Why Use Two Hub Scores?
A gene can be considered important to a co-expression module for more than one reason. Its expression may closely track the module’s overall expression pattern, or it may have strong network connections to many genes assigned to that module. These ideas are related, but they are not identical. Module membership summarizes alignment with a module eigengene; intramodular connectivity summarizes adjacency to other genes in the module.
The introductory material on module eigengenes establishes the eigengene as a summary of a module’s expression pattern. Here, assume a module assignment and its eigengene have already been obtained. The goal is to calculate and interpret gene-level scores, not to re-estimate modules or assess their association with phenotypes.
Definitions and Conditions
Suppose expression has been measured for the same \(n\) samples, and let \(M\) denote a module containing \(m\) genes. Write \(x_g=(x_{g1},\ldots,x_{gn})\) for the expression measurements of gene \(g\), and \(e_M=(e_{M1},\ldots,e_{Mn})\) for the corresponding module eigengene scores. The two vectors must refer to the same samples in the same order.
The sign needs care. A principal-component eigengene can be multiplied by \(-1\) without changing the principal component it represents. If the eigengene is reoriented, every \(kME\) sign in that module also reverses. Thus, the magnitude \(|kME_g|\) measures alignment regardless of this arbitrary orientation. When an eigengene orientation has been fixed by a stated convention, its sign can also describe direction relative to that orientation.
These definitions require different inputs. A \(kME\) calculation requires the gene expression profile and module eigengene scores. Intramodular connectivity requires the adjacency weights and module assignments. For a fair within-module comparison, use the same network construction and module definition for all genes being ranked. Connectivity also depends on module size: larger modules offer more possible connections, so raw sums need not be comparable across modules. A normalized score, such as division by \(m-1\), can aid comparison, but it is a different reported quantity and should be labelled as such.
Calculating Module Membership
The Pearson correlation is calculated after centering each vector. If \(x_{gi}\) and \(e_{Mi}\) are the measurements for sample \(i\), then
The formula is defined only when both vectors have nonzero variation. It describes linear co-variation across the observed samples. It does not establish that one gene regulates another, nor does a large correlation by itself establish biological importance.
Worked Example: Computing a Module Membership Correlation
Consider six hypothetical samples. Suppose a module eigengene has values \(e_M=(-2,-1,0,0,1,2)\), and gene \(g\) has expression values \(x_g=(-1,0,1,-1,0,1)\). Both vectors sum to zero, so their means are zero. Their centered cross-product is
The sums of squares are \(\sum_i e_{Mi}^2=10\) and \(\sum_i x_{gi}^2=4\). Therefore,
In this hypothetical module, gene \(g\) has a positive, moderately strong linear association with the eigengene across the six samples. The score is unitless. It is not a percentage of variation explained, and it does not indicate how many direct network connections the gene has.
Calculating Intramodular Connectivity
Adjacency weights depend on the network definition. In a weighted network they commonly lie between zero and one, with larger weights representing stronger network links. The precise transformation from expression similarity to adjacency depends on the chosen signed, unsigned, or signed-hybrid construction. Once that construction and module assignment are fixed, intramodular connectivity is a row sum restricted to the other genes in the same module.
Worked Example: Summing Within-Module Adjacencies
Suppose a hypothetical four-gene module contains genes A, B, C, and D. In a symmetric weighted adjacency matrix with no self-edges, gene A has weights \(a_{AB}=0.80\), \(a_{AC}=0.60\), and \(a_{AD}=0.20\). Its within-module connectivity is
If the other module genes have within-module sums \(1.45\), \(1.10\), and \(0.85\), respectively, A ranks first by raw intramodular connectivity in this module. For this four-gene module, the average adjacency from A to another module member is \(1.60/(4-1)\approx0.5333\). That average is a rescaled summary, not the raw connectivity defined above.
Suppose A also has adjacency weights to genes outside this module. Those weights do not enter this calculation. They would matter for a network-wide strength calculation, which sums the gene’s weights across the full network.
Why the Scores Can Differ
Module membership and intramodular connectivity both describe a gene in relation to a module, but they summarize distinct features. The eigengene is a multivariate summary of expression within the module. Correlation with it measures how closely a gene follows that summary pattern. Connectivity instead sums edge weights from a gene to its module peers. No general identity requires the two rankings to match: a gene can follow the dominant expression pattern closely without having the largest sum of pairwise adjacency weights, and a gene can have many strong local links without being the gene most correlated with the eigengene.
A useful justification for the first score follows from the principal-component construction. Let \(Z\) be the centered, possibly standardized, sample-by-gene expression matrix for a module. If \(v_1\) is the first principal-component loading vector, the module eigengene scores are proportional to \(Zv_1\). For a gene represented by column \(z_g\), its \(kME\) is therefore the correlation between \(z_g\) and the linear combination of module columns \(Zv_1\). This makes \(kME\) an alignment score with a module-wide expression summary. It is not, in general, a direct-edge count: the correlation uses the samples and eigengene, while connectivity uses the network’s pairwise adjacency weights.
