In the version of this Supplementary Information file originally published equations (17) (20) were missing terms. These equations should have read:

Transcription

1 Correction notice Nature Phys. 10, (2014). Avoidg catastrophic failure correlated networks of networks Saulo D. S. Reis, Yanqg Hu, Andrés Babo, José S. Andrade Jr, Santiago Canals, Mariano Sigman & Hernán A. Makse In the version of this Supplementary Information file origally published equations (17) (20) were missg terms. These equations should have read: where δ i,j is the Kronecker delta. These errors have been corrected this file 27 May 2015.

2 Avoidg catastrophic failure correlated networks of networks Reis, Hu, Babo, Andrade, Canals, Sigman, Makse I. THEORY OF CORRELATED NETWORK OF NETWORKS We first illustrate the theory to calculate the percolation threshold for a sgle uncorrelated network followg the standard calculations done by Moore and Newman [1]. We then generalize this theory to the case of two correlated terconnected networks to calculate p c under redundant and conditional modes of failures. A. Calculation of percolation threshold for a sgle network [1] The percolation problem of a sgle network can be solved by the calculation of the probability X to reach the giant component by followg a randomly chosen lk [1]. First, choose a lk of a sgle network at random. After that, select one of its ends with equal probability. The probability 1 X is the probability that, by followg this lk usg the chosen direction, we do not arrive at the giant component, but stead we connect to a fite component. Sce the degree distribution of an end node of a chosen lk is given by kp(k)/ k, one can write down a recursive equation for X as: X =1 k kp(k) k (1 X)k 1. (1) The sum is for the probability that, by followg the chosen lk, we arrive at a node with degree k which is not attached to the giant component through its remag k 1 connections. We rewrite the previous equation as follows: X =1 k kp(k) G(X), (2) k where G(X) =(1 X) k 1. (3) Once the probability X is known, we can use it to write the probability 1 S that a randomly chosen node does not belong to the giant component. Aga, this is a sum of NATURE PHYSICS 1

3 probabilities: the probability that this node has no lks attached to it, plus the probability that this node has one lk and this lk does not lead to the giant component, plus the probability that this node has two lks and none of them leads to the giant component, and so on. In other words: 1 S = k P (k)(1 X) k. (4) Aga, we can rewrite this equation as: S =1 k P (k)h(x), (5) where H(X) = (1 X) k. (6) Note that the probability S not only stands for the probability of choosg one node from the giant component at random, but also provides the fraction of nodes the network occupied by the giant component. SI-Equation (5) provides the probability of a node to belong to the giant component and is the ma quantity to be calculated by the theory from where the value of the percolation threshold can be calculated as the largest value of p c such that S(p c ) = 0. B. Analytical approach for two terconnected networks with correlations Now, we present a generalization of the above approach suited to both problems studied our work, namely, the redundant and conditional teractions of two terconnected networks with generic correlations. We have also developed an analogous theoretical framework based on the generatg approach used Ref. [2]. However, we fd that the generatg function approach [2] is more mathematically cumbersome if one wants to take to account the correlations between the networks to calculate the mutually connected giant component. Sce the size of the giant component is the only quantity needed this study, we fd that the approach of Moore and Newman is more transparent and, furthermore, allows us to take to account both modes of failure a sgle theory. Indeed, the whole theory can be cast to a few number of equations, while the generatg function approach is more volved. We defe two probabilities for network A (and their equivalents for network B). As we did for the case of a sgle network, we will take advantage of functions similar to G(X) and 2 NATURE PHYSICS

4 SUPPLEMENTARY INFORMATION H(X). By dog this, the followg recursive equations are general and can be applied to the redundant and to the conditional teraction cases dependg of the way the functions G( ) and H( ) are written for each case. Therefore, below we develop the theory for both modes of failure and later we specialize on each teraction. First, we defe the probability X A, as the probability that, by followg a randomly chosen lk of network A, we reach a node from the largest connected component of network A. The second probability, Y k A, is the probability of choosg at random a node from network A with -degree k A connected with a node from the largest component of network B. Analogously, we defe probabilities X B and Y k B for network B. Thus, if we itially remove a fraction 1 p A of nodes from network A chosen at random, and a fraction 1 p B of nodes from network B, we can write X A and X B analogy with SI-Eq. (2) [we note that when network A and network B have the same number of nodes, p =(p A + p B )/2]: X A = p A 1 k A,kA out kp ( ) A k,k A out A k A G(X A,Y k A,k,k A out) A. (7) Here, the correlations between k A and kout A from Eq. (1) the ma text are quantified by P (k,k A out), A which is the jot probability distribution of - and out-degrees of nodes from network A from where Eq. (1) the ma text can be derived. The probability function G(X A,Y k A,k,k A out) A SI-Eq. (7) is analogous to SI-Eq. (3). It stands for the probability that, by followg a randomly chosen lk from network A, we reach a node which is not part of the giant component of network A, which has -degree k A and out-degree kout A and/or is not connected with a node from the giant component network B (here and what follows, and/or refers to the nature of the two cases of study: the redundant and conditional teractions, respectively). To write down SI-Eq. (7) we use the jot - and out-degree distribution of an end node of a randomly chosen -lk kp A (k,k A out)/ k A. A Fally, the terms the squared brackets stand for the probability X A = X A (p A = 1) before removg the fraction 1 p A, which is the generalization of SI-Eq. (2). Thus, after the removal of a fraction 1 p A, the probability of followg a randomly selected -lk to reach a node which belongs to the giant cluster of A is X A (p A = 1) times the probability p A for this node beg a survival node. In a similar fashion, we write the probability X B, the jot degree distribution P (k,k B out) B and the probability function G(X B,Y k B,k,k B out) B for network B: NATURE PHYSICS 3

