consequence, the towns’ sizes change over time. In (b, bottom) further flood
advancement wipes out other roads, isolating a number of nodes (as singlets or
small clusters) from their closest town centers. In this example, at time
$t_{0}$, the size of the two towns would be $|K_{1}(t_{0})|=5$,
$|K_{2}(t_{0})|=6$. At time $t_{1}$, we would count $|K_{1}(t_{1})|=7$,
$|K_{2}(t_{1})|=4$, while at time $t_{2}$, we would have $|K_{1}(t_{2})|=3$,
$|K_{2}(t_{2})|=3$, and a total number of disconnected nodes equal to
$N_{\infty}(t_{2})=5$. This figure was produced with PowerPoint (64). Figure
4: Temporal evolution of (a) the _town’s size_ , $|K_{c}(t)|$ (Equation 5),
(b) the _town’s loss_ , $L_{c}(t)$ (Equation 7), (c) the town
_expansion/shrinking ratio_ $\zeta_{c}$ (Equation 9) and (d) the _average
detour_ $\overline{\eta}(t)$ (Equation 2) together with the _average town
loss_ $\overline{L}(t)$ (Equation 8). The local metrics $|K_{c}(t)|$,
$L_{c}(t)$ and $\zeta_{c}$ capture the dynamics of each of the $67$ towns
composing the network (with a perimeter of $5000$ meters). Due to the re-
routing mechanism illustrated in Figure 3, the towns size increases,
diminishes or remains constant. For a better visualisation we grouped the
towns with _overall size increase_ in green, (i.e. if
$|\max_{t}|K_{c}(t)|-|K_{c}(t_{0})||>|\min_{t}|K_{c}(t)|-|K_{c}(t_{0})||$),
the towns with _overall size decrease_ in fuchsia (i.e. if
$|\max_{t}|K_{c}(t)|-|K_{c}(t_{0})||<|\min_{t}|K_{c}(t)|-|K_{c}(t_{0})||$) and
the towns without size variations in gray (i.e. if
$|\max_{t}|K_{c}(t)|-|K_{c}(t_{0})||=|\min_{t}|K_{c}(t)|-|K_{c}(t_{0})||=0$).
The global metrics $\overline{\eta}(t)$ and $\overline{L}(t)$, representing
the network’s (in)efficiency and robustness, respectively, are here
illustrated for both flood scenarios. For randomly generated floods, the
average values of $\overline{\eta}(t)$ and $\overline{L}(t)$ were calculated
over $20$ realizations. This figure was produced with Matlab (63).