Supplementary Materials: Identifying critical transitions and their leading biomolecular networks in complex diseases

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1 Supplementary Materials: Identifying critical transitions and their leading biomolecular networks in complex diseases Rui Liu, Meiyi Li, Zhiping Liu, Jiarui Wu, Luonan Chen, Kazuyuki Aihara Contents A Statistical properties near a critical transition point S5 A.1 The leading networks during critical transitions S5 A.2 Variation variables S8 A.3 State transition variables S13 B Constructing the SNE S15 B.1 Markov S15 B.2 SNE S17 B.3 Robustness S19 C Constructing the transition state space and analyzing the SNE S20 C.1 Constructing the transition state space S20 C.2 Analyzing SNE S21 C.2.1 SNE for Type 1 nodes S24 C.2.2 SNE for Type 2 nodes S25 C.2.3 SNE for Type 3 nodes S27 C.2.4 SNE for Type 4 nodes S27 S1

2 C.3 The leading network and SNE network S28 D Numerical validation of leading networks detected by SNE S30 E Algorithm for calculating the SNE S35 F Application to two different diseases S40 F.1 Dataset 1: Genomic data related to liver cancer (26) S41 F.2 Dataset 2: Genomic data related to lung injury after carbonyl chloride inhalation (25) S44 G Bootstrap analysis and cross validation S46 H Functional analysis of the leading networks S50 H.1 Data preing S51 H.2 HCC induced by HCV infection S52 H.3 Acute lung injury S53 I Supplementary Table Identified extended leading network S59 J Supplementary Table KEGG Enrichment analysis S59 S2

3 In this study, we developed a novel computational method by first constructing a statetransition-based local network entropy (SNE) and then theoretically proving that the SNE can provide a general early warning indicator for identifying critical transitions and their leading biomolecular networks (or leading networks) during disease progression. SNE is based on dynamical network biomarker (DNB) (1), by using high throughput data (e.g., gene or protein expression data) even with a small number of samples. Figure S1 shows the main differences between the traditional biomarkers and a type of dynamical network biomarker (DNB), which can effectively signal the pre-disease state from its critical behavior in dynamics. Completely different from traditional static approaches, a dynamical network biomarker (DNB, or a dynamical network of biomarkers) is able to characterize pre-disease phenotypes, for which traditional biomarkers failed. Figure 1 (the main text) and Figure S1 illustrate major features of the two types of biomarkers. Note that a DNB is a group of molecules which are highly unstable but strongly correlated without consistent values for the pre-disease samples, and thus it is a new concept different from the conventional biomarkers which are required to keep consistent values for the respective disease and normal samples. A DNB is also called as the leading network because it makes the first move into the disease state, which implies that the DNB is highly related to causal or driver factors (or genes) to the disease. It is also the first theoretical result to ensure such a causality feature. Note that DNB or the leading network in this paper is not for identifying the critical transition phenomenon but for detecting the state just before the critical transition, and therefore, it is of great importance for early diagnosis of complex diseases. Hence, without confusion, identifying the critical transition in this paper means identifying the state just before the critical transition. S3

4 Potential Potential Potential a Disease Progression Normal State reversible Pre-disease State irreversible Disease State Normal state Pre-disease state Disease state (Low potential, high resilience) (High potential, low resilience) ( Low potential, high resilience) b c d Traditional biomarker score Traditional Biomarker Distinguish disease from normal state Static signal Diseased person Normal person Time Dynamical Network Biomarker Distinguish pre-disease from normal state Dynamical signal Dynamical network biomarker score Pre-diseased person Normal person Time Figure S1: Disease states and biomarkers. (a) Three stages during disease progression, i.e., a normal state, a pre-disease state, and a disease state. A normal state is a relatively heathy stage including the chronic inflammation period or the period during which the disease is under control, whereas a pre-disease state is the limit of the normal state just before the critical transition. At this stage, the pre-disease state is considered to be reversible to the normal state if appropriately treated. However, if the system passes over the critical (bifurcation) point to the disease state, it usually becomes irreversible to the normal state. (b) Two types of biomarkers, i.e., traditional biomarkers, and newly developed dynamical network biomarkers (DNBs). (c) Main targets of the two biomarkers. The traditional biomarkers are indicators on the disease state, whereas the dynamical network biomarkers signal the pre-disease state. (d) Comparison of the major features of the two biomarkers. The traditional biomarkers are static measurements on the disease, whereas dynamical network biomarkers are dynamical measurements on the pre-disease, thus providing the early-warning signals for the pre-disease state. S4

5 A Statistical properties near a critical transition point For a dynamical system or dynamical network, e.g., a protein-protein interaction (PPI) network, a gene regulatory network, or a correlation network, it is assumed that we can measure the variables at different time points or different periods (even for a small number of samples). In this section, we aim to theoretically prove several generic properties of such a dynamical network when the system approaches the critical transition point. In particular, we focus on the analysis of variation equations for the original network. A.1 The leading networks during critical transitions We consider the following discrete-time dynamical system that represents the dynamical evolution of a network: Z(t + 1) = f(z(t); P ), (S1) where Z(t) = (z 1 (t),..., z n (t)) is an n-dimensional state vector or variables at time instant t that represents gene or protein expressions, while P = (p 1,..., p s ) is a parameter vector or driving factors that represent slowly changing factors, e.g., genetic factors (SNP, CNV, etc.) and epigenetic factors (methylation, acetylation, etc.). f : R n R s R n are generally nonlinear functions. Furthermore, we assume that the following conditions hold for Eq.(S1). 1. Z is a fixed point of system (S1) such that Z = f( Z; P ). 2. There is a value P c such that one or a pair of the eigenvalues of the Jacobian matrix is equal to 1 in the modulus. f(z;p c) Z Z= Z 3. When P P c, the eigenvalues of (S1) are not always equal to 1 in the modulus. S5