5 X B = p B 1 k B,kB out kp ( ) B k,k B out B k B G(X B,Y k B,k,k B out) B. (8) For the probability Y k A of choosg at random a node from the network A with degree k A connected through an out-lk with a node from the giant component of B, we write down the followg expression: Y k A = p B 1 P ( k k) B A (1 XB ) kb. (9) k B The term side the squared brackets is the probability of choosg a node from network B which is not part of the giant component of B and it is connected with a node from network A of -degree k A. Naturally, Y k A is this probability times the probability p B of the B-node beg a survival node after the removal of a fraction 1 p B of nodes from network B. To write down this equation, we use the conditional probability P (k B k A ) of a node from network B with -degree k B beg connected with a node with -degree k A from network A, and the probability that, by followg an -lk from B, we do not reach the giant component of B, (1 X B ). The conditional probability P ( k B k A ) quantify the correlations expressed by Eq. (2) the ma text. Similar equation can be written for Y k B : Y k B = p A 1 P ( k k) A B (1 XA ) ka. (10) k A With X A, X B, Y k A, and Y k B on hand, it is possible to compute the fraction of survival nodes the giant component of network A, S A, and network B, S B, through the relations analogous to SI-Eq. (5): S A = p A 1 P (k,k A out)h(x A A,Y k A,k,k A out) A, (11) and k A,kA out S B = p A 1 P (k,k B out)h(x B B,Y k B,k,k B out) B. (12) k B,kB out The probability function H(X A,Y k A,k,k A out) A generalizes SI-Eq. (6), and stands for the probability of randomly selectg a node from network A with -degree k A and out-degree kout, A which is not the giant component of A and/or it is not connected with the giant 4 NATURE PHYSICS

6 SUPPLEMENTARY INFORMATION component of B (aga, and/or refers to redundant and conditional modes of teraction, respectively). Due to the different meangs that the probability function H(X A,Y k B,k,k A out) A may assume dependg of the mode of teraction, for this general approach the nature of the quantities S A and S B differ conceptually from the quantity S presented by SI-Eq. (5) for a sgle network. See SI-Fig. 1 for more details. For the conditional mode, a node, or a set of nodes from network A, for example, will fail if (i) it loses connection with the largest component of network A, or if (ii) it loses connection with the largest component of network B. Thus H(X A,Y k B,k,k A out) A is the probability function that describes the probability of pickg a node at random from network A that is not part of the largest component of A (due to condition (i) this node will fail) or that is not connected to the largest cluster of network B (due to condition (ii) this node will also fail). Thus, S A (and its counterpart S B for network B) is the fraction occupied by the largest component of survival node network A. For a fite size network, S A = n A /N A, where n A is the number of nodes the largest component and N A the number of nodes network A. It is important to note that due to the condition (ii) this fraction is necessarily the same as the size of the giant connected component of network A. S A may be terpreted also as the fraction from network A that is part of the mutually connected giant component S AB, as Ref. [2]. The same applies to network B. In other words, the number of nodes the mutually connected giant component belongg to B is the same as the number of nodes the giant connected component of B as calculated after the attack as if B was a sgle network. For the redundant mode, sce there is no cascadg propagation of damage due to the failure of a neighbor, H(X A,Y k B,k,k A out) A is the function that describes the probability of pickg a node at random, for example from network A, which is not connected to the largest component from its own network, network A, and is not connected to the largest component of network B via an out-gog lk. Therefore, the quantity S A provides the fraction of active nodes, or other words, the fraction of survival nodes that may be part of the largest component of network A, and addition a fraction from network A that are disconnected from that largest component of network A, but are not failed because they are still connected to the largest component of network B via an out-gog lk. Thus, the mutually connected giant component S AB has a different structure this mode compared to the conditional mode. This situation is illustrated Fig. 1c the ma text and SI- NATURE PHYSICS 5

