Abstract
Networked dynamics are essential for assessing the resilience of lifeline infrastructures. The dimensionreduction approach was designed as an efficient way to map the highdimensional dynamics to a lowdimensional representation capturing systemlevel behavior while taking into consideration network structure. However, its application to sociotechnical systems has not been considered yet. Here, we extend the dimensionreduction approach to resourceflow dynamics in multiplex networks. We apply it to the San Francisco fuel transportation network, considering the flow between refineries, terminals and gas stations. We capture the aggregated dynamics between the facilities of each type and identify macroscopic conditions for the system to supply a given demand of fuel. By considering multiple sea level rise scenarios between 2020 and 2100, we address the impact of coastal flooding due to climate change on the maximum suppliable demand. Finally, we analyze the system’s transient response to production failures, investigating the temporary interruption in production and the duration it takes for complete demand satisfaction to become unachievable after the interruption.
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Introduction
In recent years, the topic of system resilience has gained more attention from the scientific community^{1,2,3,4}, as the effects of climate change become more evident in the world around us^{5,6,7}. Climate change exposes lifeline systems to unexpected levels and types of stress, endangering their functionality. Resilience analysis considers the dynamical behavior of a system during a failure, by considering its different stages: preparation, robustness and recovery^{2}. One possible approach to assess resilience is based on Dynamical Systems theory, usually through changes in the parameters describing the system at different scenarios. However, traditional Dynamical Systems methodologies may require a detailed description of system dynamics, unfeasible in many cases. Techniques have been developed to partially overcome this limitation, by working with general, broadly defined functions, seeking to understand a system’s behavior around its stable state^{8,9}. As the system scale increases and interaction patterns become more complex, networkbased approaches have become crucial to model a system’s topology, and assess the impact of changes on its structure^{1,2,10}. By combining network structure and dynamics, the dimensionreduction approach^{11,12,13,14} provides a tool to assess system dynamics at large scale, while taking into account the patterns of interactions between the system’s constituents. Originally developed in the context of ecology, the approach focuses on generating a lowdimension set of equations capturing the macroscopic behavior of the system, through only a few equations (compared to the original system size). The parameters appearing in the reduced set of equations capture the network topology, and help analyze the impact of network structure on system dynamics.
In this work, we extend the dimensionreduction approach^{11,12,13,14} to the context of resourceflow networks. Resourceflow networks^{15} comprise those networked systems where some quantity is transported from one part of a system to another, following conservation laws. Resourceflow dynamics include many examples from urban systems, as supply chain management^{9,16}, disease propagation^{17}, water distribution^{18}, and trade relationships^{19}. Here, we consider a case study from supply chains: the San Francisco Fuel Transportation Network (SFFTN). We examine the influence of climate change on the SFFTN, and we delve into the ramifications of Sea Level Rise (SLR)^{20} as detailed in He et al.^{21}.
Supply chain dynamics can be challenging, as they are heavily dependent on various levels of human decision and environmental factors. Future demand estimation has been considered, through the use of difference equations^{16,22} finding how demand variation increases from consumers to producers (known as the bullwhip effect). Ordinary differential equations have been used to model supply chain dynamics from a coarsen perspective^{9}, since they treat flow as a continuous magnitude, and thus they are not suited to capture events in very small scales of time. Information privacy on flows between firms limits modeling, as only little data is available for comparison. Here, we study resourceflow networks under simple dynamics capturing the qualitative behavior of supply and demand, and the general conditions for stability induced by network topology. Instead of looking for a detailed description of each facility within the network, we focus on a systemlevel scale, through the dimensionreduction approach. By considering refineries, terminals, and gas stations included within the SFFTN, we write down a set of ordinary differential equations representing the flow of fuel from refineries to gas stations. Then, we obtain a dimensionally reduced description of the system by considering the average amount of fuel stored at each type of facility.
We use the approximation to analyze the impact of coastal flooding on the transportation system, by considering its ability to supply demand under different climate change scenarios. Later, we consider the ability of the system to sustain demand during a production failure, under different coastal flooding scenarios between years 2020 and 2100. The results depict the transition of the system through different stages of failure, up to the point of it being unable to satisfy any level of demand. By considering both the impact of topology alone and through simple dynamical laws capturing transportation’s qualitative behavior, our approximation provides an estimate of the maximum demand that could be sustained under multiple SLR scenarios. Our work constitutes three different contributions to the topic of resilience of networked dynamical systems. First, the dimensionreduction approach shows the conditions imposed over the space of stable flows by the network structure. Second, by studying lifeline infrastructure systems such as the SFFTN, we extend the dimensionreduction approach to the context of sociotechnical systems. Lastly, by considering the impact of SLR through climate change scenario analysis, we study a realistic example of failure from a dynamical systems perspective.
Results
The San Francisco Fuel Transportation Network and the effects of coastal flooding
The SFFTN was presented originally in He et al.^{21}. It was constructed by considering spatial information from OpenStreetMaps and Google Places, the California Energy Commission (CEC), and the companies involved in fuel production and transportation in the region. It includes two types of nodes, corresponding to transportation means (railway, sea, road, oil and product pipelines), and facilities (ports, terminals, refineries, gas stations and airports). We simplify this complex structure by representing it as a multiplex network where nodes are facilities (and each layer corresponds to a type of facility) and transportation means correspond to different types of links between the layers. For the purpose of this work, we focus on the subnetwork supporting the transportation from refineries (production points) to terminals (intermediate storage points) and gas stations (consumption points). Refineries and terminals connect through product pipelines, in a structure that allows flow between any pair of them. Both of them connect to gas stations through the road network, which allows transportation from any refinery or terminal to any gas station. The majority of the flow goes from refineries to terminals through product pipelines, and from terminals to gas stations through trucks. A smaller portion of the fuel is directly transported from refineries to gas stations through trucks. The spatial representation of the SFFTN can be found in Fig. 1a, and its network abstraction in Fig. 1d.
