## Introduction

Resilience of complex networks is one of the most studied topics of network science, with an expanding literature on spreading of failures, mitigation of damage, and recovery processes1,2,3,4,5,6. The level of functionality of a network is typically quantified by its connectedness, e.g., size of the largest component1, average path length7,8, or various centrality metrics9. Such simple topology-based metrics ensure mathematical tractability and allow us to analyze and compare networks that can be very different in nature, providing general insights into the organization of complex systems. However, such a perspective necessarily ignores important system-specific details. For example, abstracted topological models of infrastructure networks recovering from damage or catastrophic failure aim to rapidly restore the largest component10,11,12,13. But, extensive connectivity is not a necessary condition to guarantee that all supply and demand can be met. For instance, consumers of a power grid can be served if they are connected to at least one power source and that source satisfies operational constraints14. The concept of “islanding”, a technique of intentionally partitioning the network to avoid cascading failures, is actually a practical strategy used to improve security and resilience during restoration efforts in power grids15,16,17. Indeed, after the 2010 earthquake in Chile the recovery process first created five islands, which were only connected to each other in the final steps of reconstruction18.

The restoration of critical infrastructure operation after a catastrophic event, such as a hurricane or earthquake, is a problem of great practical importance and is the focus of a significant body of work in civil and industrial engineering disciplines. The goal of engineering based models of recovery is to provide system-specific predictions and actionable recommendations. This is achieved by incorporating component level details and realistic transmission dynamics into the models, often in the form of generalized formulations of network design problems (NDPs), which satisfy network flows. In this context, objective functions of such NDP-based models aim to minimize the construction and/or operational costs of recovering edges and nodes in a utility network. Basic forms of the NDP have been well-studied19,20, and have recently been combined with scheduling and resource allocation problems to model the entire restoration process21,22,23. While these models provide a principled manner to obtain optimal, centralized recovery strategies, their complexity (at least NP-complete)19 renders computation not scalable, and interpretation restricted in scope to small instances. More efficient approximate solutions for NDPs have been found using optimization meta-heuristics such as hybrid ant system24 and gradient descent25 methods. Such algorithms are generally applicable to global search problems and were designed to reduce computational complexity by not guaranteeing optimality; therefore, they provide limited insight into the mechanism of network formation during recovery. We will analyze the output of an NDP algorithm, and leveraging on these observations, we will develop a percolation-based model for network recovery with the goal of uncovering important principles of network formation and recoverability.

Percolation processes, often used for studying properties of stochastic network formation, have recently been applied to network recovery26. In the kinetic formulation of random percolation27, we start with N unconnected nodes and consider a discrete time process. At each timestep, an edge is selected from the set of all possible edges at random, and added to the network. Initially the largest connected component (LCC) is sublinear in N; above a critical edge density it spans a finite fraction of the network and the LCC is referred to as the giant component. Controlling location of the critical point is of great interest in many systems–for instance, suppressing the formation of the giant component may reduce the likelihood of virus spreading in social contact networks. This can be achieved by selecting M > 1 candidate edges at each timestep, and adding the edge that optimizes some criteria. The general class of models that results from this choice is referred to as competitive percolation or an Achlioptas process28,29. While simple, Achlioptas processes often have the benefit of being scalable, numerically analyzable, and provide a parameter, M, for tuning how close the formation process is to matching the desired criteria. Note that when M is equal to the number of possible edges, we always add the edge that is optimal with respect to the selection criteria. Previous percolation-based recovery models typically measure solution quality by how quickly the LCC grows11,12, or assume nodes which are not connected to the LCC to be nonfunctional10,13. However, empirical studies of recovery scenarios suggest that these assumptions do not apply to infrastructure networks after catastrophic scenarios18.

