- Regular article
- Open Access

# Information dynamic spectrum characterizes system instability toward critical transitions

- Kang-Yu Ni
^{1}Email author and - Tsai-Ching Lu
^{1}

**3**:28

https://doi.org/10.1140/epjds/s13688-014-0028-7

© Ni and Lu; licensee Springer 2014

**Received:**13 May 2013**Accepted:**30 September 2014**Published:**4 November 2014

## Abstract

This paper addresses the need of characterizing system instability toward critical transitions in complex systems. We propose a novel information dynamic spectrum framework and a probabilistic light cone method to automate the analysis. Our framework uniquely investigates heterogeneously networked dynamical systems with transient directional influences, which subsumes unidirectional diffusion dynamics. When the observed instability of a system deviates from the prediction, the method automatically indicates the approach of an upcoming critical transition. We provide several demonstrations in engineering, economics, and social systems. The results suggest that early detecting critical transitions of synchronizations, sudden collapse, and exponential growth is possible.

## 1 Introduction

Detection and prediction of emerging tipping points are major challenges in complex systems [1]–[5], because while self-organized emerging interactions can facilitate information exchange, they increase the risk of attack or failure. When a system operates in a high-risk, unstable region, a small perturbation can induce a critical transition that leads to catastrophic failures. Although studies of data-driven computational models, with prior knowledge of individual systems, have greatly advanced the understanding of such emerging phenomena, we remain unequipped to accurately detect and predict tipping points prior to critical transitions [2].

Current works in emerging tipping point detection focus on homogeneous structured dynamical systems. In [6], Scheffer *et al.* give a seminal review on early warning signals for detecting critical transitions in the ecological domain. Signals such as increased temporal correlation, skewness, and spatial correlation of population dynamics are used to quantify the phenomena of critical slowing down as early warning indicators. These methods do not address heterogeneously networked systems. Through the remainder of this paper, *networked dynamical system* will be abbreviated as *network*. Scheffer *et al.* in [7] highlight the need to anticipate critical transitions in heterogeneous networks with empirical indicators. Such heterogeneity typically leads to connectivity patterns that are statistically characterized by heavy tails, large fluctuations, scale-free properties, and non-trivial correlations, which cannot be described by traditional modeling techniques that rely on the homogeneity assumption of a network [8]. Consequently, network science has evolved from studying complex systems by modeling single, non-interacting networks to modeling interdependent networks [9]. In particular, Buldrev *et al.*[4] show that a network of networks is extremely vulnerable with respect to random failures: a random removal of a small fraction of the nodes from a network can trigger a catastrophic cascade of failures. The interconnectedness and interdependency of complex networks further pose hyper risks [1]. Understanding complex interactions and coupling dynamics in large-scale complex networks is undoubtedly a crucial step toward preventing dangerous system behaviors.

The spectral *early warning signals* (EWS) theory [10], [11], inspired by [6], is one of the few attempts to detect critical transitions in heterogeneous networks. The spectral EWS theory states that the covariance spectrum can quantify the phenomenon of critical slowing down in heterogeneous networks by mathematically proving the link between complex network structures and observed time series. In particular, the leading covariance eigenvalue provides a structure-invariant indicator of upcoming critical transitions. As the system reaches the tipping point, the self-organized structure reveals itself via the components of covariance spectrum. Although spectral EWS quantifies how much the elements of a system change together, the symmetric nature of covariance spectrum does not permit the analysis of directional influences among elements.

Transfer entropy (TE) [12] and symbolic transfer entropy (STE) [13] have been proposed to identify directional influence in complex systems. For instance, STE is used to analyze brain activity data for the detection and identification of asymmetric dependences of brain regions in epileptic seizure activity [13]. In [14], the transfer entropy matrix is used on financial market data to analyze the asymmetrical influence of mature markets on emerging markets. Although the transfer entropy quantified in [13], [14] showed the promising results in analyzing financial and neurophysiological data, the changing structures and dynamics of networks were not addressed. Therefore, there is a need to quantify local transfer entropy that is changing in time [15].