Worked Example: Interpreting Different Hub Rankings
Suppose a hypothetical module has four genes with the following scores, all calculated from the same expression data, eigengene orientation, adjacency definition, and module assignment.
| Gene | \(kME\) | \(k_{\mathrm{within}}\) |
|---|---|---|
| A | 0.94 | 1.20 |
| B | 0.81 | 1.75 |
| C | 0.45 | 1.05 |
| D | -0.88 | 0.60 |
By signed \(kME\), A ranks first because \(0.94\) is the largest value. If ranking by absolute module membership, A still ranks first, followed by D, B, and C, since \(0.94>0.88>0.81>0.45\). By intramodular connectivity, B ranks first because \(1.75\) is the largest within-module sum. The rankings are not contradictory: A is most aligned with the eigengene, whereas B has the greatest total adjacency to its module peers.
D illustrates why the ranking rule matters. Its negative \(kME\) indicates inverse association with the chosen eigengene orientation, but its magnitude is \(0.88\), suggesting strong alignment in the opposite direction. Whether that makes D a candidate hub depends on the scientific question and the network convention; it should not be silently treated as equivalent to a positively aligned gene. The score alone does not reveal why the inverse pattern occurs.
Comparison, Diagnostics, and Limitations
The main alternative is not a competing formula for the same property, but a different summary of network position. Degree counts a gene’s links in an unweighted network; strength sums adjacency weights across the specified set of genes. Intramodular connectivity is a module-restricted strength. Network-wide strength instead includes links to genes outside the module. Other centrality measures, such as betweenness, quantify different structural roles, for example how often a gene lies on shortest paths. A gene can rank highly by one measure and not another because the measures ask different questions.
| Score | What it uses | What a high score indicates |
|---|---|---|
| Module membership, \(kME\) | Gene expression and module eigengene across samples | Strong linear alignment with the module summary; use the sign or magnitude according to the ranking question |
| Intramodular connectivity | Adjacency weights to other genes in the assigned module | Large total within-module adjacency |
| Network-wide strength | Adjacency weights to all other network genes | Large total adjacency in the full network, including links outside the module |
Before interpreting scores, verify that sample ordering matches between expression and eigengene vectors, that neither vector is constant, and that missing values have not caused correlations to use inconsistent sample subsets. Check that the adjacency matrix is symmetric when the network is intended to be undirected, that self-edges are excluded, and that connectivity is summed over the correct module members. Compare scores only under clearly stated preprocessing and network-construction choices: changing the expression transformation, correlation measure, adjacency type, or threshold can change both scores and rankings.
Treat a ranking as a description under a specified analysis, not as proof that a gene is a causal regulator, a biological “master switch,” or a reproducible hub. Correlation is sensitive to outliers and measures linear association; connectivity inherits the choices used to construct the adjacency matrix. Modules may also contain genes with different expression patterns, and a single eigengene may not represent every subgroup equally well. Very small modules offer few possible within-module links, while raw connectivity values from modules of different sizes are not directly comparable without attention to scale. These scores are useful for prioritizing genes for further investigation, not for replacing validation or biological interpretation.
Common Mistakes
- Equating the scores. A high \(kME\) is not a high adjacency sum. Report which quantity was calculated and what it measures.
- Ignoring eigengene orientation. A negative \(kME\) is not automatically weak membership. Check the sign convention and, if appropriate, compare \(|kME|\).
- Including external edges in an intramodular sum. Sum only over other genes assigned to the same module; use a separately labelled network-wide score for all genes.
- Comparing raw strengths across module sizes without qualification. A larger module permits more terms in the sum. Report module size or a clearly defined normalized quantity.
- Calling a hub causal. These are association and network-position summaries. They do not establish regulatory direction or causation.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A gene’s \(kME\) is \(-0.90\). What does the sign indicate, and what does the magnitude indicate?
- A gene has within-module adjacency weights \(0.25\), \(0.40\), and \(0.15\). Calculate its intramodular connectivity.
- Why might the gene with the largest \(kME\) not have the largest intramodular connectivity?
- A gene has strong edges to genes outside its module but weak edges to module members. Should its intramodular connectivity be high? Explain.
- What must be checked before comparing raw intramodular connectivity across two modules of different sizes?
Solutions. (1) The gene is strongly inversely correlated with the eigengene as oriented; \(|kME|=0.90\) indicates strong linear alignment in magnitude. (2) \(0.25+0.40+0.15=0.80\). (3) \(kME\) measures correlation with the eigengene, whereas connectivity sums pairwise adjacency weights; they are distinct summaries. (4) Not necessarily: edges outside the module are excluded from intramodular connectivity. (5) Check module sizes and the scale of the sums; a larger module has more possible terms. If comparing normalized scores, define and label the normalization.