Reviving a failed network via microscopic interventions

cell complex networked systems abrupt dynamic transitions nodes/links hysteresis in the state, their damaged topology. we develop a two-step recovery scheme: ﬁrst - topological reconstruction to the point where the system can be revived, then dynamic interventions, to reignite the system’s lost functionality. Applied to a range of nonlinear network dynamics, we identify a complex system’s recoverable phase, a state in which the system can be reignited by a microscopic intervention, i.e . controlling just a single node. Mapping the boundaries of this newly discovered phase, we obtain guidelines for our two-step recovery.

Step II. Reigniting (Fig. 1f). Introducing dynamic interventions to steer the system back to functionality.

The challenge in
Step II is that in most practical scenarios we lack direct control over the dynamic activity of the majority of the nodes. Hence we seek to reignite the system via micro-interventions, i.e. controlling just a small number of components, typically -just a single or, at most, few nodes. We, therefore, need to characterize the conditions under which such single-node reigniting can be achieved, representing the primary focus of the current contribution. Along the way, we expose a new dynamic phase of complex systems -the Recoverable phase, a state in which the system can be driven towards functionality by controlling a microscopic set of nodes.
Unlike the classic controllability challenge [26][27][28][29][30][31][32] , formulated in the context of linear dynamics, in recoverability we benefit from the nonlinear nature of the interactions. Indeed, when reigniting we do not aim to steer the system to a general arbitrary state, as done in the control theoretic setting, but rather towards one of its naturally occurring fixed-points -a problem of different nature, relying on fundamentally distinct mathematical tools 21 .
The challenge of irreversible collapse Consider a complex system of N components (nodes), linked through A ij , a sparse, binary and random network with arbitrary degree distribution P (k). In A ij the links represent dynamic interactions, whose nature depends on context. For instance, in a social system A ij = 1 captures an acquaintance, and hence a potentially infectious interaction between individuals i and j, whereas in a sub-cellular network it may represent a regulatory interaction between gene i and gene j. To account for these dynamic distinctions we denote each node's activity by x i (t), quantifying, e.g., individual i's probability of infection or gene i's instantaneous expression level. We then track the dynamics of the system via [33][34][35] a general equation, characterized by three potentially nonlinear functions M 0 (x), M 1 (x) and M 2 (x). The first function, M 0 (x i ) captures node i's self-dynamics, describing mechanisms such as protein degradation 36 (cellular), individual recovery 37,38 (epidemic) or birth/death processes 39 (population dynamics). The product M 1 (x i )M 2 (x j ) describes the i, j interaction mechanism, e.g., genetic activation 9,17,40 , infection 37,38 or symbiosis 41 . The mean strength of these interactions is governed by the weight w. In the context of recoverability, we seek to revive the activity of all nodes by activating a selected set of nodes, hence we focus on constructive interactions, in which nodes positively contribute to each others activity. This is expressed in Eq. (1) through wM 1 (x)M 2 (x) ≥ 0 (see Supplementary Section 1). Such conditions naturally arise in many relevant applications, social, biological or ecological, as exemplified in Figs. 3a and 4a,f.
Setting the derivative on the left hand side of (1) to zero, we obtain the system's fixed-points, x α = (x α,1 , . . . , x α,N ) , which, if dynamically stable, represent different states, desirable or undesirable, in which the system can naturally reside. Transitions between these states often result from perturbations to A ij and ω, such as node/link deletion or weight change. When this occurs, it is often difficult to reverse the unwanted transition, since the system fails to spontaneously recover, even if we retrieve the lost nodes, links or the weights. To illustrate this difficulty we refer to a concrete example below.
Example 1. Cellular dynamics (Fig. 3). As our lead example we consider gene-regulation, capturing activation interactions between genes. Such genetic activation is often modeled within (1), using Michaelis-Menten dynamics 40 (Fig. 3a). Under this framework M 0 (x i ) = −Bx a i , describing degradation (a = 1), dimerization (a = 2) or a more complex bio-chemical depletion process (fractional a), occurring at a rate B; below we set B = 1. The activation interaction is captured by a Hill function of the form , a switch-like function that saturates to M 2 (x j ) → 1 for large x j , representing j's positive, albeit bounded, contribution to i's activity x i (t).
