to extract sij from the experimentally accessible gij , we formally link ( 1 ) and ( 2 ) via ( 3){dxidxi=1dxidxj=k=1nxixkdxkdxjij}. 
 equation ( 3 ) is exact and the sum accounts for all network paths connecting i and j ( supplementary note s.i.1 - 2 ) . it is of limited use , however , as it requires us to solve n coupled algebraic equations . in supplementary note s.i.1 we show that ( 3 ) can be reformulated as ( 4)s=(gi+d(sg))g1 , where i is the identity matrix and d(m ) sets the off - diagonal terms of m to zero . to obtain an approximate solution for s we use that fact that typically , perturbations decay rapidly as they propagate through the network , so that the response observed between two nodes is dominated by the shortest path between them . this allows us to approximate d(sg ) with d((gi)g ) ( supplementary note s.i.3 ) , obtaining ( 5)s=(gi+d((gi)g))g1 . equation ( 5 ) , our main result , provides sij from the experimentally accessible gij . it achieves this through a ' silencing effect , in which direct response terms are preserved , while indirect responses are silenced . to understand this consider a specific term in gij , documenting the response of node i to j 's perturbation . ( 3 ) , this response is a consequence of all direct and indirect paths leading from j to i. as we document below , the transformation ( 5 ) detects the indirect paths and silences them , maintaining only the contribution of the direct paths ( fig . 1d - f ) . to demonstrate the predictive power of ( 5 ) , we implemented michaelis - menten dynamics on a model network ( supplementary note s.iii ) , as commonly used to model generegulation . we obtained gij by perturbing the activity of each node and then calculated sij using ( 5 ) . figure 2a shows the gij and sij terms associated with interacting ( green ) and non - interacting ( orange ) node pairs . although gij is higher for direct interactions , the overlap between the orange and the green symbols indicates a lack of a clear threshold q that separates direct and indirect interactions . in contrast , sij displays a clear separation between direct and indirect interactions , accurately predicting each direct link . 2b , red ) has an area of auroc = 0.91 , reflecting inherent limitations in separating direct from indirect interactions based on gij only . in contrast for sij we obtain auroc = 0.997 ( blue ) , where the true positive rate ( tpr ) reaches 100% with a false positive rate ( fpr ) of less than 10 . also , although for gij precision increases gradually with the threshold q ( fig . hence , in our well controlled model system effectively any non - zero sij corresponds to a direct link . the performance of ( 5 ) is due to the silencing effect : it leaves gij unchanged if i and j are linked , while it systematically lowers all gij not rooted in a direct interaction . to quantify this effect we measured the discrimination ratio g = gijdir/gijindir ( s = sijdir/sijindir ) which captures the ratio between gij ( sij ) terms associated with direct links and those associated with indirect links ( fig . we find that sij is much more discriminative than gij owing to its silencing of indirect responses . to quantify this effect we measure the silencing ( 6)=sg , which captures the increased power of sij to discriminate between direct and indirect links compared to gij . in our model system we find that  = 15 , a silencing of more than an order of magnitude ( fig . furthermore , the longer is the distance dij between two nodes , the larger is the silencing ( fig . as an illustration , consider a linear cascade in which changes in any node result in a finite response gij by all other nodes ( fig . equation ( 5 ) silences all indirect responses , while leaving the response of direct links effectively unchanged , offering a discriminative measure that enables a perfect reconstruction of the original network . to test the predictive power of ( 5 ) on real data we used the e. coli datasets distributed by the dream5 network inference challenge . the input data include a compendium of microarray experiments measuring the expression levels of 4,511 e. coli genes ( 141 of which are known transcription factors ) under 805 different experimental conditions ( supplementary note s.iv.1 ) . we constructed three separate global response matrices gij between the 141 transcription factors and their 4,511 potential target genes , based on ( i ) pearson correlations ; ( ii ) spearman rank correlations ; and ( iii ) mutual information , which are three commonly used methods for link detection ( supplementary note s.iv.3 ) . from each of the three gij matrices we obtained sij via ( 5 ) , and compared the performance of gij with the pertinent sij . to validate our predictions we relied on the gold standard used in the dream5 challenge , consisting of 2,066 established gene regulatory interactions . measuring auroc from gij and sij , we find an improvement of 56% for pearson correlations ( fig . we further tested the discrimination ratio ,  , and the silencing ,  , for each of these methods , finding that indirect correlations are subject to an average of two - fold silencing in the transition from gij to sij ( fig . 