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Deviating from the common situation where some physical objects are considered to move in space, the present chapter deals with the spreading of failures in complex systems like power grids, communications networks and financial networks. Failures can trigger a cascade of further failures that spread to other parts of the system with, possibly, dramatic consequences and catastrophic events like blackouts and economic collapses. The focus of this chapter is on the specific mechanisms occurring in interdependent systems, where components of one system cannot function without components of the other, as in the case of a communication tower which needs power from the nearby power station. Understanding how these failures spread in both time and space is critical for ensuring that large-scale complex systems remain functional. A further challenge examined here involves the question of how to optimally repair and recover a system after it has experienced some failures—for preventing quick failure reappearance in the repaired components.

Earthquakes are one of the most devastating natural disasters that plague society. Skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional probability (CP) methods we find that memory exists not only in inter-occurrence seismic records, but also in released energy as well as in the series of the number of events per unit time. Analysis of a stan- dard Epidemic Type Aftershock Sequences (ETAS) earthquake model indicates that the empirically observed earthquake memory can be reproduced only for a narrow range of model’s parameters. This finding, therefore provides tight constraints on the model’s parameters and can serve as a testbed for existing earthquake forecast- ing models. Furthermore, we show that by implementing DFA and CP results, the ETAS model can significantly improve the short term forecasting rate for the real (Italian) earthquake catalog.

While abrupt regime shifts between different metastable states have occurred in natural systems from many areas including ecology, biology, and climate, evidence for this phenomenon in transportation systems has been rarely observed so far. This limitation might be rooted in the fact that we lack methods to identify and analyze possible multiple states that could emerge at scales of the entire traffic network. Here, using percolation approaches, we observe such a metastable regime in traffic systems. In particular, we find multiple metastable network states, corresponding to varying levels of traffic performance, which recur over different days. Based on high-resolution global positioning system (GPS) datasets of urban traffic in the megacities of Beijing and Shanghai (each with over 50,000 road segments), we find evidence supporting the existence of tipping points separating three regimes: a global functional regime and a metastable hysteresis-like regime, followed by a global collapsed regime. We can determine the intrinsic critical points where the metastable hysteresis-like regime begins and ends and show that these critical points are very similar across different days. Our findings provide a better understanding of traffic resilience patterns and could be useful for designing early warning signals for traffic resilience management and, potentially, other complex systems.

Hepatitis D virus (HDV) requires hepatitis B surface antigen (HBsAg) for its assembly and release. Current HBV treatments are only marginally effective against HDV because they fail to inhibit HBsAg production/secretion. However, monotherapy with the nucleic acid polymer REP 2139-Ca is accompanied by rapid declines in both HBsAg and HDV RNA. We used mathematical modeling to estimate HDV-HBsAg-host parameters and to elucidate the mode of action and efficacy of REP 2139-Ca against HDV in 12 treatment-naive HBV/HDV co-infected patients. The model accurately reproduced the observed decline of HBsAg and HDV, which was simultaneous. Median serum HBsAg half-life (t1/2) was estimated as 1.3 [0.9–1.8] days corresponding to a pretreatment production and clearance of ~108 [107.7–108.3] IU/day. The HDV-infected cell loss was estimated to be 0.052 [0.035–0.074] days−1 corresponding to an infected cell t1/2 = 13.3 days. The efficacy of blocking HBsAg and HDV production were 98.2 [94.5–99.9]% and 99.7 [96.0–99.8]%, respectively. In conclusion, both HBsAg production and HDV replication are effectively inhibited by REP 2139-Ca. Modeling HBsAg kinetics during REP 2139-Ca monotherapy indicates a short HBsAg half-life (1.3 days) suggesting a rapid turnover of HBsAg in HBV/HDV co-infection.