For sufficiently dense A ij or large ω the system exhibits an active state x 1 , in which all nodes have positive activity (x 1,i > 0 for all i), representing a living cell. However, perturbations to the network topology, such as link/node removal or weight reduction can cause a sharp transition to the inactive state x 0 = (0, . . . , 0) , in which all nodes have zero activity, i.e. cell death. To track this transition systematically we measured the average activityx α = (1/N ) N i=1 x α,i , which followsx α > 0 for α = 1 andx α = 0 for α = 0, as we subject it to increasing levels of topological stress : removing a q-fraction of nodes (Fig. 3b); deleting a q-fraction of links (Fig. 3d) or reducing ω by a factor q (Fig. 3c). In all cases we observe a sudden transition, as the perturbations exceed a critical threshold q c , fromx α > 0 (green), representing the active state α = 1, tox α = 0 (red), representing the inactive state α = 0. To drive the system back to the desired x 1 , the instinctive approach is to reverse the topological perturbations, namely reconstruct the deleted nodes/links or push the weight ω to its original strength. In Fig. 3f we do precisely that, finding that the system fails to recover. The reason is that while x 1 is only stable above q c , x 0 is always stable -both below and above the critical point. This leads to a hysteresis phenomenon that causes the system to remain inactive, despite the reversal of the perturbation, e.g., recovering the original weight ω.
Example 1, above, while representing a specific scenario, illustrates the family of challenges that we tackle here: system's with irreversible topologically driven transitions, e.g., from a desired x 1 to an undesired x 0 . To revive such systems, we must dynamically reignite them by exerting external control over the activities x i (t), in order to drive them back towards the basin of attraction of x 1 (Fig. 1h).

Recoverability -single-node reigniting
After restructuring A ij , ω, the most natural way to reignite the system is to drive all activities x i towards an initial condition from which the system will naturally recover to the desired x α . Namely, we must steer the system into x α 's basin of attraction (Fig. 1h) which comprises all initial conditions x(t = 0) from which Eq. (1) converges to x(t → ∞) = x α . The problem is that such level of control over the dynamics of all nodes is seldom attainable, hence we seek to recover the system's functionality by driving just a microscopic fraction f → 0 of forced nodes.
To achieve this, we consider the limit of f ∼ 1/N , in which case our reigniting is attempted through, typically, a single, randomly selected source node s. To reignite the system we artificially set s's activity to x s (t) = ∆ > 0, namely we externally hold its activity, forcing it to equal ∆. The remaining N −1 nodes continue to follow the natural system dynamics, i.e. Eq. (1), responding to the s-forcing. In technical terms, the failed state of the system, x 0 , captures Eq. (1)'s initial condition, and the forced node imposes a strict boundary condition at s. In a recoverable system, after some time, the activities will enter B 1 , at which point we can cease our external control and allow the system to naturally transition to x 1 , based on its internal dynamics. This captures a successful reigniting of the system's activity. If, however, the system is non-recoverable, such single-node reigniting is insufficient, the system remains at the basin B 0 , and once we discontinue our external forcing, it relaxes back to x 0 , a failed reigniting.
To analytically track the system's response to our forcing at s, we divide the rest of the network into shells K s (l) = {j | L sj = l}, comprising all nodes located at distance l from s (Fig. 2a). In this notation K s (0) = {s}, K s (1) is the group of s's nearest neighbors, K s (2) its next neighbors, and so on. Then, starting with x s (t) = ∆, we track the average activity of nodes in K s (l), via where |K s (l)| represents the number of nodes in K s (l). The shells adjacent to the source, i.e. small l, will be forced to respond to s's activation ∆, exhibiting a gain in their activity x s (l, t). However, such response may, under certain conditions, decay as l is increased, leaving the distant shells almost unaffected, and therefore still within the basin B 0 . Under these conditions, upon termination of our ∆-forcing, all shells retreat back to x 0 , rendering the system unrecoverable. Successful reigniting, therefore, requires that capturing a state in which the forcing at s was able to penetrate the network and impact the state of even the most distant nodes at K s (l → ∞). This represents a recoverable system that will naturally revert to x 1 once the forcing ∆ is deactivated.
In Supplementary Section 2.1 we use Eq. (1) to derive a direct set of equations for the shell states x s (l, t) in (3), providing a set of continuous time discrete space differential equations, with a set boundary condition at l = 0, capturing our s-forcing. The functions M 0 (x), M 1 (x), M 2 (x) and the weight ω in (5) are taken from Eq.