In this work, we aim to narrow the gap between topology-based recovery approaches and computationally difficult optimization approaches by incorporating features which mirror infrastructure restoration processes. We start by applying a generalized version of a well-studied NDP recovery algorithm22,23,30,31 to a small case study, and we identify that the satisfaction of demand is a key driving force in the initial periods of recovery, outranking operational efficiency and direct repair costs of network elements in importance. Motivated by this finding, we define a simple, competitive percolation-based model of recovery that aims to maximize the satisfaction of consumer demand in a greedy manner. We show that component size anti-correlates with the likelihood of further growth– leading to islanding and the suppression of the emergence of large-scale connectivity, similar to explosive percolation transitions29 and in contrast with traditional topological recovery models. We apply our recovery algorithm to synthetic power grids to systematically investigate how realistic structural features of the network affect the efficiency of the recovery process. We learn that high structural redundancy (related to the existence of multiple paths between nodes) allows for reduced total cost and faster recovery time; however, to benefit from that redundancy, an increasing amount of information needs to be considered at each step of reconstruction. We also study the role of the ratio of suppliers and consumers and find that decentralized supply generally benefits recovery efforts, unless the fraction of suppliers becomes unrealistically high. Our model deepens our understanding of network formation during recovery and of the relationship between network structure and recoverability. We anticipate that our work can lead to efficient approximations of the NDP algorithm by leveraging the important mechanisms uncovered by our competitive percolation model.

## Model

### Problem statement and the optimal recovery model

We are interested in the problem of restoring the operation of a critical infrastructure system after sustaining large-scale damage. The infrastructure network is represented by a graph $${\mathscr{G}}=({\mathscr{N}}, {\mathcal E} )$$, where $${\mathscr{N}}$$ is the set of N nodes corresponding to substations and $${\mathcal E}$$ is a set of E edges corresponding to transmission lines, e.g., power lines, water or gas pipes. We introduce the parameter di representing the commodity demand of node i: if di < 0, node i is a net consumer; if di > 0, node i is a net supplier. We normalize di such that the total consumption (or production) sums to unity, i.e., $$\frac{1}{2}{\sum }_{i=1}^{N}\,|{d}_{i}|=1$$. Following a catastrophic event, a subset of the network $${\mathscr{G}}^{\prime} =({\mathscr{N}}\,^{\prime} , {\mathcal E} ^{\prime} )$$ becomes damaged. We study a discrete time reconstruction process: in each timestep we fix one damaged component, and the process ends once the entire network is functional. Our goal is to identify a sequence in which to repair the elements such that the total cost of recovery is minimized. In this manuscript, we focus on the fundamental case where all links are damaged but nodes remain functional, i.e., $${\mathcal E} ^{\prime} = {\mathcal E}$$ and $${\mathscr{N}}\,^{\prime} =\varnothing$$.

Optimization frameworks are often used in order to explore the space of possible repair sequences and identify best solutions. We implement an algorithm for the time-dependent network design problem (td-NDP)22,23,30, which is a well known example of an optimization algorithm for network recovery developed by the civil engineering community. Out of the recovery processes we examine in this paper, td-NDP is the most realistic, and therefore the most computationally complex. It is formulated as a mixed integer problem which optimizes a cost that includes reconstruction costs of network components, operational costs, and penalties incurred for unsatisfied demand, while taking constraints on flows of commodities into account. In general, mixed integer programs are known to be NP-hard except in special cases. For the td-NDP this means that problems become exponentially harder as the size of the network to be reconstructed increases; therefore, it is common practice to break up the recovery into time windows of length T, and find the locally optimal solution in each window. Previous studies showed that often even T = 1 yields adequate approximation of the globally optimal solution32. Here, depending on the size of the network, we use up to T = 5, striking a compromise between computational efficiency and establishing a hard upper bound to the true but unknown minimum cost configuration. A formal definition of td-NDP is provided in the Methods section.

To uncover the key driving factors and properties of the recovery process, we apply the td-NDP to a representative example, the transmission power grid of Shelby County, Tennessee, which consists of 9 suppliers and 37 consumers (with 14 junction nodes where di = 0), connected by E = 76 transmission lines. Network topology and necessary parameters were obtained from refs22,23. Figure 1a shows the total repair cost as a function of time as we perform the td-NDP with T = 5 on a network that was initially completely destroyed, with cost broken down by the type of expense. We see that deficit cost (i.e., the penalty accrued for unsatisfied demand) is dominant and exponentially decreasing throughout the initial stages of recovery. Investigating how this impacts the growth of the components, Fig. 2a shows the commodity deficit or supply of each connected component throughout the recovery process, with circle sizes representing the component size, and colors representing if a component is over- (blue) or undersupplied (red). We see that the td-NDP process results in many small components initially, with relatively small surplus/deficit, and only towards the end of the process all components are joined. This is consistent with islanding techniques discussed in engineering practice. Our goal is to develop a simple and computationally efficient model of the recovery process that captures these key features.

