Genome-wide associations studies (GWAS), in which thousands to millions of SNPs across the human genome are tested for associations with disease phenotypes, have emerged as a particularly promising approach for drawing causal inferences between traits and genetic variation [2,3,7,8]. However, although GWAS have uncovered numerous disease susceptibility loci [3,8,9], the majority of them have had relatively subtle individual associations with disease risk. The success of GWAS analyzed only for individual SNP effects largely depends on fundamental assumptions about a lack of genetic complexity and a simple single-gene architecture of diseases, and becomes greatly compromised when gene-environment or gene-gene interactions modify the relationship between genotypes and phenotypes [10-13].

The genetic architecture of common human diseases is, in fact, characterized in part by interactions between genes, i.e., epistasis [13-19]. Accordingly, the focus of recent research has shifted from identifying single locus susceptibility [2,7] to quantifying interaction effects between multiple candidate loci throughout the human genome [13,16,20,21]. However, the study of epistasis faces an initial challenge arising from the existence of fundamental differences between the concepts of biological and statistical interaction (e.g. [21]). These differences imply that statistical epistasis, defined at the population level as the non-additive mathematical relationship among multiple genetic variants, cannot be literally translated into biological epistasis, which is the physical interaction among two or more molecules at the cellular level of an organism, and vice-versa [17]. Moreover, detecting gene-gene interactions and accounting for them in GWAS further represents a statistical and computational challenge [12,13,20,22]. The statistical challenge results from the prohibitive amount of data necessary to support the huge number of hypotheses involved in modeling interactions, even when considering only pairwise interactions [3,11]. The computational challenge, in turn, arises from the necessity to exhaustively evaluate all possible combinations of SNPs, which becomes infeasible when interactions involve more than two SNPs: the computational complexity, which is in the quadratic order for pairwise interactions, increases exponentially with higher-order interactions, rendering any exhaustive assessment impossible [12,13,21].

The necessity to overcome these difficulties calls for efficient tools to detect genetic interactions [2,7,23]. Methods such as machine learning [24-26] and dimensionality reduction [27,28] have recently proven useful in detecting influential interactions. However, these approaches are aimed at identifying best models consisting of several SNPs and thus ignore the broader gene-gene interaction landscape.

A particularly intuitive approach to explore the genetic architecture of common human diseases and to identify genetic interactions is to use networks. A network is generally defined as a collection of vertices joined in pairs by edges and is a powerful tool to represent and study complex systems [29,30]. In biological systems, for instance, networks can be used to characterize interactions at all levels of organization, from the molecular level with metabolic [31,32], protein-protein interaction [33], and genetic regulatory networks [34], to the macroscopic level with food webs [35].

Networks allow for a structured representation of a collection of entities and their relationships, which provides a well-suited framework for the study of epistasis. The use of networks does not resolve the dimensionality problems inherent in exploring high-order interactions amongst multiple SNPs. An intuitive solution that has previously proven helpful is to filter out the considerable noise masking the useful genotypes and to reduce the search space to a subset of high-susceptibility SNPs before constructing a network of genetic interactions.

An example of such a sequential approach is the work of McKinney et al. [36], who developed a genetic-association interaction network to visualize and interpret synergetic interactions between pairs of SNPs. Loci were initially chosen based on the strength of their main effects. Although useful, purging databases for irrelevant genetic variants and preliminarily selecting high-susceptibility SNPs inevitably comes at the risk of discarding loci comprised in significant higher order interactions. Hence, alternative solutions for reducing the space of possible interactions in GWAS are needed.

In the present study, we propose to infer genetic interaction networks that are not dependent on statistical main effects. We first rank all possible pairwise interactions between SNPs according to their relative strength and subsequently build and analyze statistical epistasis networks including only those interactions whose strength exceeds a given threshold. Hence, the approach we apply distinguishes itself from existing ones in the following ways: 1) We qualify the strength of all pairwise interactions identifiable in the complete data set rather than a subset of high main-effect SNPs; 2) We organize our genetic network around the strongest pairwise interactions rather than around the strongest main effects; 3) We analyze network topologies to systematically identify the network that best captures the genetic architecture inherent in the data; 4) In contrast to many existing techniques that aim at identifying a classification model consisting of a subset of susceptibility SNPs, our epistasis network captures a broader landscape of gene-gene interactions through exhaustively enumerating all possible pairwise interactions.