While the liver, specifically hepatocytes, are widely accepted as the main source of hepatitis C virus (HCV) production, the role of the liver/hepatocytes in clearance of circulating HCV remains unknown. Frequent HCV kinetic data were recorded and mathematically modeled from five liver transplant patients throughout the anhepatic (absence of liver) phase and for 4 hr post-reperfusion. During the anhepatic phase, HCV remained at pre-anhepatic levels (n = 3) or declined (n = 2) with t1/2~1 hr. Immediately post-reperfusion, virus declined in a biphasic manner in four patients consisting of a rapid decline (t1/2 = 5 min) followed by a slower decline (t1/2 = 67 min). Consistent with the majority of patients in the anhepatic phase, when we monitored HCV clearance at 37°C from culture medium in the absence/presence of chronically infected hepatoma cells that were inhibited from secreting HCV, the HCV t1/2 in cell culture was longer in the absence of chronically HCV-infected cells. The results suggest that the liver plays a major role in the clearance of circulating HCV and that hepatocytes may be involved.

Coupling between networks is widely prevalent in real systems and has dramatic effects on their resilience and functional properties. However, current theoretical models tend to assume homogeneous coupling where all the various subcomponents interact with one another, whereas real-world systems tend to have various different coupling patterns. We develop two frameworks to explore the resilience of such modular networks, including specific deterministic coupling patterns and coupling patterns where specific subnetworks are connected randomly. We find both analytically and numerically that the location of the percolation phase transition varies nonmonotonically with the fraction of interconnected nodes when the total number of interconnecting links remains fixed. Furthermore, there exists an optimal fraction [Formula: see text] of interconnected nodes where the system becomes optimally resilient and is able to withstand more damage. Our results suggest that, although the exact location of the optimal [Formula: see text] varies based on the coupling patterns, for all coupling patterns, there exists such an optimal point. Our findings provide a deeper understanding of network resilience and show how networks can be optimized based on their specific coupling patterns.

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., A novel network immunization strategy is proposed and studied to provide a theoretical framework for understanding and carrying out efficient immunization with limited knowledge.

While the liver, specifically hepatocytes, are widely accepted as the main source for hepatitis C virus (HCV) production, the role of the liver/hepatocytes in the clearance of circulating HCV remains largely unknown. Here we evaluated the function of the liver/hepatocytes in clearing virus from the circulation by investigating viral clearance during liver transplantation and from culture medium in vitro. Frequent HCV kinetic data during liver transplantation were recorded from 5 individuals throughout the anhepatic (AH) phase and for 4 hours after reperfusion (RP), along with recordings of fluid balances. Using mathematical modeling, the serum viral clearance rate, c, was estimated. Analogously, we monitored the clearance rate of HCV at 37°C from culture medium in vitro in the absence and presence of chronically infected Huh7 human hepatoma cells. During the AH phase, in 3 transplant cases viral levels remained at pre-AH levels, while in the other 2 cases HCV declined (half-life, t1/2~1h). Immediately post-RP, virus declined in a biphasic manner in Cases 1-4 consisting of an extremely rapid (median t1/2=5min) decline followed by a slower decline (HCV t1/2=67min). In Case 5, HCV remained at the same level post-RP as at the end of AH. Declines in virus level were not explained by adjusting for dilution from IV fluid and blood products. Consistent with what was observed in the majority of patients in the anhepatic phase, the t1/2 of HCV in cell culture was much longer in the absence of chronically HCV-infected Huh7 cells. Therefore, kinetic and modeling results from both in vivo liver transplantation cases and in vitro cell culture studies suggest that the liver plays a major role in clearing HCV from the circulation.