3d ) . silencing is especially crucial in the presence of the cascade and co - regulation motifs shown in figures 3e - f , where most inference methods indicate a spurious link between x and y owing to the indirect correlation mediated by node i. indeed , the transformation ( 5 ) silences these indirect correlations by a factor of three or more for pearson and spearman correlations and by a smaller factor for mutual information , overcoming one of the most common hurdles of inference methods , which tend to over - represent triadic motifs . as all experimental data is subject to noise , the global response matrix , gij , is characterized by some degree of uncertainty . to test the performance of our methodology in the presence of noise , we repeated the numerical experiment of figure 2 , this time adding gaussian noise to gij , which allows us to calculate silencing as a function of increasing the signal to noise ratio  ( fig . 4 ) . as expected , silencing is unaffected by small values of  , so that  features a plateau below   0.1 . for large  , silencing decays as  ~  , demonstrating that the performance of the method decreases slowly with increasing the signal to noise ratio . indeed , as opposed to a rapid exponential decay , the observed slower power - law dependence indicates that the method is rather tolerant against noise . silencing is lost only when the noise reaches the critical level c  0.75 , when the signal is almost completely overridden by noise , leading to  = 1 ( supplementary note s.v.1 ) . they represent the fact that in most cases we are unable to read the states of all nodes in the system . to illustrate the effect of the hidden nodes on the performance of the silencing method , we consider the case of a simple cascade i  k  j , where the intermediate node k is hidden . in this scenario , ( 5 ) will not be able to silence the indirect i  j link because in the observable system the gij term can not be attributed to any indirect path . hence , absent any other information about the system , it is mathematically impossible to infer the indirectness of gij , as the removal of k isolated i from j. this touches upon the fundamental mechanism of silencing : as illustrated in figure 1 ( and supplementary note s.i.2 ) the silencing transformation ( 5 ) exploits the flow of information through indirect paths . consequently , if as a result of hidden nodes the network fragments into several components such that the node pair i and j become isolated from each other , then all indirect paths between them became hidden and the pertinent gij term will not be silenced ( fig . is expected to fail only when the network breaks into many isolated components so that most node pairs become isolated . fortunately , a fundamental property of complex networks is that with average degree k > > 1 , one needs to remove a large fraction of the nodes to fragment the underlying giant connected component . therefore we can build on percolation theory , which allows us to analytically predict how the size of the largest connected component changes with the random removal of a certain fraction of nodes . the calculation shows that silencing is maintained as long as the fraction of hidden nodes is smaller than ( 7)c1k , where =2ln(2 + 2)1.7 ( supplementary note s.v.2 ) . this equation indicates that for large k the method will be reliable even if a large fraction of the nodes are hidden . to test this prediction , we revisited the numerically obtained gij analyzed in figure 2 and measured the degree of silencing after randomly removing an increasing fraction of nodes . in each case we find that , as predicted , the degree of silencing is driven mainly by  , approaching   1 ( no silencing ) when   1 , namely when the isolated pairs begin to dominate the network ( fig . ( 7 ) predicts c  0.57 , i.e. the method will fail only when almost 60% of the nodes are hidden . note that for biological networks k is expected to be in the range of k  10 , predicting c  0.8 . namely , one needs to lose access to 80% of the nodes for silencing to lose its effectiveness . to extract sij from the experimentally accessible gij , we formally link ( 1 ) and ( 2 ) via ( 3){dxidxi=1dxidxj=k=1nxixkdxkdxjij}. 