7 Fig. 1. At the end of the attack process, there is a remag node network A which is not connected to the giant component of A calculated as if it is a sgle network. Such a node is still on sce it is connected to B via an out-gog lk. Thus, the mutually connected giant component contas this node. Furthermore, a node that has lost all its out-gog lk will fail the conditional teraction, even if it is still connected to its own giant component. However, the redundant mode, a node without out-gog lks may still function as long as it is still connected to the giant component of its own sgle network. For stance, many nodes are still functiong Fig. 1c ma text, redundant mode, even though they are not terconnected. However, conditional teraction Fig. 1b ma text, all stable nodes needs to have out-gog lks. That is, redundant mode, the nodes can still receive power via the same network or the other network, while the conditional node, they need out-gog connectivity all the time. Takg to account these considerations, the value of p c is obtaed from the behavior of the giant component of either of the networks the conditional mode, while the redundant mode, the value of p c is obtaed from the size of the mutually giant connected component. However, this last case, it is statistically the same to obta p c from the giant components of one of the networks as well. In what follows the calculations of the giant components are done by considerg two networks of equal size N and damagg each network with a fraction 1 p of nodes. Next, we explicitly write the probability functions G and H for both, conditional and redundant teractions, respectively, to occur on teractive networks after a random failure of 1 p A and 1 p B nodes. It is important to note that the probabilities G and H describe the probability of randomly choosg a node which is not part of the giant component of one network and/or is not connected to a node from the giant component of the adjacent network. In other words, this node picked at random is not part of the giant component of the whole network. We test the general case where both networks are attacked: p A 1 and p B 1. The theory can be used to attackg only one network by settg p B = 1. Redundant teraction: We consider the total fraction 1 p of nodes removed from the two networks. If network A and network B have the same number of nodes, then p =(p A + p B )/2. For redundant teraction two events are important. Both events are defed as follows. The first is the probability that, by followg a randomly chosen lk of a network, we do not reach the giant component of that network. For network A, this 6 NATURE PHYSICS

8 SUPPLEMENTARY INFORMATION probability can be written as (1 X A ). The second is the probability of choosg at random a node from one network, say network A, with -degree k A which is not connected with a node from the giant component of network B. This probability can be written as (1 Y k A ). In the case of redundant teraction (with no cascadg due to conditional mode) these two probabilities are dependent, sce the lack of connectivity with network B does not imply failure of a node from network A. Thus, the probability function G(X A,Y k A,k,k A out) A that, by followg a randomly selected lk we arrive at a node with -degree k A and out-degree kout A which is not part of the giant cluster of its own network and is not connected with a node from the giant cluster of the adjacent network can be written as: G(X A,Y k A,k,k A out) A =(1 X A ) ka 1 (1 Y k A ) ka out. (13) Similarly, the probability function H(X A,Y k A,k,k A out) A of pickg a node, at random, with -degree k A and out-degree kout A from one network which is not part of the giant cluster of its own network and is not connected with a node from the giant cluster from the adjacent network is: H(X A,Y k A,k,k A out) A =(1 X A ) ka (1 Yk A ) ka out. (14) Aga, we can write equivalent expressions for G(X B,Y k B,k,k B out) B and H(X B,Y k B,k,k B out) B as G(X B,Y k B,k,k B out) B =(1 X B ) kb 1 (1 Y k B ) kb out, (15) and H(X B,Y k B,k B,k B out) =(1 X B ) kb (1 Yk B ) kb out. (16) Conditional teraction: This teraction leads to cascadg processes. In the conditional teraction process, we are terested the cascadg effects on the coupled networks, A and B, due to an itial random failure of a portion of nodes both networks, where p A 1 and p B 1. In the case of attackg network A only, the fraction p B is set to be equal to one, such that a node from network B can only fail due to the conditional teraction. For the conditional teraction, G(X A,Y k A,k,k A out) A depends on the probability that, by followg a lk from network A, we do not arrive at a node with -degree k connected to the giant component of its own network, (1 X A ) k 1, and on the probability of randomly choosg a node from network A with k out outgog lks towards network B, (1 Y k A ) kout. Also, we have the probability H(X A,Y k A,k,k A out) A of pickg up a node from one network NATURE PHYSICS 7

9 which is not part of the giant component of its own network or pickg up one node from one network which is not connected with one node from the giant component of the adjacent network, which is also dependent of the probabilities (1 X A ) and (1 Y k A ). Different from the redundant mode, these probabilities, (1 X A ) and (1 Y k A ), are not mutually exclusive the conditional teraction. Thus: G(X A,Y k A,k A,k A out) = (1 X A ) ka 1 + (1 Y k A ) ka out (1 XA ) ka 1 (1 Y k A ) ka out (17) +δ k A out,0[(1 X A ) ka 1 1], and H(X A,Y k A,k A,k A out) = (1 X A ) ka + (1 Yk A ) ka out (1 XA ) ka (1 Yk A ) ka out (18) +δ k A out,0[(1 X A ) ka 1], where δ i,j is the Kronecker delta. We can write the equivalent expressions for G(X B,Y k B,k,k B out) B and H(X B,Y k B,k,k B out) B as follows: G(X B,Y k B,k,k B out) B = (1 X B ) kb 1 + (1 Y k B ) kb out (1 XB ) kb 1 (1 Y k B ) kb out (19) +δ k B out,0[(1 X B ) kb 1 1], and H(X B,Y k B,k B,k B out) = (1 X B ) kb + (1 Yk B ) kb out (1 XB ) kb (1 Yk B ) kb out (20) +δ k B out,0[(1 X B ) kb 1]. With the set of SI-Eqs. (13)-(14) and SI-Eqs. (18)-(19), and their equivalents for network B, SI-Eqs. (15)-(16) and SI-Eqs. (20)-(21), it is possible to solve both problems, the redundant and the conditional teractions, on a system of two coupled networks terconnected through degree-degree correlated outgog nodes. The correlation between the coupled networks is represented by the - out-degree distribution P (k,k A out) A and by the conditional probability P ( k k) B A. In the followg section, we present the network model used to generate a system of two networks terconnected with correlations described by power law functions with the exponents α and β. These networks are used on the calculations of 8 NATURE PHYSICS