(1), and the parameter κ is extracted from A ij via capturing the average neighbor's residual degree 10,42 . Starting from an initial condition, where x s (l, t = 0) ∈ B 0 for all l > 0, Eq. (5) helps us track the response of the initially failed system to our reigniting.
To extract insight from (5), we seek its steady state x s (l) = x s (l, t → ∞), by setting the derivatives on the left hand side to zero. We show that we can approximate this state via the recurrence relation (Supplementary Section 2.3) where is its inverse function. The parameterx 0 in the right hand side of (8) represents the mean activity of nodes in K s (l > 1) under the failed state x 0 . Hence, we arrive at a direct equation for the l-shell steady states x s (l) under any given forcing ∆.
Equations (7) and (8) represent our key result. They reduce the recoverability of (1), a multidimensional nonlinear dynamic equation, into a manageable first order recurrence relation. This recurrence takes the system's topology and weights (ω, κ) and its nonlinear dynamic mechanisms (M 0 (x), M 1 (x), M 2 (x) and hence R(x)) as input, and predicts the system's recoverability as output. For any given forcing ∆, the recurrence (7) either leads to x s (l → ∞) ∈ B 1 , satisfying the recoverability condition (4), or to x s (l → ∞) ∈ B 0 , and hence indicating a failed recovery.
To obtain x s (l → ∞), and observe whether it is in B 0 or B 1 , we seek the potential fixed-points of the recurrence equation (7). These points can be extracted from the solutions of capturing stationary states in which x s (l) = x s (l −1) 43,44 . Therefore, to analyze the system's recoverability we extract the intersections of the two functions F (x) and M 2 (x) in (9), observing two characteristic behaviors ( Fig. 2): if, for example, F (x) and M 2 (x) have only a single intersection x ∈ B 0 , then the series inevitably converges to that point (Fig. 2c). This represents a non-recoverable system, that, regardless of the magnitude of ∆, resorts back to x 0 . If, on the other hand the functions have several intersections, then its convergence depends on the boundary condition x s (0) = ∆. For ∆ < ∆ c it will approach B 0 , a failed reigniting, and for ∆ ≥ ∆ c it will reach B 1 , capturing a successful reigniting (Fig. 2d,e; see also Supplementary section 2.3). Therefore, our formalism predicts both whether the system is recoverable or not, and in case it is -it predicts the required critical forcing ∆ c for reigniting. Most crucially, the patterns of intersection of (9) depend on the values of κ and ω in (8). This means that the system can transition between the recoverable and the non-recoverable phases following appropriate manipulations of ω and κ -precisely the desired guidelines for the restructuring step, which must be carried out prior to reigniting.
Next we demonstrate our framework on a range of relevant systems, starting from gene regulation (Example 1), then advancing to neuronal and ecological dynamics. We show that tracking the l → ∞ limit of Eq. (7) can help us obtain direct insights into reviving collapsed systems, specifically: • Recoverability. Given the state of the network, i.e. κ, ω, we predict whether the system can be revived by single-node reigniting or not.
• Reigniting force. In case the system is recoverable, we obtain the minimal magnitude of the forcing ∆ ≥ ∆ c required for recovery.
• Restructuring. If non-recoverable, the equations offer guidelines on relevant structural interventions affecting the topology/weights, to steer ω and κ towards the recoverable phase.

Applications
Cellular dynamics ( Fig. 3; see also Supplementary Section 3.1). As our first application we return to whose roots (x) determine the potential fixed points of the reignited system. Clearly, x = 0 represents a solution to (10), capturing the fact that the failed state x 0 = (0, . . . , 0) ∈ B 0 is always stable. Hence, the question is, under what conditions do we observe a second solution x > 0, representing a potential convergence to B 1 . To answer this, we first note that while the right hand side of Eq. (10) is independent of the network topology, its left hand side, F (x), is affected by A ij through κ and by its link weights through ω. Therefore, in Fig. 3f,h we plot M 2 (x) vs. x (yellow) and observe its intersections with F (x) (purple) as we vary the values of these two structural parameters. This allows us to observe, graphically, the potential convergence of the system to B 0 or B 1 .