Due to the effect of climate change, coastal regions in the San Francisco Bay Area (SFBA) are likely to flood frequently in the next hundred years^{23}. Through the use of computational modeling, scenario analysis has been used to assess the impact of coastal flooding on the systems located within the region^{23}. Using the flooding scenarios constructed in Radke et al.^{23}, we consider four different time horizons (2020–2040, 2040–2060, 2060–2080, and 2080–2100), under two Representative Concentration Pathways (RCPs, 4.5 and 8.5)^{24}. For each, four global climate or earth system models (GCM) are used to produce predictions of typical, high, and extreme SLR at the SFBA, corresponding to the 50, 95, and 99.9 percentiles of the SLR predicted by each model and for each location at the SFBA. The models are CanESM2, MIROC5, CNRMCM5 and HadGEM2ES, corresponding to the CMIP5 suite of models^{25}. Water column height estimations are constructed through the 3Di hydrodynamic model^{26}. Further details are provided in the Methods section. Figure 1a–f depicts the impact of SLR on time horizons 2060–2080 and 2080–2100 for RCP 8.5 and the 99.9 SLR percentile, where it is more appreciable. A region is considered flooded under a given scenario if the water column at that region is higher than 15cm. Coastal flooding can be appreciated in the spatial representation (Fig. 1a–c). Nodes at flooded locations are removed from the network, as they are considered failed. In the abstract representation (Fig. 1d–f), it is evident how coastal flooding largely reduces the number of terminals and product pipelines, while also reducing the number of refineries in the last time horizon. The effect of coastal flooding is not only to remove facilities, but also to disconnect them from the different layers. This impacts the flow capacities between the layers, reducing the ability of the refineries to transmit the produced fuel to the terminals and gas stations. See Methods section for more detail on the change of the network structure due to coastal flooding.
Dynamical representation
We start by considering a set of ordinary differential equations describing fuel transportation, produced at refineries, transported to terminals, and then to gas stations (the main path of flow) or directly to gas stations (a secondary path of flow). Usually, production, consumption, and flow between facilities will depend on the stock level they have at a given moment. To describe the state of the system, we consider variables \({x}_{i}^{q}\in [0,1]\), indicating the stock level at facility i = 1, …. N_{q} in layer q = 1, 2, 3 (corresponding to refineries, terminals, and gas stations, respectively). We consider that nodes in layer q have stock capacity C^{q}, and so the resource stored at it is \({C}^{q}{x}_{i}^{q}\). We assume that production at a refinery and consumption at a gas station only depend on their individual stock levels, and that flow between facilities only depends on the stock levels of the facilities involved. Then the dynamics of the stock levels are described by
where:

\(\Pi ({x}_{i}^{1})\in [0,1]\) is the production level at refinery i, and P is the production capacity (maximum production possible) of a refinery.

\(\Delta ({x}_{i}^{3})\in [0,1]\) is the demand level at gas station i, and D is the maximum demand that can be supplied by a gas station.

\(\Psi ({x}_{i}^{q},{x}_{j}^{r})\) is the flow level from facility i in layer q to facility j in layer r, and \({W}_{ij}^{qr}\) is the flow capacity between them.
Notice the similarity of Eq. (1) with the BarzelBarabasi equation^{27}, considered in the original application of the dimensionreduction approach^{11}. However, in the context of resourceflow systems, the interaction term appears with a different sign at the different ends of the flow, and thus the interaction term is different at each layer. One of the main difficulties when working with supply chains connecting multiple firms is that detailed information on their transportation policies and capacities is often unavailable. In the case of the SFFTN, reasonable values for the system’s parameters can be found in an aggregated fashion through the CEC, as described in the Methods section. To address the uncertainty in the parameter values, we consider ranges for some of them. These can be found in Table 1.
Dimensionreduction
We construct a reduced representation of the SFFTN, induced by flow capacities \({W}_{ij}^{qr}\) and production and demand capacities, P and D. The dimensionreduction approach considers the average dynamics by constructing an effective state, representative of the system as a whole. A set of equations for this average state is derived, taking into account the original equations and the network structure. Thus, the resulting set of equations can be directly analyzed by means of traditional Dynamical Systems tools. In the context of this problem, we regard the layer stock levels as the relevant variables, \({N}_{q}{C}^{q}{y}^{q} = {\sum }_{i = 1}^{{N}_{q}}{C}^{q}{x}_{i}^{q}\). Notice that in the case of equal stock capacities, the layer stock level matches the layer average fill level. We choose to use the layer stock level as systemstate variable for two reasons. First, the resulting meanfield estimator preserves the important property of flow conservation across layers. Considering different weights based on facility connectivity would lead to distortions in that property, as flows would be weighted differently depending on where they arrive to or depart from. Second, in our case study information on flow capacities at the facility level is not available, and thus it wouldn’t be possible to construct an estimator based on them in the first place. However, as it will be shown below, aggregated information on flow capacities can be estimated, and thus dynamical behavior for the layer stock levels can be analyzed. By calculating the time derivatives \({\dot{y}}^{q}\), and following the typical approximations used for dimension reduction (that stock, production, demand and flow capacities are not correlated to each other, and that flow capacities and stock levels are not correlated) we obtain the set of differential equations for the layer averages y^{q}:
where:

p = P/C^{1} is the normalized production capacity, and d = D/C^{3} is the normalized maximum suppliable demand.

\({s}_{qr}={\sum }_{i = 1}^{{N}_{q}}{\sum }_{j = 1}^{{N}_{r}}{W}_{ij}^{qr}/{N}_{q}{C}^{q}\) is the normalized average flow capacity from layer q to layer r.

α_{qr} = N_{r}C ^{r}/N_{q}C ^{q} is the stock capacity ratio between layers q and r.
In the Methods section, we detail the deduction of the approximation for the general case where facilities may have different stock, production, demand and flow capacities. The approximation assumes low correlation between the system parameters and the values that the level function Π, Δ and Ψ take, similar to the assumptions in^{11,13}. Numerical testing of the approximation for different connectivity patterns between gas stations and the other two layers is provided in the Supplementary Note 1, while different connectivity levels and flow capacities are considered in the Supplementary Note 2. In particular, it is worth to mention that the the approximated set in Eq. (2) works well even when stock, production, demand and flow capacities are perturbed up to a 10% of their expected value. The accuracy of the approximation increases as the interlayer connectivity and the number of nodes increase.