Mathematical modeling of Ebola virus (EBOV)&ndash, host dynamics during infection and treatment in vivo is in its infancy due to few studies with frequent viral kinetic data, lack of approved antiviral therapies, and limited insight into the timing of EBOV infection of cells and tissues throughout the body. Current in-host mathematical models simplify EBOV infection by assuming a single homogeneous compartment of infection. In particular, a recent modeling study assumed the liver as the largest solid organ targeted by EBOV infection and predicted that nearly all cells become refractory to infection within seven days of initial infection without antiviral treatment. We compared our observations of EBOV kinetics in multiple anatomic compartments and hepatocellular injury in a critically ill patient with Ebola virus disease (EVD) with this model&rsquo, s predictions. We also explored the model&rsquo, s predictions, with and without antiviral therapy, by recapitulating the model using published inputs and assumptions. Our findings highlight the challenges of modeling EBOV&ndash, host dynamics and therapeutic efficacy and emphasize the need for iterative interdisciplinary efforts to refine mathematical models that might advance understanding of EVD pathogenesis and treatment.

Links in most real networks often change over time. Such temporality of links encodes the ordering and causality of interactions between nodes and has a profound effect on network dynamics and function. Empirical evidence has shown that the temporal nature of links in many real-world networks is not random. Nonetheless, it is challenging to predict temporal link patterns while considering the entanglement between topological and temporal link patterns. Here, we propose an entropy-rate-based framework, based on combined topological–temporal regularities, for quantifying the predictability of any temporal network. We apply our framework on various model networks, demonstrating that it indeed captures the intrinsic topological–temporal regularities whereas previous methods considered only temporal aspects. We also apply our framework on 18 real networks of different types and determine their predictability. Interestingly, we find that, for most real temporal networks, despite the greater complexity of predictability brought by the increase in dimension, the combined topological–temporal predictability is higher than the temporal predictability. Our results demonstrate the necessity for incorporating both temporal and topological aspects of networks in order to improve predictions of dynamical processes., An entropy-based framework solved the long standing problem of quantifying the regularities in any temporal network whose links change over time, and unraveled the limit of predictability for 18 real temporal networks in diverse fields.

Cascading failure is a potentially devastating process that spreads on real-world complex networks and can impact the integrity of wide-ranging infrastructures, natural systems, and societal cohesiveness. One of the essential features that create complex network vulnerability to failure propagation is the dependency among their components, exposing entire systems to significant risks from destabilizing hazards such as human attacks, natural disasters or internal breakdowns. Developing realistic models for cascading failures as well as strategies to halt and mitigate the failure propagation can point to new approaches to restoring and strengthening real-world networks. In this review, we summarize recent progress on models developed based on physics and complex network science to understand the mechanisms, dynamics and overall impact of cascading failures. We present models for cascading failures in single networks and interdependent networks and explain how different dynamic propagation mechanisms can lead to an abrupt collapse and a rich dynamic behavior. Finally, we close the review with novel emerging strategies for containing cascades of failures and discuss open questions that remain to be addressed., Comment: This review has been accepted for publication in the Journal of Complex Networks Published by Oxford University Press

Over the past few decades the world underwent several major economic and financial bubbles, such as the dot-com bubble of early 2000s and the global crisis following the collapse of the US housing market in 2008. Here we review the progress made in network theory as applied to economics and highlight some important insights complex networks allow into the highly interconnected economic system. Richness of phenomena that appears once we increase our complexity beyond a single network is explored and main results, as well as future research are discussed.

Detecting and characterizing community structure plays a crucial role in the study of networked systems. However, there is still a lack of understanding of how community structure affects the systems' resilience and stability. Here, we develop a framework to study the resilience of networks with community structure based on percolation theory. We find both analytically and numerically that the interlinks (connections between the communities) affect the percolation phase transition in a manner similar to an external field in a ferromagnetic-paramagnetic spin system. We also study the universality class by defining the analogous critical exponents $\delta$ and $\gamma$, and find that their values for various models and in real-world co-authors networks follow fundamental scaling relations as in physical phase transitions. The methodology and results presented here not only facilitate the study of resilience of networks but also brings a fresh perspective to the understanding of phase transitions under external fields., Comment: 15 pages,4 figures

Young people in the USA who inject drugs, particularly those at a risk of residence instability, experience the highest incidence of hepatitis C (HCV) infections. This study examined associations between geographic mobility patterns and sociodemographic, behavioral, and social network characteristics of 164 young (ages 18–30) persons who inject drugs (PWID). We identified a potential bridge sub-population who reported residence in both urban and suburban areas in the past year (crossover transients) and higher-risk behaviors (receptive syringe sharing, multiple sex partners) compared to their residentially localized counterparts. Because they link suburban and urban networks, crossover transients may facilitate transmission of HIV and HCV between higher and lower prevalence areas. Interventions should address risk associated with residential instability, particularly among PWID who travel between urban and suburban areas.