10 SUPPLEMENTARY INFORMATION the distribution P (k,k A out) A and P ( k k) B A for each pair of (α, β). The fal result is the probability for a node to belong to the giant component of network A or B as given by SI-Eq. (11) and SI-Eq. (12) as a function of the fraction of removed nodes 1 p (with p A = p B = p) from where the percolation threshold p c can be evaluated from S A (p c ) = 0 and S B (p c ) = 0 as a function of the three exponents defg the networks: γ, α and β, and the cutoff the degree distribution k max. We use two networks of equal size N = 1500 nodes, each. C. Network model. Test of theory In order to test the percolation theory usg the above formalism, we need to generate a system of teractg networks with the prescribed set of exponents and degree cutoff. The first step of our network model is to generate two networks, A and B, with the same number N of nodes and with the desired -degree distribution P (k ) as defed by γ and the maximum degree k max. To do this we use the standard configuration model which has been extensively used to generate different network topologies with arbitrary degree distribution [3]. The algorithm of the configuration model basically consists of assigng a randomly chosen degree sequence to the N nodes of the networks such a way that this sequence is distributed as P (k ) k γ with 1 k k max and P (k )=0fork >k max. After that, we select a pair of nodes at random, both with k > 0, and we connect them. The next step of the model is to connect networks A and B such a way that their outgog nodes have degree-degree correlations that can be described by the parameters α and β as defed Eqs. (1) and (2) the ma text. In order to do this, we use an algorithm spired by the configuration model. First, we assign a sequence of out-degrees k out to the nodes of each network. This process is performed dependently to each network by addg the same number of outgog lks. Each outgog lk is added dividually to nodes chosen at random with a probability that is proportional to k α. Thus, an out-degree sequence is assigned to the nodes each network such a way that k out k α accordg to Eq. (1) ma text. This process results a set of outgog stubs attached to every node network A and B. The next step is to jo these stubs such a way that we satisfy the correlations given by Eq. (2) ma text. The next step is to choose two nodes, one from each network, such that k nn = A k β, NATURE PHYSICS 9

11 and then, we connect them if they have available outgog lks. Here, we choose the factor A such that k nn = 1 for k = 1 when β = 1, and k nn = k max for k = 1 and β = 1. Thus, we write the value of the factor as A = A(k max,β)=k max (1 β)/2. The algorithm works as follows: we randomly choose one node i from one network. After that, we choose another node j, from the second network, with -degree k j with probability that follows a Poisson distribution P (k j,λ), where the mean value λ = knn. We connect nodes i and j if they are not connected yet. It should be noted that Eqs. (1) and (2) the ma text may not be self-consistent for all values of α, β. For stance, for very low values of β, e.g., β = 1, the degree correlations between coupled networks are not always self-consistent with the structural relations between k and k out described by α. Sce β measures the convergence of connections between networks, when β is negative hubs prefer to connect with low-degree nodes. To better understand these features, consider β = 1, and for nodes with k = 1 and k = k max. With this configuration, nodes with k = 1 are likely to be connected with nodes from the adjacent network with k = k max. When α = 1, most of the lks are attached to the highly active nodes, notably, nodes with k = k max, and less likely to nodes with k = 1. In this regime, there are not enough low-degree nodes with outgog lks to be connected with the high-degree nodes, thus the desired relation between k nn versus k cannot be realized. The other possible situation is when α is negative. In this regime, most of the outgog lks are attached to low-degree nodes, consequently, the few hubs from the network are unlikely to receive an outgog lk, and even when it happens, one hub does not have enough outgog lks to be connected to the stubs of the low-degree nodes. For these reasons we limit our study to α> 1 and β> 0.5 where the relations are found to be self-consistent. For every itial pair (α, β), we generate a network with the above algorithm and then we recalculate the effective values of (α, β) which are then used to plot the phase diagram p c (α, β) Figs. 2 and 4 the ma text. D. Calculation of the giant components and percolation threshold p c (γ,α,β,k max ) With the networks generated the previous section we are able to compute the functions P (k A,k A out) and P ( k B k A ). Then we apply the recursive equations derived previously to calculate the size of the giant components S A and S B from SI-Eqs. (11) and (12). We do 10 NATURE PHYSICS

