Abstract
Critical events in society or biological systems can be understood as largescale selfemergent phenomena due to deteriorating stability. We often observe peculiar patterns preceding these events, posing a question of—how to interpret the selforganized patterns to know more about the imminent crisis. We start with a very general description — of interacting population giving rise to largescale emergent behaviors that constitute critical events. Then we pose a key question: is there a quantifiable relation between the network of interactions and the emergent patterns? Our investigation leads to a fundamental understanding to: 1. Detect the system's transition based on the principal mode of the pattern dynamics; 2. Identify its evolving structure based on the observed patterns. The main finding of this study is that while the pattern is distorted by the network of interactions, its principal mode is invariant to the distortion even when the network constantly evolves. Our analysis on realworld markets show common selforganized behavior near the critical transitions, such as housing market collapse and stock market crashes, thus detection of critical events before they are in full effect is possible.
Network selforganization as an indicator of critical transition
Selforganization has been understood as a nonequilibrium phenomenon of physical systems. (Here, we restrict our notion of selforganization to the ‘spontaneous order emergence’ due to a nonequilibrium phase transition.) Such phenomenon is characterized by the system's population transitioning from isolated behaviors to a persistent coordination due to the increasing instability near the transition. Ecosystems and human society also go through sudden regime shifts. T kinds of systematic shifts pose challenges to maintaining the stability of the nature or human society. Sudden disappearance of natural species could cause further disruption in the ecosystem. Unexpected opinion swings or market collapse poses hard challenges to the society. Efforts to detect and respond to such events before the onset of transitions are especially beneficial, because the measures to deal with any undesirable changes can be more effective before the system's full evolution toward a highly nonlinear regime^{1}.
It has been speculated that catastrophic changes in nature are often preceded by peculiar signs^{2,3,4}, such as regularshaped patches of vegetation before desertification; recent studies^{5,6,7} investigated catastrophic population changes observed in ecosystems, and derived general quantitative indicators—increased temporal correlation, skewness, and spatial correlations of the population dynamics. Another study^{8} characterized a largescale dynamical systems going through a bifurcation revealing selforganized spatial patterns as early signs. These approaches consider only the homogeneous lattice as the model of interactions. It would be a natural next step to investigate the selforganization of evolving complex networks—consisting of a large number of entities exchanging heterogeneous influences that can model a wide range of real world systems.
Despite its broad applicability, however, the model of ‘a heterogeneously networked dynamical system going through a phase transition’ has not been treated properly. The formidable issue of the complexity of connectivity has been mainly handled by statistical approaches—degree distributions, random network models^{9,10,11}, etc. Order emergence due to spin dynamics^{12} or synchronization^{13,14} has been extensively studied without considering the evolving connectivity and phase transition at the same time. Only a recent investigation^{15} paid attention to the Turing patterns of complex networks governed by the specific case of activatorinhibitor dynamics, and another^{16} studied the interaction between network connectivity and dynamics.
The effective early warning signals introduced in Refs. 5,6,7 motivates us to develop a statistical measure that summarizes the collective dynamics of a networked population. In their model, each network node has been modeled as following a common dynamics, but the heterogeneous interactions in consideration may render highly complex patterns. The complexity naturally poses challenges in assessing where the system is and where it is heading. The key question is: how does the network's structure affect its pattern of behavior at catastrophic regime shifts? Our investigation on the complex network model leads to a very general indicator of a critical transition that is also not affected by its structure. While the heterogeneous network structure distorts the emergent pattern, the transition indicator (the pattern's principal mode) is invariant to the distortion even when the network constantly evolves with added/removed nodes and changing connectivity and weights. Moreover, the same analysis enables the estimation of the network's connectivity regardless of the particular system dynamics.