A reliable log system is a prerequisite for security applications. One of the first actions a hacker takes upon penetrating a system is altering the log files. Maintaining redundant copies in a distributed manner in a Byzantine setting has always been a challenging task, however it has recently become simpler given recent advances in blockchain technologies. In this work, we present a tamper-resistant log system through the use of a blockchain. We leverage the immutable write action and distributed storage provided by the blockchain and add an additional encryption layer to develop a secure log system. We assess the security and privacy aspects of our solution. Finally, we implement our system over Hyperledger Fabric and demonstrate the system's value for several use cases.

The detection and prediction of risk in financial markets is one of the main challenges of economic forecasting, and draws much attention from the scientific community. An even more challenging task is the prediction of the future relative gain of companies. We here develop a novel combination of product text analysis, network theory and topological based machine learning to study the future performance of companies in financial markets. Our network links are based on the similarity of firms’ products and constructed using the Securities Exchange Commission (SEC) filings of US listed firms. We find that several topological features of this network can serve as good precursors of risks or future gain of companies. We then apply machine learning to network attributes vectors for each node to predict successful and failing firms. The resulting accuracies are much better than current state of the art techniques. The framework presented here not only facilitates the prediction of financial markets but also provides insight and demonstrates the power of combining network theory and topology based machine learning.

We study a greedy forager who consumes food throughout a region. If the forager does not eat any food for S time steps it dies. We assume that the forager moves preferentially in the direction of greatest smell of food. Each food item in a given direction contributes towards the total smell of food in that direction, however the smell of any individual food item decays with its distance from the forager. We study both power-law decay and exponential decay of the smell with the distance of the food from the forager. For power-law decay, we vary the exponent α governing this decay, while for exponential decay we vary λ also governing the rate of the decay. For power-law decay we find, both analytically and through simulations, that for a forager living in one dimension, there is a critical value of α , namely , where for the forager will die in finite time, however for the forager has a nonzero probability to live infinite time. We calculate analytically the critical value, , separating these two behaviors and find that depends on S as . We find analytically that at the system has an essential singularity. For exponential decay we find analytically that for all λ , the forager has a finite probability to live for infinite time. We also study, using simulations, a forager with long-range decaying smell in two dimensions (2D) and find that for this case the forager always dies within finite time. However, in 2D we observe indications of an optimal α (and λ ) for which the forager has the longest lifetime.

BACKGROUND & AIMS Acute hepatitis C (AHC) is not frequently identified because patients are usually asymptomatic, although may be recognized after iatrogenic exposures such as needle stick injuries, medical injection, and acupuncture. We describe an outbreak of AHC among 12 patients who received IV saline flush from a single multi-dose vial after intravenous contrast administration for a computerized tomography (CT) scan. The last patient to receive IV contrast with saline flush from a multi-dose vial at the clinic on the previous day was known to have chronic HCV genotype 1b (termed potential source, PS). Here we sought to confirm (via genetic analysis) the source of infection and to predict the minimal contaminating level of IV saline flush needed to transmit infectious virus to all patients. METHODS In order to confirm the source of infection, we sequenced the HCV E1E2 region in 7 CT patients, in PS, and in 2 control samples from unrelated patients also infected with HCV genotype 1b. A transmission probabilistic model was developed to predict the contamination volume of blood that would have been sufficient to transmit infectious virus to all patients. RESULTS Viral sequencing showed close clustering of the cases with the PS. The transmission probabilistic model predicted that contamination of the multi-dose saline vial with 0.6-8.7 microliters of blood would have been sufficient to transmit infectious virus to all patients. CONCLUSION Analysis of this unique cohort provides a new understanding of HCV transmission with respect to contaminating volumes and viral titers.