### Competitive percolation optimizing for LCC growth

Previous topology-based recovery processes prioritize the rapid growth of the largest connected component (LCC)10,11,12,13. Models vary in the details, such as the type of failure (random, localized, catastrophic) and additional secondary objectives (such as prioritizing nodes based on population), but the metric for the quality of the solution is directly related to how quickly the LCC grows.

As a representative example of topology driven recovery strategies, we implement an Achiloptas process using a selection rule that maximizes the sum of the resulting component, which we refer to as LCC percolation. In this process, we randomly select M > 0 candidate edges out of the set of damaged edges at each discrete timestep. We then examine the impact that repairing each individual edge would have and select the edge that, when added, creates the largest connected component. More specifically, let si denote the size of the component to which node i belongs. If nodes i and j belong to separate components, repairing edge (i, j) creates a component with size Sij = si + sj; if they belong to the same component, the size of the component does not change and we set Sij = 0. Out of the M candidate edges, we repair the one that maximizes Sij; if multiple candidate links have the same maximal Sij, we select one of them uniformly at random. If M = 1, the process is equivalent to traditional percolation. If M = E, the process is largely deterministic, we always repair an edge that is optimal with respect to the selection criteria.

We now apply LCC percolation as a model for the recovery of the Shelby County power grid and compare the results to the benchmark td-NDP process. Figure 2b shows that if we use growth of the LCC as our objective, the LCC, represented by the largest circle, grows rapidly throughout the process as expected. However, the deficit/surplus of this component fluctuates greatly. As indicated in Fig. 1b by the magnitude of total commodity deficit D, i.e., the total unsatisfied demand in the network, such a recovery algorithm is costly and leaves large portions of the grid without power until the final steps of the recovery algorithm. To conclude, we find that a recovery process based on quick growth of the LCC is neither cost efficient nor effective for satisfying consumer’s demand as quickly as possible. As a result, we do not consider this algorithm further.

### Competitive percolation optimizing for demand satisfaction

We have shown using an example power grid that the key driving factor in the recovery process is the reduction of the total commodity deficit, i.e., the total unsatisfied demand, and that optimizing LCC fails to capture this. We anticipate that this also holds for the recovery of other critical infrastructure networks, such as gas and water supply networks. To capture the essence of real recovery strategies, we propose a competitive percolation process which we refer to as recovery percolation that, instead of optimizing for LCC growth, aims to directly reduce the unsatisfied demand. In addition to network topology, this recovery process also takes into account the net demand or production of the individual nodes.

We define Di as the commodity deficit of the connected component to which node i belongs. We assume that capacity constraints of the transmission lines are sufficient and thus do not limit the flow of commodities during the recovery process, a common practice in infrastructure recovery literature14,22,23. Therefore the commodity deficit of a component is the sum of demand or supply of individual nodes belonging to the component, i.e., $${D}_{i}={\sum }_{j\in {{\mathscr{C}}}_{i}}\,{d}_{j}$$, where $${{\mathscr{C}}}_{i}$$ is the set of nodes belonging to the component containing node i.

We use commodity deficit of the components as a selection criteria for the competitive percolation model to account for the goal of balancing supply and demand. Similar to LCC percolation, we randomly select M > 0 damaged candidate edges, from which one is chosen to be repaired and added to the network at each timestep as follows. We first consider how much demand would be met by adding each of the M edges individually to the network. More specifically, if nodes i and j belong to components such that DiDj < 0, then repairing edge (i, j) reduces the total commodity deficit by ΔD = min (|Di|, |Dj|); if DiDj ≥ 0, then there is no commodity deficit reduction, i.e., ΔD = 0. Out of the M candidate edges, we repair the one that maximizes ΔD; if multiple candidates have the same maximal ΔD, we repair one of them chosen uniformly at random. If M = E the process always selects an edge that is optimal with respect to the selection criteria; if M is reduced, the process becomes more stochastic as it approaches 1. Overall, the computational complexity of this algorithm is O(EM), since E links are repaired and to repair each link M candidates are considered. This polynomial runtime is in contrast with the exponential td-NDP algorithm.

