Crises in economic complex networks: Black Swans or Dragon Kings?

Economic Analysis and Policy 62 (2019) 105–115

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Economic Analysis and Policy journal homepage: www.elsevier.com/locate/eap

Recent trends in economic research

Crises in economic complex networks: Black Swans or Dragon Kings? ∗

Marisa Faggini a , , Bruna Bruno a , Anna Parziale b a b

Department of Economics and Statistics - University of Salerno, Italy Department of Law Sciences,University of Salerno, Italy

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Article history: Received 13 December 2017 Received in revised form 25 January 2019 Accepted 27 January 2019 Available online 2 February 2019 JEL classification: D85 H12 F62

a b s t r a c t After the global financial crisis of 2008, the literature paid new attention to economic crises, analysing them according to a network perspective. Assuming this perspective, this paper is aimed at offering an overview of the functioning of the global economy as a complex network, characterized by cascading failures when an extreme event (EE) occurs, and showing that the economic crisis of 2008 was an extreme event with well-identified features and consequently was forecastable. © 2019 Economic Society of Australia, Queensland. Published by Elsevier B.V. All rights reserved.

Keywords: Economic crises Complex networks Extreme events

1. Introduction Taking into consideration the fact that the traditional economic models are mainly focused on the analysis of static and linear patterns, it is worth noting that the economic theory needs to shift towards the analysis of dynamic systems, characterized by non-linearity, asymmetries, distortions, and discontinuities (Bar-Yam, 1997). This, coupled with the problems generated by the global crisis of 2008, calls for rethinking the inner functioning of economic systems as well as that of the economic models used to describe them. The interactions1 occurring between economic agents change over time, making the behaviour of an agent open to influence from the ties that join it to other agents and enabling it always to adapt to their changes and to environmental changes. It follows that the structure arising from the connections between the parts shapes the emerging distinctive and dynamic features of a system. Due to this dynamic nature, societies, organizations, and even individuals should no longer be analysed just regarding their individual resilience but rather within the network built on their mutual interdependencies and their disposition towards co-operation. Not only are networks everywhere but also the organization of individual agents into networks and groups plays an important role in the determination of the outcome of many socio-economic interactions.2 ∗ Corresponding author. E-mail addresses: [email protected] (M. Faggini), [email protected] (B. Bruno), [email protected] (A. Parziale). 1 The interactions occurring between economic entities involve several factors, ranging from competition for capital, labour, and resources intended to change government and policy to wars, for example (Bar-Yam, 1997), and, because many of these connections are not only non-linear but also driven by socio-economic events, the traditional models are unable to capture them. 2 For instance, networks of personal contacts are fundamental to obtain information about job opportunities (see, among others, Boorman, 1975; Montgomery, 1991). Networks also play a pivotal role in goods trading and exchange in non-centralized markets (see, among others, Tesfatsion, 1997, https://doi.org/10.1016/j.eap.2019.01.009 0313-5926/© 2019 Economic Society of Australia, Queensland. Published by Elsevier B.V. All rights reserved.