6 The above three conditions with other transverse conditions (2) imply that the system undergoes a phase change at Z or a codimension-one bifurcation when P reaches the threshold P c. From a mathematical perspective, the bifurcation is generic, i.e. almost all of the bifurcations in a general system satisfy these conditions. It is notable that most of the systems described by differential equations can be generally discretized and transformed into Eq.(S1), e.g., using methods such as the Euler scheme and the Poincaré section. Thus, we focus on difference equations (S1) during our theoretical analysis in this section. It is known that the dynamics of complex disease progression is highly complex before or after a sudden deterioration, so the state equations of systems are generally constructed in a very high-dimensional space using a large number of variables and parameters (1 5). However, if a system driven by known or unknown parameters approaches a critical point, which is a very special phase during the dynamical progression, it is theoretically guaranteed that the system will eventually be constrained to one- or two-dimensional space (i.e., the center manifold), which can be expressed in a simple form around a codimension-one bifurcation point (1, 6, 7). This is generally guaranteed by the bifurcation theory and center manifold theory (6 9). Thus, we can detect the signal of any dynamical system only during this special phase and not in other periods (i.e., neither the normal state before a transition nor a disease state after the transition), which is the theoretical foundation of this study (1). For system (S1) near Z and before P reaches P c, we assume that the system is at a stable fixed point Z so all of the eigenvalues are within (0, 1) in the modulus. The parameter value P c when the state shift of the system occurs, is known as a bifurcation parameter value or a critical transition value. This theoretical result was derived based on a consideration of the linearized equations for Eq.(S1) and the noise perturbations near Z. Specifically, by introducing new variables S6

7 Y (t) = (y 1 (t),..., y n (t)) and a transformation matrix S, i.e. Y (t) = S 1 (Z(t) Z), we have Y (t + 1) = ΛY (t) + ζ(t). where Λ(P ) is the diagonalized matrix of f(z;p ) Z (S2) ζ(t) = (ζ Z= Z. 1 (t),..., ζ n (t)) are small Gaussian noises with zero means. We denote σ i as the small standard deviation of ζ i for all k. Without any loss of generality, the diagonalized matrix Λ(P ) = diag(λ 1 (P ),..., λ n (P )) for each λ i is between 0 and 1. (In fact, three typical cases arising during the diagonalization (1), but we only illustrate the diagonal case with different real eigenvalues for simplicity. The derivations of the other two cases are similar). Of the eigenvalues of Λ, the largest one (in the modulus), say λ 1, approaches 1 in the modulus when parameter P P c. The eigenvalue λ 1 characterizes the system s rate of change around a fixed point and it is known as the dominant eigenvalue. The normal state corresponds to a period where λ 1 < 1, whereas the pre-disease stage corresponds to the period with λ 1 1. Without loss of generality, we assume that the first variable y 1 in Y corresponds to λ 1, namely (y 1, 0,..., 0) is the eigenvector of λ 1. Close to a fixed point, we have shown that there is a dominant group or a DNB that satisfies the following conditions when the system approaches a critical transition point (1). Theorem 1 We consider a stochastically perturbed linear system for Eq.(S1), i.e., Eq.(S14). When P approaches the bifurcation point, the following results hold. If both i and j are in the dominant group, then PCC(z i, z j ) 1, while SD(z i ) and SD(z j ) ; if i is in the dominant group but j is not, then PCC(z i, z j ) 0, S7

8 while SD(z i ), and SD(z j ) approaches a bounded value; if neither i nor j is in the dominant group, then PCC(z i, z j ) approaches a constant, while both SD(z i ) and SD(z j ) approach bounded values, where PCC is Pearson s correlation coefficient and SD is the standard deviation. This theorem is the theoretical basis for whole the analysis of this work (1). In other words, all of the results in this paper are derived mainly based on this theorem. The DNB is the subnetwork that makes the first move from one state toward another state at the critical transition point, so we refer to the DNB as the leading network in this critical transition. For a general discrete-time dynamical system in this paper, the critical transition includes all codimension-one bifurcations except the super-critical Neimark-Sacker bifurcation, i.e., it includes saddle-node bifurcation (including transcritical and pitchfork bifurcation) if the dominant eigenvalue is equal to 1; period-doubling (or flip) bifurcation if the dominant eigenvalue is equal to 1; sub-critical Neimark-Sacker bifurcation if there is a pair of pure imaginary complex conjugate eigenvalues with modulus 1. In this work, we do not study the super-critical Neimark-Sacker bifurcation which is not the catastrophic-type bifurcation. In other words, i.e., there is no drastic change of the state for the super-critical Hopf bifurcation because the amplitude of the periodic solution after the critical transition point increases with the bifurcation parameter in a polynomial manner, rather than an exponential or drastic manner. A.2 Variation variables In contrast to the analysis of the original variables Z in (1) or Theorem 1, in this work we focus on the variation equation for Eq.(S1) with variation variables Z. We note that z i (t) = s i1 y 1 (t) + + s in y n (t) + z i, (S3) S8

9 and we let the variation variables be Z(t) = Z(t) Z(t 1), (S4) then from Eq.(S3) we have z i (t) = s i1 y 1 (t) + + s in y n (t), (S5) where Y (t) = Y (t) Y (t 1). (S6) We refer to z i (t) and y i (t) as the variation variables for z i (t) and y i (t), respectively. From Eq.(S2) and Eq.(S6), it clearly holds that Y (t + 1) = Λ Y (t) + ξ(t), (S7) where ξ(t) = ζ(t) ζ(t 1) are Gaussian noises with zero means and covariances κ ij = Cov(ξ i, ξ j ). It is clear that the standard deviation of ξ i (t) is 2σ i for all t. Obviously, variable y 1 corresponds to the dominant eigenvalue λ 1. For any integer T > 0, by iteration we have Y (t + T ) =Λ Y (t + T 1) + ξ(t + T 1) =Λ[Λ Y (t + T 2) + ξ(t + T 2)] + ξ(t + T 1)... =Λ T Y (t) + Λ T 1 ξ(t) + Λ T 2 ξ(t + 1) + + Λξ(t + T 2) + ξ(t + T 1). Clearly, the summation of the coefficients for the covariance matrices for T Gaussian noises is (1 Λ T )(I Λ) 1, S9

10 where I is the n-dimensional identity matrix. Note that when the system is in a normal state, λ i < 1. Because when T is large enough, it holds that Y (t + T ) = ε(t), (S8) where ε(t) = (ε 1 (t),..., ε n (t)) are small Gaussian noises with zero means. Based on the law of large numbers (10), the covariances are κ ij /(1 λ i ) and the deviation of ε i (t) is κ ii /(1 λ i ) for all t, which is a bounded value when λ i 1. In the original variables Z from Eq.(S5) and Eq.(S8), it holds that z i (t + T ) = s i1 y 1 (t + T ) + + s in y n (t + T ) = s i1 ε 1 (t) + + s in ε n (t). (S9) Therefore, when the system is in a normal state, or equivalently λ i < 1, any variation variable z i (t + T ) is statistically independent of its initial variable z i (t) for a sufficiently long T, which generally holds because biochemical reactions occur within a very short time interval (i.e., sub-ms time). Thus, any two samples can be considered as having a long T because there are large numbers of biochemical reactions during their observation intervals. Hence, the variation variables for any two samples are statistically independent when the system is in its normal state. Next, we discuss the case close to a critical transition when the dominant eigenvalue λ 1 1 (for λ 1 1, the derivation is similar so it is omitted). Note that the variation variable y 1 is related to the dominant eigenvalue λ 1. Since y 1 (t + T ) = λ 1 y 1 (t + T 1) + ζ 1 (t + T 1) (S10) S10