*associative transfer entropy*(ATE) measure which decomposes directional influence of transfer entropy into associative states of the influences. We transform multivariate time series of a complex system into the spectrum of the

*transfer entropy matrix*(TEM) and the spectrum of the

*associative transfer entropy matrix*(ATEM) in order to capture information dynamics of the system. We develop novel

*spectral radius measures*of TEM and ATEM to detect early warning signs of source-driven instability and to reveal the sources and dynamics of directional influences. Motivated by the TE and ATE behaviors observed in different critical transition examples, we then develop a method to automate the generation of early warning indicators. In particular, the nature of convex growth of spectral radius of TEM and ATEM (TE and ATE trajectories), prior to critical transitions, enables us to generate probabilistic light cones of instability trajectories using natural logarithmic curve modeling. This method enables us to analyze system instability trajectories, which in turn provides early indications of critical transitions. We demonstrate our methods on four types of transitions: (1) oscillatory synchronizations of networked non-Foster circuits; (2) pitchfork bifurcations of chaotic systems with various canonical network structures; (3) abrupt falls of directional influences in Latin America stock indices and Dow Jones during the 2008 financial crisis; and (4) exponential growth in Wikipedia editing behaviors. Our results suggest that one can analyze the trajectories of system instability to indicate the nature of upcoming critical transitions prior to their tipping points (see Table 1).

**Examples of critical transitions demonstrated in this paper**

System/Application | Nature of transitions | What to observe |
---|---|---|

Non-Foster circuits | Oscillatory synchronizations | ATE+ cross TE, rapid increase of TE and ATE+ |

Canonical chaotic networks | Pitchfork bifurcations | Rapid decrease of TE |

Latin American stock market | Abrupt falls | Rapid decrease of TE |

Dow Jones | Abrupt falls | Rapid decrease of TE |

Wikipedia editing behaviors | Exponential growth | ATE+ cross ATE−, rapid increase of TE and ATE+ |

## 2 Information dynamic spectrum

We propose an information dynamic spectrum framework for monitoring complex systems. The assumptions in this work are (1) the outputs of the elements in the complex system can be observed as time progresses, i.e. time series are available, and (2) the outputs are the results of internal dynamics within individual elements and external interactions with other elements. An example is the time series of stock indices over a period of time. The fluctuations of a stock index depend on its own dynamics, as well as the hidden interactions with other indices. One example application of the proposed framework is to monitor the dynamics of this complex network of stock indices.

We will here establish the notations. Given the time series of a system of *m* elements: $X(t)={[{x}_{1}(t),{x}_{2}(t),\dots ,{x}_{m}(t)]}^{T}$, where *t* is time and $t=1,2,\dots ,N$, ${x}_{i}(t)$ is the time series of the *i* th element, and *T* represents the transpose of a vector, an interesting problem is to analyze the information dynamics of the system. In the following subsections, we will first describe the definition of transfer entropy, and then introduce a novel measure, associative transfer entropy, followed by the proposed information dynamic spectrum framework.

### 2.1 Transfer entropy

Transfer entropy (TE) is a directional measure of information flow between a pair of time series. A time series is a sequence of time-ordered observations of a node in a network, or an element of a system. TE quantifies how much information is transferred from the current state into the future from one time series to another. The concept of TE extends from the concept of mutual information, which is a measure of mutual dependence between two time series and more mutual information means less uncertainty if knowing one time series. However, mutual information is symmetric and does not inform the direction the information flows. To address this, TE takes into account the dynamics of information transport and provides how much information is transferred from the current state into the future.