Broken behavioral symmetry near phase transition causing selforganization
There are many examples of complex networks showing selforganization. The formation and collapse of speculative market bubble have been largely regarded as the consequence of herd behavior. We postulate that in many cases the herd behavior emerges due to the broken balance between autonomous behavior and peer influence. Figure 1 illustrates the emergence of coordinated behavior. Every entity revolves around its own stable state while exchanging influence with peers or environments, maintaining the balance between these two effects. When a population of such entities goes through a systemwide change that weakens the intrinsic dynamics, the balance is broken and the effect of exchange propagates and dominates, which results in largescale phenomena. It would then be possible to decode the emergent pattern to identify both the ongoing change and the network of influence.
Natural or social systems can be modeled as consisting of a population of entities (x_{1}(t), x_{2}(t), …, x_{N}(t)) = X(t), exchanging material or informational influences through networked interactions. Each network node—an animal/plant or a person—constantly goes through changes in its state, such as metabolism, movement, or simply a daily cycle. It is natural to decompose the state changes x_{i} of the ith node into two different components—one due to local intrinsic dynamics f_{i}(x_{i}) (F(X) = (f_{1}(x_{1}), f_{2}(x_{2}), …, f_{N}(x_{N}) for the whole network) and another due to a peer influence (The model only deals with undirected connectivity as in related studies^{5,6,7,8,11,12,13,14}.) , where L = {L_{ij}} is the combinatorial n × n Laplacian matrix of the adjacency relations (degrees of interactions) A. L is computed by L = DegreeMatrix(A) − A. Each f_{i}(x_{i}) is a gradient to an equilibrium —an attractor (We only handle the mathematically more manageable case of a point attractor.) describing a stable behavior of x_{i}. For instance, may represent a stable metabolic cycle or a person's wellestablished opinion about a certain topic. On the other hand, X is independently perturbed by a random Brownian motion σdW, which represents a fluctuation that constantly kicks each x_{i} slightly away from the . A phase parameter C augments the f_{i}(.) to f_{i}(.,C), capturing the network's evolution from a stable regime () toward a different regime (C > C_{crit}) via a transition point (C = C_{crit}). The stability slowly decreases () as C → C_{crit}.
Some networks may evolve by an internal feedback, for example, the population adjusting their behavior based on the perceived system state. Then the phase parameter C is affected by the state X: , where H(X) represents the degree of the network's global order. For instance, media reports of rapidly increasing asset price can further drive the consumers to the market; H(X) may represent the price increase reflecting more consumers entering the market.
When the phase transition weakens the intrinsic dynamics, the effect of peer influence dominates and a selforganized population behavior emerges. Far from the transition, the dynamics F(X,C) dominate over the relatively weaker interaction LX—exhibiting isolated motions. Approaching the transition, the returning force F′(X, C) become weaker than the persisting LX. The shifting balance from the dynamics to structure causes the motions to become highly correlated, resulting in largescale patterns. This is the wellknown phenomenon of spontaneous symmetry breaking, which can also explain the selforganization of complex networks.
Covariance spectrum revealing both instability and structure
Our main results are derived from this specific model (studied by many authors^{5,6,7,8,12}), but are applicable to a broad range of complex networks where each node revolves around a stable state and exchange influences with other nodes—animal or human population is a prime example. The combination of the evolving dynamics, the peer influence, and the random perturbation leads to a network of stochastic differential equations: dX = (F(X,C) − LX)dt + σdW. The network's total deterministic dynamics around the equilibrium can be linearly approximated by the Jacobian of F(X, C) − LX at X^{eq}:
The degree of coordination is measured by the covariance matrix over some time window [t, t+Δt]: Cov_{c}(t, X) = E_{t,t+Δt}[X − Mean(X)][X − Mean(X)]^{T} for a fixed C. (In reality, C is assumed to change sufficiently slower than X.) A combination of the perturbation equation^{17} and its covariance equation^{18} derives [Supplementary Information 1] a closedform solution of the Cov_{c} in terms of the Jacobian and the Laplacian:
Let's assume that the returning forces become more homogeneous near the transition (, so that the covariance eigenvalues μ_{1},μ_{2},…,μ_{N} can be expressed using the k(C) and the Laplacian eigenvalues λ_{1} = 0, λ_{2}, …, λ_{N}:
For a rigorous proof, please see [Supplementary Information 1].