First we consider ω = 0.8, κ = 4 (Fig. 3g), conditions under which the system is bi-stable, and hence, in principle, can reside both in x 0 or x 1 . Despite this bi-stability, we find that (10) exhibits only one solution, represented by the single intersection at x = 0. This guarantees that (7) converges to x s (l → ∞) = 0, independently of ∆. Consequently, the system is non-recoverable, regardless of the strength of our forcing. The meaning is, that even though the system can potentially feature a stable x 1 , it cannot be reignited from an x 0 initial condition by single-node activation. Indeed, this prediction is confirmed in Fig. 3h, which indicates that the system fails to recover, despite the forcing ∆ at s.
Increasing the network density to κ = 7, however, changes the picture, as now (10) features three intersection points (Fig. 3i): an unstable intermediate point (grey) and the two stable points at x = 0 (red) and at x > 0 (green), representing convergence to B 0 and B 1 , respectively. This predicts a critical forcing ∆ c (vertical grey dashed line), above which all x i reactivate to B 1 , and below which they remain within B 0 . If ∆ > ∆ c the system will be successfully reignited by controlling just a single node, as all shells will, eventually be driven into B 1 , and then naturally converge to the desired x 1 . Fig. 3j shows just that, as now the system responds to our forcing, and, as predicted, recovers its lost functionality.
The results above uncover the existence of a novel, previously overlooked, dynamic phase. Indeed, the regulatory system of Fig. 3a has been previously shown to follow two phases, inactive, where only x 0 is stable vs. bi-stable where both x 0 and x 1 are stable, depending on initial conditions ( Fig. 3b-d). Our analysis here uncovers a third phase: recoverable, a subspace of the bi-stable phase, in which the system can be reignited from x 0 to x 1 by controlling a microscopic set of nodes, here just one single node. The parameters driving this three state phase space are ω, κ and ∆, which together determine the existence (and if exists -the value) of the critical forcing required to revive the inactive system. In Fig. 3k we show these phases in the ω, κ space, as obtained from numerically simulating the states of the system for ∼ 10 3 different combinations of ω and κ (Supplementary Section 4.3). The white solid line represents our theoretical prediction, based on analyzing the intersections of (10), indicating that the boundaries of recoverability can be well-approximated by our analytical framework. We also present the ω, ∆ and κ, ∆ phase diagrams, further indicating the agreement between simulation and theory (Fig. 3l,m).
To further test our predictions in an empirical setting, we collected data on two real biological networks, capturing protein interactions in Human 45 (κ = 21.7) and Yeast 46 (κ = 10.7) cells. In these networks the interaction topology A ij is extracted from empirical data, but the average weight ω, determining the interaction rates can change due to environmental conditions. We, therefore set, for each network, two different weights (Fig. 3n): for Human we set ω 1 = 0.2 (Human UR orange) and ω 2 = 1 (Human R blue), the former in the unrecoverable phase, and the latter -recoverable. Similarly, for Yeast we set ω 1 = 0.4 (Yeast UR, orange, unrecoverable) and ω 2 = 1 (Yeast R, blue, recoverable). As predicted, we find in Fig.  3n,o that, indeed, Yeast UR and Human UR remain inactive under reigniting. This is while Yeast R and Human R, both predicted to reside in the recoverable phase, can be successfully revived via single-node reigniting.
Neuronal dynamics (Fig. 4a-e, Supplementary Section 3.2). As our second example we consider the Cowan-Wilson neuronal dynamics 47,48 , in which (1) follows the form shown in Fig. 4a. The system naturally exhibits three dynamic phases. The inactive state x 0 , in which all activities are suppressed, is obtained when the network is extremely sparse, i.e. small κ, ω. The active x 1 , in which x i are relatively large, is observed when ω, κ are high. In between these two extremes the system features a bi-stable phase, in which both x 0 and x 1 are potentially stable. In Fig. 4b,c we observe these phases on a a set of random networks with varying ω, obtaining two critical points ω c,1 < ω c,2 (dashed grey lines), in which the system transitions between active, inactive and bi-stable. This phase-space predicts a hysteresis phenomenon: if ω is driven below ω c,1 and the system fails, it will not spontaneously recover unless we retrieve ω to be above ω c,2 .