11 holds for any integer T, we have [ y 1 (t + T ) + y 1 (t + T 1) + + y 1 (t + 1)] =[y 1 (t + T ) y 1 (t + T 1)] + [y 1 (t + T 1) y 1 (t + T 2)] + [y 1 (t + T 2) y 1 (t + 2)] + [y 1 (t + 1) y 1 (t)] =λ 1 [ y 1 (t + T 1) + + y 1 (t + 1) + y 1 (t)] + [ζ 1 (t + T 1) ζ 1 (t + T 1) + ζ 1 (t + T 1) ζ 1 (t) + ζ 1 (t) ζ 1 (t 1)] =λ 1 [ y 1 (t + T 1) + + y 1 (t)] + [ζ 1 (t + T 1) ζ 1 (t 1)]. Therefore, y 1 (t + T ) + y 1 (t + T 1) + + y 1 (t + 1) =λ 1 [ y 1 (t + T 1) + + y 1 (t)] + [ζ 1 (t + T 1) ζ 1 (t 1)], or y 1 (t + T ) =(λ 1 1) y 1 (t + T 1) + + (λ 1 1) y 1 (t + 1) + λ 1 y 1 (t) + [ζ 1 (t + T 1) ζ 1 (t 1)]. Thus, when λ 1 1 we have y 1 (t + T ) = y 1 (t) + [ζ 1 (t + T 1) ζ 1 (t 1)], (S11) which means that y 1 (t + T ) depends greatly on y 1 (t) for a small noise. In other words, the dominant variables y 1 (t) are highly dependent on each other (or dependent on their previous state) for any two samples near the critical transition. It is clear that the same result holds when λ 1 1. However, because λ i < λ 1, i = 2, 3,..., n, the other variables y i (t + T ) satisfy Eq.(S8), i.e., y i (t + T ) = ε i (t), i = 2, 3,..., n. (S12) S11

12 Note that the variable y 1 is related to the dominant eigenvalue λ 1. There is a special group of variables z j, in which the variables z j are related to y 1, i.e., z j in Eq.(S5) with s j1 0, which is known as the dominant group. These variables z j are known as the dominant group members or DNB members (1). For any two DNB members z j and z i with s j1 0 and s i1 0 in Eq.(S5), when λ 1 1, from Eq.(S11) and Eq.(S12) we have z j (t + T ) =s j1 y 1 (t + T ) + s j2 y 2 (t + T ) + + s jn y n (t + T ) =s j1 [ y 1 (t) + (ζ 1 (t + T 1) ζ 1 (t 1))] + s j2 ε 2 (t) + + s jn ε n (t) = s j1 s i1 z i (t) + s j1 (ζ 1 (t + T 1) ζ 1 (t 1)) + (N(t) + s j1 s i1 (s j2 s i2 )ε 2 (t) + + s j1 s i1 (s jn s in )ε n (t)) = s j1 s i1 z i (t) + N(t) + ρ j (t), (S13) where ρ j (t) = s j1 (ζ 1 (t+t 1) ζ 1 (t 1))+( s j1 s i1 (s j2 s i2 )ε 2 (t)+ + s j1 s i1 (s jn s in )ε n (t)) is Gaussian noise which is assumed to be small, N(t) = ( s j1s i2 s i1 y 2 (t)+...+ s j1s in s i1 y n (t)). From Eq.(S13), it is clear that when λ 1 1 for any two DNB members, the variable z j (t + T ) is correlated to z i (t), where it also holds for i = j, i.e., for any DNB member, that the variable z j (t + T ) is correlated to its previous z j (t). By contrast, as suggested by Eq.(S9), for any non-dnb member z k, z k (t + T ) is statistically independent of z k (t). Thus, near any fixed point, we derive the following theorem for any two samples (i.e., at time t and time t + T ). Theorem 2 For a stochastically perturbed linear system of Eq.(S1): Z(t + 1) = A(P )Z(t) + ε(t), (S14) where ε(t) is the Gaussian noise and P is a parameter vector that controls the Jacobian matrix A. We denote the variation variable as z i (t) = z i (t) z i (t 1) and assume T to be sufficiently large. S12

13 1. When P is not in the vicinity of a critical transition point or a bifurcation point, the following holds. For any i and j including i = j, z i (t + T ) is statistically independent of z j (t), where i, j = 1, 2,..., n. 2. When P approaches to a critical transition point, the following holds. If both i and j are in the dominant group, or DNB members, then there is a strong correlation between z i (t + T ) and z j (t); If neither i nor j is in the dominant group, then z i (t + T ) is statistically independent of z j (t). Note that this also holds for i = j. Remark 1 For a nonlinear case (S1) at a fixed point, the dynamical behavior has the same trend described in Theorem 1 and Theorem 2. Clearly, the conditions in Theorem 1 or 2 can be used to detect the DNB (or the leading network) and the critical transition. In particular, the expression of each node in a network may change stochastically at any time instant due to perturbation, but the above theorem and remark guarantee that, for a group of nodes, some statistical indices will change drastically whenever the system approaches a critical transition point, which provides a reliable early warning signal for identifying the pre-disease state. A.3 State transition variables Next, we further discretize the variable Z i (t) as x i (t) to derive similar theoretical results to Theorem 2. S13