The approximation reduces the original set of over 3400 equations to only 3, allowing the use of traditional Dynamical Systems techniques to analyze the dynamics of the system in terms of global, average parameters. In turn, we change our focus of analysis from the details of each facility to the macroscopic dynamics between layers. The dimensionally reduced system captures the macroscopic behavior of the demand, without focusing on the details at the level of the facilities. By considering the layer stock level, the system’s dynamics are approximated by a reduced set of equations, universal for all systems with equal stock capacity structure and average production, demand, and flow capacities. We approximate each layer independently, to account for the transportation between them.
A relevant magnitude for describing the behavior of the system is the total amount of resource stored in it, denoted as U. In terms of normalized parameters, it can be written as U = y^{1} + α_{12}(y^{2} + α_{23}y^{3}) ∈ [0, 1 + α_{12}(1 + α_{23})], where its value is normalized by the total capacity of the first layer N_{1}C^{1} (and thus the total stock in Mgal is equal to N_{1}C^{1}U). The ranges of values for the normalized parameters can be found in Table 2.
Stable states
In resourceflow dynamics, it is usual to have a continuous range of possible stable flow levels^{9}, depending on the amount of resource entering or exiting the system. Thus, we look for conditions that consider stable demand or production as variable parameters. We define Π^{*}, Δ^{*}, and Ψ^{*}, as the stable production, demand, and flow levels. By setting \({\dot{y}}^{q}=0\) in Eq. (2), we find the two following conditions linking them:
The first condition requires stable production and demand levels Π^{*} and Δ^{*} to be proportional to each other. Notice that p = P/C^{1} is the total production capacity and that α_{12}α_{23}d = N_{3}D/N_{1}C^{1} is the total demand capacity, both in units of total stock capacity of refineries. The second condition indicates that the average demand has to be lower than the addition of the flow capacities from layer 1 to layer 3 for that demand level to be stable. For the purpose of this work, we assume that the first condition in Eq. (3) is always satisfied, meaning that the total production is equal to the total demand, and thus the stability of the system only depends on the flow capacities. The second condition from Eq. (3) is equivalent to the maximum flow theorem between the layers^{28}. By considering the constraints over the stable flows imposed by this condition, we find that they are confined to a line in a 3dimensional space (see Methods section for the full deduction of Eq. (3)) for each value of the stable demand Δ^{*}. Thus, network structure shapes the stable state space through the maximum flow condition, limiting stable flows to those satisfying it. However, notice that the stable stock levels depend on the functions Π, Δ and Ψ being considered. Comparing these results with the findings from^{11}, we can see two examples of how network structure shapes the stable state spaces of the ordinary differential equations. In^{11}, the mutualistic interaction considered has a positive effect on each pair of species involved (i.e. it increases the number of individuals of both species). As network connectivity increases, there is a critical point above which the interaction percolates, making the system resilient to sudden reductions in the abundance of each species. In the case of resourceflow networks, the topology limits the stable flows to those satisfying the maximum flow condition^{28}. Future advances in this topic could help understand further how the evolution of different systems is shaped by the intertwine between dynamics and network structures of various types.
Numerical example
While the stability of the system can be addressed in general, assessing many dynamical characteristics (as stable stock levels) requires specifying functions Π, Δ, and Ψ. However, only aggregated information is available for the SFFTN. Thus, we look to model its dynamics by considering three simple characteristics. First, according to the CEC, the production is held constant at maximum capacity, independently of the stock level of the refineries. Thus, we set Π(x) to be approximately constant except for stock levels near x ≈ 1, when a refinery would not have enough stock space to produce more. We consider a similar behavior for Δ(x), being constant aside from stock levels near x ≈ 0, where a gas station would not be able to supply any demand due to the absence of resources. For the flow level Ψ(x_{1}, x_{2}), we assume that it will increase as the sender x_{1} has a higher stock level (more to send), or the receiver x_{2} has a lower stock level (more space to receive). Figure 2a, b and c depict the functions Π(x), Δ(x) and Ψ(x_{1}, x_{2}) used for this work. Under these functions, the system dynamics present a smooth evolution to a stable point (see Fig. 2d). The approximation captures well the layer stock level, while the different facilities show dispersion around it due to the differences in the flow capacities. The stable values of the layer stock levels y^{q} depend on the initial amount of resource saved in the system, U(t = 0) (Fig. 2e). For low amounts of total resources, most of them are located at refineries, leaving the third layer empty below a certain point. The flow capacities can change the size and shape of the stable state space. For example, at a given value of U(t = 0), increasing the value of s_{12} reduces the stable stock level at refineries (y^{1}), and increases the stable stock level at gas stations (y^{3}). This is due to the increased flow, that moves resource to the third layer faster than it is consumed. Notice that the stable stock levels y^{q} are bounded between two values (different for each layer), dependent on s_{12}. Thus, the total stable resource U(t) is also bounded. As s_{12} reduces, the resource is more concentrated in the first layer, leaving the third layer empty. Similar effects can be observed by changing the value of the other two normalized average flow capacities.
Impacts of SLR on the system dynamics
Next, we consider the impact of SLR on the maximum stable demand that the system would be able to sustain. We assume that in all the scenarios the total production and demand remain constant. Thus, as refineries and gas stations are removed from the system due to flooding, average capacities p and d increase to maintain constant (and equal) total production and demand. Figure 3a shows the average percentage changes of the parameters due to flooding, for the two RCP and the 4 time horizons considered at the 99.9 (worst case) SLR percentile. The majority of the parameters reduce their values, except for the average production capacity p and the total capacity ratio α_{23}. The biggest change occurs in the flow capacity from refineries to terminals, with a reduction of 40% (RCP 4.5) and 75% (RCP 8.5) of s_{12} (representing the pipeline average flow capacity from refineries to terminals). This is followed by a similar decrease for s_{13}, the flow capacity from refineries to gas stations directly. The value of p increases when the number of refineries reduces to keep the total production constant. α_{23} increases as proportionally more terminals are affected than gas stations. More detail on the changes of the parameters can be found in Table 3 in the Methods section.