12 SUPPLEMENTARY INFORMATION this calculation for different values of p for cases of study and then extract the percolation threshold p c at which the giant components S A and S B vanish conditional mode. SI-Figure 2 shows the predictions of the theory the conditional mode for a network with γ =2.5, α =0.5, β =0.5 and k max = 100. We plot the relative size of the giant components A and B, S A and S B, as predicted by SI-Eqs. (11) and (12). As one can see SI-Fig. 2, there is a well-defed critical value at which the A-giant component vanishes which defes the percolation threshold p c (γ,α,β,k max )=0.335 for these particular parameters. SI-Figure 2 also presents the comparison between theoretical results and direct simulations. We test the theory by attackg randomly the generated correlated networks and calculatg numerically the giant components versus the fraction of removed nodes 1 p. The results show a good agreement corroboratg the theory. After testg the theory, a full analysis is done spanng a large parameter space by changg the four parameters defg the theory: (γ,α,β,k max ). The results are plotted the ma text Fig. 2 and 4 for the stated values of the parameters. Beyond the calculation of p c (α, β), we also identify regimes of first-order phase transitions the conditional teraction, found specially when p c is high, beyond the standard second-order percolation transition; a result that will be expanded subsequent papers. II. EXPERIMENTS: ANALYSIS OF INTERCONNECTED BRAIN NETWORKS Our functional bra networks are based on functional magnetic resonance imagg (fmri). The fmri data consists of temporal series, known as the blood oxygen leveldependent (BOLD) signals, from different bra regions. The bra regions are represented by voxels. In this work we use data sets gathered two different and dependent experiments. The first is the NYU public data set from restg state humans participants. The NYU CSC TestRetest resource is available at trt/. The second data set was gathered a dual-task experiment on humans previously produced by our group [4] and recently analyzed Ref. [5]. The bra networks analyzed here can be found at: Both datasets were collected healthy volunteers and usg 3.0T MRI systems equipped with echoplanar imagg (EPI). The first study was approved by the stitutional review boards of the New York University School of Medice and New York University. The second study NATURE PHYSICS 11

13 is part of a larger neuroimagg research program headed by Denis Le Bihan and approved by the Comité Consultatif pour la Protection des Personnes dans la Recherche Biomédicale, Hôpital de Bicêtre (Le Kreml-Bicêtre, France). Restg state experiments: A total of 12 right-handed participants were cluded (8 women and 4 men, mean age 27, rangg from 21 to 49). Durg the scan, participants were structed to rest with their eyes open while the word Relax was centrally projected white, agast a black background. A total of 197 bra volumes were acquired. For fmri a gradient echo (GE) EPI was used with the followg parameters: repetition time (TR) = 2.0 s; echo time (TE) = 25 ms; angle = 90 ; field of view (FOV) = mm; matrix = 64 64; 39 slices 3 mm thick. For spatial normalization and localization, a high-resolution T1-weighted anatomical image was also acquired usg a magnetization prepared gradient echo sequence (MP-RAGE, TR = 2500 ms; TE = 4.35 ms; version time (TI) = 900 ms; flip angle = 8 ; FOV = 256 mm; 176 slices). Data were processed usg both AFNI (version AFNI , and FSL (version 5.0, and the help of the batch scripts for preprocessg. The preprocessg consisted on: motion correctg (AFNI) usg Fourier terpolation, spatial smoothg (fsl) with gaussian kernel (FWHM=6mm), mean tensity normalization (fsl), FFT band-pass filterg (AFNI) with 0.08Hz and 0.01Hz bounds, lear and quadratic trends removg, transformation to MIN152 space (fsl) with a 12 degrees of freedom aff transformation, (AFNI) and extraction of global, white matter and cerebrospal fluid nuisance signals. Dual task experiments: Sixteen participants (7 women and 9 men, mean age, 23, rangg from 20 to 28) were asked to perform two consecutive tasks with the struction of providg fast and accurate responses to each of them. The first task was a visual task of comparg a given number (target T1) to a fixed reference, and, second, an auditory task of judgg the pitch of an auditory tone (target T2) [4]. The two stimuli are presented with a stimulus onset asynchrony (SOA) varyg from: 0, 300, 900 and 1200 ms. Subjects had to respond with a key press usg right and left hands, whether the number flashed on the screen or the tone were above or below a target number or frequency, respectively. Full details and prelimary statistical analysis of this experiment have been reported elsewhere [4, 5]. Subjects performed a total of 160 trials (40 for each SOA value) with a 12 s ter-trial 12 NATURE PHYSICS