 equation ( 3 ) is exact and the sum accounts for all network paths connecting i and j ( supplementary note s.i.1 - 2 ) . it is of limited use , however , as it requires us to solve n coupled algebraic equations . in supplementary note s.i.1 we show that ( 3 ) can be reformulated as ( 4)s=(gi+d(sg))g1 , where i is the identity matrix and d(m ) sets the off - diagonal terms of m to zero . to obtain an approximate solution for s we use that fact that typically , perturbations decay rapidly as they propagate through the network , so that the response observed between two nodes is dominated by the shortest path between them . this allows us to approximate d(sg ) with d((gi)g ) ( supplementary note s.i.3 ) , obtaining ( 5)s=(gi+d((gi)g))g1 . equation ( 5 ) , our main result , provides sij from the experimentally accessible gij . it achieves this through a ' silencing effect , in which direct response terms are preserved , while indirect responses are silenced . to understand this consider a specific term in gij , documenting the response of node i to j 's perturbation . ( 3 ) , this response is a consequence of all direct and indirect paths leading from j to i. as we document below , the transformation ( 5 ) detects the indirect paths and silences them , maintaining only the contribution of the direct paths ( fig . to demonstrate the predictive power of ( 5 ) , we implemented michaelis - menten dynamics on a model network ( supplementary note s.iii ) , as commonly used to model generegulation . we obtained gij by perturbing the activity of each node and then calculated sij using ( 5 ) . figure 2a shows the gij and sij terms associated with interacting ( green ) and non - interacting ( orange ) node pairs . although gij is higher for direct interactions , the overlap between the orange and the green symbols indicates a lack of a clear threshold q that separates direct and indirect interactions . in contrast , sij displays a clear separation between direct and indirect interactions , accurately predicting each direct link . 2b , red ) has an area of auroc = 0.91 , reflecting inherent limitations in separating direct from indirect interactions based on gij only . in contrast for sij we obtain auroc = 0.997 ( blue ) , where the true positive rate ( tpr ) reaches 100% with a false positive rate ( fpr ) of less than 10 . also , although for gij precision increases gradually with the threshold q ( fig . hence , in our well controlled model system effectively any non - zero sij corresponds to a direct link . the performance of ( 5 ) is due to the silencing effect : it leaves gij unchanged if i and j are linked , while it systematically lowers all gij not rooted in a direct interaction . to quantify this effect we measured the discrimination ratio g = gijdir/gijindir ( s = sijdir/sijindir ) which captures the ratio between gij ( sij ) terms associated with direct links and those associated with indirect links ( fig . we find that sij is much more discriminative than gij owing to its silencing of indirect responses . to quantify this effect we measure the silencing ( 6)=sg , which captures the increased power of sij to discriminate between direct and indirect links compared to gij . in our model system we find that  = 15 , a silencing of more than an order of magnitude ( fig . furthermore , the longer is the distance dij between two nodes , the larger is the silencing ( fig . consider a linear cascade in which changes in any node result in a finite response gij by all other nodes ( fig . equation ( 5 ) silences all indirect responses , while leaving the response of direct links effectively unchanged , offering a discriminative measure that enables a perfect reconstruction of the original network . to test the predictive power of ( 5 ) on real data we used the e. coli datasets distributed by the dream5 network inference challenge . the input data include a compendium of microarray experiments measuring the expression levels of 4,511 e. coli genes ( 141 of which are known transcription factors ) under 805 different experimental conditions ( supplementary note s.iv.1 ) . we constructed three separate global response matrices gij between the 141 transcription factors and their 4,511 potential target genes , based on ( i ) pearson correlations ; ( ii ) spearman rank correlations ; and ( iii ) mutual information , which are three commonly used methods for link detection ( supplementary note s.iv.3 ) . from each of the three gij matrices we obtained sij via ( 5 ) , and compared the performance of gij with the pertinent sij . to validate our predictions we relied on the gold standard used in the dream5 challenge , consisting of 2,066 established gene regulatory interactions . measuring auroc from gij and sij , we find an improvement of 56% for pearson correlations ( fig . we further tested the discrimination ratio ,  , and the silencing ,  , for each of these methods , finding that indirect correlations are subject to an average of two - fold silencing in the transition from gij to sij ( fig . 