The network G_0.013 allows for some interesting observations about the structure of the pairwise interaction space of the genetic data. First, SNPs aggregate to form connected components, which may indicate that multiple SNPs jointly modify disease outcome. In G_0.013, SNPs are grouped into 79 connected components of size ranging from 2 to 39. These connected components show various structural patterns, also known as motifs, including lines, crosses, and stars. The largest connected component has a tree-like structure. This may imply the existence of unique interaction patterns among groups of SNPs.

Second, the network has an approximately scale-free topology and an ensemble of particularly high-degree vertices, which suggests that it may be exceptionally robust. Scale-free networks permeate natural and social sciences [47-49]. The most well-known scale-free networks are the backbone of the Internet and social networks. In biology, scale-free topologies have been found in metabolic networks [31], protein-protein interaction networks [33], and gene-regulatory networks [34]. Those various scale-free networks share an intriguing property: the value of γ in the degree distributions p(d) = c × d ^-γ mostly satisfies 2 ≤ γ ≤ 3 [47], which is also the case for G_0.013 (γ = 2.01). As more scale-free networks are being discovered in a variety of fields, a question remains: how can systems as fundamentally different as the cell and the Internet have a similar architecture and obey the same laws [47]? Scale-free networks typically have many vertices with low degrees and a few vertices with high degrees, also known as hubs [30]. This essentially differentiates scale-free networks from random networks where the majority of vertices have average degrees. The probability p(d) of degree d in the Poisson distribution decreases exponentially as d increases, and thus random networks are very unlikely to have hubs with degrees much larger than the average. The existence of hubs in a scale-free network implies strong robustness against failures. Because random vertex removal is very unlikely to affect hubs, the connectivity of the network most likely remains intact. In biological networks, this robustness translates into the resilience of organisms to intrinsic and environmental perturbations. For instance, in protein-protein interaction networks [33], most proteins interact with only one or two other proteins but a few are able to interact to a large number. Such hub proteins are rarely affected by mutations and organisms can remain functional under most perturbations. The simultaneous emergence of scale-free topologies in many biological networks suggests that evolution has favored such a structure in natural systems. Moreover, it suggests that the robustness of natural systems does not only result from inherent genetic redundancy but also, and maybe more importantly, from the topological organization of entities and interactions [33]. Although our epistasis network is developed based on statistical rather than on real bio-chemical interactions, it is interesting to observe similar topologies between biological and statistical networks.

Third, the existence of main effects does not necessarily correlate with the occurrence of interactions. This, in turn, suggests that many current main-effect-prioritized methods might have overlooked SNPs contributing to the disease susceptibility through their interactions with other SNPs rather than through their main effects. As shown in the graph, large main-effect SNPs do not necessarily associate with strong pairwise interactions or interact with many other SNPs. Instead, SNPs involved in potential important pairwise interactions, such as those located on the central path of the largest connected component, often have relatively small main effects.

The statistical epistasis network approach has many advantages. 1) Networks allow for efficiently visualizing both main and epistatic effects and how they interplay. 2) Networks serve as a very intuitive tool to study pairwise interactions and to characterize the entire epistatic interaction space. Moreover, they may also help identify higher-order interactions by grouping SNPs into connected components. High-order epistasis does not necessarily require detectable pairwise interactions between SNPs. However, given that current computational power allows only for exhaustively enumerating pairwise interactions in moderate-size data sets, pairwise interaction networks may serve as a useful guide to explore higher-order epistasis among SNPs that exhibit lower-order interactions. 3) Our network model is assembled using the entire set of available SNPs without limiting ourselves to only high main-effect ones. This reduces the risk of overlooking candidate SNPs that are involved in important interactions but with low main effects. 4) Network topological analyses are used to systematically determine the best network that captures the genetic architecture of a data set. 5) Networks, along with graph theory, are well-developed fields, and many advanced techniques and analytical tools are likely to benefit future network-based epistasis studies. In particular, additional topological properties such as motif distribution and network diameter [30,42] are worth investigating.

Among the limitations of this approach is that it is still under development and no user-friendly interface is available yet. Different data sets may require different analytical tools and a fully automated analysis software may therefore not be appropriate. Moreover, since the approach aims at highlighting pairs of SNPs with strong pairwise interactions, it is likely to overlook SNPs that are only involved in higher-order interactions. As mentioned previously, strong three- or higher-order interactions may exist despite the absence of pairwise interactions.