*x*at time

*t*with length-

*ℓ*history. The amount of TE from source

*x*to destination

*y*with a time lag

*τ*in the future is defined as [12],

where $p(\cdot ,\cdot ,\cdot )$ represents the joint probability and $p(\cdot |\cdot ,\cdot )$ and $p(\cdot |\cdot )$ represent the conditional probabilities. This definition is derived from the Shannon entropy: $H=-{\sum}_{z}p(z)logp(z)$, a measure of uncertainty. The definition of ${TE}_{x\to y}$ in Eq. (1) can be expressed as $H(y(t+\tau )|{y}^{(k)}(t))-H(y(t+\tau )|{y}^{(k)}(t),{x}^{(\ell )}(t))$. This is the uncertainty in *y* at time $t+\tau $ given its history ${y}^{(k)}(t)$, minus the uncertainty in *y* at time $t+\tau $ given its history ${y}^{(k)}(t)$ and *x*’s history ${x}^{(\ell )}(t)$. Therefore, TE measures the net amount of information transferred from *x* to *y*. An important feature of TE is that it is asymmetric, in contrast to mutual information, which is a symmetric measure. More details about TE, including some interesting examples, can be found in the original paper [12].

### 2.2 Associative transfer entropy

*x*that leads to a likely increase in

*y*in the future can be the same as the amount of influence for an increase in

*x*that leads to a likely decrease in

*y*. There is no distinction between these two very different outcomes using TE. Our proposed ATE addresses such a difference and is defined by:

where *S* is the associate state, a subset of the set of all possible states of $(y(t+\tau ),{y}^{(k)}(t),{x}^{(\ell )}(t))$. The set *S* represents a certain associated state between *x* and *y*. The purpose of ATE is to capture information transfer between two variables for a particular association of their states. For example, for binary time series, TE can be decomposed into a positive association ATE+ and a negative association ATE−, in order to distinguish two situations where the amount of information transferred may be the same but with opposite effects: in one case the value of the source *x* drives the value of the destination *y* in the same direction, and in the other case the source drives the destination in the opposite direction.

*x*at

*t*, the probability that there will be an increment at $t+1$ is equally likely. Keeping these conditional probabilities fixed at 0.5, the only bias will come from $p(\dot{y}(t+1)=0|\dot{x}(t)=0)=p(\dot{y}(t+1)=1|\dot{x}(t)=1)=P$. When $P<0.5$, an increment in

*x*is more likely to cause a decrement in

*y*, and a decrement in

*x*is more likely to cause an increment in

*y*. Figure 1(a)-(f) shows a few such simulated time series of length 1000 from arbitrary initializations with $P=0,0.2,0.4,0.6,0.8,1$, respectively.

*P*, for $0\le P\le 1$, averaged over 100 trials for each

*P*. In this binary case, we decompose TE into positive association ATE+ and negative association ATE−, summing over the sets ${S}_{+}=\{(\dot{y}(t+1),\dot{x}(t),\dot{y}(t))=(0,0,0),(0,0,1),(1,1,0),\text{or}(1,1,1)\}$ and ${S}_{-}=\{(\dot{y}(t+1),\dot{x}(t),\dot{y}(t))=(0,1,0),(0,1,1),(1,0,0),\phantom{\rule{0.25em}{0ex}}\text{or}(1,0,1)\}$, respectively. Note that ATE+ sums over positive associations, which means that $\dot{y}(t+1)$ and $\dot{x}(t)$ are either both 0 or both 1. ATE− sums over negative associations, which means that one of $\dot{y}(t+1)$ and $\dot{x}(t)$ is 0 and the other is 1. In Figure 2(a), as

*x*is the only influence for

*y*in the future, we see that when $P>0.5$ and increases, ATE+ increases; similarly, when $P<0.5$ and decreases, ATE− increases. One the other hand, TE does not distinguish the types of influence between the two time series. In this case, the expectation values for ATE+ and ATE− can be derived as functions of

*P*: $\text{ATE}+=P{log}_{2}(P/0.5)$ and $\text{ATE}-=(1-P){log}_{2}[(1-P)/0.5]$. These are consistent with the numerical results shown in the figure. To understand more about ATE+ and ATE−, we show two examples in which the influence on