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The aforementioned phenomena have supported the ongoing and massive growth of network science, which has attracted the attention of several interdisciplinary sciences (Kennett and Havlin, 2015) and developed tools for analysing complex phenomena, such as the world economic system. A branch of discrete mathematics, the graph theory, has traditionally investigated networks, and the random network model proposed by Erdös and Renyi (1959) dominated this field for many decades. More recently, Watts (2003) developed the related research further, proposing small-world networks. The introduction of the scale-free network model (Albert and Barabási, 1999) also advanced network science, introducing a brand new perspective on the role of coevolving interdependency (Mitchell, 2006). In this way, network science has grown as a tool to study complex systems (Kennett and Havlin, 2015), the structural characteristics of which allow them to be described as decentralized, non-hierarchical, dispersed, and distributed networks. One of the main advantages of this complex network approach is that it enables researchers to grasp the different dimensions of how each node in a network is connected, allowing a focus not only on the mere size of the flows passing through a node but also on the connectivity and/or the importance of each node in the network. According to Newman (2003), there are two different but interrelated approaches to the modern analysis of complex networks. The first one is aimed at grasping the statistical properties of the structural and dynamic behaviour of networked systems, while the second one points to modelling networks to explain and support the understanding of the way in which they are created and evolve to face better any possible failures due to extreme events (hereafter EEs). Because EEs can often have catastrophic consequences and, especially in complex networks, can lead to redesigning the structure of networks, developing effective control strategies for understanding, assessing, mitigating, and, if possible, predicting these events is of fundamental importance and practical interest. However, it is worth noting that this is possible only when all their possible underlining causes are understood. In fact, since the financial crisis shattered the world economy, the economic literature has paid more attention to understanding and analysing its impacts and consequences rather than its causes. This study embraces a different perspective, focusing mainly on the investigation of the causes of the world economy crisis rather than on its impacts and consequences. To this end, the present paper approaches and discusses the functioning of the global economy as a complex network, in which the occurrence of EEs triggers cascading failures. To accomplish this task, three main categories of networks and their characteristics are reviewed to select the class of networks that best fits the global economy. Moreover, to assess the risk of the occurrence of EEs, different statistical approaches are analysed. One of the most significant is the Self-Organized Criticality (SOC) approach, which describes an EE that can be predicted through the dynamics of previous small events as a Dragon King, while outliers’ events, the risks of which cannot be anticipated, are referred to as Black Swans. Starting from this, the causes of the 2008 world economic crisis are identified. The paper is organized as follows. Section 2 provides an overview of the network theory applied to economics, describing the different network models. Section 3, after focusing on Complex Networks, provides a definition of EEs and an overview of the approaches used to assess their occurrences in these networks. The subsequent Section 4 offers a new reading of the 2008 global crisis as an extreme event in the framework of complex networks. Finally, Section 5 presents some implications and concluding remarks. 2. Economics of networks Economic scientists, who approach the economic system as a static system – characterized by linear relationships, equilibrium, and connections that fit relatively simple equations – have to shift towards new economic theories to understand better the real economic functioning and the way in which governments should effectively manage it. Thus, the literature has strived to approach the economy as a complex network, oriented towards evolving and changing rather than to the mere competition and to the search for efficiency and growth. This emergent approach is more than an extension of mainstream economics, viewing it in a holistic way. According to this view, the economy is a system in which time evolution is fundamental, actions and strategies always evolve, structures are always formed and reformed, phenomena are not often visible to the standard equilibrium analysis, and meso-layers, bridging the micro- and the macro-layers, are even more important Static equilibrium and perfect rationality, neglecting innovation and downplaying institutions, and the assumption of zero-sum market transactions pave the way for ‘‘an economy made up of millions of overlapping activities, in which individuals, businesses and other institutions are highly connected and constantly interact, where preferences change and markets shift in unpredictable ways’’ (Kay, 2012). In short, the modern world is highly networked. Economic theory has only integrated networks into its analysis in recent times.3 In more detail, the first approach to network theory mostly investigated balanced networks and the mechanisms at the core of their formation, mainly based on the maximization of utility. The related researches approached networks as interconnected cooperative and competitive games, focusing mainly on the investigation of the efficiency–stability trade-off (Jackson and Wolinsky, 1996). Nevertheless, 1998; Weisbuch et al., 1995) and in providing mutual assurance in developing countries (Fafchamps and Lund, 2003). Splitting societies into groups is important in many contexts, such as the provision of public goods and the formation of alliances, cartels, and federations (see, among others, Boccaletti et al., 2006; Guesnerie and Oddou, 1981; Tiebout, 1956). 3 This is due to two different methodological aspects: ‘‘first, social networks require rather sophisticated mathematical tools; secondly, networks that were traditionally analysed in mathematics were inappropriate for describing economic networks’’ (Baskaran and Bruck, 2005).