Figure 1b shows that for the Shelby County power grid the total commodity deficit during the recovery percolation for M = E = 76 well approximates the td-NDP, especially at the beginning of the recovery process when costs are much higher. We also see that even when M = 10, corresponding to only ~13% of the total edges considered at each timestep, the approximation remains very effective. Figure 2c shows similar dynamical behavior in recovery percolation as in the td-NDP solution (cf. Fig. 2a): larger components delay formation, and tend to have smaller commodity deficit.

## Results

In the following, we apply the recovery percolation model to various synthetic network topologies with realistic features to identify important mechanisms driving network formation and to understand how network structure affects the efficiency of recovery efforts. For each synthetically generated network, the demand distribution is chosen to approximate the demand observed in real power grids (details are provided in the Methods section).

### Recovery percolation on complete networks

We have shown that recovery percolation follows our benchmark td-NDP solution closely on a real-world topology. We also observed that the growth of connected components is suppressed via recovery percolation as compared to LCC percolation. To understand this behavior, we study large systems with N = 104 nodes and we allow potential edges to exist between any node pair, removing underlying topology constraints. Note that the td-NDP process is intractable for networks of this size.

Figure 3 (main figure) shows the growth of the LCC for a range of M values. For M = 1, the model reduces to random percolation which has a second-order phase transition at t/N = 0.5, and above this critical point the LCC becomes proportional to N. As we increase M, the apparent transition point shifts to higher values and approaches 1, indicating that the appearance of large-scale connectivity is suppressed; however, once the transition point is reached, the growth of the LCC becomes increasingly abrupt. This observation is analogous to explosive percolation, where links are chosen to be constructed explicitly to delay component growth29. In contrast, in recovery percolation it is an indirect consequence of a practical restoration strategy.

To understand the underlying mechanism of component growth, we plot the average component sizes and their corresponding average undersupply at various points during the reconstruction process in the bottom row of plots in Fig. 3. Note that average oversupply behaves in a similar manner, but is omitted for clarity. The left column of plots in Fig. 3 shows the same quantities such that the size of the LCC is fixed. The main trend we observe at any given point in time is that for large enough M there is negative correlation between component size and undersupply, and this correlation becomes stronger as M increases. This means that as components grow their commodity deficit is reduced and therefore the likelihood of further growth is also reduced, ultimately suppressing the appearance of large scale connectivity.

The observed two features also describe islanding, an intentional strategy in resilience planning and recovery in real-world power grids. This islanding behavior is already observed in early stages of the restoration process, becoming more apparent as t approaches the transition point.

### Recovery percolation on synthetic power grids

So far we investigated the recovery process on an underlying graph without topological constraints. We also wish to analyze more realistic networks and turn our attention to synthetic power grids33,34. This allows us to systematically investigate how typical structural features of power grids affect the efficiency of the recovery process.

Power grids are spatially embedded networks, and physical constraints, particularly at the high-voltage transmission level, limit the maximum number of connections a node can have. Their degree distributions, therefore, have an exponential tail, in contrast to many complex networks that display high levels of degree heterogeneity. Transmission power grids typically have average degree 〈k〉 between 2.5 and 334,35. An important requirement of power grids is structural redundancy, meaning that the failure of a single link cannot cause the network to fall into disconnected components. A network without redundancy has tree structure, has average degree 2 and all node pairs are connected through a unique path. Any additional link creates loops and improves redundancy. Structural redundancy can be characterized locally by counting short range loops. For example, power grids have a high clustering coefficient c, typically ranging between 0.05 and 0.134. The algebraic connectivity, denoted by λ2, is the second smallest eigenvalue of the network’s Laplacian matrix and captures a measure of global redundancy: it is related to the number of links that have to be removed in order to break the network into two similarly sized components, with high values corresponding to high redundancy. The exact value of λ2 depends on system size, where for a given number of nodes, λ2 is minimal for tree structure, and monotonically increases as further links are added36.