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 $\gamma$ and $\delta$, related to the external field can also be defined for first-order 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 $\gamma$ 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, $\delta - 1 = \gamma / \beta$ 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., Comment: 6 pages and 4 figures

Until recently, network science has focused on the properties of single isolated networks that do not interact or depend on other networks. However it has now been recognized that many real-networks, such as power grids, transportation systems, and communication infrastructures interact and depend on other networks. Here, we will present a review of the framework developed in recent years for studying the vulnerability and recovery of networks composed of interdependent networks. In interdependent networks, when nodes in one network fail, they cause dependent nodes in other networks to also fail. This is also the case when some nodes, like for example certain people, play a role in two networks, i.e. in a multiplex. Dependency relations may act recursively and can lead to cascades of failures concluding in sudden fragmentation of the system. We review the analytical solutions for the critical threshold and the giant component of a network of n interdependent networks. The general theory and behavior of interdependent networks has many novel features that are not present in classical network theory. Interdependent networks embedded in space are significantly more vulnerable compared to non-embedded networks. In particular, small localized attacks may lead to cascading failures and catastrophic consequences. Finally, when recovery of components is possible, global spontaneous recovery of the networks and hysteresis phenomena occur. The theory developed for this process points to an optimal repairing strategy for a network of networks. Understanding realistic effects present in networks of networks is required in order to move towards determining system vulnerability.

Much work has been devoted to studying percolation of networks and interdependent networks under varying levels of failures. Researchers have considered many different realistic network structures such as scale-free networks, spatial networks, and more. However, thus far no study has analyzed a system of hierarchical community structure of many networks. For example, infrastructure across cities is likely distributed with nodes tightly connected within small neighborhoods, somewhat less connected across the whole city, and even fewer connections between cities. Furthermore, while previous work identified interconnected nodes, those nodes with links outside their neighborhood, to be more likely to be attacked or to fail, in a hierarchical structure nodes can be interconnected in different layers (between neighborhoods, between cities, etc.). We consider here the case where the nodes with interconnections at the highest level of the hierarchy are most likely to fail, followed by those with interconnections at the next level, etc. This is because nodes at higher levels of the hierarchy have the longest links as well as having more flow passing through them. We develop an analytic solution for percolation of both single and interdependent networks of this structure and verify our theory through simulations. We find that, depending on the number of levels in the hierarchy, there may be multiple transitions in the giant component (fraction of connected nodes), as the network separates at the various levels. Our results show that these multiple jumps are a feature of hierarchical networks and can affect the vulnerability of infrastructure networks.

Earthquakes are one of the most devastating natural disasters that plague society. Skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional probability (CP) methods, we find that memory exists not only in interoccurrence seismic records but also in released energy as well as in the series of the number of events per unit time. Analysis of a standard epidemic-type aftershock sequences (ETAS) earthquake model indicates that the empirically observed earthquake memory can be reproduced only for a narrow range of the model's parameters. This finding therefore provides tight constraints on the model's parameters and can serve as a testbed for existing earthquake forecasting models. Furthermore, we show that by implementing DFA and CP results, the ETAS model can significantly improve the short-term forecasting rate for the real (Italian) earthquake catalog.