The system is able to sustain the original demand in the time horizons 20202040 and 20402060 for all the scenarios considered. However, demand failure is observed in 20802100 for RCP 4.5 and 20602080, 20802100 for RCP 8.5 and 95 and 99.9 SLR percentiles (Fig. 3b, c). This is due to two effects combined, observed in Fig. 3b. On one hand, the flow capacities from refineries to the other facilities reduce, limiting the maximum flow that the system is able to establish. On the other hand, the reduction in the number of terminals (expressed through the decrease of α_{12} and increase of α_{23}) requires a higher average flow to and from each terminal to keep supplying the total demand. This is represented in Fig. 3b, where we consider a phaselike diagram for the maximum stable demand constructed with the dynamics from Fig. 2a. The diagrams show the maximum stable demand as a function of s_{12} and s_{13}, with the remaining parameters fixed at their average value for the two considered RCPs and the last two time horizons, for the worst case scenario (99.9 SLR percentile). Each white dot in the demand diagrams corresponds to the average value of the flow capacities for each of the four GCM. Flooding reduces the flow capacities, (white dots move to the lower left of each diagram). At the same time, the region with maximum stable demand below 1 increases (red region increases). Interestingly, the results are similar for RCP 4.5 within 20802100 and RCP 8.5 within 20602080, where the flow capacities are at the border of the lowdemand region. For RCP 8.5 within 20802100, the estimated flow capacities are completely inside the lowdemand region, indicating that the system is far from being able to supply the full demand.
Topological constraints on the stable demand
While the diagrams in Fig. 3b depict the effect of coastal flooding on the maximum stable demand, they depend on the specific dynamics considered (Fig. 2a). Inspired by the generalized modeling framework^{8,9}, we consider a metric that only takes into account the capacity structure of the network, and thus consider the minimum requirements over the stable demand (Eq. (3)), necessary over any functions Π, Δ and Ψ. We estimate the percentage of scenarios under which the system is able to provide a given demand level. For this purpose, we take into account the ranges of values for each of the original parameters alongside the different SLR scenarios. We use Eq. (3) to measure the ability of the system to sustain a given stable demand level Δ^{*}, and calculate the percentage of scenarios where the condition is satisfied. The results are presented in Fig. 3c, for the three SLR percentiles, the last two time horizons and the two RCPs. While for RCP 4.5 within 20602080 there is only a slight chance of failing to supply 100% of the original demand (Δ^{*} = 1), the percentage of scenarios unable to sustain 100% of the demand increases to 60% (for the 99.9 percentile) within 20802100. For RCP 8.5 the percentage is around 60% (for the 99.9 percentile) for 20602080, increasing to 100% for any demand over 50 of the original demand.
Production interruption under IPCC scenarios of Sea Level Rise
The previous analysis depicts the impact of permanent, longterm changes that affect the topology of a resourceflow network by modifying the size relation between layers and their flow capacities. These longterm changes may interplay with other, shortterm, failure events. As an example, we consider production interruptions, that could be associated with a stoppage of supply to the refineries. We model this failure event by setting p = 0 in the model during a finite amount of time ΔT, ranging from half week to three weeks, and then reestablishing its original value. We initialize the system in a stable state (associated with a particular value of U), and set p = 0 during a timelapse ΔT. As the demand is constant, the third layer decreases its stock level at rate d. Depending on the duration of the failure, we observe three outcomes from the production failure (Fig. 4a). For small ΔT (Fig. 4a, ΔT = 0.5 and 1.5 weeks), the total amount of resources is reduced, but none of the layers reach nearzero stock levels. Notice how, independently of which layer reaches the lowest stock level during the failure, once the failure resumes, the stock levels reorganize based on the new value of U. For intermediate ΔT (Fig. 4a, ΔT = 2 weeks), the failure may reduce U up to a point where the y^{3} ≈ 0, even after production is restored. This is a side effect of the redistribution of resources across layers, as demonstrated in Fig. 2e, where it is shown how at low values of U, y^{3} ≈ 0. As expected, for large ΔT, all the layers are depleted before the failure ends (Fig. 4a, ΔT = 5 weeks). Notice that when y^{3} ≈ 0, Δ(y^{3}) starts to decrease, and thus we observe a demand failure. As the system is initially producing at maximum capacity, when production is restored the system stabilizes in a lower level of resource, instead of going back to the original state. To increase the total resource U the system would require to have a higher production than demand. Under our assumptions, this only happens if demand fails, allowing production to accumulate in the different layers.
We analyze two aspects of production failure: the time that it takes the system to reach demand failure τ (that is, the time it takes for the third layer to reach y^{3}(τ) ≈ 0), and the average demand level during failure Q_{D}:
Where T is the time period under consideration. Notice that it is not required for T to be equal to the failure duration, as the aftermath can be of relevant to. These two metrics capture part of the dynamic aspects of the failure, moving one step forward from stable state analysis. To study the behavior of the system under this transient failure, we simulate the evolution of the system using the average values of each parameter (for each considered scenario), and initialize the stock levels y^{q} at stable states with different values of U. Recall that, as depicted in Fig. 2c, stable stock levels y^{q} depend on the initial resource stored in the system U(t = 0), and are limited by the flow capacities s_{qr}. For each coastal flooding scenario, we consider failure duration ΔT from 0.5 to 3 weeks. We use this set of simulations to measure the time to demand failure τ and the average demand level Q_{D}.