14 Definition 1 For any original variables z i in Eq.(S1) at time point t, we define the transition state variable x i : z i (t) {0, 1} as x i (t) = { 1, if zi (t) > d i, 0, if z i (t) d i, (S15) where the threshold d i R + is a constant or the threshold for discretization. Then, x i (t) {0, 1} is defined for measuring whether node i experiences a large change at sampling point t, i.e., if z i (t) z i (t 1) is sufficiently large (> d i ), then x i (t) = 1, otherwise x i (t) = 0. Thus, X(t) = (x 1 (t),..., x n (t)) is the transition state for the network at t. We will discuss how we determine d i to construct the transition state space in Section C. Based on Theorem 2, we derive the following result. Theorem 3 We consider a stochastically perturbed linear system for Eq.(S1), i.e., Eq.(S14). When P approaches the bifurcation point, the following results hold for a sufficiently large T. If both i and j are in the dominant group or DNB members, the correlation between the transition states x i (t + T ) and x j (t) increases drastically and p(x i (t + T ) = 1 x j (t) = γ) 1, p(x i (t + T ) = 0 x j (t) = γ) 0, where γ {0, 1}. If neither i nor j is in the dominant group, or DNB members, then the transition state x i (t + T ) is statistically independent of x j (t) and p(x i (t + T ) = γ i x j (t) = γ j ) = p(x i (t + T ) = γ i ) a, where γ i, γ j {0, 1}, a is a constant within (0, 1). In other words, there is no significant change in the conditional probability. S14

15 Next, we construct a SNE to identify the critical transition and its leading network (or DNB) based on Theorem 3. B Constructing the SNE B.1 Markov In general, it is difficult to analyze a large network directly because of its complexity. Then, we will focus on the local structure of a network to analyze its dynamical properties. Specifically, there is a local network for each node, i.e., the local network is centered on node i with its m linked first-order neighbor nodes i 1, i 2,..., i m. The local network centered on node i has the local transition state X i (t) = (x i (t), x i1 (t),..., x im (t)) at time t (see Fig.S2 a). The following description and derivation are all based on a local node, i.e., i, to simplify the notation, so we omit i and denote X i (t) as X(t), while we also denote the transition state simply as a state. Given the current state X(t) at time t for this local network, then at the next time point t + 1 there is a total of 2 m+1 possible state transitions (or possible transition states) for state X(t + 1) (see Fig.S2 b), each of which is a stochastic event that is denoted, respectively, as {A u } u=1,2,...,2 m+1, where A u = {x i = γ 0, x i1 = γ 1,..., x im = γ m }, (S16) with γ l {0, 1} for l {0, 1, 2,..., m}. The next state depends on the current state and its transition functions related to Eq.(S14). Obviously, for this local subnetwork, the discrete stochastic {X(t + i)} i=0,1,... = {X(t), X(t + 1),..., X(t + i),...} (S17) with X(t + i) = A u, u {1, 2,..., 2 m+1 }, S15

16 is a stochastic Markov during a period or phase of the system (see Fig.S3), i.e., during the normal stage or during the pre-disease stage. This stochastic is defined or given by a Markov matrix P = (p u,v ), which describes the transition rates from state v to state u as follows p u,v (t) = Pr(X(t + 1) = A u X(t) = A v ), (S18) where u, v {1, 2,..., 2 m+1 } and u p u,v (t) = 1. This can actually be determined based on the topological structure of the local network, or Eq.(S14). Figure S2: Local network centered on node i with m linked first-order neighbor nodes. The outline shows the local network structure for node i, which is linked to m neighbor nodes i 1, i 2,..., i m. Each edge i i k of k = 1, 2,..., m indicates how the neighbor i k may influence i directly and lead to a state transition in i. (a) is the state of this local network at time t. (b) is the next time point t + 1, which has 2 m+1 possible states {A 1, A 2,..., A 2 m+1} for this local structure. S16

17 B.2 SNE For a local structure centered on node i and its m linked first-order neighbor nodes i 1, i 2,..., i m, we already know that its state transition is a stochastic Markov, as shown in (S17). Within a period or phase, we assume that there is no change in the transition matrix, i.e., the transition probabilities p u,v (t) in (S18) between any two possible states A v and A u. Thus, the {X(t)} t [t1,t 2 ] is a stationary stochastic Markov during a specific period, i.e., the normal stage or the pre-disease stage. Thus, there is a stationary distribution π = (π 1,..., π 2 m+1) that satisfies π v p u,v = π u. Therefore, we can define the local network entropy, i.e., the SNE, as v H i (t) = H(χ) = u,v π v p u,v log p u,v, (S19) where the subscript index i indicates the center node i of this local network, while χ represents the state-transition X(t), X(t + 1),..., X(t + T ),... of the local network. Figure S3: Outline of the state transition. The outline shows the essential stochastic Markov of the state transition in a local structure. S17

18 Next, we prove that the entropy Eq.(S19) is equivalent to the so-called entropy rate defined in previous studies (11), (12), i.e., 1 H(χ) = lim H(X(t), X(t + 1),..., X(t + T )), T T (S20) when the limit exists. Note that the stochastic X(t), X(t + 1),... is a stochastic Markov during a period, where the step size between any two adjacent sampling times is a small unit δt. Based on the Markov chains property, we have H(χ) = π v p u,v log p u,v = u,v v = H(X(t + 1) X(t)) π v ( u p u,v log p u,v ) = H(X(t + T ) X(t + T 1))) where T is sufficiently large. Therefore, = H(X(t + T ) X(t + T 1), X(t + T 2),..., X(t) ), H(χ) = lim H(X(t + T ) X(t + T 1),..., X(t)), (S21) T provided that the limit exists. Based on the chain rule (13), we have 1 H(X(t), X(t + 1),..., X(t + T ) ) T = 1 T H(X(t + i) X(t + i 1), X(t + i 2),..., X(t) ), T i=0 which means that the entropy rate is the time average of the conditional entropies. (S22) Based on Cesáro s mean (14), because H(X(t + T ) X(t + T 1),..., X(t)) H(χ) as T (see (S21)), from (S22) we derive 1 H(X(t), X(t + 1),..., X(t + T ) ) H(χ) as T T, which proves Eq.(S20), i.e., 1 H i (t) = H(χ) = lim H(X(t), X(t + 1),..., X(t + T )). T T (S23) S18