The leading eigenvalue depends only on the k_{1}(C) (but not on λ_{2}, …, λ_{N})—implying that it can serve as a structureinvariant indicator of the transition. The invariance is a consequence of the peer interaction LX not contributing to the synchronization manifold^{17} v_{sync} = (1,1,…1), but has nontrivial implications:
Property 1. The degree of the global order μ_{1} of the population indicates the network's instability due to a phase transition regardless of its connectivity.
Property 2. When the evolution is driven by an internal feedback , the resulting changes to the instability does not depend on the scale N or the connectivity L.
We conjecture that the first property would still hold without assuming the homogeneous returning forces , but we have been only able to derive the closedform solution (3) and prove the invariance under this stronger assumption. Increasing order parameter is known to indicate instability due to a nonequilibrium phase transition, but our particular form of order parameter has the desirable property of structureinvariance. Though the ultimate goal is to forecast catastrophic events, instability seems more scientifically identifiable condition from which events may break out in combination with other factors.
So far we have assumed that the network Laplacian L is constant. The network itself may be allowed to change with the timescale much slower than the X (L is effectively fixed within the time window [t, t + Δt]). Still all of our results hold under the relaxed assumption. Property 1 leads to a corollary that μ_{1} is invariant to the evolution of the network—when it adds or removes nodes or edges (with changing weights) during the transition, μ_{1} will still provide a structureinvariant measure of the phase C. This property is especially powerful, because most real world networks go through constant changes in degrees of interactions. Property 2 holds because the ‘global order’ H(X) only contributes to the direction of v_{sync}. It has an interesting interpretation in the context of the influence of newsfeeds—that the ‘destabilizing effect’ of global news (e.g., about the current state of the stock market) does not depend on the social network. (This property is verified in the Supplementary Information 2)
On the other hand, the rest of the covariance eigenvalues reveals the structure of the network. The function (3) is monotonically decreasing with the increasing Laplacian eigenvalues: . Therefore, the second largest mode μ_{2}(C) represents the overall degree of the structure amplification. From equation (3), the instability will amplify the Laplacian spectrum λ_{2}, …, λ_{N} as k(C) → 0:
Property 3. The network reveals its underlying structure through the amplified Laplacian modes at the transition, facilitating the estimation of the network structure.
In fact, L can be recovered from the covariance matrices Cov_{C}(t − Δt, X) and Cov_{C}(t, X) computed over two successive time windows. Equation (3) derives:
F′(x^{eq}, C) is the only unknown factor. As can be seen from the expression (1), the heterogeneity of will distort the covariance eigenmodes. Near the transition, F′(x^{eq}, C) → 0 so that the evolving connectivity L can be estimated more reliably independent of the particular F. That is, the distortion effect is zero at the transition—the dynamic network loses its individual property and its more genuine structure emerges.
Instability and structure estimation of model networks at bifurcations
The derived relation between the network connectivity and the covariance matrix in equation (2) holds regardless of the F(.,C)), as long as it goes through a local bifurcation—a general mathematical model of phase transition. The fold bifurcation models^{5,6,7} the catastrophic changes of logistically changing population, with the shifting grazing rate C, the growth rate r, and the carrying capacity K. The pitchfork bifurcation F(X, C) = CX − X^{3} models the transition from a singlewell potential to a doublewell potential—an emergence of alternative regimes. There are other forms of bifurcation, but the diversity is reduced near the C = C_{crit} to exhibit the same qualitative behavior. Figure 2 shows the growth of covariance spectra for a linearly connected 9node network (bottom) and a fully connected 9node network (top), all going through fold bifurcations. The μ_{1}(C) evolves identically for both networks, but the rest of the spectrum follows different curves as expected by the equation (3). The growth of covariance spectra from ten different networks having varied connectivity confirms the invariance property of the μ_{1}(C) and the variance property of the μ_{2}(C).