14 SUPPLEMENTARY INFORMATION terval five blocks of 384 s with a restg time of 5 m between blocks. In our analysis we use all scans, that is, scans comg from all SOA. Sce each of the 16 subjects perform four SOA experiments, we have a total of 64 bra scans. The experiments were performed on a 3T fmri system (Bruker). Functional images were obtaed with a T2*- weighted gradient echoplanar imagg sequence [repetition time (TR) 1.5 s; echo time 40 ms; angle 90; field of view (FOV) mm; matrix 64 64]. The whole bra was acquired 24 slices with a slice thickness of 5 mm. Volumes were realigned usg the first volume as reference, corrected for slice acquisition timg differences, normalized to the standard template of the Montreal Neurological Institute (MNI) usg a 12 degree affe transformation, and spatially smoothed (FWHM = 6mm). High-resolution images (threedimensional GE version-recovery sequence, TI = 700 mm; FOV = mm; matrix = ; slice thickness = 1 mm) were also acquired. We computed the phase and amplitude of the hemodynamic response of each trial as explaed M. Sigman, A. Jobert, S. Dehaene, Parsg a sequence of bra activations of psychological times usg fmri. Neuroimage 35, (2007). We note that the present data contas a standard preprocessg spatial smoothg with gaussian kernel (FWHM=6mm), which was not applied Ref. [5]. Such smoothg produces smaller percolation thresholds as compared with those obtaed Ref. [5]. Construction of bra networks: In order to build bra networks both experiments, we follow standard procedures the literature [5 7]. We first compute the correlations C ij between the BOLD signals of any pair of voxels i and j from the fmri images. Each element of the resultg matrix has value on the range 1 C ij 1. If one considers that each voxel represents a node from the bra network question, it is possible to assume that the correlations C ij are proportional to the probability of nodes i and j beg functionally connected. Therefore, one can defe a threshold T, such that if T <C ij the nodes i and j are connected. We beg to add the lks from higher values to lower values of T. This growg process can be compared to the bond percolation process. As we lower the value of T, different clusters of connected nodes appear, and as the threshold T approaches a critical value of T c, multiple components merge formg a giant component. In random networks, the size of the largest component creases rapidly and contuously through a critical phase transition at T c, which a sgle cipient cluster domates and spans over the system [8]. Instead, sce the connections bra networks are highly NATURE PHYSICS 13

15 correlated rather than random, the size of the largest component creases progressively with a series of sharp jumps. These jumps have been previously reported Ref. [5]. This process reveals the multiplicity of percolation transitions: percolatg networks subsequently merge each discrete transition as T decreases further. We observe this structure the two datasets vestigated this study: for the human restg sate Fig. 3a ma text and for the human dual task Fig. 3b ma text. For each dataset we identify the critical value of T, namely T c, which the two largest components merge, as one can notice Fig. 3 the ma text. While the anatomical projection of the largest component varied across experiments, this mergg pattern at T c was clearly observed each participant of the two experiments analyzed here, two examples are shown Figs. 3a-b ma text. The transition is confirmed by the measurement of the second largest cluster which shows a peak at T c, see SI-Fig. 3. For T values larger than T c the two largest bra clusters are disconnected, formg two dependent networks. Each network is ternally connected by a set of strong-lks, which correspond to k [5] the notation of systems of networks. By lowerg T to values smaller than T c, the two networks connect by a set of weak-lks, which correspond to k out [5], i.e. the set of lks connectg the two networks. Our analysis of the structural organization of weak lks connectg different clusters is performed with T 0 <T <T c. Here, T 0 is chosen such a way that the average k out of outgog degrees of the nodes on the two largest clusters is k out = 1. For lower values of T 0, where k out = 2 and = 5, we found no relevant difference with the studied case of k out = 1. As done previous network experiments based on the dual task data [5] we create a mask where we keep voxels which were activated more than 75% of the cases, i.e., at least 48 stances out of the 64 total cases considered. The obtaed number of activated voxels the whole bra is N 60, 000, varyg slightly for different dividuals and stimuli. The activated or functional map exhibits phases consistently fallg with the expected response latency for a task-duced activation [4]. As expected for an experiment volvg visual and auditory stimuli and bi-manual responses, the responsive regions cluded bilateral visual occipito-temporal cortices, bilateral auditory cortices, motor, premotor and cerebellar cortices, and a large-scale bilateral parieto-frontal structure. In the present analysis we follow [5] and we do not explore the differences networks between different SOA 14 NATURE PHYSICS