To generate networks that exhibit the features of typical transmission power grids, we use a simplified version of a practical model developed by Schultz et al.34. The model generates spatially embedded networks mimicking the growth of real-world power grids. The process is initiated by randomly placing N0 nodes on the unit square and connecting them with their minimum spanning tree. To increase redundancy, qN0 (0 ≤ q ≤ 1) number of links are added one-by-one connecting a random node i to node j, such that it minimizes the redundancy-cost trade-off function

$$f(i,j)=\frac{{({d}_{{\rm{net}}}(i,j)+1)}^{r}}{{d}_{{\rm{euc}}}(i,j)},$$
(1)

Changing parameters q, r, and s allows us to systematically explore how these parameters impact the structure of these model power grids (Fig. 4): q controls the average degree 〈k〉 = 2(1 + q) and adds redundancy to the network; r controls how loops are formed, where small r favors short distance connections leading to high c and low λ2, while large r favors long loops leading to low c and high λ2; and s increases typical distances in the network, lowering both c and λ2. Extreme choices of these parameters, however, may produce networks with unrealistic properties. On Fig. 4, we highlighted a realistic regime corresponding to the range spanned by a set of real networks with more than 500 nodes provided in ref.34.

Although later we focus on larger networks, for completness we note that using this network generation procedure we can create synthetic networks with approximately the same properties as the Shelby County dataset. The Shelby County transmission grid has number of nodes N = 60, average degree 〈k〉 = 2.53, clustering coefficient c = 0.078, algebraic connectivity λ2 = 0.073, and average shortest path length davg = 5.35. Setting the parameters to, for example, N = 60, N0 = 8, q = 0.27, r = 1, and s = 0.4, generates networks with c = 0.078 ± 0.038, λ2 = 0.059 ± 0.019, and davg = 5.13 ± 0.46, where the reported values are an average of 103 realizations and the error is the standard deviation.

#### Comparing recovery percolation and td-NDP

We first consider a set of parameters that yield typical topologies and compare the performance of recovery percolation to that of the locally optimal td-NDP recovery. We choose the parameters to create networks similar to the Western US grid following the specifications of ref.34 (N = 103, q = 0.33, r = 1, and s = 0). For the td-NDP analysis, we reduce the time window from T = 5, as used in the Shelby County model, down to T = 2 for tractability reasons since our synthetic networks are much larger (increasing T causes an exponential increase in complexity). Note that lower values of T lead to more localized search, resulting in a suboptimal solution. However, T = 2 is still a useful touchstone since (i) it represents the best estimate with the available tools and (ii) even T = 1 provides a reasonable approximation of the optimal solution (see Fig. 1b and ref.32). Figure 5a shows the growth of the LCC for the td-NDP process and recovery percolation varying M from 1 to 100. For recovery percolation we find similar behavior to what we observed for complete networks: as M is increased the growth of the LCC is suppressed and the formation of large-scale connectivity is delayed, but when it forms it grows more rapidly. For large M, the recovery percolation closely resembles the td-NDP recovery in terms of LCC formation.

As the dominant cost factor in recovery of infrastructure networks is the total commodity deficit D(t), this is the most important metric in network recoverability, beyond the size of the LCC. Figure 5b shows D(t) reduction throughout the recovery process. As M increases, we see a closer fit with td-NDP, especially in the more expensive early stages of recovery. We observe that M = 10 approximates total commodity deficit quite well, which is significantly less computationally intensive than td-NDP or even the deterministic version of recovery percolation (M = E). This is consistent with what we found for the power grid of Shelby County (Fig. 1b), although the difference between M = 10 and E is less drastic in that case.

To better understand how the choice of M affects the quality of recovery percolation, we calculate the total cost CM, defined here as the area under the curve D(t) over time (i.e., CM = ∑tD(t)) as a function of M. Figure 5c shows that CM rapidly approaches C, its value at M = E. For this particular case, we only need to consider M = 20, that is less than 2% of edges, at each timestep to get within 10% of the optimal cost.

It is worth highlighting that, for sufficiently large M (but still small compared to the total number of edges E), recovery percolation captures the essential properties of the td-NDP process for T = 2, despite the fact that recovery percolation only considers commodity deficit, while td-NDP takes into account such details as heterogeneous repair costs of individual power lines, operational costs, performs network flows, and selects optimal recovery actions considering two timesteps.

#### Effect of network structure on recovery percolation

Recovery percolation together with the synthetic transmission power grids provide a stylized model to extract key network features that impact the efficiency of the restoration process. For this we systematically investigate how typical structural features affect the following quantities:

1. 1.

Total optimal cost of recovery C, which is the minimum cost obtainable with recovery percolation (M = E).

2. 2.