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the application of game theory to networks analysis was intended to grasp equilibrium outcomes that are strategically stable and to compare them with efficiency requirements. This implied the development of simple analytical models, the simplicity of which makes them able to be modelled analytically. This simplicity also characterized the topologies used to describe networks: a star or a complete network, in which one node interacts with all the others. However, when the size of a network increases and the topology becomes complex, these models become ever more difficult to solve. In recent times, economic networks have been investigated at various levels, such as stocks (Emmert-Streib and Dehmer, 2010), brokers (Shih and Greene, 1998), inter-firm transaction networks (Luo et al., 2012; Luo and Magee, 2011) and product markets (White, 2002), highlighting their capability of self-organizing under the competing influences of negative and positive feedback mechanisms as well. Therefore, because some economic networks display non-trivial topological features, the connection patterns of which are neither regular nor random, an approach that stresses their complexity is needed. In this direction, a new and emerging science of complex networks,4 shedding light on the way in which added value rises from the connections between elements, could offer a fundamental insight into system’s dynamics and into the way in which they can be traced back to the structural properties of the underlying interaction network. Moreover, because empirical results have pointed out that several large-scale complex economic networks5 are scalefree6 (Schweitzer et al., 2009a,b), a change in the traditional way of representing the structure of the economy is required; thus, it can be now considered as being made up of thousands or millions of individual entities with different degrees of network connectivity. Economic networks are characterized by a few nodes with a high degree of connectivity, meaning that a few economic agents hold the power and in so doing dominate markets and influence transactions, and by several nodes with low connectivity, which are agents linked just to a smaller group of other agents. The interactions occurring between all those agents are characterized by an evolutionary process that might happen at multiple and different levels. Thus, the agents can change their status, being influenced by neighbouring nodes, new agents can be added to the network, as well as others might move away from it, creating or destroying the links across vertices (Gomes, 2014) and shaping new collective behaviours. These collective behaviours do not require the coordination of people to perform the same action; rather, they result from the convergence of selfish interests coupled with the impact of interactions occurring between people through the complex network to which they belong. Such collective behaviours may be robust to external interventions, arising from bottom-up mechanisms and from the selfish nature of individuals. When positive feedback forces dominate, deviations from equilibrium lead to crises. Such instabilities can be seen as intrinsic endogenous progenies of the dynamic organization of the system. In this view, the origin of crashes is much subtler than has often been thought, as it is constructed progressively by the markets as a whole as a self-organizing process. The true cause of a crash is this systemic instability. Therefore, characterizing the structure of networks and their relationships is a key issue in understanding this instability and its consequences. 2.1. From random graphs to network thinking According to Wasserman and Faust (1994), a network is a system in which the elements are somehow connected. The aforementioned elements are the nodes (also known as actors or vertices), and the connections between interacting nodes are the links, also defined as ties, edges, or arcs. Nodes might take the form of neurons, individuals, groups, organizations, or even countries, whereas ties can take the form, among others, of communication, friendship, collaboration, alliance, or trade. The different degrees of connection being an important element for understanding how different shocks affect each node, the selected network models using this property can be divided into three major classes, which will be discussed in the following sections. 2.1.1. Random and small-world networks The first of these classes includes so-called homogeneous networks, which are Erdös and Rényi’s random and small-world networks (Fig. 1), characterized by the fact that connectivity probability P(k) peaks at an average ⟨k⟩ and exponentially decays for a large k. The systematic study of these networks, following Erdös and Renyi’s (1959) model, intends to generate random graphs with N nodes and K links. Starting from N disconnected nodes, Erdös and Rényi (hereafter ER) random graphs are generated connecting couples of randomly selected nodes to avoid multiple connections. The term ‘‘random’’ refers to the nature of the arrangement of links, which are chosen to bridge different nodes in a completely casual way. Thus, each couple of nodes is connected by a link with a uniform probability p, which follows a Poisson distribution (Hein et al., 2006). In their seminal work, Erdös and Rényi analysed the properties of these networks using probabilistic methods as a function of the increasing number of random connections (Boccaletti et al., 2006). 4 For a survey of applications, see da Fontoura Costa et al. (2011). 5 For example, in financial contexts, the degree of distribution is seen to scale as a power law for the connections of banks in an interbank network, in which the fat tail indicates that a few banks interact with many others. Then, banks with similar investment behaviour form clusters in the network. Similar regularities can also be traced for the international trade network (ITN), albeit with some important differences. For instance, the total value of country trade is not power law distributed but scales as a log-normal density. This hints at an underlying null model of uncorrelated trade flows, which in turn poses interesting questions about the purported complexity of the ITN (Schweitzer et al., 2009a,b). 6 ‘‘Scale-free networks adhere to power laws and, as such, offer dramatically different representations of value distributions to those presumed in the strong field theory cases preferred in much of neoclassical economics’’ (Foster, 2005).

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Fig. 1. Transition from a locally ordered structure to a random graph, dependent on p.

One of the foundational characteristics of such networks is that the length of the shortest chain that connects two vertices (or members) grows very slowly together with the size of the whole network. This implies important consequences for the speed of the spreading of possible contagion (Watts and Strogatz, 1998). In these networks, the spreading of contagion mainly depends on a critical value. In fact, because the distribution degree of the ER network peaks at the average value and decays exponentially, it has been demonstrated that the spread of disease is characterized by a critical non-zero value λc . Taking into account the fact that the infection’s path is characteristic for λ = δ/ν ,7 if λc >λ the contagion dies, while if λc <λ it spreads in a persistent way. Even though they do not exhibit the strong clustering of nodes and the degree of their distributions is approximately Gaussian, the related behaviours are different from the power-law degree distributions often observed in real-world networks.8 This implies that ER networks are not a well-suited framework for modelling highly connected architectures of emergent patterns in the economic world. The small-world network suggested by Watts and Strogatz (1998) enables the reconciliation of the properties of regular networks9 with random ones, introducing a certain number of random long-range connections in an initially regular network. In this type of network, most nodes are not close to one to another, but most of them can be reached from every other with just a few steps. In these networks, the distance between two nodes chosen randomly grows proportionally to the logarithm of the number of nodes in the network. This means that, even in networks with many nodes, the shortest path length between two individual nodes is likely to be relatively small, hence the ‘‘small-world’’ designation. Otherwise, in a regularly connected network, the path length linearly scales with the size of the network. Therefore, a small-world network is:

• numerically large; thus, the world contains n≫1 people; • sparse; thus, each person is connected to an average of only k other people; • decentralized; thus, there is no dominant central vertex to which almost all the other vertices are directly connected. This implies a stronger condition than sparseness: not only the average degree k must be lower than n but also the maximum degree Kmax over all the vertices must be lower than n. However, these models typically result in degree distributions that do not match most of the investigated real-world networks. 2.1.2. Scale-free networks: The rich get richer Albert and Barabási (1999) developed the second and alternative network model, according to which growth via ‘‘preferential attachment’’ produces so-called scale-free networks (Fig. 2), in which the P(k) decays as a power law. The main differences between these and the previous models are related to:

• Their power-law degree distribution. In particular, it has been found that the degree distribution follows a power-law tail that assumes the same functional form at all scales. The following functional form p(k) : kγ defines the probability of a node having k edges; 7 How an infection spreads depends on λ = δ/ν , where δ , in the case of the contagion model, is the probability of changing the status from susceptible to infected, while ν denotes the probability of changing the status from infected to removed (for further details, see Kermack and McKendrick, 1927). 8 In the theory of probability, the normal or Gaussian distribution is a continuous probability distribution, which is often used as a first approximation to describe real-valued random variables that tend to concentrate around a single average value. Indeed, a power law is a functional relationship between two quantities, one of which varies as a power of the other. For instance, the number of cities with a certain population size is found to vary as a power of the size of the population. Empirical power-law distributions hold only approximately or over a limited range. 9 Networks are ‘‘regular’’ when each node has exactly the same number of links. Regular networks are highly ordered (see Fig. 1).

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Fig. 2. The birth of scale-free networks (Barabási, 2009).

• The spread of contagion. It follows a different path, because there is no critical value λc below which the contagion does not spread. The infection spreads to all the parts of the network, making it very sensitive and vulnerable to this spreading (Hein et al., 2006); • The nodes do not have a typical number of neighbours. Thus, the term scale-free refers to the distribution principle of the number of links per node. In a network with a p degree of distribution (Fig. 1), most nodes have approximately the same number of links. Therefore, it is difficult to find large deviations from the average. In contrast, in a scale-free network, there are large variations in the number of links per node. However, it is also worth noting that there is a small number of nodes with many links, which coexist with a larger number of nodes with a smaller number of links. As a consequence, the network grows in such a way that nodes with a higher degree receive more new links than nodes with a lower degree. This implies the existence of hubs, nodes with a high degree of centrality. ‘‘Many—perhaps most—real-world networks that have been studied seem to be scale-free; that is, they have power law rather than Gaussian degree of distributions’’ (Mitchell, 2006). ‘‘Scale free network theory tells us that we should not expect value to be normally distributed’’ (Foster, 2005). In this new view, for example, firms are not the same or, from a stochastic perspective, differentiated by normally distributed variations. 2.1.3. Complex systems: ‘‘Network thinking’’ Up to this point, the description of networks has mainly focused on their structural properties. Recently, network research has focused not only on the analysis of their topology but also on the investigation of the process dynamics that take place inside them. In fact, the network theory witnessed the birth of a new movement of interest and new research paths, analysing networks as complex systems, which are irregular structures that evolve dynamically over time. Even though the literature lacks a widely accepted definition, in most cases a complex system is defined as a large network of relatively simple components that interact with and adapt to each other and, in so doing, affect their current and future individual environment. Complex networks are made up of irregular structures that involve thousands or millions of nodes, the shape of which changes as time elapses (Boccaletti et al., 2006). High-dimensionality, heterogeneity, and evolution are thus the main features that characterize complex networks. These networks are able to self-organize, not needing deliberate management or control of any singular entity. The emergent behaviour arises from the interactions of parts that are in turn influenced by the overall state of the system. Complex systems generate dynamics, enabling the unexpected transformation of their elements through adaptation, mutation, transformation, and so on. In more detail, it is the environment that triggers changes in the entities that are settled in it and that in turn boost the change in the environment or vice versa. This is what is meant by co-evolution and adaptation. Therefore, a deeper understanding of this kind of co-evolution as well as of the place in which the system’s change occurs over time is crucial. In a context characterized by forces able to maintain the current order, co-exist other forces that push the system itself too far from equilibrium in an unpredictable way, which can also lead to a possible crisis. The exact future may not be predictable, but anticipating directions or trends is possible. Moreover, the level of connectivity in a network shapes the complexity of the network itself. Thus, increasing connectivity triggers a corresponding increment of interrelationships, represented by chains of agents that are linked together. At low and high levels of connectivity, the number of new interrelationships rises more slowly than the connectivity increases. Complex networks show three kinds of behaviour other than those described previously.

• Small behaviour states – which represent the diameter of a network (the average number of links between two nodes), which tends to grow more slowly than the number of nodes. This means that a relatively small number of ‘‘hops’’ is needed to connect each node of the network in pairs. In other words, a small-world network is relatively tightly connected.