The major route of hepatitis C virus (HCV) transmission in the United States is injection drug use. We hypothesized that if a HCV vaccine was available, vaccination could impact HCV transmission among people who inject drugs by reducing HCV titers after viral exposure without necessarily achieving sterilizing immunity. To investigate this possibility, we developed a mathematical model to determine transmission probabilities relative to the HCV RNA titers of needle/syringe sharing donors. We simulated sharing of two types of syringes fitted with needles that retain either large or small amounts of fluid post-expulsion. Using previously published viral kinetics data from both naïve subjects infected with HCV and reinfected individuals who had previously cleared an HCV infection, we estimated transmission risk between pairs of serodiscordant injecting drug users, taking into account syringe type, rinsing, and sharing frequency. We calculated that the risk of HCV transmission through syringe sharing increased ~10-fold as viral titers (log(10) IU/mL) increased ~25-fold. Cumulative analyses showed that, assuming sharing episodes every 7 days, the mean transmission risk over the first 6 months was >90% between two people sharing syringes when one had an HCV RNA titer >5 log(10) IU/mL, regardless of infection outcome. For those with pre-existing immunity that rapidly controlled HCV, the cumulative risk decreased to 1–25% depending on HCV titer and syringe type. Our modeling approach demonstrates that even with transient viral replication following exposure during injection drug use, HCV transmission among people sharing syringes could be reduced through vaccination if a HCV vaccine were available.

We study secure message-passing in the presence of multiple adversaries in modular networks. We assume a dominant fraction of nodes in each module have the same vulnerability, i.e., the same entity spying on them. We find both analytically and via simulations that the links between the modules (interlinks) have effects analogous to a magnetic field in a spin system in that for any amount of interlinks the system no longer undergoes a phase transition. We then define the exponents $\delta$, which relates the order parameter (the size of the giant secure component) at the critical point to the field strength (average number of interlinks per node), and $\gamma$, which describes the susceptibility near criticality. These are found to be $\delta=2$ and $\gamma=1$ (with the scaling of the order parameter near the critical point given by $\beta=1$). When two or more vulnerabilities are equally present in a module we find $\delta=1$ and $\gamma=0$ (with $\beta\geq2$). Apart from defining a previously unidentified universality class, these exponents show that increasing connections between modules is more beneficial for security than increasing connections within modules. We also measure the correlation critical exponent $\nu$, and the upper critical dimension $d_c$, finding that $\nu d_c=3$ as for ordinary percolation, suggesting that for secure message-passing $d_c =6$. These results provide an interesting analogy between secure message-passing in modular networks and the physics of magnetic spin-systems.

Recently a simple military exercise on the Internet was perceived as the beginning of a new civil war in the US. Social media aggregate people around common interests eliciting a collective framing of narratives and worldviews. However, the wide availability of user-provided content and the direct path between producers and consumers of information often foster confusion about causations, encouraging mistrust, rumors, and even conspiracy thinking. In order to contrast such a trend attempts to \textit{debunk} are often undertaken. Here, we examine the effectiveness of debunking through a quantitative analysis of 54 million users over a time span of five years (Jan 2010, Dec 2014). In particular, we compare how users interact with proven (scientific) and unsubstantiated (conspiracy-like) information on Facebook in the US. Our findings confirm the existence of echo chambers where users interact primarily with either conspiracy-like or scientific pages. Both groups interact similarly with the information within their echo chamber. We examine 47,780 debunking posts and find that attempts at debunking are largely ineffective. For one, only a small fraction of usual consumers of unsubstantiated information interact with the posts. Furthermore, we show that those few are often the most committed conspiracy users and rather than internalizing debunking information, they often react to it negatively. Indeed, after interacting with debunking posts, users retain, or even increase, their engagement within the conspiracy echo chamber.

Networks interact with one another in a variety of ways. Even though increased connectivity between networks would tend to make the system more robust, if dependencies exist between networks, these systems are highly vulnerable to random failure or attack. Damage in one network causes damage in another. This leads to cascading failures which amplify the original damage and can rapidly lead to complete system collapse.