Time to recovery t90, the number of timesteps needed to reduce total commodity deficit by 90% percent.

3. 3.

Characteristic M*, which captures the approximability of the process. It is defined as the smallest M value for which CM ≤ 1.2C.

Simulations show that redundancy q, which controls the average degree, has a strong effect (Fig. 6a–c). Increasing redundancy lowers both optimal total cost C and recovery time t90; however, it increases M*, meaning that to approximate the optimal solution more edges need to be sampled. This observation is robust to the choice of other parameters. Redundancy increases possible ways to reconnect the network, allowing less costly reconstruction strategies, but this also means that more paths must be explored to pick out the optimal one. The effect of r is more subtle, we find that long range shortcuts (r = 10) further decrease C and t90; while short cycles (r = 0) have the opposite effect. The value of r has little effect on M*.

The effect of line splitting depends on both the fraction of suppliers ps and the redundancy q (Fig. 6d–f). For centralized supply (ps = 0.05), we find that in case of low redundancy, s increases cost C and recovery time t90; while in case of high redundancy, s has the opposite effect, reducing C and t90. Line splitting s increases the characteristic M*, and this increase is particularly significant for low values of q. For distributed supply (ps = 0.3), we find that both C and t90 are decreased by s independently of the value of q. While the value of M* is increased by s for low q, and decreased for high q.

Finally, the fraction of suppliers ps also strongly influences the recovery process (Fig. 6g–i). Total optimal cost C and recovery time t90 are high for very centralized (low ps) and very distributed (high ps) supply, with a minimum in between. If the demand and supply follow the same distribution, the minimum is at $${p}_{{\rm{s}}}^{\ast }=0.5$$. For our choice, the demand is more heterogeneously distributed than the supply, resulting $${p}_{{\rm{s}}}^{\ast } < 0.5$$. Increasing ps, also allows easier approximation of the optimal solution, i.e., M* decreases with increasing ps (with the exception of a limited regime with high q and low ps and low q and high ps).

Overall, we find that high structural redundancy reduces the optimal cost and time of recovery; however, higher edge sampling M is needed to benefit from this reduction. Long range shortcuts in the network further reduce the cost, without significantly increasing M. We also benefit from distributed supply, reducing both cost and recovery time, and depending on the level redundancy, may also improve approximability.

## Discussion

We investigated the problem of optimal cost reconstruction of critical infrastructure systems after catastrophic events. We started by analyzing realistic recovery strategies for a small-scale case study, the power grid of Shelby County, TN. We identified the penalty incurred for over- and undersupply of commodities as the main contribution to the cost, outranking operational and repair costs by orders of magnitude in the initial periods of recovery. Motivated by this observation, we introduced the recovery percolation model, a competitive percolation model that in addition to network structure also takes the demand and supply associated with each node into account. The advantage of our stylized model is that it is computationally tractable and easy to interpret compared to the complex optimization formulations used in the engineering literature, while adequately reproducing important features of realistic recovery processes. This allows us to identify underlying mechanisms of the recovery process. For example, we showed that component size anti-correlates with the unsatisfied demand, which suppresses the emergence of large-scale connectivity through a process analogous to explosive percolation. Such a suppression of large-scale connectivity can be in fact observed in real recovery events18. The model also allowed us to systematically investigate the effect of typical network characteristics on the efficiency of the recovery process using synthetic power grids, finding that high structural redundancy, long range connections, and decentralized supply benefit recovery efforts.

Although our analysis focused on transmission power grids, recovery percolation is readily applicable to other transportation networks where the commodity transported is interchangeable (i.e., demand could be satisfied from different sources) such as gas and water infrastructure or supply-chain networks. Possible future work may extend recovery percolation to networks where the items transferred each have a specific destination, for example, passengers in human transportation networks37.

The computational complexity of identifying actionable reconstruction strategies is an open issue, especially in the case of interdependent and decentralized recovery scenarios, where systems are larger, and the optimization problem must be solved numerous times22,30,38. Our stylized model is efficient, but as such, it ignores certain details. For example, our approach assumes that link capacities are sufficiently large to service the network flows during the recovery process. Future work may explore how to extend recovery percolation to take into account such constraints. Similar to td-NPD, these strategies provide scenarios that may be useful for developing recovery operator based approaches to mathematically model the dynamics of recovery and enable development of data-driven control approaches39. Further work is needed to extend our model to simultaneous recovery of multiple critical infrastructure systems explicitly taking into account interdependencies between the systems. Competitive percolation strategies in general, can provide opportunities for modeling real-world processes. For instance, in addition to this application to recovery, there is recent work of applying competitive percolation strategies to suppress the outbreak of epidemics via targeted immunization40.