To study the time to demand failure τ, we consider the longest failure duration ΔT = 3 weeks and measure at which time the system stops being able to supply the required demand. We use ΔT = 3 weeks as all the considered settings reach failure within that time. Notice that this corresponds to the bottom right of Fig. 4a. Figure 4b shows the value of τ as a function of the initial total resource in the network U(t = 0). Obviously, τ increases with U(t = 0) (if the system has more resources stored, it will take more time for it to get depleted). As the network is impacted by coastal flooding, we see two effects. First, for the same value of U(t = 0), the value of τ is lower as we consider farther time horizons. At the same time, the change in parameters reduces the range of stable values for U, producing a smaller range of values for U(t = 0), undermining the possibility of starting from a more stock full stable state. As we consider scenarios with higher levels of coastal flooding, we see a decrease of the higher τ from ≈ 3 weeks, to approximately 1.5 weeks in 20802100 with RCP 4.5 and the 50 SLR percentile. A similar value is found for 20602080 at RCP 8.5 and the 50 SLR percentile and RCP 4.5 with the 95 percentile of SLR. Once the network is unable to provide full demand, the range of U(t = 0) concentrates at a single point, and τ ≈ 0 for that value. This can be seen in Fig. 4b, for 20802100 at RCP 8.5 and 50 SLR percentile, and for the 20602080 at RCP 4.5 and 8.5 for the 95 SLR percentile. Comparing with Fig. 2e, we see that as the flow capacities are reduced, the stable stock level of refineries reaches 1, and the stock level of terminals stabilizes at an intermediate value, dependent on the specific value of the flow capacities. Fuel is not able to reach the gas stations, and thus the system is essentially at demand failure.
While τ estimates how long the system will last without failing, Q_{D} estimates its performance during a given failure. Notice that while τ measures directly a characteristic of the production failure (when the system stops being able to supply full demand), Q_{D} provides an overview on the restoration of the demand as well, as it takes into account the value after the demand has been restored. As all the considered scenarios reach failure after 3 weeks, we choose T = 3 weeks in Eq. (4). This way, we compare all the failure evolutions under the same time period. As demand failure depends on the value of U at the start, we average Q_{D} over the possible values of initial resource U. Figure 4c shows the values of Q_{D} for the different failure duration ΔT and scenarios considered. We observe a common behavior: for short failures (ΔT < 1 week), Q_{D} ≈ 1. Then, as ΔT increases, Q_{D} transitions to a linear decreasing function of ΔT. This means that for short failures the demand is not affected, and for long failures the decrease is linear in the failure duration, in accordance with Fig. 4a. For all the scenarios considered, the total decrease in Q_{D} is approximately 10% at ΔT = 3 weeks. When comparing Q_{D} as a function of ΔT for the different SLR scenarios, we observe very similar curves, with a lower Q_{D} at a given ΔT as we consider further time horizons. The curve only changes slightly for system capable of providing full demand, with the highest difference around 5% for ΔT = 3 weeks. For failed systems, Q_{D} is noticeable below 1 for ΔT = 0.5 week, as the demand is heavily impacted by the coastal flooding.
Limitations and advantages of dimensionreduction for sociotechnical systems
We have presented on the application of dimensional reduction to resource flow systems, particularly exemplified by the transportation of fuel in the San Francisco Bay Area. The approximation aggregates the facilities based on their stock capacities, producing systemlevel variables that capture layerlevel dynamics. It allows studying a very complex networked system, including thousands of different facilities from an aggregated perspective, even with very limited information. As it is discussed in the Supplementary Note 1, the approximation still works well under random perturbation of the different types of parameters. However, by aggregating the system we ignore internal correlations that may exist in it (for example, correlations between production and flow capacities). Systems with significant correlations between their parameters (i.e. stock, production and demand capacities) might not be well captured by this aggregated approximations. The Supplementary Note 1 includes testing of the approximation for different in degrees of gas stations, varying to how many terminals and refineries they connect to. The test shows that as long as gas stations connect to more than 10 terminals, the approximation has less than 5% error. Even more, the approximation still works well under small variations of the parameters (while still maintaining the same average value).
From the perspective of the applications to real systems, we consider that dimensionreduction supposes two contributions to the analysis of large, interconnected sociotechnical systems. First, it allows providing estimates of the impact of failures even with very limited information. Of course, in the case of trying to precisely estimate the impact of extreme events on a particular network, researchers should look for the most detailed information possible. Nevertheless, the procedure discussed here is independent on the specific functional forms considered. Second, dimensional reduction can be extremely useful as we consider systems of systems, where each system is already complex on its own. Simplifying the subsystems can serve as starting point for assessing the behavior of a system of systems, by including them from an analytically tractable point of view, and assess their interactions as a whole.
Discussion
We presented an analysis of the demand levels that the San Francisco Fuel Transportation Network is able to satisfy under different coastal flooding scenarios. We focused on the demand stability under the perturbations induced by coastal flooding, and its dynamic behavior under a production failure. We approximate the system’s dynamics by reducing its dimension, focusing on the aggregated flow between refineries, terminals and gas stations. By working with the dynamical representation of the system, changes in network topology can be directly related to its role as resource supplier, helping to reduce the gap between traditional network measures (like number of connected components or global efficiency) and realistic policies (like demand rationing), more accessible to decisionmakers.
By considering layer dynamics through the dimensionreduction approach, we find that the space of stable macroscopic flows is constrained by a relationship equivalent to the maximum flow theorem between layers. This result adds up to the findings for mutualistic interactions in^{11}, which demonstrate how the stability of the system is connected to the percolation of the mutualistic interaction through the network, allowing the system to recover from reductions in the abundance of each species. Further analysis on other network structures can expand the understanding on how network topology shapes the dynamics of networked ordinary differential equations.
In the case study considered here, the assumptions used for the approximation are reasonable as the pipeline system and road networks allows transporting fuel essentially from any facility to any other, while at the same time we do not have access to greater detail on the flows between facilities. However, in more complex structures, with higher levels of heterogeneity, other systemstate variables may be preferred. The goal of the dimensionreduction framework is to map the highdimensional dynamics to a smaller set of relevant system variables, which may be different depending on the problem, and thus they should be adjusted depending on the case under consideration. For example, the statevariable considered here preserves flow properties typical from resourceflow systems, while the estimator used in^{11} is specially designed to put emphasis on hubs, which are crucial for percolation of interactions.
Our results indicate that the structure of the San Francisco Fuel Transportation Network would be able to sustain the current demand during the period 20202060. Yet, the current structure will likely start being affected after that period, specially under high levels of greenhouse gas emissions. Through failure simulation, we find that the maximum survival time without production decreases from three weeks to one and a half even in the bestcase scenario (for a three weeks production stoppage). However, as only very limited information is available on the network dynamics, these values should be considered as broad estimations. It would be interesting to consider the dimensionreduction approach in other transportation networks, where more information is available, and the effect of differences on production, stock, demand and flow capacities can be compared with data.