16 SUPPLEMENTARY INFORMATION conditions. Rather, we consider them as dependent equivalent experiments, generatg a total of 64 different scans, one for each condition of temporal gap and subject. The followg emergent clusters are seen restg state: medial prefrontal cortex, posterior cgulate, and lateral temporoparietal regions, all of them part of the default mode network (DMN) typically seen restg state data and specifically found our NYU dataset [9]. A. Computation of parameters γ, α, β, and k max Once T c is determed, we are able to compute the degree distribution of the bra networks. For a given bra scan we search for all connected components of strong lks with C ij >T c, where T c is the first jump the largest connected component as seen Fig. 3 ma text. We then calculate P (k ) usg all bra networks for a given experiment; the results are plotted Fig. 3 ma text. We consider all nodes with k 1 at T c from all the connected clusters. As one can see Fig. 3b ma text, for all data sets, we found degree distributions which can be described by power laws P (k ) k γ with a given cut-off k max. For the restg state, we found γ =2.85±0.04 and k max = 133 while for the dual task we found γ =2.25 ± 0.07, k max = 139 (see Table I ma text). We use a statistical test based on maximum likelihood methods and bootstrap analysis to determe the distribution of degree of the networks. We follow the method of Clauset, Shalizi, Newman, SIAM Review 51, 661 (2009) of maximum likelihood estimator for discrete variables which was already used our previous analysis of the dual task data [5]. We fit the degree-distribution assumg a power law with a given terval. For this, we use a generalized power-law form P (k; k m,k max )= k γ ζ(γ,k m ) ζ(γ,k max ), (21) where k m and k max are the boundaries of the fittg terval and the Hurwitz ζ function is given by ζ(γ,α) = i (i + α) γ. We set k m = 1. We calculate the slopes successive tervals by contuously creasg k max. For each one of them we calculate the maximum likelihood estimator through the numerical solution of ( γ = argmax γ ) M ln k i M ln [ζ(γ,k m ) ζ(γ,k max )], (22) i=1 NATURE PHYSICS 15

17 where k i are all the degrees that fall with the fittg terval and M is the total number of nodes with degrees this terval. The optimum terval was determed through the Kolmogorov-Smirnov test. For the goodness-of-fit test, we use KS test generatg 10,000 synthetic random distributions followg the best-fit power law. Analogous analysis is performed to test for a possible exponential distribution to describe the data. We use KS statistics to determe the optimum fittg tervals and also the goodness-of-fit. In all the cases where the power law was accepted we ruled out the possibility of an exponential distribution, see [5]. In order to compute the correlation of k, k out and k nn we consider the followg statistics for the weak lks and the degrees of the external nearest neighbors of an outgog node. This correlation is gathered from the calculation of the average -degree, k nn of the external neighbors of a node with -degree k. The strong-lks are those lks added to the network for T >T c. The weak lks are those added to the network for values of T 0 <T <T c until the average out-degree reaches k out = 1. For statistical determation of the scalg properties of weak-lks, we consider that they connect two nodes different networks, or even nodes the same component. To calculate the statistical scalg properties of weak lks, we consider the out-weak-degree k out of a node as the number of all lks added for T 0 <T <T c. Figure 3f ma text shows that the scenario for the correlation between k nn and k is consistent with Eq. (2) the ma text. For the restg state experiments (Fig. 3f the ma text) there is a positive correlation between the k of outgog nodes placed different functional networks. correlation is also positive. For the dual-task human subjects (Fig. 3f ma text) the Moreover, when analyzg the relation between k and k out for the same outgog nodes, they are described by the correlations presented Fig. 3e ma text usg power laws. Figures 3e-f the ma text depict the power-law fits usg Ordary Least Square method with a given terval of degree. We assess the goodness of fittg each terval via the coefficient of determation R 2. We accept fittgs where R The exponents measured are presented Table I the ma text. Figures 4a and b the ma text show the results we found when we apply the theory presented Section I of this Supplementary Information on two coupled networks of degree exponent γ =2.85 and 2.25, respectively with the cut-off given by k max = 133, 139, 16 NATURE PHYSICS

18 SUPPLEMENTARY INFORMATION respectively as given by the values for human restg state and dual task. For γ =2.25 and γ =2.85, the value associated with the data gathered from humans, the results are similar with those presented on Fig. 2 ma text both theoretical cases, the conditional (left panels ) and redundant (right panels) teractions. The ma differences between the results for γ =2.25 and 2.85 are the values found for p c, where the values found for γ =2.25 are systematically smaller than the values found for γ =2.85, gog from p c 0.1 to 0.6 for γ =2.25, and from p c 0.1 to 0.8 for γ =2.85. These results can be understood from the knowledge gathered on the percolation of sgle networks [10]. For lower values of the degree exponent γ the hubs on scale-free networks become more frequent, protectg the network from breakg apart. When comparg the two cases of Fig. 4 ma text with the theoretical case of γ =2.5 (Fig. 2 ma text), one can notice that the broader the distribution (as lower the value of γ), the more robust is the system of coupled networks. There general trends are consistent with the calculations of p c for unstructured terconnected networks with one-toone connections done Ref. [2]. The white circles Fig. 4 the ma text correspond to the values of α and β measured from real data. As one can see, the experimental values are placed on the region that represents the best compromise between the predictions for optimal stability under conditional and redundant teractions. It is also terestg to note that the extreme vulnerability predicted Ref. [2] can be somehow mitigated by decreasg the number of one-to-one terconnections as shown Ref. [11]. However, this case, the system of networks may be rendered non-operational due to the lack of terconnections. Indeed, by connectg both networks with one-to-one outgog lks and by makg these terconnections at random, there is a high probability that a hub one network will be connected with a low degree node the other network. These low degree nodes are highly probable to be chosen a random attack, thus the hubs become very vulnerable due to the conditional teraction with a low degree node the other network. This effect leads to the catastrophic cascadg behavior found [2]. Another way to protect a network the conditional mode is to crease the number of out-gog lks per nodes, sce the failure of a node occurs when all its ter-lked nodes have failed. Thus, by just creasg the number of terlks from one to many out-gog lks emanatg from a given node, larger resilience is obtaed. If these lks are distributed at random, then this situation corresponds to α = β = 0 our model. However, this random conditional case, the network may be rendered non-operational due to the random NATURE PHYSICS 17