Our formalism, however, predicts an additional, previously unknown dynamic phase, in which the system is recoverable. This phase, shown in Fig. 4d (blue) represents a sub-space of the bi-stable regime, in which the failed system, if untouched, remains at x 0 as per the above hysteresis. Yet, under single-node reigniting, it can be forced back towards activity (x 1 ). To demonstrate this we collected data on the structure of a human brain network 49 to construct A ij (κ = 6.6), and simulated neuronal dynamics using ω 1 = 2, in the unrecoverable phase (Brain UR, orange), and ω 2 = 4, which is recoverable (Brain R, blue). Indeed, Fig. 4e confirms that Brain UR remains inactive, while Brain R's activity is successfully revived.
Mutualistic population dynamics (Fig. 4f-j, Supplementary Section 3.3). As our final example we examine symbiotic interactions in ecological networks, such as plant-pollinator relationships 50 . Once again, we find a window of recoverability, in which one can steer a collapsed system towards activity via singlenode reigniting. Here we used an empirical plant-pollinator network 51 , Eco, to examine recoverability ( Fig. 4i,j), a low ω unrecoverable Eco UR (orange) and an increased ω recoverable Eco R (blue).
Taken together, these examples demonstrate the predictive power of our framework, which allows to systematically map the conditions for recoverability. Next, we show how to use the obtained recoverability maps to provide direct insights into our proposed two-step recovery, extracting guidelines for reviving a collapsed network.

Restructuring guidelines
To successfully reignite a failed system we must first steer it from the non-recoverable to the recoverable phase. This can be achieved through appropriate restructuring interventions to impact ω or κ, for instance, increasing weights or adding nodes/links. The phase diagrams of Figs. 3k-m and 4d,i can provide guidelines for such restructuring, as, indeed, they indicate what interventions can potentially push us closer to the recoverable phase.
To illustrate this, in Fig. 5a-d we simulate a cellular network (Yeast) that has been driven towards inactivity due to major topological perturbations, such as node or link deletion (grey nodes/links). Some of the removed components are inaccessible (red), and hence when restructuring we cannot retrieve them. To revive the system, under these constraints, we incorporate our proposed two-step recovery: • Step I. Restructuring. First we conduct topological interventions, to bring the system to the recoverable phase. As explained above, certain nodes or links are inaccessible to us (red), hence our potential interventions are restricted. The challenge is, therefore, to design a set of accessible interventions that will enable us to revive the system's dynamic activity. As recoverability is driven by two relevant parameters, κ and ω, we map all potential intervention to their effect on these two parameters (Fig. 5e). Indeed, a sequence of such interventions represents a path in the κ, ω space. Therefore, in the restructuring step we seek paths of accessible interventions originating in the present state of the system, and delivering the network into the bounds of the recoverable phase (Fig. 5f). Our goal, we emphasize, is not to simply retrieve the lost nodes and links, but to achieve recoverability. This designates, not a single point, but rather an entire sub-space in κ, ω (Fig. 5f, blue), affording us some level of restructuring flexibility. As a result, despite the constraints imposed by the network's irretrievable components, we are able to design three distinct restructuring paths, all originating in the current state of the failed system (Yeast perturbed, white), but leading to different destinations -Net 1,2 or 3 -within the recoverable sub-space. • Step II. Reigniting. Once in the recoverable phase we can revive the system via single-node reigniting, and retain its lost activity, as shown in Fig. 5g for the three restructured networks Net 1,2 and 3.
This example illustrates how the phase diagrams of recoverability provide direct guidelines for restructuring. For example, in Fig. 5f path 1 builds mainly on controlling the interaction strength (ω), but assumes little freedom to add nodes or links (κ). In contrast path 3 involves a significant component of adding nodes/links to A ij , affecting not just ω but also κ. The optimal restructuring path is, therefore, determined by the nature of our constraints, for example, the relative difficulty in adding weights as in path 1, vs. adding nodes/links, a là path 3. The crucial point is, that, knowing the phase-boundaries of recoverablity, we were able to set different end-points for each path (Net 1,2,3), providing us with a spectrum of potential interventions -either focusing on ω, and hence aiming for end-point 1, or focusing on κ, and thus seeking to reach end-point 3. Such flexibility, enabled thanks to our phase-diagrams, is crucial for real-world restructuring.
While the potential space of structural interventions in Step I is incomprehensibly vast, our phase diagrams reduce this space into just two relevant control parameters -κ, characterizing A ij 's density, and ω, capturing its link weights. This allows us to asses the contribution of all potential interventions by quantifying their effect on these two parameters -enabling to seek optimal pathways for crossing the recoverability phase boundary.