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• Self-organized criticalities – which refer to a network’s state of precarious stability, in which one of several paths are imminently possible.

• Scale-free dynamic states – which refer to the state of some nodes that are more connected than others; thus, the additional links are more likely to connect to those nodes that are already well connected. This behaviour describes the so-called ‘‘rich get richer effect’’, whereby the preferential10 attachment of new users is a real element of networks. Merging these three characteristics offers important insights into how and why complex networks, like modern economies, perform their behaviours. 3. Complex networks In complex networks, the most general perception of extremity/exceptionality is related to the existence and functioning of hubs, for example nodes with a huge number of connections. Although such nodes are quite rare, they usually shape the structure of the overall network and many of the dynamic processes that happen on top of the network. In these networks, the links between actors are not equally distributed, but a large number of nodes, characterized by just a few links and a few hubs with a large number of links, can be detected. In such structures, the inequality is the price of efficiency, because such hubs have smaller distances and higher diffusion speeds within the entire network than an equally distributed structure. At the same time, such unequally distributed networks are less vulnerable to the random loss of a node. Hence, the high interdependence between nodes makes them much more vulnerable to random failures and targeted attacks, revealing cascading failures when these events occur. Therefore, on the one hand, an extreme event is more likely to strike one of the many less central nodes than one of the hubs that keep the entire structure together. On the other hand, equally distributed networks collapse faster if one of the hubs collapses. 3.1. Extreme events in complex networks An event that is responsible for the failure of complex networks, and for this reason is extreme, results from large fluctuations in the load and the limited throughput capacity of the nodes. If huge fluctuations occur in the flow that exceed the nodal or link capacities, that is, if at time T x(t=T)≥xee, where xee 11 is the threshold that identifies the occurrence of an event, then this event is extreme. These exceedances can be triggered both by external disturbances (external attacks) and by some inherently emergent behaviour (e.g. transport processes). This implies that, when they occur in networks, extremes events (EEs) are not isolable phenomena but rather properties of interactions emerging within or between complex and dynamic systems. In particular, in this case, EEs are characterized and created by context. It is possible to manipulate the occurrence probabilities of EEs and their distribution over the nodes of scale-free networks by tuning the nodal capacity. However, increasing the nodal capacity is an expensive proposition; thus, this can be achieved only if this increment leads to a proportionate decrease in the likelihood of an EE. Moreover, because, beyond a critical value of this growth (Kishore et al., 2013), there are no significant changes in the number of EEs, to understand the limitations of capacity addition to alleviate the effect of an EE, a benchmark is needed. This leads to important implications, especially when increasing the capacity is chosen as the route to mitigate the effect of an EE on networks. For example, building bridges at a road intersection is a method intended to add the capacity that might increase the throughput across the node. In communication networks, at each node, additional servers and switches might be needed to handle efficiently the exceeding traffic that flows through it. This tuning often being too expensive, when an EE occurs, scientists try to understand the related risk and effects. It has to be noted that many approaches have been developed to assess this risk, and the most important are:

• SEVT (Statistical Extreme Value Theory); • POT (Peak Over Threshold) variant; • SOC (Self-Organizing Criticality) approach. The SEVT provides firms with a theoretical foundation on which statistical models can be built to describe and represent an EE. A powerful statistical theory of extremes has been developed, which, in its most familiar version, specifies that the successive values of the recorded variable are independent and identically distributed random variables, without being constrained by the number and the nature of the other variables that may be involved in the system under investigation. On the contrary, the application of the SEVT emphasizes the relaxation of the independence condition in favour of multivariate and spatial frameworks and deeper use of regular variation and point process approaches. Since this framework is based on Gaussian or its derived distributions, EEs are very improbable events that react to exogenous causes. 10 The existence of a preferential attachment denotes that the new nodes entering the network prefer to connect themselves to the nodes with a higher degree of connectivity; this is often described as a ‘‘rich-get-richer’’ process. Preferential attachment is one of the main reasons for the generation of ‘‘hubs’’ and of the power-law that shapes the degree distribution of complex networks (Gomes, 2014). 11 In a recent work, the threshold for extreme events, x , was taken to be proportional to the typical size of the flux passing through a node (see Kishore ee

et al., 2013).