19 Therefore, the SNE is actually the conditional entropy while it also describes the average transition entropy, depending on the state transition. Thus, we note that the SNE is the conditional entropy, i.e., H i (t) = H(X(t) X(t 1)) = H(X(t), X(t 1)) H(X(t 1)). We also note that X(t) (or Z(t) Z(t 1)) are variation variables. Clearly, in a normal state (or a disease state), a system recovers from a small perturbation quickly because of high resilience, i.e., X(t) and X(t 1) are almost independent. Thus, we have H i (t) H(X(t)) due to H(X(t), X(t 1)) H(X(t)) + H(X(t 1)) > 0, which results in a high SNE. By contrast, the system has difficulty recovering from a small perturbation in a pre-disease state because of low resilience, i.e., X(t) and X(t 1) are strongly correlated, which implies that H i (t) rapidly approaches the minimum, H i (t) 0 due to H(X(t), X(t 1)) H(X(t 1)) (Fig. 1 g in the main text). We combine the SNEs for all nodes and define the average network entropy for the whole network with n nodes as the average SNE (i.e., ANE) as follows: H(t) = 1 n n H i (t). i=1 (S24) B.3 Robustness Next, we show that the SNE is ly correlated with the robustness of the network. Thus, the SNE is used to quantify the robustness of a network in this study. We denote p ɛ (t) as the probability that a sample deviates (in the mean sense) by more than ɛ from its unperturbed value at any sampling time t. If t is sufficiently large, the perturbed observables may recover from small perturbations and return to their equilibrium state, which means that p ɛ (t) will converge to zero as t. We define the fluctuation decay rate R as the rate of this convergence on a logarithmic scale as follows: ( R = lim 1 ) t t log(p ɛ(t)). (S25) S19

20 A large value of R results in small deviations of the observables from the equilibrium state whereas a small value of R corresponds to large fluctuations around the mean value. Thus, R may characterize the insensitivity to perturbations, which represents the capacity to withstand random changes, i.e., robustness. According to the fluctuation theorem (15), the R used in (S25) is ly correlated with the SNE defined by Eq.(S19). Therefore, the system is highly robust in its normal state when the average network entropy (S24) reaches a relatively high level. However, when the system approaches the critical transition point in the pre-disease state, the SNE reaches a low level and the robustness of the system is reduced accordingly (see Fig. 1g in the main text). This critical phenomenon coincides with the critical slowing down or weak resilience (16,17), which is a generic dynamical phenomenon that occurs near a bifurcation point when the system becomes increasingly slow at recovering from small perturbations and reverting to its equilibrium state (18), (19). Clearly, the SNE can be used to quantify robustness and identify this critical stage. C Constructing the transition state space and analyzing the SNE We construct the transition state space x i from (S15) by determining the threshold d i and we then analyze the SNE by decomposing the network based on the DNB or the leading network. C.1 Constructing the transition state space In a local network centered on node i with m linked first-order neighbors i 1, i 2,..., i m, we first need to determine the threshold parameters d = {d i, d i1,..., d im }, which measures whether the state z i (t) has a large change or a state transition from its former state z i (t 1). Note that a system is in a stable state during its normal state (see Fig. 1(b) in the main text), whereas a system is sensitive to perturbation in its pre-disease state (see Fig. 1(c) in the main text). Thus, S20

21 d i should be set to distinguish the small changes in normal state from the large changes in pre-disease state. For this local network, therefore, we select d = {d i, d i1,..., d im } when the system is in a normal state such that for each node k, p( z k (t 0 ) > d k ) = α, if time point t 0 is in a normal state (see Fig.S4). Obviously, for such thresholds d, it holds that p( z i (t 0 ) > d i, z i1 (t 0 ) > d i1,..., z im (t 0 ) > d im ) α. (S26) With such a distribution, genes with large deviations clearly correspond to high probabilities. Figure S4: Outline of the choice of threshold d. Given thresholds d i and d j for DNB node i and non-dnb node j, respectively, there is a large difference between p( z i > d i ) and p( z j > d j ) when the system is in a pre-disease state. (a) For a DNB node i, the standard deviation of z i increases drastically as the system approaches a critical point, so it is more likely to be z i > d i, indicating a large state transition in i in a pre-disease state. (b), For a non-dnb node j, the standard deviation of z j is small, so there is no significant difference in the probability of a random event { z j > d j }, even when the system is in a pre-disease state. C.2 Analyzing SNE Based on the transition state space defined above, we can now discuss what happens to the SNE in Eq.(S19) when the system approaches a critical transition point from a normal state. For a local network centered on node i with m linked first-order neighbors i 1, i 2,..., i m (see Fig.S2 a), let χ denote the state-transition of the local network, p nor u,v (t) denotes the S21

22 transition probabilities in the normal state, and p pre u,v (t) those in the pre-disease state. Correspondingly, let H nor (χ) = u,v πv nor pu,v nor log pu,v nor and H pre (χ) = u,v πv pre pu,v pre log pu,v pre represent the SNEs in the normal state and the pre-disease state, respectively. Based on the transition state defined in Eq.(S15), for node i when the system is in a normal state, it holds that p nor (x i (t) = 1) = p nor ( z i > d i ) = α, (S27) which represents the probability of large state transition at t, namely, p nor (x i (t) = 0) = p nor ( z i d i ) = 1 α. Here, node i can be either a DNB member or a non-dnb node. When the system is in a pre-disease state and there is at least one DNB member in the local network, e.g., node j, then based on Theorem 1, there is SD(z j ), which means SD( z j ). (S28) This is straightforward because Z(t) = Z(t) Z(t 1), and SD( z j ) = 2 SD(z j ). Therefore, it holds that p pre (x j (t) = 1) = p pre ( z j (t) > d j ) 1, (S29) for a given constant threshold d j, and p pre (x j (t) = 0) = p pre ( z j (t) d j ) 0. (S30) S22

23 Note that in Section B we defined the dominant-group, or the DNB, as a group of nodes that make the first move toward the disease state, thereby indicating a sudden deterioration. Then, the nodes in the network can be categorized into four groups according to the local structure of the DNB or the leading network: Type 1: A DNB core node is a DNB node that is linked with DNB nodes only, i.e., in Fig.S2a if node i and its linked neighbors i 1, i 2,..., i m are all DNB members, then i is a Type 1 node (see the red nodes shown in Fig. 2a in the main text). Type 2: A DNB boundary node is a DNB node that is linked with at least one non-dnb node, i.e., in Fig.S2a if node i is a DNB node and some of its linked neighbors are non- DNB nodes, then i is a Type 2 node (see the orange nodes shown in Fig. 2a in the main text). Type 3: A non-dnb core node is a non-dnb node that is linked with at least one DNB node, i.e., in Fig.S2a if node i is a non-dnb node and some of its linked neighbors are DNB nodes, then i is a Type 3 node (see the blue nodes shown in Fig. 2a in the main text). Type 4: A non-dnb boundary node is a non-dnb node that has no links with DNB nodes, i.e., in Fig.S2a if node i is a non-dnb node, and its linked neighbors i 1, i 2,..., i m are all non-dnb members, then i is a Type 4 node (see the purple nodes shown in Fig. 2a in the main text). Next, we prove that the SNE has the following generic properties in terms of its dynamics, which correspond to these four types of nodes when the system is near a critical transition (see Table S1). S23