19 nature of the terlk connectivity. A functional real network is expected to be operatg with correlations and therefore the most efficient structure when there are many correlated lks connectg the networks is the one found for the bra networks vestigated the present work. In other words, assumg that a natural system like the bra functions with trsic correlations ter-network connectivity, then the solution found here (large α and β>0) seems to be the natural optimal structure for global stability and avoidance of systemic catastrophic cascadg effects. Another problem of terest is the targeted attack of terdependent networks as treated Ref. [12]. It would be of terest to determe how the present correlations affect the targeted attack to, for stance, the highly connected nodes. 18 NATURE PHYSICS

20 SUPPLEMENTARY INFORMATION [1] Moore, C. & Newman, M. E. J. Exact solution of site and bond percolation on small-world networks. Phys. Rev. E 62, (2000). [2] Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havl, S. Catastrophic cascade of failures terdependent networks. Nature 464, (2010). [3] Dorogovtsev, S. N. Lectures on Complex Networks (Oxford Univ. Press, Oxford, 2010). [4] Sigman, M. & Dehaene, S. Bra mechanisms of serial and parallel processg durg dual-task performance. J. Neurosci. 28, (2008). [5] Gallos, L. K., Makse, H. A. & Sigman, M. A small world of weak ties provides optimal global tegration of self-similar modules functional bra networks. Proc. Natl. Acad. Sci. USA 109, (2012). [6] Eguiluz, V. M., Chialvo, D. R., Cecchi, G. A., Baliki, M. & Apkarian, A. V. Scale-free bra functional networks. Phys. Rev. Lett. 94, (2005). [7] Bullmore E. & Sporns O. Complex bra networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience 10, (2009). [8] Bollobás, B. Random Graphs (Academic Press, London, 1985). [9] Shehzad, Z., Kelly, A. M. C. & Reiss, P. T. The restg bra: unconstraed yet reliable. Cereb. Cortex 10, (2009). [10] Cohen, R., Ben-Avraham, D. & Havl, S. Percolation critical exponents scale-free networks. Phys. Rev. E 66, (2002). [11] Parshani, R., Buldyrev, S. V. & Havl, S. Interdependent networks: reducg the couplg strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105, (2010). [12] Huang, X., Gao, J., Buldyrev, S. V., Havl, S. & Stanley, H. E. Robustness of terdependent networks under targeted attack. Phys. Rev. E 83, (2011). NATURE PHYSICS 19

21 a b c d e failure A B A B A B A B A B Redundant Conditional FIG. 1: Pictorial representation of the a-e conditional and a-b redundant modes of teraction. a, One node is removed, or fails, network A, b, as a regular percolation process this node is removed together with its lks. In the redundant mode of teraction, the neighbors of this node are not removed, because they still mata connection with the giant component from network B, but c, for the conditional mode of teraction the two nodes are removed, sce they do not belong to the giant component of network A. d, As a consequence of the removal of the nodes network A all the nodes from network B that lose connectivity with network A are also removed. e, Fally, the last node from network B is removed once it loses connectivity with the giant component of network B. In the end, for the conditional mode of teraction, only the mutually connected component remas. 20 NATURE PHYSICS

22 SUPPLEMENTARY INFORMATION S, A S B α = 0.5 β = 0.5 Theory Network A Theory Network B Simulation Network A Simulation Network B p FIG. 2: Giant component of network A and B the conditional mode of failure. We present the prediction of the theory for values of N A = N B = 1500, γ =2.5, α =0.5, β =0.5 and k max = 100 and compare with computer simulations of the giant component obtaed numerically by attackg the same network. We perform average over 100 different realizations. We attack a fraction 1 p of both networks and calculate the fraction of nodes belongg to the correspondg giant components. The results show a very good agreement between theory and simulations. NATURE PHYSICS 21

23 a b Fraction of the component Restg State Largest component 2nd largest component T Dual task Fraction of the component Largest component 2nd largest component T FIG. 3: First and second largest component the bra networks correspondg to restg state and dual task. The largest component shows a jump while the second largest component shows a peak, dicatg a percolation transition at T c. a, Restg state. b, Dual task NATURE PHYSICS