We show that the dimensionreduction approach can help understand better the dynamics of lifeline systems, by constructing analytically tractable representations that capture their macroscopic behavior. While these simplifications average the details on finer scales, they allow studying the effect of changes in the infrastructure from a system level perspective. In future analysis, it would be interesting to explore through the dimensionreduction approach how different lifeline infrastructures interact with each other, forming systems of systems. The reduction can help by simplifying the dynamics of each subsystem, while keeping tract of their interactions. Steps in this direction have already been done in the context of Ecology^{12}, by considering the interaction between network communities. Increasing our understanding of how dynamics of interconnected lifeline infrastructure are affected by climate extreme events is crucial for preparation for the changes our world is currently facing.
Methods
The San Francisco Fuel Transportation Network
The network topology of the SFFTN has been analyzed in a previous project by He et al.^{21}. The spatial elements conforming the network were obtained from multiple sources. Location of refineries and terminals was provided by the U.S. Energy Information Administration. The location of gas stations was obtained through Google Places. Pipeline structure was obtained through the U.S. National Pipeline Mapping, and roads were downloaded from OpenStreetMaps^{29}. To connect pipelines and roads with the three different facilities, the nearest node in each transportation network was connected to each facility (only roads in the case of gas stations). This provides the spatial representation necessary to assess the impact of coastal flooding.
Layered dimension reduction
Following previous work from Gao et al.^{11} we construct an approximation that captures the average behavior of the different layers in the transportation network. The dimensionreduction approach was originally developed in the context of the BarzelBarabasi equation,
that considers an homogeneous interaction across all the elements of the system. In the Eq. (5), F and G represent self and pair interactions, and A_{ij} is the adjacency matrix associated to the interaction graph. Effective variables are constructed through weighting each variable based on their connectivity in the network. However, in the context of resourceflow networks, interactions have to take into account incoming and outgoing flow to adjacent nodes. Thus, an extension of the original approach is necessary. From the point of view of system stability analysis, dimensionreduction operates by constructing a single variable representing the system’s global state. This variable is a function of the constituents of the systems. In^{11}, the systemlevel variable is constructed by weighting each variable based on its connectivity. This leads to a systemaverage that puts emphasis on more connected nodes. While this makes sense for mutualistic interactions, it is not the only option possible. For the purpose of this work, we will consider that the importance of a node is dependent on their stock capacity C^{q}, which is a property of the facility instead of its connectivity. While weighting each node by its degree (or strength, equivalent to its total flow capacity in the context of resourceflow) puts the focus on the node that has the highest connectivity, weighting by node’s stock capacity we preserve the connection between our system variables and the total amount of resource stored in the system (essential in the context of flow of resources). At the same time, if we wanted to weight each node based on their connectivity, it would be necessary to distinguish between incoming and outgoing flow capacities. This would lead to an asymmetric estimator, which considers each layer differently depending on which side we are looking at. By weighting by the node’s stock capacity, we obtain a symmetric weighting, independent of the flow direction.
In the dynamical description of the transportation network, the state of each facility is described by \({x}_{i}^{q}\), the stock level of node i in layer q. Here, we derive the equations departing from a slightly more general point of view than the one described in section Dynamical representation. We consider that every facility has its own stock capacity \({C}_{i}^{q}\), production capacity P_{i} (in refineries) and demand capacity D_{i} (in gas stations). We derive the equations with more generality looking to make the proposed approximation more clear, while for the application discussed in the Results we simplify by considering every capacity equal due to the lack detail in the data. The goal is to obtain a set of equations capturing the dynamics of the fuel stored at each of the three facility layers. The stock level of facility i in layer q is represented by \({x}_{i}^{q}\). The evolution of the variables \({x}_{i}^{q}\) is described by the set of ordinary differential equations:
where the \({C}_{i}^{q}\) is the stock capacity of facilities in layer q, P_{i} the production capacity of refinery i in layer 1, D_{i} the demand capacity of gas station i in layer 3 and \({W}_{ij}^{qr}\) is the flow capacity between node i in layer q and node j in layer r. The total amount of resource in layer q is \({\sum }_{i = 1}^{{N}_{q}}{C}_{i}^{q}{x}_{i}^{q}\), and thus the stock level of layer q is \({y}^{q}=\frac{1}{{N}_{q}{C}^{q}}{\sum }_{i = 1}^{{N}_{q}}{C}_{i}^{q}{x}_{i}^{q}\), where \({C}^{q}={\sum }_{i = 1}^{{N}_{q}}{C}_{i}^{q}\). To obtain a differential equation for the layer stock level y^{q}, we take the derivative and approximate Eq. (6). Then, we have the three equations:
Following the common steps of the meanfield approximation, we assume that the correlation between stock, production and flow capacities, and the values of the functions Π(x), Δ(x) and Ψ(x, y) are low. Then, we can approximate the average of the products with the product of the averages. For the production term, we have:
To link the average of functions \(\Pi ({x}_{i}^{1})\) with the layer stock level y^{1}, we recall the results from^{13}, where the authors propose to make a polynomial decomposition of the functions, and subsequently repeat the approximation over the expansion coefficients. In that case, we have (to first order, the three dots represent the next terms in the expansion):
Where in the third line we approximated \({\sum }_{i = 1}^{{N}_{1}}{x}_{i}^{1}\approx {\sum }_{i = 1}^{{N}_{1}}{y}^{1}{C}^{1}/{C}_{i}\). Thus, the approximation leads to a layer production function \({\sum }_{i = 1}^{{N}_{1}}{P}_{i}\Pi ({x}_{i}^{1})/{N}_{1}\approx P\tilde{\Pi }({y}^{1})\), where P is the refinery’s average production capacity and \(\tilde{\Pi }({y}^{1})\), which is an average of the production levels, weighted by the inverse of the stock capacity of each layer. To first order, and under the assumption of low correlation between the polynomial expansion coefficients and the stock levels, the coefficient involving the capacities is \({C}^{1}{\sum }_{i = 1}^{{N}_{1}}1/{C}_{i}^{1}\). This rescaling accounts for the variability of the capacities across facilities (and thus, for the different scales of each production function). In the context of our work, where we consider that all the refineries have the same stock capacity, the function \(\tilde{\Pi }\) matches exactly the function Π. In general, the quality of the approximation should be tested for other functions Π and stock capacities based on the case under study. The derivation discussed is equivalent to the one for \({\sum }_{i = 1}^{{N}_{3}}{D}_{i}\Delta ({x}_{i}^{3})/{N}_{3}\approx D\tilde{\Delta }({y}^{3})\).