The property 1 supports the invariance of the leading eigenvalue while network nodes and edges are being added or removed, and the Figure 3 verifies it. The left plot is provided as a reference: the spectrum changes of a 9node linear network when its structure is fixed while going through a pitchfork bifurcation. The right plot shows the spectrum changes of a growing network, when it grows from a 4node linear network to a 9node linear network by adding one node and one edge at a time. Both leading eigenvalues μ_{1} show identical growths.
The structure revealing property at phase transition is shown in Figure 4. In the bottom inset figures, three distinct geometric network structures in twodimensional 20 × 20 grids are compared to illustrate different pattern emergence. When each network is far from the pitchfork bifurcation, its 20 × 20 = 400 nodes (represented as pixel values) show only slight difference. However, when they approach the bifurcation point, the pattern clearly reveals the individual differences in geometry. (The same property for largescale random networks is verified in the Supplementary Information 3.) In the top panel, 9node linear network dynamics was simulated to go through a pitchfork bifurcation. Equation (4) is used to estimate the Laplacian matrix from the state covariance matrices. The Frobenius norm between the estimated Laplacian matrix and the true Laplacian matrix is plotted (upper plot); the difference achieves a minimum at the bifurcation point, verifying the optimal structure recovery property at the bifurcation. The next two sections reveal that realworld housing and financial markets show common selforganized behaviors near phase transitions, and are measurable using the method derived from the same generic network model.
Consumer herd behavior in housing market
Human society is being modeled and studied as a complex system to identify trends or crises, further encouraged by the recent surge of publicly available data. One study^{19} focused on the behavior of mimicry in trading decisions based on price fluctuations, while another^{20} utilized informationgathering behavior of traders by monitoring Web search volumes. These behaviors fit the profile of our behavior model; however, they utilize conventional scalar quantity (price index, search volume, etc.) as the measurement of the population behavior. Our analysis interprets multiple measurements based on an underlying dynamic model to detect trends or crises.
Housing bubble is often attributed to consumers' herd behavior. Their buying or selling decisions are made based on both personal circumstances and peer effects. We postulate that external influence (e.g., low mortgage rates) destabilize individuals to base their purchase decisions less on personal circumstances but more on the others' decisions. In Figure 5, the changes in housing market indices (CaseShiller index^{21}) since 1987 of 14 major US cities have been utilized as reflecting the consumer behavior^{22}. CaseSchiller Index covers 20 cities, but the study excluded 6 cities due to incomplete data. The curve in red dashes depicts the μ_{1}, computed from the 14 indices (colored plots). The coordinated price increase and collapse coincide with a very high degree of instability measured by the μ_{1}, especially during the latest housing boom (2003–2006) and bust (2007–2009). Though each index is a somewhat noisy measurement of the local consumer behavior, the μ_{1} does show very high degrees of market correlation. On the other hand, we expect that an unstable market would reveal the underlying network structure more reliably than a stable and decorrelated market would. Each inset shows the projections of the 14 cities into the twodimensional space during a different period, where each city's coordinate is the corresponding first and the second eigenvector components. The distribution of the cities during the market instability (three top panels) indeed shows expected clusters of markets. In contrast, the periods when the market stabilized (1992) or the market tried to find a direction (2006), the projections (two bottom panels) do not show any meaningful clusters. Most interestingly, Washington DC stayed close to Los Angeles and San Diego during three different periods (1988 bubble, 2004 bubble, and 2008 collapse). We suspect that San Francisco behaved differently during the 2004 peak, because of the lasting impact of the earlier dotcom collapse. (San Francisco, Los Angeles, and San Diego belong to the same state of California.) The rest of the cities form another cluster. New ‘bubble cities’—Las Vegas, Miami, and Tampa—disrupt the cluster structure during the recent housing bubble. These three cities went through the most severe bubble cycles during the 2003–2008 housing bubble and crash. We are not concerned about explicit causal influences between the markets; the proximity may indicate true correlations or just similar local market characteristics.