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According to the so-called ‘‘peak over threshold’’ (POT) method, the distribution of exceedances over a certain threshold is considered. This implies a focus on an (unknown) distribution function F(u) of a random variable X, which estimates the distribution function F(u) of x values above a certain threshold u. A crucial step in the development of the POT method is to select the best-fitting distribution function for the data set and the appropriate threshold value, which is the estimation of the extreme events that are sensitive to its changes. Even though this approach is different from the SEVT, it leads to the same conclusion; thus, for the POT method, an extreme event is also exceptional. Emergent trends of research have argued differently. An EE should be considered as somewhat frequent and arising from the same organizational principle as those that generate other events12 or as consequence of Self-Organizing Criticality (SOC). SOC systems are able to self-organize getting higher levels of complexity. As the complexity increases, they could reach a critical state, triggering an event that generates a qualitative change in the system. After this, the process starts again. If the relationship between event magnitude and frequency is plotted in an SOC system, the L-shape of a power law distribution can be obtained. The power law distribution is N(s) ≈ s−t (where N is the number of observations at scale s and t > 0 is a parameter). The scaling parameter typically lies in the range 2
such cases, we say that the tail of the distribution follows a power law. 14 Before the discovery of Australia, people living the so-called old world were convinced that all swans were white, an unassailable belief, as it seemed to be completely confirmed by the empirical evidence. The sighting of the first black swan might have been an interesting surprise for a few ornithologists (and others extremely concerned with the colouring of birds), but that is not where the significance of the story lies. It illustrates a severe limitation to our learning from observations or experience and the fragility of our knowledge. One single observation can invalidate a general statement derived from millennia of confirmatory sightings of millions of white swans (Taleb, 2007).

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Fig. 3. Share of wages and private consumption in the gross domestic product (GDP) of the United States + the European Union + Japan. Source: Michel Husson (http://hussonet.free.fr/toxicap.xls). Sornette and Woodard (2010).

Sornette (2009) embraced a different perspective, showing that EEs are even ‘‘wilder’’ than predicted, extrapolating them from the power law. In some cases, small events satisfy power-law scaling, but one or more EEs are significantly larger and have a greater probability of occurring than a power law would mandate. However, the most extreme events often do not belong to a scale-free distribution, because they are statistically and mechanistically different from the rest of their smaller siblings, being related to the existence of transient organization. Most of the complex systems of interest exhibit qualitative changes of regimes and dynamics, known as ruptures, phase transition, bifurcations, and catastrophes. These qualitative changes due to smooth variations of some ‘‘control’’ parameters or a function of the network’s topology and/or metric are often at the roots of the so-called Dragon Kings.15 In fact, they are the result of the same system properties that give rise to the power law, but they violate the power law because those properties have been arranged in such a way as to create severe instability, producing a systemic risk. Moreover, the presence of a positive feedback mechanism ‘‘creates faster-thanexponential growth, making them larger than expected. When in the system all feedbacks are harmonized in the same self-reinforcing direction, a small, seemingly non-causal disruption to the system can lead to massive failure. Therefore, Dragon-kings are related to critical transitions of this underlying dynamics. This triggers the rising of specific properties and signatures that may be unique characteristics of that event’’ (Sornette, 2009). These properties make dragon kings not only identifiable in real time but also predictable from observing the spatio-temporal dynamics of previous small events, allowing for the diagnosis of the maturation of a system towards a crisis. Of course, for the identification of these phenomena, multiple methodologies or tools16 are needed. However, it has to be underlined that these are complementary methods, which, of course, cannot be applied in the same way to all systems. 4. The economic crisis of 2008: Black swan or Dragon King? The economic crisis is a well-known event. It started with mortgage dealers, who issued mortgages with terms that were unfavourable to borrowers, who were often families that did not qualify for ordinary home loans. This situation led the world economy to the worst crisis since the Great Depression of the 1930s. In 2001, the Federal Reserve responded to the dotcom bubble burst by cutting the interest rates to 1 per cent. In this scenario, the low interest rates encouraged people to buy houses; consequently, the prices started to rise and mortgage companies relaxed their lending criteria, trying to capitalize on the booming property market. The main consequence of the availability of easy credit was that too many people took out a loan to buy properties that they could not afford. The US and other countries were living beyond their means. Fig. 3 depicts the situation in the early 1980s until 1984, when consumption was mainly funded by wages, while after this period the gap between consumption and wages started to grow exponentially. This meant that consumption had to be funded by other sources of income, for example the increasing profits from investments (Sornette and Woodard, 2010). Therefore, the increased level of consumption was funded by extracting wealth from financial profits, which played a crucial role in increasing household consumption. As Campbell and Cocco (2007) maintained, this led to a rise in housing prices.17 15 The term ‘‘king’’ refers to the fact that kings in certain countries are much wealthier than the wealthiest inhabitants of their countries in a precise sense. While the wealth distribution over the whole population excluding the king obeys the well-known power law Pareto distribution, the king (including his family) is an ‘‘outlier’’, with wealth many times larger than predicted by the extrapolation of the Pareto distribution. This ‘‘king’’ regime may result from cumulative historical developments, that is, specific geo-political–cultural mechanisms enhancing the special king status in the wealth distribution and in other associated attributes that also contributed to the accumulation of wealth by the royal family (Sornette and Ouillon, 2012). 16 For further information, see Sornette (2009) and Sornette and Ouillon (2012). 17 ‘‘[..] we have found that consumption responds to predictable changes in house prices, an effect which is consistent with an increase in house prices relaxing borrowing constraints, but that may also be explained by reverse causality from financial liberalization to house prices, or by a precautionary savings motive’’ (Campbell and Cocco, 2007, p. 25).