## Methods

### Time-dependent NDP

Here, we define our benchmark model for network recovery: the time-dependent network design problem (td-NDP). Our version follows the more general formulation developed by Gonzalez et al.22,23,30. The td-NDP takes a graph $${\mathscr{G}}=({\mathscr{N}}, {\mathcal E} )$$, where $${\mathscr{N}}$$ is a set of nodes, and $${\mathcal E}$$ is the set of edges connecting nodes. At the beginning of the recovery process the td-NDP uses the destroyed graph, $${\mathscr{G}}^{\prime} =({\mathscr{N}}\,^{\prime} , {\mathcal E} ^{\prime} )$$, where $${\mathscr{N}}\,^{\prime}$$ and $${\mathcal E} ^{\prime}$$ represents the nodes and edges that are not functioning, respectively. The objective function (cf. Eq. (2a)) minimizes the total reconstruction cost over a given time domain $${\mathscr{T}}$$ with $$t\in {\mathscr{T}}$$, which includes the cost to repair nodes, qit, cost to repair edges, fijt, cost of flow on each edge, cijt, and oversupply and undersupply penalties for each node, $${M}_{it}^{+}$$ and $${M}_{it}^{-}$$. These costs usually depend on multiple factors, such as the level of damage, the type and size of the components to be restored, their geographical accessibility, the amount of labor and resources required, and the social vulnerability of the affected areas, among others22,41,42. To keep track of demand satisfaction, each node also has a supply capacity (demand if negative), bit. In the most general formalization of the problem, node supply bit can depend on time t, but in this paper we only consider constant values. The variables $${\delta }_{it}^{+}$$ ($${\delta }_{it}^{-}$$) account for oversupply (or undersupply) of node i. We refer to the sum of the absolute values of oversupply and undersupply ($$|{\delta }_{it}^{+}|+|{\delta }_{it}^{-}|$$) as the commodity deficit of node i. The td-NDP includes as decision variables the amount of flow on each edge, xijt, whether or not a node i [edge (i, j)] is chosen to be recovered at timestep t, $${\tilde{w}}_{it}$$ ($${\tilde{y}}_{ijt}$$), and whether or not a node i [edge (i, j)] is functional at timestep t, wit (yijt). Constraints 2b2o are imposed to ensure that conservation of flow properties are held and that only recovered and functional nodes can produce or consume flow.

The td-NDP formulation is a mixed integer program, which has been shown to be, in general, NP-hard (and becomes exponentially harder as $$|{\mathscr{T}}\,|$$ and $$|{\mathscr{G}}^{\prime} |$$ grows). The number of variables and constraints also become larger as the input graph becomes larger. For many reasonable size problems, computing a global optimal (i.e., where $${\mathscr{T}}$$ contains the entire time horizon for recovery) is intractable. Therefore, heuristics are used to restrict the size of $${\mathscr{T}}$$ by dividing the total recovery time into smaller windows, and finding the locally optimal solutions within these windows22. It has been shown that such heuristic finds solutions very close to the optimal; however, the computational complexity is still relatively high as a result of the underlying mixed-integer program, underscoring the need for efficient approximate methods.

### Supply and demand distribution

For our computational experiments, we generate our demand distribution by following the load distribution of the European power grid42. This dataset was chosen due to its large system size (N = 1463, E = 2199) and its high resolution. Our goal is not to identify the true analytic form of the load distribution, but to generate statistically similar samples through bootstrapping. We found that an exponentiated Weibull distribution of the form f(x, a, c) = ac(1 − exp (−xc)(a−1)) exp (−xc)x(c−1)), where a = 3.59 and c = 0.8 well approximates the features of the demand distribution. Suppliers’ capacities are uniformly distributed to balance the total demand. We set the ratio of suppliers (0.3) to consumers (0.7) according this dataset, unless otherwise noted. We assume that the link capacities are sufficiently large to service the network flows in normal operation and during the recovery process.