*y*from itself is positive, instead of a neutral influence in the case of Figure 2(a). In Figure 2(b), $p(\dot{y}(t+1)=a|\dot{y}(t)=a)=0.6$, while in Figure 2(c), $p(\dot{y}(t+1)=a|\dot{y}(t)=a)=0.9$, which has a stronger positive influence. Near $P=1$, one can observe that ATE+ in Figure 2(c) is much smaller than ATE+ in Figure 2(b), even though the positive influence from $\dot{x}(t)\to \dot{y}(t+1)$ is high. The explanation of this is that TE and ATE± negate its own influence, $\dot{y}(t)\to \dot{y}(t+1)$, and the self-influence in Figure 2(c) is 0.9, much larger than that in Figure 2(b), which is 0.6. In other words, TE and ATE± measures the net influence, which is the amount of external influence minus the internal influence.

### 2.3 ATE symbolization technique

The definition of TE and ATE in Eq. (1) and Eq. (2) respectively are based on continuous random variables. To estimate probability distributions from a finite number of observations, one can discretize the observed values. We adopt the symbolic transfer entropy method [13] that is based on the symbolization technique for permutation entropy [16]. We adopt this method to compute ATE because ATE sums over a subset of all the possible discrete states that TE sums over. Therefore, this is a natural extension of the computation. The key step is to transform the continuous-valued time series ${\{x(t)\}}_{t=1}^{N}$ into a symbolic time series with a suitable length *n*, where $n\ge 2$ is an integer. First, for each *t*, the *n* consecutive values $\{x(t),x(t+1),\dots ,x(t+n-1)\}$ are ordered in the ascending order. The sequence of the indices that correspond to such permutation is recorded and denoted by $\stackrel{\u02c6}{x}(t)$, the associated permutation of order-*n* symbol of $x(t)$ at *t*. Then we estimate the ATE of ${\{x(t)\}}_{t=1}^{N}$ by calculating the ATE of ${\{\stackrel{\u02c6}{x}(t)\}}_{t=1}^{N}$. This now can be done efficiently because the number of states to sum over in the ATE equation (2) is finite and the probabilities and conditional probabilities can be directly calculated based on the discrete-valued symbolic time series ${\{\stackrel{\u02c6}{x}(t)\}}_{t=1}^{N}$.

### 2.4 Local TE and ATE

*x*has a strong positive influence, then a strong negative influence during the middle period, and finally a strong positive influence at the end. For simplicity, we fix $p(\dot{x}(t+1)=0|\dot{y}(t)=0)=p(\dot{x}(t+1)=1|\dot{y}(t)=1)=p(\dot{x}(t+1)=0|\dot{y}(t)=1)=p(\dot{x}(t+1)=1|\dot{y}(t)=0)=0.5$. The simulated data can be seen in the top graph of Figure 3. The local TE and ATE± curves are plotted in the bottom graph, with sliding window size $W=100$. One can see that initially ATE+ has large values, then ATE− has large values, and in the final stage ATE+ has large values. This is consistent with our setup of the simulated data. Near the switching of information flow at $t=300$ and $t=600$, the ATE± curves are not step functions, but instead have linear transitions. The reason is because the calculation of local ATE relies on a sliding window, which moves continuously and can include different dynamics within the window. Nonetheless, we observe the effectiveness of local ATE, compared to local TE, which does not distinguish between positive and negative influence.

### 2.5 Spectral radius of TEM and ATEM

Since TE and ATE are pairwise measurements, they do not give a single measurement of the total amount of entropy transferred within a system. To extend pairwise TE and ATE, we first define the ATE Matrix (ATEM) of a system of *m* elements as an $m\times m$ matrix with *ij* th entry ${({M}^{S})}_{ij}={ATE}_{{x}_{i}\to {x}_{j}}^{S}$, where the superscript *S* is the associative state that is defined for Eq. (2). Similarly, the *ij* th entry of the $m\times m$ TE Matrix (TEM) is ${(M)}_{ij}={TE}_{{x}_{i}\to {x}_{j}}$. TEM has been used in [14] to reveal the asymmetric influences from mature markets to emerging markets, but was not used to give the total amount of TE of the entire system, nor in a sliding window fashion to measure the transient influence dynamics. Note that since TE and ATE are directional, TEM and ATEM are nonsymmetric, and thus their eigenvalues are complex-valued.