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The bursting of the housing price bubble, leading to the acceleration of defaults on loans, quickly depreciated the value of mortgage-backed security (Doms et al., 2007). Essentially, the US economy was characterized by unsustainable deficits, and its subprime mortgage market set the foundation for financial leverage and engineering, which ultimately financed consumers’ expenditure in excess of their capacity. The turnaround of the housing bubble and problems in the subprime mortgage market were the triggers that pricked the US bubble in 2007 (Sheng, 2010). In this sense, Demyanyk and van Hemert (2009) explained that the 2001 subprime mortgages were highly risky, although the fast house price appreciation hid their real riskiness. Moving the focus to the bankers, they bundled up these loans and sold them to investors of other national and international banks, generating a contagion effect. In late 2007, the prime mortgage markets also started to show higher than normal default rates. Thus, once US borrowers started defaulting on their mortgages, many people lost their houses and international investors, including banks and hedge funds, lost their investments. Due to the multitude of relationships between investors and international financial institutions, the problems from the US markets spread rapidly to Europe, creating the so-called ‘‘domino effect’’ (Markwat et al., 2009). In particular, the French bank BNP Paribas18 suspended three investment funds, justifying this decision on the grounds of the US housing sector’s problems. In September 2007, Northern Rock, which was the largest British mortgage bank, was close to insolvency, a situation that led to the fast migration of customers to other banks. Successively, Bear Stearns, which is one of the big five Wall Street investment banks, also approached bankruptcy, which the intervention of JP Morgan Chase, the Federal Reserve, and the US Treasury averted, offering protection to those economic agents that became bankrupt. In September 2008, another large investment bank, Lehman Brothers, collapsed due to its massive risky positions financed with extraordinary leverage. This led to a drastic reduction of investors’ confidence in the business environment, due among other issues to the collapse of assets’ prices and credit channels. Consequently, US and European major stock indexes suffered huge losses (Moldovan, 2011). The US economy was essentially characterized by unsustainable deficits, while its subprime mortgage market was the basis for financial leverage and engineering that ultimately financed consumers’ expenditure in excess of capacity. The turnaround of the housing bubble and problems in the subprime mortgage market were the triggers that burst the US bubble in 2007 (Sheng, 2010). The speed at which the problems of the US financial markets spread worldwide are a reminder that, in times of crisis, financial markets tend to move together (Moldovan, 2011). The collapse of Lehman Brothers showed that, due to its complexity, depth, speed of contagion and transmission, and scale of loss, the modern financial crisis was unprecedented (Stiglitz, 2009). This was mainly due to the characteristics of the actual world economy, which is highly interconnected in a complex way; thus, in this specific context, the global financial market represents a complex network of constantly changing networks. The network architecture played a fundamental role in making the global financial market fragile and vulnerable. ‘‘Network concentration created a number of large, complex financial institutions that dominate global trading and are larger than even national economies. However, they are regulated by an obsolete regulatory structure that is fragmented into national segments and further compartmentalized into department silos, none of which has a system wide view of the network that allows the identification of system wide risks’’ (Sheng, 2010). From this viewpoint, the United States might be considered a super hub of global financial markets, while the London market is another. Therefore, London and New York may account for more than half of global market transactions, because most of the wholesale banks, investment houses, and asset management funds are operationally located. The failure of one hub, such as Lehman Brothers, revealed unexpected interconnections, determining cascading failures that spread throughout the whole system and creating large-scale collective failures19 due to the removal of nodes with large loads. Nevertheless, the Lehman Brothers failure was considered to be the casus belli, while the real cause of the collapse was the instability of financial markets (Sornette, 2009) that arose in the previous months and years. This was mainly due to the slow build-up of powerful hidden forces that came together in one critical instant during which an explosion to infinity of a normally well-behaved quantity was manifested (Sornette and Johansen, 1997). In this specific case, the progressive upsurge of market cooperativity or effective interactions occurring between investors caused the market price to rise, creating many different bubbles. Therefore, according to Sornette (2009), five bubbles were responsible for the 2008 financial crisis: (1) The ‘‘new economy’’ ICT bubble, which started in the mid-1990s and ended with the crash of 2000. (2) The real-estate bubble, mainly triggered by the easy access to a large amount of liquidity as a result of the active monetary policy of the US Federal Reserve, which lowered the Fed rate to relieve successfully the consequences of the 2000 crash. (3) The innovations in financial engineering with CDOs (collateralized debt obligations), other derivatives of debts, and loan instruments issued by banks and eagerly bought by the market. (4) The commodity bubbles for food, metals, and energy. (5) The stock market bubble peaking in October 2007. 18 For further considerations, see Forte and Pesce (2009). 19 Many research studies have examined the connections among countries by exploring correlations of various financial time series data. Moreover, many studies have analysed the relationships between stock and foreign exchange markets, given the significant increase in global capital flows in the last two decades. Other studies have focused on global stock market return predictability, offering diverse findings across different regions and time periods (Vodenska et al., 2016).