Selforganized trader behavior causing largescale stock price moves
Financial markets are also driven by collective behaviors of traders. Each trader makes decisions after absorbing information—not only the information about external factors—economic picture, political change, world events, etc.—but also the information about the state of the market, which is the sum of all traders' decisions. Any information of predictive values quickly feeds back to the system, affects trading, and is ultimately reflected to the price (EMH^{23}). If the market is truly efficient, such instant information feedback strips the price dynamics of any trends, allowing only random fluctuations; random walk is a popular model of financial market dynamics. If stock markets do follow pure random walks, then prediction is essentially not possible. However, there have been counter arguments and efforts to compromise the EMH with behavioral economics^{24}. Some of these investigations^{25} observe that there are transient temporal correlations during financial bubbles or crashes, whereas others^{26,27} investigate increasing crosscorrelation within share movements. More recent work^{20,22,28,29} employs physics based models of selforganized emergent trader behavior to explain extreme market moves.
This type of nonstationary dynamic models could compromise the random behavior (due to the efficiency) and ordered behavior of market dynamics. Our analysis also supports the same conclusion that the market shows increased correlation among share prices before and during extreme market events. Our dynamic model provides computational tools to capture such emergent market behaviors. While increased market synchronization is commonly observed before and during market crashes^{26,27}, our work firmly establishes the first covariance eigenvalue μ_{1} of share prices as a mathematically grounded measure of market instability. Another study^{30} found empirical evidence that comovements of share prices over longterm periods (over a month) precede market crashes. For the empirical evaluations shown below, a combination of two μ_{1}′s from both shortterm and longterm market correlations. The [Supplementary Information 4] provides details of the evaluation methodology.
Figure 6 summarizes the result where the share price correlation of DJIA (Dow Jones Industrial Average) measured by μ_{1} is used as an indicator of significant ‘market events’, defined as rare, largescale daily changes of DJIA. The daily returns of the closing prices of each share (such as IBM, GE, etc. that belongs to DJIA) are the input time series to the computation. Figure 6 (a) compares the timeline of Dow Jones market events (daily change over 4%, marked with blue spikes) from 1990 to 2014 and the computed μ_{1} (red plot). Most of the large spikes during the 27year period occurred while the μ_{1} is high.
We then used the μ_{1} as a simple decision function for predicting market events: after each trading day, if the computed μ_{1} crosses a certain threshold, we predict that a DJIA change (drop or jump) larger than 4% will occur in a fixed prediction horizon (one day, one week, or a month). Figure 6 (b: top) show the ROC (Receiver Operating Characteristics) curve for the oneweek (five trading days) prediction performance. The horizontal axis represents the false positive rate (FPR) and the vertical axis represents the true positive rate (TPR). The plot shows that μ_{1} detects more than 80% of the market events while tolerating less than 20% false positives. It compares different event definitions, in terms of magnitude of index changes: 3%, 4%, and 5% jump/drop events. It indicates that extreme events (5% events) are more predictable than more common events (3% events). The Figure 6 (c: top) compares different prediction horizons: oneday, oneweek, and onemonth (22 trading days as an approximation) predictions. While the accuracy degrades for longer prediction horizons, onemonth prediction still shows much higher detection rates than random guesses (the curve will be reduced to the diagonal line of TPR = FPR).
One can observe that the historic crash of 2008 involved a series of largescale events, as also shown in the Figure 6 (a), where the market panic due to the financial sector trouble caused mass selling across the board, further causing multiple large scale stock moves. The first a few days of significant drops surprised investors, but following market instability may have been well expected. Therefore, we have removed the period of extreme market moves and computed the ROC (Figure 6 (e)) for the period of 1990–2007 to verify if the method works for more ‘modest crashes’. The ROC curve for the period shows detection performance for the 4% and 5% events not much different from the whole period including the 1987 and 2008 crashes, verifying the generality of the method.
We also made a comparison to the VIX (Chicago Board Options Exchange Market Volatility Index^{31}), the most popular and successful market volatility measure. The VIX represents the market's expectation of stock market volatility over the next 30day period. The green curve of the Figure 6 (a) is the timeline of VIX, scaled for a better comparison to the μ_{1}. Overall these two independently developed measures closely follow each other, especially during highly volatile periods. Figure 6 (b,c: bottom) shows the ROC curves for VIX are plotted below the corresponding μ_{1} ROCs. For better comparisons they are shown in the same plots in (d). (d: top) shows that the two methods are comparable for 1week prediction performances over different event classes, but μ_{1} still does slightly better for the 4% and 5% event classes. (d: bottom) reveals that μ_{1} performs noticeably better for 1month prediction. Given that VIX achieves slightly better 1day performance, we could conjecture that VIX is more reactive than predictive, compared to our method.