Many real systems such as, roads, shipping routes, and infrastructure systems can be modeled based on spatially embedded networks. The inter-links between two distant spatial networks, such as those formed by transcontinental airline flights, play a crucial role in optimizing communication and transportation over such long distances. Still, little is known about how inter-links affect the structural resilience of such systems. Here, we develop a framework to study the structural resilience of interlinked spatially embedded networks based on percolation theory. We find that the inter-links can be regarded as an external field near the percolation phase transition, analogous to a magnetic field in a ferromagnetic–paramagnetic spin system. By defining the analogous critical exponents δ and γ, we find that their values for various inter-links structures follow Widom's scaling relations. Furthermore, we study the optimal robustness of our model and compare it with the analysis of real-world networks. The framework presented here not only facilitates the understanding of phase transitions with external fields in complex networks but also provides insight into optimizing real-world infrastructure networks.

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with (i) unconstrained dependency links and (ii) dependency links restricted to a maximum Euclidean length $r$. Analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations. For the case of two fully interdependent spatially embedded networks it was found [Li et al., Phys. Rev. Lett. 108, 228702 (2012)] that the system undergoes a first-order phase transition only for $rg{r}_{c}\ensuremath{\approx}8$. We find here that for treelike networks of networks (composed of $n$ networks) ${r}_{c}$ significantly decreases as $n$ increases and rapidly $(n\ensuremath{\ge}11)$ reaches its limiting value of 1. For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations that there is a certain fraction of dependent nodes, ${q}_{\mathrm{max}}$, above which the entire network structure collapses even if a single node is removed. The value of ${q}_{\mathrm{max}}$ decreases quickly with $m$, the degree of the random regular network of networks. Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse. The theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically.

Many real-world phenomena can be modelled using networks. Often, these networks interact with one another in non-trivial ways. Recently, a theory of interdependent networks has been developed which describes dependency between nodes across networks. Interdependent networks have a number of unique properties which are absent in single networks. In particular, systems of interdependent networks often undergo abrupt first-order percolation transitions induced by cascading failures. Here we present an overview of recent developments and significant findings regarding interdependent networks and networks of networks.

Summary of the networks. Presented here : N , the number of nodes in the network ; L , the number of links ; k , the average degree ;CV (k) the coefficient of variation of degree ;CV (w) , the coefficient of variation of weight ; ρ = L/ N 2 , the network density and r , the degreeassortativity coefficient. The coefficients of variation CV (w) and CV (k) are defined as the ratioof the standard deviation , σ , to the mean , μ , of the weight and degree populations , respectively Network NL k ρ CV (k) CV (w) r Airport 1227 18050 29.42 0.024 1.29 2.25 −0.06Migration 3056 71551 46.83 0.015 1.94 6.13 −0.06Cargo 951 25819 54.30 0.057 1.22 6.85 −0.14Neural 297 2141 14.46 0.049 0.89 1.35 −0.16Food Web 121 1763 29.14 0.24 0.45 11.77 −0.10Metabolic 311 1304 8.39 0.027 1.79 7.91 −0.25Random 1000 15028 30.06 0.03 0.177 1.44 −0.005greater availability of large datasets [6]. The brain’s neurons have been mapped using these methods[7], as have air traffic patterns [8], and the flow of cargo throughout the world [9].The explosion of research on complex networks in recent years has led to the discovery of variousproperties of networks and has allowed us to find ways of reducing the complexity while preservingcertain key features. Many of these methods focus on reducing the number of nodes in the network.Aside from simple thresholding, more sophisticated coarse-graining techniques have also been used[10] to reduce the number of distinct entities in the network. Here, we will focus on methods of reducingthe number of links in the network while preserving the nodes. This is advantageous since it reduces thecomplexity of the system while still preserving scale-free properties.Further, there has been considerable effort in understanding how networks as a whole respond todamage [11–14]. These studies have explored different methods of perturbing the network such asintentional attack and random failure.Despite the significant amount of research in both of these areas separately, there has been littlework in combining the study of backbone and skeleton methods with stress applied to the system. Here,we examine how skeletons and backbones respond to different methods of stress applied to the system.1.1