24 Table S1: SNE and node types Type Node State transition for the center node SNE for local network 1 DNB p( z(t) z(t 1) > d) 1 decreases drastically to 0 2 DNB p( z(t) z(t 1) > d) 1 decreases 3 non- DNB 4 non- DNB *β (0, 1) is a constant. p( z(t) z(t 1) > d) β p( z(t) z(t 1) > d) β decreases has no significant change C.2.1 SNE for Type 1 nodes We assume that DNB node i is of Type 1 and all of its neighbors are DNB members i 1, i 2,..., i m. From Eq.(S29), we know that when a system is near a critical point, it is more likely that DNB node i has a large state transition at any time t. The same property holds for other DND nodes i l (l = 1, 2,..., m). We define the event A 1 = {x i = 1, x i1 = 1,..., x im = 1} (see Fig.S2). When the system approaches the critical point, from Theorem 3 we have p pre 1,v = p pre (X(t + 1) = A 1 X(t) = A v ) 1, (S31) while the probability of other transitions approaches 0, because p pre s,v = p pre (X(t + 1) = A s X(t) = A v ) = p pre (x il (t + 1) = 0,..., X(t) = A v ) p pre (x il (t + 1) = 0 X(t) = A v ) 0, for any l {1,..., m}, u {1, 2,..., 2 m+1 }, s {2, 3,..., 2 m+1 }. Equation (S31) implies that if any DNB node has a state transition, then its DNB neighbors will probably have a state transition as well. They change dynamically in a highly collective manner. S24

25 Therefore, when the system approaches a critical point, the SNE for type 1 node H pre = u,v πv pre pu,v pre log pu,v pre 0, suggesting that the SNE decreases drastically for Type 1 nodes. C.2.2 SNE for Type 2 nodes We assume that node i is a Type 2 node, i.e., a DNB boundary node linked with at least one non-dnb node. Thus, we assume that some of its neighbors, i.e., i 1, i 2,..., i h, are DNB nodes whereas others i h+1, i h+2,..., i m are non-dnb nodes. Therefore, when a system approaches a critical point, Eq.(S29) holds for i and i l (l = 1, 2,..., h). For i h+r (r = 1, 2,..., m h), Theorem 1 shows that even when the system is in a predisease state, SD(z ih+r ) approaches bounded values. Thus, it holds that p pre (x ih+r (t) = 1) β ih+r, (S32) where β ih+r are constants. Definition 2 Let B 1 represent the event {x i = 1, x i1 = 1,..., x ih = 1}. Let {B s } s=2,3,...,2 h+1 denote random events {x i = γ 0, x i1 = γ 1,..., x ih = γ h } with γ l {0, 1} for l {0, 1, 2,..., h} and at least one γ l = 0. Let {C k } k=1,2,...,2 m h represent random events {x ih+1 = γ h+1,..., x im = γ m } with γ h+r {0, 1} for r {1, 2,..., m h}. Obviously, A u = {B s, C k }, where u {1, 2,..., 2 m+1 }, s {1, 2,..., 2 h+1 }, k {1, 2,..., 2 m h }. For any random event B s (s 2), there is at least one DNB node, i.e., i l, l {0, 1, 2,..., h}, that satisfies x il = 0 (x i0 = x i ). Because p pre (x il = 0,..., ) p pre (x il = 0) 0 from Eq.(S30), we know that there are (2 m+1 2 m h ) probabilities vanishing, i.e., p pre (B s, C k ) = 0, s = 2, 3,..., 2 h+1, k = 1, 2,..., 2 m h, (S33) S25

26 and only 2 m h non-zero probabilities remaining, i.e., that satisfy 2m h k=1 p pre k = 1. p pre k := p pre (B 1, C k ), In a pre-disease state, note that PCC(z il, z ih+r ) 0, l = 1, 2,..., h; r = 1, 2,..., m h, and the correlations among non-dnb nodes i h+r (r = 1, 2,..., m h) are similar to those in a normal state. For any random event C k, from Eq.(S33) we have Thus, p pre k = 2h+1 s k =1 p pre (B 1, C k ) = p pre (B 1, C k ) + 2 h+1 s=2 = p pre (C k ) = p nor (C k ) = 2 h+1 s=1 ps nor k, which suggests p pre k log(p pre k ) = p nor (B s, C k ). = > 2 h+1 s k =1 2 h+1 s k =1 2 h+1 s k =1 p nor s k p nor s k ( p nor s k p pre (B s, C k ) 2 h+1 log log s k =1 2 h s k =1 log ( p nor s k )). p nor s k p nor s k S26

27 Therefore, < 2 m h k=1 p pre k 2 m h k=1 2 m+1 = k=1 log p pre k 2 h+1 p nor k s k =1 ( p nor s k log p nor k, ) logps nor k which infers H pre < H nor. Thus, for any Type 2 node, the SNE will decrease when the system approaches a critical point. C.2.3 SNE for Type 3 nodes We assume that node i is a Type 3 node, i.e., a non-dnb node linked with non-dnb nodes and at least one DNB one. Therefore, we assume that some of the neighbors i 1, i 2,..., i h are DNB nodes, whereas others i h+1, i h+2,..., i m are non-dnb nodes. As with Type 2 nodes, we know that the SNE will decrease when the system approaches a critical point. C.2.4 SNE for Type 4 nodes We assume that node i is a Type 4 node, i.e., i is a non-dnb node linked with non-dnb nodes alone, i 1, i 2,..., i m. Therefore, Eq.(S32) holds for i and i l (l = 1, 2,..., m). There are no significant changes in the correlations among i and its neighbors even in a predisease state. Thus, we may assume that the correlations between i and i h+r (r = 1, 2,..., m h) are the same as those in a normal state. Therefore, p pre (X(t + 1) = A u, X(t) = A v ) = p nor (X(t + 1) = A u X(t) = A v ), where u, v {1, 2,..., 2 m+1 }. Thus, the SNE of a Type 4 node remains the same as that in its normal state and it can be viewed as invariant. S27