For the flow terms, we follow a similar procedure, with the minor difference that summation is done over the two layers:
Where \({W}^{qr}={\sum }_{i = 1}^{{N}_{q}}{\sum }_{j = 1}^{{N}_{r}}{W}_{ij}^{qr}\) is the total flow capacity between layers q and r. To approximate function Ψ and obtain function \(\tilde{\Psi }\), the approximation steps are as follows (again to first order, with dots representing higher orders in the Taylor expansion):
In this case, the approximation yields the term \({W}^{qr}\tilde{\Psi }({y}^{q},{y}^{r})/{N}_{q}\). This term includes function \(\tilde{\Psi }\), the average flow level from layer q to layer r, and the average total flow capacity of a facility in layer q to the whole layer r. For the case in which we consider incoming flow instead of outgoing, the expression slightly changes, as we are summing over layer r instead of over layer q. This leads to the expression \({W}^{qr}\tilde{\Psi }({y}^{q},{y}^{r})/{N}_{r}\), and thus the global flow level is multiplied by the average (in)flow capacity perceived by nodes in layer r, considering all nodes in layer q. As in the previous case, when we consider all the nodes in the same layer to have the same stock capacity, function \(\tilde{\Psi }\) matches Ψ. Thus, we obtain the final set of equations:
where we recall that, in the case of equal stock capacities (\({C}_{i}^{q}={C}^{q}\)), \(\tilde{\Pi }=\Pi ,\tilde{\Delta }=\Delta\) and \(\tilde{\Psi }=\Psi\). For the rest of the Methods section, we assume that \({C}_{i}^{q}={C}^{q}\) and thus \(\tilde{\Pi }=\Pi ,\tilde{\Delta }=\Delta\) and \(\tilde{\Psi }=\Psi\).
To take this set of equations to a more compact form involving less parameters, we divide each equation by the corresponding stock capacity, and rename the combined parameters. This leads to:
where:

p = P/C^{1} is the normalized production capacity, and d = D/C ^{3} is the normalized maximum suppliable demand (which we call demand capacitiy).

\({s}_{qr} = {\sum }_{i = 1}^{{N}_{q}}{\sum }_{j = 1}^{{N}_{r}}{W}_{ij}^{qr}/{N}_{q}{C}^{q}\) is the normalized average flow capacity from layer q to layer r.

α_{qr} = N_{r}C^{r}/N_{q}C ^{q} is the stock capacity ratio between layers q and r.
The quotient between average normalized flow capacities and total capacity ratios represents the different perspective on the average depending on the layer: while s_{12} can be seen as the average outflow capacity of a refinery, s_{12}/α_{12} = W^{12}/N_{2}C^{2}, the normalized average inflow capacity at terminals.
The (normalized) total resource stored in the system is U = y^{1} + α_{12}(y^{2} + α_{23}y^{3}), while N_{1}C^{1}U is the total resource stored in the original units. Thanks to the internal balance of flows within the system, we have:
Which represents the global balance of resource in the whole system. Notice that the balance of resource stops changing (\(\dot{U}=0\)) when production and demand balance each other, and that the change in time of U does not depend on the stock level of the intermediate layer.
Stable state
To identify the stable state we set y^{q} = 0 in Eq. (13), and identify the conditions for stability. As we haven’t precised functional forms for Π, Δ and Ψ yet, we focus on the conditions over these functions first. If we call y^{q*} the values of the (average) stable stock levels at layer q, and we define Π(y^{1*}) = Π^{*}, Δ(y^{2*}) = Δ^{*} and \(\Psi ({y}^{q* },{y}^{r* })={\Psi }_{qr}^{* }\), then:
The solution to these equations only exists if the production and demand are balanced,
Under this condition, the stable flows are define by Δ^{*} (or equivalently by Π^{*}).
We can see that all the stable flows are located within a line in the 3dimensional space of flows for a fixed value of Δ^{*}. This ambiguity comes from the existence of two possible paths through which the flow can go. As total flow balance is the only condition for stability, many internal flows are possible. Notice that the range of values for \({\Psi }_{12}^{* }\) is dependent on the different capacities and the demand level Δ^{*}. As each flow level \({\Psi }_{qr}^{* }\) is constrained to be between 0 and 1, we obtain the constraint for \({\Psi }_{12}^{* }\):
We can see that increasing Δ^{*} moves the lower range up, and increasing the flow capacity s_{12} moves the upper limit down. If we look for the border condition at which there is only a single possible value for the stable flow (and thus maximum and minimum in Eq. (18) are equal), we find the requirement:
This equation is equivalent to the maxflow theorem condition between the three considered layers. In this case, the total demand α_{12}α_{23}d is equal to the sum of the flow capacities of the direct path from refineries to gas stations, s_{13}, and the long path from refineries to terminals, and terminals to refineries, \(\min ({s}_{12},{\alpha }_{12}{s}_{23})\). Comparing these result with the findings from^{11}, we can see two examples of how network structure shapes the stable state spaces of the differential equations. In^{11}, the mutualistic interaction considered has a positive effect on the pair of species involved (i.e. it increases the number of individuals of both species). The dimensionreduction captures the connectivity of the network in a single coefficient, and there is a critical point above which the mutualistic interaction percolates, making the system resilient to sudden reductions in the abundance of each species. In the context of resourceflow, the dimensionreduction captures the maximum flow condition limiting the stable flows^{28}.