3d ) . silencing is especially crucial in the presence of the cascade and co - regulation motifs shown in figures 3e - f , where most inference methods indicate a spurious link between x and y owing to the indirect correlation mediated by node i. indeed , the transformation ( 5 ) silences these indirect correlations by a factor of three or more for pearson and spearman correlations and by a smaller factor for mutual information , overcoming one of the most common hurdles of inference methods , which tend to over - represent triadic motifs . as all experimental data is subject to noise , the global response matrix , gij , is characterized by some degree of uncertainty . to test the performance of our methodology in the presence of noise , we repeated the numerical experiment of figure 2 , this time adding gaussian noise to gij , which allows us to calculate silencing as a function of increasing the signal to noise ratio  ( fig . 4 ) . as expected , silencing is unaffected by small values of  , so that  features a plateau below   0.1 . for large  , silencing decays as  ~  , demonstrating that the performance of the method decreases slowly with increasing the signal to noise ratio . indeed , as opposed to a rapid exponential decay , the observed slower power - law dependence indicates that the method is rather tolerant against noise . silencing is lost only when the noise reaches the critical level c  0.75 , when the signal is almost completely overridden by noise , leading to  = 1 ( supplementary note s.v.1 ) . they represent the fact that in most cases we are unable to read the states of all nodes in the system . to illustrate the effect of the hidden nodes on the performance of the silencing method , we consider the case of a simple cascade i  k  j , where the intermediate node k is hidden . in this scenario , ( 5 ) will not be able to silence the indirect i  j link because in the observable system the gij term can not be attributed to any indirect path . hence , absent any other information about the system , it is mathematically impossible to infer the indirectness of gij , as the removal of k isolated i from j. this touches upon the fundamental mechanism of silencing : as illustrated in figure 1 ( and supplementary note s.i.2 ) the silencing transformation ( 5 ) exploits the flow of information through indirect paths . consequently , if as a result of hidden nodes the network fragments into several components such that the node pair i and j become isolated from each other , then all indirect paths between them became hidden and the pertinent gij term will not be silenced ( fig . is expected to fail only when the network breaks into many isolated components so that most node pairs become isolated . > > 1 , one needs to remove a large fraction of the nodes to fragment the underlying giant connected component . therefore we can build on percolation theory , which allows us to analytically predict how the size of the largest connected component changes with the random removal of a certain fraction of nodes . the calculation shows that silencing is maintained as long as the fraction of hidden nodes is smaller than ( 7)c1k , where =2ln(2 + 2)1.7 ( supplementary note s.v.2 ) . this equation indicates that for large k the method will be reliable even if a large fraction of the nodes are hidden . to test this prediction , we revisited the numerically obtained gij analyzed in figure 2 and measured the degree of silencing after randomly removing an increasing fraction of nodes . in each case we find that , as predicted , the degree of silencing is driven mainly by  , approaching   1 ( no silencing ) when   1 , namely when the isolated pairs begin to dominate the network ( fig . ( 7 ) predicts c  0.57 , i.e. the method will fail only when almost 60% of the nodes are hidden . note that for biological networks k is expected to be in the range of k  10 , predicting c  0.8 . namely , one needs to lose access to 80% of the nodes for silencing to lose its effectiveness . ( 5 ) is scalable and requires no assumptions about the network topology . by silencing indirect effects , it turns the raw correlation data into a predictive sij matrix , dominated by direct interactions . it is especially suited to treat perturbation data , such as genetic perturbation experiments , in which case gij describes the response of all genes ( dxi ) as a consequence of the perturbation of the source gene ( dxj ) . in practice , however , gij could be the result of a broader set of experimental realizations where other measures are used to evaluate the association between nodes , typically statistical measures such as pearson or spearman correlation coefficients . still , our empirical results ( fig . 