28 Based on these four cases, we conclude that the average SNE (S24) decreases drastically as the system approaches a critical transition point. This critical phenomenon coincides with the physical meaning of the conditional entropy in terms of the low resilience or robustness in a predisease state, i.e., the next state transition depends greatly on its current state, which implies a low conditional entropy due to this dependence near the pre-disease state. We adopt an efficient strategy for estimating the average SNE using only those nodes with decreasing SNEs, rather than all of the nodes in the network. Using such a scheme, the average SNE is more sensitive to a pre-disease state. Remark 2 Near a critical transition point, the decreasing value of an SNE depends on the ratio of the number of DNB nodes to the number of non-dnb nodes in the local network. If the ratio of DNB nodes is higher, the SNE is smaller in the local network. In particular, the SNE decreases drastically for a Type 1 node because all of the nodes in the local network that are centered on a DNB core node are DNB nodes. The decrease in SNEs for Type 2 or Type 3 nodes results from the DNB nodes in the local network. The derivation of the SNE is based on the properties of the DNB, but there are several advantages in using the SNE to detect a critical transition. (1) It is easier to detect DNB nodes directly, rather than a dominant group in the overall network, which is generally a difficult task because of the scale of datasets. (2) We only need to focus on the local structure of a network node by node, which significantly reduces the computational complexity. Therefore, predictions based on the SNE method are DNB-free. C.3 The leading network and SNE network After we calculate the SNE for each node, it is reasonable to ignore any nodes with increasing SNEs because this may be caused by noise and data errors. In a pre-disease state, the remaining S28

29 nodes with decreasing SNEs form a subnetwork, which we use as our index or criterion for detecting a critical transition and the leading network in the transition to a disease state. Our analysis and computations are based on local structure, i.e., one node and its linked neighbors, but the final subnetwork is of the most critical information for determining the dynamics of the original overall network. Clearly, this method simplifies the computation and it also helps us to avoid the effect of noise, thereby providing a more accurate and reliable early warning signal. In this study, we adopted the following definitions of different networks. An overall network is formed of Type 1, Type 2, Type 3, and Type 4 nodes, i.e., a network includes all genes. An SNE network is formed of Type 1, Type 2, and Type 3 nodes, i.e., the DNB and its first-order neighbors. A leading network is formed of Type 1 and Type 2 nodes, i.e., the DNB. A leading network is formed of Type 1 and Type 2 nodes, so critical behavior in their dynamics drives the system toward a phase transition. Therefore, Type 1 and Type 2 nodes that are directly related to the dominant eigenvalue behave in a strongly collective manner, carrying most of the dynamical information when the system is in a pre-disease state. In addition, their critical characteristics are little distorted by noise because of their overwhelming trend as a system approaches a critical point. Type 3 and Type 4 nodes, in contrast, are irrelevant to the dominant eigenvalue, and loses the correlations with Type 1 and Type 2 nodes as a system approaches a critical point, and they are much more likely to be influenced by the widespread noise and data errors, resulting in the fluctuations (i.e., possible increasing or decreasing) SNEs. Therefore, in the computational algorithm, we ignore those Type 3 and Type 4 nodes (in particular, Type 4 nodes) that form most of nodes in the network, with increasing SNEs. S29

30 In our algorithm, therefore, we only select nodes with decreasing SNEs (i.e., Type 1 and Type 2, and some Type 3) to calculate the average SNE as our index or criterion for detecting critical transitions and the leading network in an efficient and accurate manner. Based on the definition of the four types of nodes that form the entire network, the DNB or leading network is composed of Type 1 and Type 2 nodes, while the SNE network is composed of Type 1, Type 2, and Type 3 nodes because of the decrease in their SNEs near the critical transition. D Numerical validation of leading networks detected by SNE Figure S5: A model of a six-gene regulatory network. In this sketch of a regulatory network, the arrow represents regulation, whereas the blunt line denotes regulation. In this section, we use a six-gene network (see Fig.S5 or Fig.3a in the main text) to conduct a numerical simulation and theoretically demonstrate the detection of early-warning signals using the SNE near a critical point. These types of gene regulatory networks are often used to study transcription, translation, diffusion, and translocation es that affect gene regulatory activities (2,5,20 22). The following six differential equations represent the gene six genes in a network where gene regulation is represented in a Michaelis-Menten form, with S30

31 the exception of the degradation rates, which are linearly proportional to the concentrations of the corresponding genes. dz 1 (t) = (5 2 P )z 2 + (5 2 P )z 3 2 P dt 5 (1+z 2 (t)) 5(1+z 3 (t)) dz 2 (t) = (2 P )z 1 + (2 P )z 3 P +2 dt 5(1+z 1 (t)) 5(1+z 3 (t)) dz 3 (t) = ( 2 P (2 P ) 1) + dt 5 5(1+z 1 (t)) dz 4 (t) dt = (1+z 2 (t)) dz 5 (t) = z 6(t) 21 dt 10(1+z 6 (t)) 5 z 1 (t) + ζ 1 (t), z 5 2 (t) + ζ 2 (t), (3 P ) + 7 P 5(1+z 2 (t)) 10(1+z 3 (t)) + 10(1+z 6 (t)) 8 5 z 4(t) + ζ 4 (t), z 10 5(t) + ζ 5 (t), dz 6 (t) = z 5(t) 21 z dt 10(1+z 5 (t)) 10 6(t) + ζ 6 (t), 3z 5(t) 10(1+z 5 (t)) 5 z 3 (t) + ζ 3 (t), (S34) where P is a scalar control parameter and ζ i (t) (i = 1, 2,..., 5) are Gaussian noises with zero means and covariances κ ij = Cov(ζ i, ζ j ). z i (i = 1,..., 5) represent the concentrations of mrna-i. In Eq.(S34), the degradation rates of mrnas are ( 2 P P +2,, 7 P, 8, 21, 21) There is a stable equilibrium point Z = ( z 1, z 2, z 3, z 4, z 5, z 6 ) = (0, 0, 0, 0, 0, 0). The differential equations Eq.(S34) can be transformed into the difference equations Z(k + 1) = f(z(k), P ) using the Euler scheme (23), i.e., [ ] z 1 (k + 1) = z 1 (k) + (5 2 P )z2 + (5 2 P )z 3 2 P 5 (1+z 2 (t)) 5(1+z 3 z (t)) 5 1 (t) + ζ 1 (t) t, [ ] z 2 (k + 1) = z 2 (k) + (2 P )z1 + (2 P )z 3 P +2 5(1+z 1 (t)) 5(1+z 3 z (t)) 5 2 (t) + ζ 2 (t) t, [ ] z 3 (k + 1) = z 3 (k) + ( 2 P (2 P ) (3 P ) 1) + 5 5(1+z P (t)) 5(1+z 2 z (t)) 5 3 (t) + ζ 3 (t) t, [ z 4 (k + 1) = z 4 (k) z 5(t) 2 10(1+z 2 (t)) 10(1+z 3 (t)) 10(1+z 5 (t)) ] 3 + 8z 10(1+z 6 (t)) 5 4(t) t, [ ] z z 5 (k + 1) = z 5 (k) + 6 (t) 21 z 10(1+z 6 (t)) 10 5(t) t, [ ] z z 6 (k + 1) = z 6 (k) + 5 (t) 21 z 10(1+z 5 (t)) 10 6(t) t, (S35) with a small time interval t. Note that Z(k) is the vector of Z(t) at the time instant k t. We denote the Jacobian matrix of Eq.(S35) as J = f(z(k);p ) Z where Z= Z, J = e t A (S36) S31