Interestingly, our phase diagrams indicate that κ and ω may play unbalanced roles: for example Fig. 5f shows that the phase boundary from unrecoverability to recoverability in the κ, ω space becomes almost flat in the limit of large κ (white solid line). This implies that while it may only require a small change in ω, i.e. pushing the system in the vertical direction, to transition a network towards recoverability, doing so through κ, along the horizontal axis, is by far more difficult. Indeed, in Supplementary Section 3.1.3 we show that in the limit of large h in Fig. 3a, the phase boundary becomes almost independent of κ, and the effect of increasing ω becomes dominant. Hence, under cellular dynamics, increasing weights, when possible, is a preferred strategy over adding links. This offers a quite general restructuring guideline, that can help design efficient κ, ω-paths in our two-step recovery scheme.

Discussion and outlook
While the structure of complex networks has been deeply investigated over the years, our understanding of their dynamics is still emerging. The challenge is often focused on prediction, aiming to foresee a network's dynamic behavior. Here, we go a step further, and focus on influence, showing how to steer a system towards a desired behavior.
At first glance, this challenge seems to be associated with network controlability. There are, however two crucial distinctions between recoverability and controlability: (i) Control theory revolves around linear dynamics, and is therefore fully characterized by the network structure/weights (A ij , ω). Recoverability, in contrast, is tailored for nonlinear systems within the form (1), hence driven by the interplay of this structure with the system's intrinsic dynamics. Indeed, the phase boundaries of recoverability strongly depend on the system's interaction mechanisms M 0 (x), M 1 (x), M 2 (x), as observed in Figs. 3k-m and 4d,i.
(ii) On the other hand, while controlability aspires to drive a system between any two arbitrary states, an unsolved challenge under nonlinear dynamics, recoverability is limited to the system's naturally occurring fixed-points.
The main advantage of item (ii) above, is that it allows us to revive a failed system using an extremely simple control signal, namely the constant forcing ∆. Had we needed a more delicate fine-tuned input, our strategy would have likely been too sensitive and impractical. Fortunately in recoverability, one only needs to kick the system out of its undesired basin, then let it relax to x 1 independently 21 (Fig. 1h).
Such crude level of control, we find, is possible even under the rather broad conditions captured by Eq. (1).
The microscopic behavior of complex networks is driven by countless parameters, from the fine-structure of A ij to the specific rates of each node's dynamic processes. Our analysis, however, shows, that their large-scale functionality can be traced to a manageable set of relevant parameters, e.g., κ, ω and ∆. Such dimension reduction is the fundamental premise of statistical physics, allowing to analyze systems with endless degrees of freedom using a limited set of statistical entries. We believe, that a similar approach to network dynamics, can help us understand, predict, and ultimately influence the behavior of complex multi-dimensional systems. In this presentation, which we also use throughout the paper, the network nodes are laid out on the x, y plane, and their activities xi are captured by the z coordinate. Hence, an active system has all nodes spread out along the positive z-axis, while a failed network is laid out around z = 0. We also use color coding from red (small xi) to blue (large xi) as visual aid. (d) Perturbations to the topology, such as node/link removal or weight reduction, result in a collapse to the inactive x0, here having all activities vanish (z = 0). (e) Restructuring. To revive the failed system we must first restructure its topology to a point where it can recover, namely a point where x1 is potentially stable. This, however, is insufficient, as often the system will not spontaneously retain its lost activity. (f) Reigniting. After restructuring we reignite the active state x1 by controlling a microscopic set of nodes, here the single node s. The controlled node is forced to sustain a constant activity xs(t) = ∆. This activating force then spreads to impact all other nodes in the network and push them towards x1. (g) The state of the network in three different time-points following single-node reigniting, forcing xs(t) = ∆ (black node at center). The reigniting force gradually impacts the network, until the system's activity x1 is restored. (h) In this process we use the natural basin structure of our dynamics. The basin Bα captures all initial states, from which the system (1) converges to the fixed-point xα. Therefore, to reignite x1 we seek to steer the system from B0 (red) to any point within B1 (green). Once in B1, we can cease our forcing, and the system will spontaneously transition to the desired x1 (dashed arrows).  