We use the spectral radius of the TEM (resp. ATEM) to measure the total amount of TE (resp. ATE) in the entire network. The spectral radius of a matrix is the supremum of the absolute values of its eigenvalues. Furthermore, to use the spectral radius of a TEM or ATEM to monitor the system, we use a time sliding window to calculate local TEM or ATEM and therefore the spectral radius of a TEM or ATEM becomes a function that depends on time.

*σ*. The scaling function in our data simulation is chosen to be $c(t)=tanh(3t-10)$. We chose this pitchfork bifurcation equation because it generates data that will transition from one state to another rather quickly for individual nodes. With the graph Laplacian term in the equation, as an external force for each node, one can get different interesting bifurcation results. Our purpose of using this equation is to show that for a variety of pitchfork bifurcations, our local TE method is able to detect the transition in advance. For instance, the adjacency matrix

*A*and Laplacian matrix

*L*of the upward binary tree are:

*j*to node

*i*, and ${a}_{ij}=0$ means there is not a connection from node

*j*to node

*i*. The Laplacian matrix $L={D}_{\mathrm{in}}-A$, where ${D}_{\mathrm{in}}$ is the in-degree matrix whose diagonal entry ${d}_{ii}$ is the sum of

*i*th row of

*A*, the number of connections going into node

*i*. Figure 4 plots the simulated time series and their spectral radii of the local TEM. We observe that before transitioning or bifurcation, the spectral radius decreases rapidly, which provides an early indication of system transitioning. The spectral radius of TEM in these cases drops to the lowest point during transitioning due to the strong internal dynamics of individual nodes, and TE measures the net influence dynamics from one node to another.

### 2.6 Detect critical transitions with ATE and TE spectral radius curves

In this subsection, we explain how to use the TE and ATE curves defined in the previous subsection to characterize critical transitions prior to their occurrences (see Table 1). We give our observations on behaviors of TE and ATE in four different critical transition scenarios. The first critical transition scenario is from a stable state to oscillations in a network of non-Foster circuits. In this case, as a network starts to synchronize in phase, the pairwise TEs will likely start to increase because of the growing amount of interacting dynamics between circuits. The TE and ATE+ spectral radius curves may therefore start to increase. If the synchronization is in the negative mode, in which the phases differ in 180^{∘}, the ATE− spectral radius curve will start to increase.

The second scenario is the case where the time series of a system bifurcate, switching from one state to alternative states. The examples in Figure 4 demonstrate the nature of such transitions. Since the phenomenon of critical slowing down is observed prior to such transitions, the TE and ATE curves may decrease rapidly prior to the tipping point.

The third scenario is when there is a sudden drop in the time series of the system. It has been suggested that there is a critical slowing phenomenon before the tipping point of a crash in many real world systems, such as stock markets, housing market, etc. In this scenario, right before the crash, the interacting dynamics between nodes may start to decrease, which can be explained by the uncertainty of the information carried by other nodes. Therefore, the TE and ATE curves are likely to decrease rapidly before the crash.

The fourth scenario is the exponential growth in system activities. In this case, the TE and ATE+ curves are likely to increase, as a result of an increase in the interacting dynamics between nodes. Given one of the above scenarios, we can look out for critical transitions using TE and ATE curves.

## 3 Numerical results of information dynamic spectrum

### 3.1 ATE indicates instability of non-Foster circuit

#### 3.1.1 Data

The circuit data used for TE and ATE evaluation is simulated from a non-Foster network described in [17]. A non-Foster network is an active network containing a power source, therefore it may not obey Foster’s reactance theorem. The non-Foster circuit data analyzed here contains three coupled circuits with internal feedback loops. The interface points include two antennas (Ant1 and Ant2) and two ports (i1 and i2) to record time series of the voltage. The circuit is operated in the stable (no oscillation) region, but near the unstable region. A perturbation is then added at timestep = 250 with 1 milli-Amp. When a small perturbation is added, the circuits may become oscillatory. The circuits will either become synchronized in oscillation, which is considered unstable, or return to the original stable state.