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These argumentations imply some important reflections about the economic crisis of 2008:

• Its origin was subtler than one might often think, because it was progressively built by the market as a whole as a self-organizing process. Essentially, its endogenous or internal origin was triggered by exogenous or external shocks.

• It represents the result of amplifying mechanisms not necessarily and completely active for the rest of the population. This event was the outcome of system dynamics that progressively approached instability, leading to a transition to another state. • Its occurrence might well be defined as a dragon king and not as a black swan. Taleb (2007) supported this statement. He confirmed that the crisis was a predictable event, counteracting those who tried to approach it with the notion of an unexpected black swan event (Morrison, 2008). It worth noting that these reflections lead to important results for the management of risks. Thus, the ‘‘Dragon king’’ Theory, in contrast to ‘‘Black Swans’’ theory, suggests that certain extreme events can be predictable through observing the spatiotemporal dynamics of preceding small events, such as the aforementioned bubbles, jointly with the critical levels of some important and standard economic indicators.20 5. Conclusions One of the most important lessons learned from the 2008 economic crisis is that the world is more complex than many people have thought and, certainly, it is more complex than many economic models have depicted. The global economy is a network made up of local networks, in which the weakest link tends to be the weakest node, cluster, hub, or local network. However, why or where the system is weak is not so clear, being a challenging topic that still calls for further investigation. Because crashes are often due to system instability, the crisis’s origin was subtler than expected, being progressively shaped by the network as a self-organizing process. Hence, to identify the weakness, the instability of economic systems has to be investigated holistically according to the complex systems theory (Harmon et al., 2015). To understand many economic behaviours, spanning from the dynamics of product adoption to financial contagion, the patterns of interactions necessarily have to be considered. Thus, failing to include the network theory in economic analysis can lead to a poor understanding of observed behaviours as well as to a poor policy design. Therefore, the clear definition of networks’ structure and their relationships is a core issue, which should be addressed to gain a better understanding of systems’ instability and its related consequences. This framework led to the assumption of a different perspective for approaching the events that trigger dysfunctions, affecting their performance and the occurrence of extreme events (EEs). Extremes, taking place in networks, are a common experience. Atypical large fluctuations in the flow, exceeding its nodal or link capacities, can lead to EEs, which can be triggered by external disturbances or can be a type of inherently emergent behaviour. It worth noting that, in many situations, they occur suddenly and can spread through the network as shockwaves. Their global consequences strongly depend on the propagation dynamics, the contagion effects, and the capacity of each network element to withstand sudden changes. Because EEs lead to losses, ranging from financial and productivity and even to life and property losses, the estimation of the probabilities of their occurrence and the achievement of some form of control by manipulating their occurrence are fundamental. New trends of research have argued that an EE is generally notable, rare, unique, profound, and significant in terms of impacts, effects, or outcomes but at the same time, predictable. In this sense, extreme events, such as economic crises, could be modelled differently using the dragon king methodology. Therefore, only one question arises: why was the economic profession unable to distinguish very large losses (global crisis) from the rest of the population of smaller losses (US financial market), highlighting the occurrence of Black Swan or the Dragon King? Acknowledgements The authors gratefully acknowledge an anonymous referee and Clevo Wilson, the Editor-in-Chief of this journal, for their useful comments for improving the manuscript. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for profit sectors. References Albert, R., Barabási, A.L., 1999. Emergence of scaling in random networks. Science 286 (5439), 509–512. Bak, P., 1996. How Nature Works; The Science of Self-Organized Criticality. Copernicus, New York. Bar-Yam, Y., 1997. Dynamics of Complex Systems. Perseus Press, Reading. Barabási, A.L., 2009. Scale-free networks: a decade and beyond. Science 325, 412–413. Baskaran, T., Bruck, T., 2005. Scale-free networks in international trade. DIW Discussion Paper No. 493. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U., 2006. Complex networks: structure and dynamics. Phys. Rep. 424 (4), 175–308. Boorman, S.A., 1975. A combinatorial optimization model for transmission of job information through contact networks. Bell J. Econ. 6, 216–249.

20 For the United States, for example, the level of asset price inflation, rising leverage, large sustained current account deficits, and a slowing trajectory of economic growth exhibited all the signs of a country on the verge of a financial crisis (Reinhart and Rogoff, 2009).

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