The average steady-state activityx as obtained from cellular dynamics following topological perturbation: deletion of a q-fraction of nodes; factor q reduction of weights and removal of a q-fraction of links. While the active x1 is only stable for q < qc (green), the inactive x0 is always stable (red). Hence the system undergoes an irreversible collapse at q = qc. (e) We track three specific states, capturing the steady-state activities xi under varying levels of weight loss q. When q is small (left) all xi are positive and the system is in the active state x1. xi gradually decline as q increases (center), until the system collapses into the inactive state x0, in which all xi vanish (right). (f) Reversing the perturbation does not revive the system. Here we retrieve the lost weights, however, since x0 is always stable, the system remains in the collapsed state, avoiding spontaneous recovery. To revive the failed system we apply single-node reigniting: (g) The functions F (x) (purple) and M2(x) (yellow) as obtained from the left/right hand sides of Eq. (10), setting κ = 4, ω = 0.8. The system follows Case 1 of Fig. 2c, exhibiting a single intersection (B0, red), and hence it is unrecoverable. (h) Indeed, forcing node s to a permanent ∆ activity (black node), the remaining shells fail to reignite. (i) Under a denser Aij with κ = 7 the function F (x) takes a different form and the system now follows Case 3. It is now recoverable if the forcing ∆ ≥ ∆c = 0.9 (vertical dashed line). (j) As predicted, now the system is successfully activated via single-node reigniting. (k) The phase diagram in the κ, ω plane (Supplementary Section 4.3). We observe the three predicted phases: inactive (red), where only x0 is stable, unrecoverable (yellow), where x1 is potentially stable, but unattainable via single-node reigniting, and recoverable (blue), where the system can be revived by a single node. The theoretical prediction obtained from the roots of Eq. (10) is also shown (white solid line). (l) The phase diagram in the ω, ∆ space. Here, for ω < 0.9 the system is never recoverable. Above that, the higher is ω the smaller is the forcing ∆ required for reigniting. Our theoretical prediction is also shown (white solid line). (m) The κ, ∆ phase diagram. The slight discrepancy between theory (white solid line) and simulation (yellow-blue transition) is likely due to the discrete nature of κ. (n) To examine our prediction in an empirical setting we used the human protein interaction network 45 (κ = 21.7) to construct Aij under two values of ω: Human UR with ω = 0.2, in the unrecoverable phase, and Human R, with ω = 1, which is recoverable (see panel k, orange vs. blue squares). Indeed, while Human UR cannot be revived (top), Human R is successfully reignited by a single node (bottom). (o) Similar results are observed also for the Yeast network 46 (κ = 10.7, orange vs. blue circles in panel k) under ω = 0.4 (Yeast UR, unrecoverable) and ω = 1 (Yeast R, recoverable). Step I. Restructuring Step II. Reigniting The dynamic states xi of all nodes. As expected, the unperturbed network is in the active state x1, hence all xi > 0. (c) -(d) Following extensive perturbation in which 30% of nodes and 30% of links (grey and red) were deleted the state of the network collapses to the inactive x0, i.e. all xi = 0. The challenge is that some of the deleted components (nodes/links) are inaccessible, and hence cannot be retrieved (red). This captures the restructuring constraints that are, indeed, inevitable in realistic scenarios. Circle at center -we focus on the same sub-network shown in (a), the unperturbed network components are highlighted, the deleted nodes links appear in grey and red. (e) Step I. Within the given constraints we restructure the network by reintroducing nodes/links or strengthening link weights. Such structural interventions can be mapped to their impact on the two relevant control parameters κ (blue) and ω (orange).
Here we track a sequence of restructuring steps. For illustration, we show the highlighted sub-network in (c) as it restructures, acquiring nodes, links and increased weights (sub-networks along the x-axis). (f) Each such sequence of restructuring steps can be mapped into a path in the κ, ω phase diagram. A successful restructuring path must lead the network from the collapsed phase (red) into the recoverable phase (blue). Our predicted phase diagram helps us design several alternative paths, affording us the flexibility to, e.g., focus mainly on increasing weights ω (Net 1, light blue path) or also on enhancing network density κ (Net 3, purple path), all depending on the nature of our constraints. The resulting networks Net 1 -3 may differ from the original unperturbed network, as indeed, our goal is not to simply reverse the damage, but to revive the system's dynamic activity. (e) Step II. Once the network is brought to the recoverable phase, we can revive it via single-node reigniting, demonstrated here on each of our restructured networks, Net 1 -3.     Step I. Restructuring Step II. Reigniting Figure 5. Two-step recovery for reviving a failed system.