#### 3.1.2 Analysis

### 3.2 TEMs infer directional influences in Latin America stock indices

#### 3.2.1 Data

**Latin America stock indices**

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|

BVPSBVPS | Chile65 | COLCAP | CRSMBCT | IBOV | IBVC | IGBVL | Merval | MEXBOL |

Panama | Chile | Colombia | Costa Rica | Brazil | Venezuela | Peru | Argentina | Mexico |

#### 3.2.2 Analysis

### 3.3 ATE reveals Wikipedia motifs that drive the changes

#### 3.3.1 Data

#### 3.3.2 Analysis

The bottom row of Figure 8 from left to right shows global ATEM+, ATEM−, and TEM from 2001 to 2011 to identify important motifs that drive the changes in other motifs. In particular, ATEM+ shows that motifs 1, 8 and 12 have the most positive influence on other motifs, observing that consecutive ‘minor add’, ‘revert’, and ‘major add’ by registered users encourage Wikipedia’s content growth. ATEM− shows that motif 9 has the most negative influence on other motifs. TEM shows the asymmetric influence among the motifs but the contrast is not as strong as ATEM+ and ATEM−. The bottom of Figure 9 shows the TE, ATE+, and ATE−, i.e. spectral radii of local TEM and ATEM, in blue, red, and green, respectively. There is a significant increase in the ATE+ and TE curves near midyear 2005 and ATE+ crosses over ATE−. These suggest an early indication of the growth in Wikipedia’s contents. Interestingly, the peaks of TE and ATE+ in late 2008 occur at the beginning of the stabilization of the editing behaviors. The parameter choices for this analysis are symbolic length $n=5$ and sliding window size $W=30$.

### 3.4 TE and ATE curves as early indicators of stock market crash

#### 3.4.1 Data

We collected the share prices of Dow Jones Industrial Average (DJIA) from June 1 to December 31, 2008, using 25 shares that persisted throughout the years near the October 2008 Crash. The stock indices are: MMM, AA, AXP, T, BA, CAT, KO, DD, XOM, GE, HPQ, HD, INTC, IBM, JNJ, JPM, MCD, MRK, MSFT, PFE, PG, UTX, VZ, WMT, and DIS. The time series resolution is one day, except when the market is closed during the weekend.

#### 3.4.2 Analysis

### 3.5 Discussions

Our empirical data analysis suggests that early detection of critical transitions using local ATE curves is possible. The local ATE curve, spectral radius of the local ATEM over a sliding time window, is an indicator of the amount of net entropy transferred in the entire system at each time. On the other hand, by looking at the entries of ATEM, one can identify the most positive or negative influential nodes of the system, as well as the most positively or negatively influenced nodes. From the observations of the TE, ATE+, and ATE− curves in different examples that corresponds to different types of critical transitions, we found that (1) ATE+ increases rapidly before synchronization in the positive mode. ATE− increases rapidly before synchronization in the negative mode. (2) TE, ATE+, and ATE− drop rapidly before abrupt falls, because TE, ATE+, and ATE− are the net information flow and external information flow begins to decrease before the sudden collapse. (3) ATE+ increases rapidly before an exponential growth in a system.

## 4 Probabilistic light cones for trajectory prediction

We propose a model-based probabilistic light cone method for predicting the trajectories of the spectral radius of TEM or ATEM. As observed in the previous section, the behaviors of TE and ATE curves can be interpreted manually as early indications of critical transitions. Here, we provide an automated method to generate early indicators. Using a natural logarithmic model, our method first computes probabilistic light cones for TE and ATE curves. This gives a prediction value with a confidence interval. If the actual TE and ATE trajectories are outside the confidence interval, the method declares instability is detected, which serves as an early indicator of a critical transition.

