Network deconvolution as a general method to distinguish direct dependencies in networks

Soheil Feizi, Daniel Marbach, Muriel Médard, Manolis Kellis
2013 Nature Biotechnology  
a n a ly s i s Recognizing direct relationships between variables connected in a network is a pervasive problem in biological, social and information sciences as correlation-based networks contain numerous indirect relationships. Here we present a general method for inferring direct effects from an observed correlation matrix containing both direct and indirect effects. We formulate the problem as the inverse of network convolution, and introduce an algorithm that removes the combined effect of
more » ... all indirect paths of arbitrary length in a closed-form solution by exploiting eigen-decomposition and infinite-series sums. We demonstrate the effectiveness of our approach in several network applications: distinguishing direct targets in gene expression regulatory networks; recognizing directly interacting amino-acid residues for protein structure prediction from sequence alignments; and distinguishing strong collaborations in co-authorship social networks using connectivity information alone. In addition to its theoretical impact as a foundational graph theoretic tool, our results suggest network deconvolution is widely applicable for computing direct dependencies in network science across diverse disciplines. Network science has been widely adopted in recent years in diverse settings, including molecular and cell biology 1 , social sciences 2 , information science 3 , document mining 4 and other data mining applications. Networks provide an efficient representation for variable interdependencies, represented as weighted edges between pairs of nodes, with the edge weight typically corresponding to the confidence or the strength of a given relationship. Given a set of observations relating the values that elements of the network take in different conditions, a network structure is typically inferred by computing the pairwise correlation, mutual information or other similarity metrics between each pair of nodes. The resulting edges include numerous indirect dependencies owing to transitive effects of correlations. For example, if there is a strong dependency between nodes 1 and 2, and between nodes 2 and 3 in the true (direct) network, high correlations will also be visible between nodes 1 and 3 in the observed (direct and indirect) network, thus inferring an edge from node 1 to node 3, even though there is no direct information flow between them (Fig. 1a) . Moreover, even if a true relationship exists between a pair of nodes, its strength may be over-estimated owing to additional indirect relationships, and distinguishing the convolved direct and indirect contributions is a daunting task. As the size of networks increases, a very large number of indirect edges may be due to second-order, third-order and higher-order interactions, resulting in diffusion of the information contained in the direct network, and leading to inaccurate network structures and network weights in many applications 1,5-11 . Several approaches have been proposed to infer direct dependencies among variables in a network. For example, partial correlations have been used to characterize conditional relationships among small sets of variables 12-14 , and probabilistic approaches, such as maximum entropy models, have been used to identify informative network edges 10, 15, 16 . Other works use graphical models and message-passing algorithms to characterize direct information flows in a network 17,18 , or variations of Granger causality 19 to capture the dynamic relationships among variables 20-22 . Alternative approaches formulated the problem of separating direct from indirect dependencies as a general feature-selection problem 23-25 , using Bayesian networks 26-28 , or using an information-theoretic approach to eliminate indirect information flow in the network 29 . These methods are limited to relatively low-order interaction terms 29 , or are computationally very expensive 12-14 , or are designed for specific applications 10, [15] [16] [17] 30, 31 , thus limiting their applicability. In this paper, we formulate the problem of network deconvolution in a graph-theoretic framework. Our goal is a systematic method for inferring the direct dependencies in a network, corresponding to true interactions, and removing the effects of transitive relationships that result from indirect effects. When the matrix of direct dependencies is known, all transitive relationships can be computed by summing this direct matrix and all its powers, corresponding to the transitive closure of a weighted adjacency matrix, which convolves all direct and indirect paths at all lengths (Fig. 1b) . Given an observed matrix of correlations that contains both direct and indirect effects, our task is to recover the original direct matrix that gave rise to the observed matrix. For a weighted network where edge weights represent the confidence, mutual information or correlation strength relating two elements in the network, the inverse problem seeks to recognize the fraction of the weight of each edge attributable to direct versus indirect contributions, rather than to keep or remove unit-weight edges. This inverse problem is dramatically harder than the forward problem of transitive closure, as the original matrix is not known.
doi:10.1038/nbt.2635 pmid:23851448 pmcid:PMC3773370 fatcat:eq6vz6yc4vbl7kvwjlkvza2asy