$$\begin{array}{rcl}\mathrm{td} \mbox{-} \mathrm{NDP} & = & {\rm{minimize}}\,\sum _{t\in {\mathscr{T}}\,|t > 0}\,(\sum _{(i,j)\in {\mathcal E} ^{\prime} }\,{f}_{ijt}{\tilde{y}}_{ijt}+\sum _{i\in {\mathscr{N}}^{\prime} }\,{q}_{it}{\tilde{w}}_{it})\\ & & \,\,\,\,\,\,+\,\sum _{t\in {\mathscr{T}}}\,(\sum _{i\in {\mathscr{N}}}\begin{array}{c}({M}_{it}^{+}{\delta }_{it}^{+}+{M}_{it}^{-}{\delta }_{it}^{-})\end{array}+\sum _{(i,j)\in {\mathcal E} }\,{c}_{ijt}{x}_{ijt})\end{array}$$
(2a)

subject to,

$$\begin{array}{cc}\sum _{j:(i,j)\in {\mathcal E} }\,{x}_{ijt}-\sum _{j:(j,i)\in {\mathcal E} }\,{x}_{jit}={b}_{it}-{\delta }_{it}^{-}+{\delta }_{it}^{+}, & \forall \,i\in {\mathscr{N}},\forall \,t\in {\mathscr{T}},\end{array}$$
(2b)
$$\begin{array}{cc}{x}_{ijt}\le {u}_{ijt}{w}_{it}, & \forall \,(i,j)\in {\mathcal E} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2c)
$$\begin{array}{cc}{x}_{ijt}\le {u}_{ijt}{w}_{jt}, & \forall \,(i,j)\in {\mathcal E} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2d)
$$\begin{array}{cc}{x}_{ijt}\le {u}_{ijt}{y}_{ijt}, & \forall \,(i,j)\in {\mathcal E} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2e)
$$\begin{array}{cc}\sum _{i\in {\mathscr{N}}^{\prime} }\,{\tilde{w}}_{it}+\sum _{(i,j)\in {\mathcal E} ^{\prime} }\,{\tilde{y}}_{ijt}\le 1 & \forall \,t\in {\mathscr{T}}\,|t > 0,\end{array}$$
(2f)
$$\begin{array}{cc}{w}_{it}\le \mathop{\sum }\limits_{\tilde{t}=1}^{t}\,{\tilde{w}}_{i\tilde{t}}, & \forall \,i\in {\mathscr{N}}\,^{\prime} ,\forall \,t\in {\mathscr{T}}\,|t > 0,\end{array}$$
(2g)
$$\begin{array}{cc}{y}_{ijt}\le \mathop{\sum }\limits_{\tilde{t}=1}^{t}\,{\tilde{y}}_{ij\tilde{t}}, & \forall \,(i,j)\in {\mathcal E} ^{\prime} ,\forall \,t\in {\mathscr{T}}\,|t > 0,\end{array}$$
(2h)
$$\begin{array}{cc}{\delta }_{it}^{+}\ge 0, & \forall \,i\in {\mathscr{N}},\forall \,t\in {\mathscr{T}},\end{array}$$
(2i)
$$\begin{array}{cc}{\delta }_{it}^{-}\ge 0, & \forall \,i\in {\mathscr{N}},\forall \,t\in {\mathscr{T}},\end{array}$$
(2j)
$$\begin{array}{cc}{x}_{ijt}\ge 0, & \forall \,(i,j)\in {\mathcal E} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2k)
$$\begin{array}{cc}{w}_{it}\in \{0,1\}, & \forall \,i\in {\mathscr{N}},\forall \,t\in {\mathscr{T}},\end{array}$$
(2l)
$$\begin{array}{cc}{y}_{ijt}\in \{0,1\}, & \forall \,(i,j)\in {\mathcal E} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2m)
$$\begin{array}{cc}{\tilde{w}}_{it}\in \{0,1\}, & \forall \,i\in {\mathscr{N}}^{\prime} ,\forall \,t\in {\mathscr{T}},\end{array}$$
(2n)
$$\begin{array}{cc}{\tilde{y}}_{ijt}\in \{0,1\}, & (i,j)\in {\mathcal E} ^{\prime} ,\forall \,t\in {\mathscr{T}}.\end{array}$$
(2o)