Our computation of probabilistic light cone is inspired by [23], where an MCMC method is used to predict citation growth based on the preferential attachment. However, critical transition is not discussed in [23]. Our method, on the other hand, aims to generate early indicators for an upcoming critical transition as time progresses. We choose natural logarithmic curves to model the growth rate of the TE trajectory as information transfer approaches its maximum prior to a critical transition as observed in the non-Foster circuit example in the previous section. We apply a moving time window over the observed spectral radius (TE or ATE±) time series to derive the unknown coefficients and constants for natural logarithm curves. For a given prediction time point, we generate the probabilistic light cone based on 95% confidence intervals of predicted trajectories with fitted natural logarithm curves.

### 4.1 Model-based forecasting

*a*and a constant

*c*:

*a*of the logarithmic function in (5), instead of fitting the TE and ATE curves deterministically, we estimate the rate of change with various discrete time steps. This generates a probabilistic light cone, which consists of a collection of fitted curves using different discrete time steps. Taking the derivative of Eq. (5), we have ${g}^{\prime}(t)=\frac{a}{t}$. We obtain the following discretized version for the unknown

*a*:

*t*in a window $[{T}_{\mathrm{start}},{T}_{\mathrm{end}}]$, we use the least squares method to solve the unknown

*a*. First, we write the following matrix equation:

*t*is then obtained by

### 4.2 Probabilistic light cone and error estimation

Given the method to estimate the constants *c* and *a* in previous section, we generate a probabilistic light cone at each time as we vary △*t* to obtain multiple estimates of *c* and *a*. We define a probabilistic light cone at a given time consists of a collection of natural logarithm curves starting from that point.

where $c(i)=g({t}_{i-1})-{a}_{\u25b3t}ln({t}_{i-1})$.

*E*be the collection of all ${error}_{\u25b3t}$ over all timesteps △

*t*and let $\sigma =\text{standard deviation}\phantom{\rule{0.25em}{0ex}}\text{of}E$. Let

*G*be the collection of all predicted value ${G}_{\u25b3t}({t}_{d})$ over all timesteps △

*t*and $\mu =mean(G)$. Therefore, the 95% confidence interval for the future time ${T}_{\mathrm{end}}+{t}_{d}$ is

where *L* is the size of *E*.

### 4.3 Numerical results

*These two points indicate the instability of the TE curve and detect the full synchronization in advance*. Figure 13 shows two snapshots of the probabilistic light cone video (see Additional file 1) produced according to the method described above and the predicted trajectory.

## 5 Conclusions

We propose a novel information dynamic spectrum framework for automated detection of critical transitions and identification of directional influences. We have shown that the framework is able to: (1) provide an effective measure for quantifying associative, asymmetric directional influence rather than symmetric influence, (2) provide an effective formulation that captures system-wise directional influence, rather than pair-wise influence and (3) provide an effective measure for detecting instability in systems with directional influence dynamics. Within this framework, we further propose the probabilistic light cone method that predicts the ATE trajectories and indicates instabilities toward critical transitions. This model-based prediction method captures information that cannot be discerned by looking at individual time series signals alone and is novel compared to conventional deterministic methods or complicated probability state space predictions. The analyses performed in this work are based on historical data, and the rapid decrease or increase in TE is not a sufficient condition for all critical transitions. Rather, as shown in Table 1, the nature of upcoming critical transitions is revealed by the behaviors of ATE curves, in conjunction with examining the trends in the time series of a system. Future work requires a complete analysis of the proposed framework, as well as an adaptive parameter selection method for the probabilistic light cone analysis.

## Electronic Supplementary Material

Below are the links to the electronic supplementary material.

## Declarations

### Acknowledgements

The authors would like to express their gratitude to Hankyu Moon for helpful discussions about EWS and Carson White for generating the non-Foster circuit data. The authors would also like to thank the two anonymous reviewers for their helpful and constructive comments that greatly helped improve the quality of the paper.

## Authors’ Affiliations

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