32 with A = 2 P 1 2 P 1 2 P 5 5 (2 P ) (2+ P ) (2 P ) 5 5 (2 P ) (3 P ) (7 P ) From Eq.(S36), we obtain six distinct eigenvalues (0.74 P, 0.55, 0.37, 0.20, 0.14) by taking t = 1. Thus, the equilibrium point Z is stable when P (0, 1]. Obviously, there is a critical value P c = 0, where the system loses stability and undergoes a critical transition. We aimed to detect early warning signals that indicate the critical transition as a control parameter P approaches a critical value 0 from P > 0. We then diagonalize the Jacobian matrix J using matrix S S = , (S37) which satisfies S 1 J S = Λ where Λ = diag(0.67 P, 0.45, 0.37, 0.20, 0.14, 0.09) is a diagonal matrix. We denote (y 1, 0, 0, 0, 0, 0), (0, y 2, 0, 0, 0, 0), (0, 0, y 3, 0, 0, 0), (0, 0, 0, y 4, 0, 0), (0, 0, 0, 0, y 5, 0), (0, 0, 0, 0, 0, y 6 ), respectively, as the eigenvectors of Λ with eigenvalue λ i (i = 1, 2,..., 6). Note that the variable y 1 is related to the largest eigenvalue 0.67 P, which is nearest to 1. Thus, as P approaches 0, the dominant eigenvalue 0.67 P tends to 1, which leads to a critical transition in the system Eq.(S35). Thus, from the relationship. Y (k) = S 1 (Z(k) Z) S32

33 Figure S6: Early warning signals in a six-gene network based on a dynamical network biomarker. (a) Standard deviation (SD) curves for (z 1, z 2, z 3, z 4, z 5, z 6 ). The horizontal axis represents a control parameter P that varies from 0.4 to The vertical axis represents the standard deviation (SD). (b) PCC curves. Each curve is drawn for a pair of variables from (z 1, z 2, z 3, z 4, z 5, z 6 ). The horizontal axis represents a control parameter P that varies from 0.4 to The vertical axis represents the PCC. In simulations based on Eq.(S35) with additive noise, all of the initial conditions are randomly set within the interval [0, 1]. The simulations were performed in MATLAB(R2009a) using the Euler-Maruyama integration method with the Ito calculus (23). The system undergoes a bifurcation at P = 0. S33

34 Figure S7: Comparison of a DNB node z 2 and a non-dnb node z 5. We use a DNB node z 2 and a non-dnb node z 5 as examples. (a) In a normal state (P = 0.4), the distribution of DNB node z 2. (b) In a pre-disease state (P = 0.01), the distribution of DNB node z 2. (c) In a normal state (P = 0.4), the distribution of non-dnb nodes z 5. (b) In a pre-disease state (P = 0.01), the distribution of non-dnb node z 5. or z 1 z 1 = 2 y 1 6 y 3, z 2 z 2 = y 1 y 2, z 3 z 3 = y 1 +y 2 6y 3, z 4 z 4 = y 3 y 4 y 6, z 5 z 5 = y 5 + y 6, z 6 z 6 = y 5 y 6, it is clear that among (z 1, z 2, z 3, z 4, z 5, z 6 ), the three variables z 1, z 2, and z 3 are related directly to y 1, which corresponds to the dominant eigenvalue. Therefore, according to the theoretical S34

35 results in Section B, {z 1, z 2, z 3 } constitute the dynamical dominant group or the DNB of the system when P (0, 1] and this will reflect the breakdown of the system as P 0. We simulated the SD and PCC curves for (z 1, z 2, z 3, z 4, z 5, z 6 ), as shown in Fig.S6. In addition, we compared the distributions of the expressions for a DNB member z 2 and a non-dnb member z 5, as shown in Fig.S7. Based on the theoretical model, we collected time-course data of the six-gene expressions. Then, using the time-course data, we identified the DNB as well as the PCCs and SDs of genes shown in Figures S6 and S7. Finally, we simulated the SNE curves for z i (i = 1, 2,..., 6), as shown in Fig.3e in the main text. The mean entropy curve is shown in Fig. 3f in the main text. E Algorithm for calculating the SNE We assume a set of samples for one individual in each sampling period (or time window) from a practical viewpoint, whereas there are many thousands of measurements for each sample in high-throughput data, i.e., high-throughput expression data or high-throughput sequencing data. The overall period is divided into [period - 1,..., period - T ]. The interval between two consecutive sampling periods (e.g., a month or a week) may be long, but the interval between two samples (or sampling time points) at each period (e.g., a day or an hour) should be shorter so the dynamical features can be reflected in the measured data. In other words, we assume that there are few samples in each period and the intervals need not be equal, although they should be short so that dynamic correlations remain among genes or proteins. For each time period, we evaluate the composite index or the DNB based on a few samples to check whether it is in a pre-disease state in this time period. We assume that there are two groups of samples, i.e., the case samples and the control samples. The algorithm to detect a DNB and the leading network contains the following steps, using high-throughput data throughout the whole study period. S35

36 Figure S8: Flowchart for the algorithm. To define the algorithm clearly, we denote the total number of measurements (genes or proteins) for each sample as N. The flowchart of the algorithm is shown in Fig.S8, which is described as follows. 1 Selection of differentially expressed genes in high-throughput biological data. 1.1 To reflect the state transition in the local network, we determine the differences between samples at two adjacent time points. At each sampling point (or period), we use the Student s t-test with a significance level of p < 0.05 to select genes where the expression changes significantly (in the sense of mean values) between the case samples (microarray data in the case group) and the control samples (microarray data in the control group). 1.2 A false discovery rate (FDR) is used to correct multiple comparisons or multiple t-tests of genes selected in 1.1 during each period. Next, at each sampling time point t 1, t 2,..., t T, we identify the differentially expressed S36