As we have discussed the mathematical model of pattern formation equation (4) and provided evidence in housing market example, stock market also does show peculiar patterns before largescale crashes—not only the increase of overall correlation as captured by μ_{1}. Figure 7 demonstrates the correlation pattern as a potential crash indicator during the 2008 crash. The biggest singleday crash (among other crashes during the period) happened on October 15, 2008, leading to the DJIA losing 7.9%. The figure shows an interesting market behavior prior to the crash and during the crash. The square blocks visualize the sequence of covariance matrices from twelve share prices (subset of the DJIA constituents) before (10/1/2008–10/14/2008) and during/after (10/15/2008–10/28/2008) the crash; darker pixels represent higher pairwise correlations between the share price movements. The diverse and changing correlations as shown in the top row represent typical market behavior weeks before the event. However, staring 10/13 (the second row) not only the overall correlation increased (the μ_{1} plot in Figure 6 captures it) but the pattern persisted through the following week (the third row) until the crash date (10/15). We do observe similar behavior of persistent pattern dynamics during the 1987 Black Monday crash, and other significant market events (Supplementary Information 5). The market behavior after the 9/11/2011 terrorist attack is also provided as an interesting example, where the purely exogenous crash, while large in magnitude, does not show the signature selforganized patterns. The observed behavior strongly agrees with our theory finding summarized in equation (4)—that a phase transition does amplify the underlying structure so that the emergent pattern will be more explicit and persistent. The principal components from market correlation have been investigated previously for identifying business sectors^{32} and discovering evolution of market influences^{26}. We have investigated the same phenomena at finer time scales to reveal that the emergent patterns not only become prominent during crises, but also predictive and persistent.
Discussion
We have developed a mathematical model of a complex network going through a phase transition to derive a predictive measure of such transition based on the observed time series from the network. Under this model, the first eigenvalue of the covariance matrices (computed from the time series) is mathematically proven as a networkinvariant indicator of instability. The same model also suggests another indicator—persistent patterns in time series correlation. Experiments on historical housing and stock market data confirm that the eigenvalue significantly increases before largescale market events, and persistent pattern dynamics are also observed as preceding signs. Many literatures have discussed collective behaviors as potential signs of catastrophic events, but only recent work^{5,6,7} developed quantitative indicator to be applicable to realworld data. We also believe that increased correlation and pattern dynamics are general behaviors of complex networks; therefore, the proposed computational tool should find broader applications. Our framework is a very early effort to establish a mathematically grounded indicator for analyzing time series data from networked entities, and to verify applicability to realword data. As our initial success indicates, the computational method in the proposed form should be applicable to data from different domains. On the other hand, the model is general enough to allow domainspecific dynamic models to be incorporated to develop more specialized indicators of instability.
Further investigation should also focus on deriving a quantitative measure of the pattern dynamics. In general, spectral analysis of fluctuations separates the dynamics of the magnitude (captured by eigenvalues) from the modes (captured by eigenvectors). Therefore, there is a potential to have an improved predictor by combining these two separate quantities. Additionally, further investigations to the whole spectrum of the covariance (not just the first eigenvalue) may provide insights into the scaling behavior of complex networks—how correlation strengths at different scales that is precisely measured by the spectrum are related to instability of complex networks. We also hope to find suitable spatiotemporal data collected from natural systems to test our method, as in the related work^{5,6,7} applied to ecological data.
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Acknowledgements
Partially supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center (DoI/NBC) contract number D12PC00285. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBE, or the U.S. Government.
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 Hankyu Moon
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H.M. and T.L. wrote the main manuscript text and prepared all figures. All authors reviewed the manuscript.
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