Network capacities
Precise information on production, flow and demand capacities (\(P,{W}_{ij}^{qr}\) and D in Eq. (1)) is not openly available. However, CEC provides total production for Northern California and average daily number of trucks^{30}. We use 97.5 Mgal as a reasonable estimation of the total production capacity N_{1}P at the five refineries in Northern California. To estimate the demand capacity, we consider its average value as N_{1}P/N_{3}, the total production divided the number of refineries. This equates to the assumption that, on average, the system is stable (total production is equal to total demand). To obtain an estimation of the average stock capacity C^{1} of the refineries, we consider the maximum total stock reported by Northern California at the refineries, provided by CEC. The maximum total stock observed is equal to 191 Mgal, or 38.2 Mgal on average per refinery. To explore a wide range of values, we consider C^{1} ∈ [38.2, 57.3] Mgal. For the terminal’s stock capacity C^{2}, we consider the average of the stock capacities provided by Kinder Morgan^{31}, equal to 31.1 Mgal, and consider the range C^{2} ∈ [31.1, 62.2] Mgal. For the gas stations stock capacity, we consider a single value of C^{3} = 0.035 Mgal, based on the information publicly provided by U.S. Oil Spill Prevention organization.
Direct estimation of individual flow capacities would require knowing policies regarding the number of trucks that a each facility can contact. However, under the dimensionreduction approximation, we can provide estimates for the total flow capacities W^{12}, W^{13} and W^{23}. According to CEC, there are approximately 5000 tanker truck trips per day. We assume all them occur between terminals and gas stations. We consider that the storage capacity of a tanker truck ranges between 3 and 7 thousands of gallons. Then, we consider the range for the total flow capacity between terminals and gas stations, W^{12} ∈ [105, 245] Mgal week^{−1}. Flow from refineries to gas stations directly is also possible, while we assume it is comparatively smaller as in general it would be more costly. Thus, we consider a smaller range W^{13} ∈ [21, 81] Mgal week^{−1}. For the pipeline flow capacity between refineries and terminals, we calculate the minimal cut to separate all the refineries from all the terminals, finding it would require to remove 5 independent segments of pipeline in the original (unperturbed) network. We consider the flow capacity of a single pipeline to be between 2 and 4 millions of gallons per day, and thus we take as range for the weekly total flow capacity W^{12} ∈ [70, 140] Mgal week^{−1}.
Sea level rise and coastal flooding
By using sea level and climate projections between 2020 and 2100 from California’s Fourth Climate Change Assessment^{32} flood inundation time series maps for two Representative Concentration Pathways (RCPs) and four general climate or earth system models (GCMs), three probabilistic percentile estimates of SLR were created for a previous project^{21,23}. The four GCM are CanESM2, MIROC5, CNRMCM5 and HadGEM2ES, corresponding to the CMIP5 suite of models^{25}. The two RCPs considered are 4.5 and 8.5, representing a mild and a high level of greenhouse gas emissions. The four GCM are mathematical models that represent physical processes in the atmosphere, ocean, cryosphere, and land surface. The three probabilistic percentile estimates of SLR correspond to the 50, 95 and 99.9 estimates of SLR for a given time horizons and RCP. The SLR estimates are used as input for the 3Di hydrodynamic model^{26}, taking into account the elevation of the region considered and the tidal movement. The hydrodynamic model outputs the water column at 50m spatial resolution produced by tidal movement at each point of space during a high sea level event (i.e. a 72hour storm event with the highest sea level at a given scenario) for each of the scenarios considered (corresponding to a combination of SLR percentile, GCM, RCP and time horizon). A detailed description of the modeling process is well documented in reference^{23} under Appendix A. For the purpose of flooding impact analysis, we considered that a node was failed if the water height at its location was 15cm or higher, based on information provided by the fuel companies of the region.
Impact of coastal flooding
By overlaying flooding maps and network spatial representation, we identified failed nodes, removed them, and recalculated every aggregated parameter for each flooding scenario. We consider that stock and flow capacities are not affected by flooding, and thus changes in the parameters are due solely to changes in network structure. The changes in network structure are captured by the number of facilities of each type, the minimal pipeline cut to separate refineries from terminals, and the fraction of gas stations accessible from the average refinery and terminal through the road network, as explained below. A range of values for each one of the magnitudes is provided in Table 3 to provide an idea of the impact of SLR.
Total stock capacity ratios change solely due to changes in the number of nodes of each type, as stock capacities are considered fixed. For the pipeline flow capacities (associated with parameters W^{12} and s_{12}), we calculate the minimal cut between refineries and terminals under each flooding scenario, and consider as aggregated flow capacity the minimum cut times the flow capacity of a typical product pipeline. Minimum cut is reported in Table 3.
To estimate the reduction in flow capacity from terminals to gas stations (done by truck), we calculate the number of accessible gas stations for each terminal (through the road network). If the average terminal has fN_{3} gas stations accessible under a given flooding scenario (0 < f < 1), then we estimate the new total flow capacity \({W}^{23 \, {\prime} }=f{W}^{23}\) (with W^{23} being the original unperturbed flow capacity from terminals to gas stations). An equivalent calculation is done for the flow capacity from refineries to gas stations.
Data availability
The original network was constructed with privacyprotected information from the fuel companies involved in fuel transportation and thus cannot be shared. The flooding maps and results from the simulations used to make the plots are available at github.com/humnetlab/sfftndr. All other data can be made available on reasonable request.
Code availability
Simulations are based on ODE solver package deSolve (cran.rproject.org/web/packages/deSolve). The code used for analysis the simulations can be found at github.com/humnetlab/sfftndr.
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Acknowledgements
A.S., A.G. and M.G. acknowledge the support of the US DoD SERDP under grant Networked Infrastructures under Compound Extremes (NICE) (RC201183), and J.R. and M.G. acknowledge the support of C3.ai under grant Multiscale analysis for Improved Risk Assessment of Wildfires facilitated by Data and Computation.
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A.S., A.G. and M.G. conceived the project. Y.H. and J.R. reconstructed the original network and coastal flooding maps. A.S. did the analytical calculations, dynamical simulations and scenario analysis. A.S. and M.G. were the lead writers of the manuscript. All authors read and revised the manuscript.
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Salgado, A., He, Y., Radke, J. et al. Dimension reduction approach for understanding resourceflow resilience to climate change. Commun Phys 7, 192 (2024). https://doi.org/10.1038/s4200502401664z
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DOI: https://doi.org/10.1038/s4200502401664z
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