Different from the direct contact in epidemics spread, overload failures propagate through hidden functional dependencies. Many studies focused on the critical conditions and catastrophic consequences of cascading failures. However, to understand the network vulnerability and mitigate the cascading overload failures, the knowledge of how the failures propagate in time and space is essential but still missing. Here we study the spatio-temporal propagation behaviour of cascading overload failures analytically and numerically on spatially embedded networks. The cascading overload failures are found to spread radially from the centre of the initial failure with an approximately constant velocity. The propagation velocity decreases with increasing tolerance, and can be well predicted by our theoretical framework with one single correction for all the tolerance values. This propagation velocity is found similar in various model networks and real network structures. Our findings may help to predict the dynamics of cascading overload failures in realistic systems.

Targeted immunization of centralized nodes in large-scale networks has attracted significant attention. However, in real-world scenarios, knowledge and observations of the network may be limited thereby precluding a full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current COVID-19 epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized. This process can globally immunize a network. We find that even for small n (≈10) there is significant improvement in the immunization (quarantine) which is very close to the levels of an immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P(k). In the limit of n → ∞ we recover prior work on targeted immunization, whereas for n = 1 we recover the known case of random immunization. Between these two extremes, we observe that as n increases, pc increases quickly towards its optimal value under targeted immunization with complete information. In particular, we find a new general scaling relationship between |pc(∞) − pc(n)| and n as |pc(∞) − pc(n)| ∼ n−1exp ( − αn). For Scale-free (SF) networks, where P(k) ∼ k−γ, 2 < γ < 3, we find that pc has a transition from zero to non-zero when n increases from n = 1 to O(log N) (where N is the size of network). Thus, for SF networks, having knowledge of ≈log N nodes and immunizing them can dramatically reduce epidemic spreading. We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks, pc increases significantly even for small n.

From mass extinction to cell death, complex networked systems often exhibit abrupt dynamic transitions between desirable and undesirable states. Such transitions are often caused by topological perturbations, such as node or link removal, or decreasing link strengths. The problem is that reversing the topological damage, namely retrieving the lost nodes/links or reinforcing the weakened interactions, does not guarantee the spontaneous recovery to the desired functional state. Indeed, many of the relevant systems exhibit a hysteresis phenomenon, remaining in the dysfunctional state, despite reconstructing their damaged topology. To address this challenge, 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 twostep recovery.Complex systems, biological, social or technological, often experience perturbations and disturbances, from overload failures in power systems 1-4 to species extinction in ecological networks [5][6][7] . The impact of such perturbations is often subtle, the system exhibits a minor response, but continues to sustain its global functionality 8-10 . However, in extreme cases, a large enough perturbation may lead to a large-scale collapse, with the system abruptly transitioning from a functional to a dysfunctional dynamic state [11][12][13][14][15][16] . (Fig. 1a-d). For instance, in cellular dynamics, genetic knockouts, beyond a certain threshold, lead to cell death 17,18 ; in ecological systems, changes in environmental conditions may, in extreme cases, cause mass-extinction [5][6][7]19 ; and in infrastructure networks, a cascading failure, at times, results in a major blackout 11,20 . When such collapse occurs, the naïve instinct is to reverse the damage, retrieve the failed nodes and reconstruct the lost links. Such response however is seldom efficient, as (i) we rarely have access to all system components 21 , limiting our ability to reconstruct the perturbed network; (ii) even if we could reverse the damage, due to hysteresis [22][23][24] , in many cases, the system will not spontaneously regain its lost functionality 25 .To address this challenge, we consider here a two-step recovery process:Step I. Restructuring (Fig. 1e). Retrieving the network topology and weights to a point where the system can potentially regain its functionality.

Targeted immunization or attacks of large-scale networks has attracted significant attention by the scientific community. However, in real-world scenarios, knowledge and observations of the network may be limited thereby precluding a full assessment of the optimal nodes to immunize (or remove) in order to avoid epidemic spreading such as that of current COVID-19 epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central between these n nodes is immunized (or attacked). This process is continued repeatedly until 1 − p fraction of nodes are immunized (or attacked). We develop an analytical framework for this approach and determine the critical percolation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P (k). In the limit of n → ∞ we recover prior work on targeted attack, whereas for n = 1 we recover the known case of random failure. Between these two extremes, we observe that as n increases, pc increases quickly towards its optimal value under targeted immunization (attack) with complete information. In particular, we find a new scaling relationship between |pc(∞) − pc(n)| and n as |pc(∞) − pc(n)| ∼ n −1 exp(−αn). For Scale-free (SF) networks, where P (k) ∼ k −γ , 2 < γ < 3, we find that pc has a transition from zero to non-zero when n increases from n = 1 to order of log N (N is the size of network). Thus, for SF networks, knowledge of order of log N nodes and immunizing them can reduce dramatically an epidemics.All rights reserved. No reuse allowed without permission.

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we study if and how the effects of such interconnections can be described as an external field for interdependent networks experiencing first-order percolation transitions. We find that the critical exponents γ and δ, related to the external field can also be defined for firstorder transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents can be found even within a single model of interdependent networks, depending on the dependency coupling strength. Specifically, the exponent γ in the first-order regime (high coupling) does not obey the fluctuation dissipation theorem, whereas in the continuous regime (for low coupling) it does. Nevertheless, in both cases they satisfy Widom's identity, δ − 1 = γ/β which further supports the validity of their definitions. Our results provide physical intuition into the nature of the phase transition in interdependent networks and explain the underlying reasons for two distinct sets of exponents.