3 ) clearly show that the transformation ( 5 ) successfully applies to these empirically accessible measures as well . the method 's broad applicability is rooted in the fact that it does not depend on the value of each specific term in gij , but rather on the global relationships between them . indeed , the global structure of gij reflects the patterns of propagation of the perturbations along the network . equation ( 5 ) helps uncover these paths from the raw data , disentangling the direct from the indirect effects . these patterns of information flow are inherent to the underlying network structure , and should not depend on the specific experimental realization of ( 1 ) . for instance , a cascade i  j  k will be characterized by a decreasing correlation propagating along the arrows , a large correlation between i and j and a weaker one between i and k. although the magnitude of these correlations might depend on the size or the form of i 's perturbation as well as on the statistical measure we used to evaluate them , the decay pattern required to infer the structure of the cascade is an inherent property of the network flow and can be successfully detected by the silencing method ( supplementary note s.i.4 ) . hence it is expected to apply under rather general conditions . however , as equation ( 5 ) indicates it requires that the input matrix , gij , is invertible . for instance in the empirical results of figure 3a we constructed gij from pearson correlations , using the states of 4,511 nodes measured under 805 experimental conditions . in general , if the number of experimental conditions is smaller than the number of nodes the resulting pearson correlation matrix may be singular . in this case additional processing will be required before ( 5 ) could be applied . here , following the dream5 protocol , we only focused on the correlations between the 141 known transcription factors and the rest of the nodes , which lead to an invertible gij ( supplementary note s.iv ) . other means to ensure gij 's invertibility are discussed in supplementary note s.iv.4 . isolating indirect effects in correlation data , a fundamental challenge of network inference , in contrast , the success of the silencing method is rooted in its exploitation the global network topology : it relies on the fundamental principles of network structure and dynamics to identify and silence the effects of indirect paths . the ability to extract sij from gij could also have implications for our understanding of network dynamics . indeed , gij is a global network measure , as its magnitude is determined by the numerous indirect paths connecting i and j. hence , for a given dynamics , the gij matrix will take a different form depending on the network topology , making it a poor predictor of the system 's dynamics . by eliminating indirect effects , sij measures the effect gene i would have on gene j it thus helps us quantify the dynamical mechanism that governs individual pairwise interactions , avoiding the convolution of dynamical and topological effects present in experimental data . the structure of gij reflects the microscopic mechanisms that govern the pairwise interactions , e.g. genetic regulation and biochemical processes . it is difficult , however to extract this information from gij since its terms are a convolution of many interactions , reflecting the many paths leading from i to j. the transition to sij , via ( 5 ) , allows us to treat each isolated interaction on its own , providing a direct observation into the microscopic interaction mechanism . direct application of this fact could be the derivation of a rate equation that governs the system 's dynamics from gij , as well as predicting the universality class and the scaling laws governing the system 's response to perturbations . hence ( 5 ) helps translate the ever - growing amount of data on global correlations into valuable local information .
predicting physical and functional links between cellular components is a fundamental challenge of biology and network science . 
 yet , correlations , a ubiquitous input for biological link prediction , are affected by both direct and indirect effects , confounding our ability to identify true pairwise interactions . here 
 we exploit the fundamental properties of dynamical correlations in networks to develop a method to silence indirect effects . 
 the method receives as input the observed correlations between node pairs and uses a matrix transformation to turn the correlation matrix into a highly discriminative silenced matrix , which enhances only the terms associated with direct causal links . achieving perfect accuracy in model systems , we test the method against empirical data collected for the escherichia coli regulatory interaction network , showing that it improves on the best preforming link prediction methods . 
 overall the silencing methodology helps translate the abundant correlation data into valuable local information , with applications ranging from link prediction to inferring the dynamical mechanisms governing biological networks .