Signed and signed hybrid: what’s the difference?

In a previous post I gave my recommendation to use signed rather unsigned networks. This post will describe the two slightly different formulas that WGCNA offers for building signed networks from a correlation matrix. As a quick reminder, constructing a network really means calculating its adjacency matrix aij. Elements of this matrix encode the connection strengths between nodes of the network and must lie between 0 (unconnected) and 1 (fully connected). The “signed” network adjacency 010-a_ij_signed scales the correlation to lie between 0 and 1, then raises it to a power:

020-a_ij_signed-equation

A salient feature of the signed adjacency is that zero correlation gives rise to a (theoretically) non-zero adjacency (connection strength). Although at first sight counter-intuitive, this can sometimes be desirable, for example in sample networks. However, in most applications to large, high-throughput data, the power β is sufficiently large (usually at least 10-12) that negative and small positive correlations lead to negligibly small adjacencies. The figure below illustrates this point.

signed
Signed adjacency (y-axis) as a function of correlation (x-axis) for two soft thresholding powers β. When the power is sufficiently high (say 12, red line), the adjacency is negligibly small for all negative and small positive correlations.

Unlike the “signed” adjacency, the “signed hybrid” adjacency is exactly zero for all negative (or zero) correlations:

030-signedHybrid-equation

We call this adjacency “hybrid” because it uses a combination (hybrid) of hard and soft thresholding: there is a hard threshold at 0, and soft thresholding above zero. The figure below illustrates the signed hybrid adjacency for soft thresholding powers 1 and 6.

signedHybrid
Signed hybrid adjacency (y-axis) as a function of correlation (x-axis). This adjacency function leads to zero adjacencies for all negative correlations and all choices of β. Choosing β=6 also suppresses low positive correlations.

And which one should you use? In high throughput data, with more variables (genes) than observation (samples), one would typically use a fairly high soft thresholding power β, say β=12 for a signed or β=6 for a signed hybrid network. While the formulas look quite different, the results for these two powers will be extremely similar:

both
Solid red line shows signed adjacency obtained with power β=12, dashed cyan line shows signed hybrid adjacency obtained with power β=6. The two adjacencies are nearly indistinguishable.

Thus, in the end the two signed network variants result in nearly identical networks as long as (1) the signed network uses twice the soft thresholding power of the signed hybrid network, and (2) the power is suitable for analysis of high-throughput, more-variables-than-samples, data: at least 4 for signed hybrid, and at least 8 for signed networks. In other words, use either but remember to double the power for the signed, compared to the signed hybrid.

Signed or unsigned: which network type is preferable?

How should pairs of nodes with strong negative correlations be treated in a correlation network analysis? One option is to consider them connected, just as if the correlation were positive. A network constructed in this way is an unsigned network, because the sign of the correlation does not matter. On the other hand, strongly negatively correlated nodes can also be considered unconnected. This leads to a signed network, so called because the sign of a strong correlation value makes all the difference between the pair of nodes being strongly connected or not connected at all. To avoid any confusion, I want to emphasize that the resulting adjacency matrix (the matrix that contains the connection strengths between nodes) is always non-negative.

Should you use a signed or unsigned network? By and large, I recommend using one of the signed varieties, for two main reasons. First, more often than not, direction does matter: it is important to know where node profiles go up and where they go down, and mixing negatively correlated nodes together necessarily mixes the two directions together. Second, negatively correlated nodes often belong to different categories. For example, in gene expression data, negatively correlated genes tend to come from biologically very different categories. It is true that some pathways or processes involve pairs of genes that are negatively correlated; if there are enough negatively correlated genes, they will form a module on their own and the two modules can then be analyzed together. (For the advanced practitioner, another option is to use the fuzzy module membership measure based on the module eigengene to attach a few strongly negatively correlated genes to a module after the modules have been identified).

By and large does not mean always, and there may be applications in which an unsigned network is preferable. In principle there’s also nothing wrong with carrying out both types of analysis, but working with two related yet distinct analyses of the same data may quickly get confusing and tiring.

For historical reasons (compatibility with old calculations), the defaults in the current implementation of WGCNA R package disregard my own recommendation and imply unsigned networks. To work with signed networks, use arguments type or networkType with value “signed” or “signed hybrid” (more on their difference later) whenever calling a function that in some shape or form constructs a network. If in doubt, help in R is always only a few keystrokes away.

Read on: Two types of signed networks in WGCNA