Detecting Omics Block Dominance

Why Diagnose Dominance?

A fused patient network or kernel is intended to combine information across omics views. But a fused result can be driven mainly by one view: perhaps that view has much larger similarity values, receives a larger fusion weight, or distinguishes patient pairs in a pattern that the other views do not. In that case, the result may look multi-omics while reflecting mostly one source of information.

Dominance is not a single formal property with one universal diagnostic. It is a question about how the fused object is constructed and what it preserves. A useful assessment separates scale dominance (one view has larger numerical similarities), contribution dominance (one view supplies more of the fused values under a stated decomposition), and pattern dominance (the fused patient-pair similarities or rankings resemble one view more than the others). These are related but not interchangeable.

Definition: Let $K^{(v)}$ be the $n \times n$ similarity matrix for omics view $v$, with the same $n$ patients in the same order in every view. A view is dominant under a specified diagnostic when it contributes disproportionately to the fused similarities or when the fused patient-pair pattern more closely reflects that view. Always name the diagnostic: dominance is not established by a single scale summary alone.

For a network, use its edge weights in place of kernel similarities. In either case, compare off-diagonal entries: the diagonal of a kernel often contains self-similarities and is not a patient-to-patient edge. The matrices must refer to the same patients and to comparable notions of similarity. If some views use distances rather than similarities, convert them consistently before making these comparisons.

Scale Summaries and What They Can Show

A first check is to summarize off-diagonal values in each view before fusion. Two simple summaries are the mean off-diagonal similarity and the root mean square (RMS) off-diagonal similarity. For symmetric matrices, it is convenient to list each unordered patient pair once, rather than counting both $(i,j)$ and $(j,i)$.

\bar{K}^{(v)}_{\mathrm{off}}= \frac{1}{\binom{n}{2}}\sum_{i<j}K^{(v)}_{ij}, \qquad \operatorname{RMS}^{(v)}_{\mathrm{off}}= \sqrt{\frac{1}{\binom{n}{2}}\sum_{i<j}\left(K^{(v)}_{ij}\right)^2}.

The mean reflects the overall level of similarity; the RMS also responds to large edge values. Neither alone tells whether a view determines the fused pattern. A high mean can reflect a nearly constant similarity matrix, which provides little differentiation among pairs. Always inspect dispersion or the actual pair values as well. Comparisons of scale are meaningful only when the construction and normalization of the matrices make their values comparable.

Worked Example: Comparing Raw Similarity Scales

Suppose two hypothetical views have three off-diagonal similarities for the same three patient pairs. View A has $0.90, 0.60, 0.30$; view B has $0.60, 0.40, 0.20$. Assume an unweighted additive fusion, so each fused value is the average of the two view values.

The off-diagonal means are $(0.90+0.60+0.30)/3=1.80/3=0.600$ for A and $(0.60+0.40+0.20)/3=1.20/3=0.400$ for B. Their RMS values are $\sqrt{(0.90^2+0.60^2+0.30^2)/3}=\sqrt{1.26/3}=\sqrt{0.42}\approx0.6481$ and $\sqrt{(0.60^2+0.40^2+0.20^2)/3}=\sqrt{0.56/3}\approx0.4320$, respectively.

The fused values are $0.75, 0.50, 0.25$. Under this equal-weight construction, A supplies $1.80/(1.80+1.20)=0.600$, or 60.0%, of the summed off-diagonal edge mass, while B supplies 40.0%. This supports a scale-based description of A as the larger contributor to total edge mass. It does not show that A contains more biological information or that it is more valid. In this example both views have the same ordering of the three pairs, so the scale difference does not mean their patterns disagree.

Contribution Measures for Additive Fusion

For an additive kernel fusion $F=\sum_v a_vK^{(v)}$, where $a_v\geq0$ are specified weights, each matrix $C^{(v)}=a_vK^{(v)}$ is an exact component of the fused matrix. When all off-diagonal similarities are nonnegative and their total is positive, an edge-mass share can be defined as

q_v= \frac{\sum_{i<j}a_vK^{(v)}_{ij}} {\sum_{w}\sum_{i<j}a_wK^{(w)}_{ij}}.

These shares add to one, and $q_v$ describes the fraction of total off-diagonal mass assigned to view $v$ by this particular measure. It is not a fraction of explained biological signal. If entries can be negative, positive and negative values can cancel, so this mass-share interpretation is generally unsuitable; report the signed contributions or use a separately justified measure instead.

Another useful summary is the Frobenius norm of each weighted component, calculated over the off-diagonal entries. A large component norm indicates large weighted deviations in magnitude, but the component norms do not generally add to the norm of the fused matrix. The reason is overlap or cancellation between components.

Justification: For two additive components, the squared Frobenius norm of their sum is $$ \|a_1K^{(1)}+a_2K^{(2)}\|_F^2 =a_1^2\|K^{(1)}\|_F^2+a_2^2\|K^{(2)}\|_F^2 +2a_1a_2\langle K^{(1)},K^{(2)}\rangle_F. $$ The inner product term measures overlap in the entries. It is positive when the components tend to have similar signed patterns and negative when they tend to oppose one another. Thus, comparing component norms alone cannot determine how much each component shapes the fused matrix: the cross-term also matters.

If the matrices are centered before fusion, this same reasoning applies to the centered matrices. Centering can make pattern comparisons more informative by removing overall levels, but it changes what is being compared. State whether summaries use raw, centered, or normalized matrices. A coefficient $a_v$ also cannot be interpreted in isolation from the scale of $K^{(v)}$: multiplying a kernel by a constant changes its contribution unless the fusion procedure compensates for that change.

Worked Example: Reading Contribution Shares

Consider hypothetical, nonnegative off-diagonal similarities for three patient pairs. Two views have values A: $0.90, 0.70, 0.50$ and B: $0.10, 0.20, 0.10$. The fusion is $F=0.5K^{(A)}+0.5K^{(B)}$.

The weighted contributions on the three pairs are A: $0.45, 0.35, 0.25$ and B: $0.05, 0.10, 0.05$. Their summed masses are $1.05$ and $0.20$, respectively. The total fused mass is $1.25$, so $q_A=1.05/1.25=0.840$ and $q_B=0.20/1.25=0.160$. Thus A supplies 84.0% of the fused off-diagonal mass under this diagnostic, despite receiving the same coefficient as B.

The fused edge values are $0.50, 0.45, 0.30$. They retain the ordering in A, but a mass share does not prove that A alone determines every edge or that the two views offer no complementary information. Here B adds $0.05, 0.10, 0.05$ to the respective fused similarities.

Patient-Pair Pattern Comparisons

Scale summaries and contribution shares should be paired with a direct check of patient-pair patterns. For each view and the fused object, form a vector of the $\binom{n}{2}$ off-diagonal entries, using the same order of patient pairs throughout. One descriptive measure is the Pearson correlation between a view’s vector and the fused vector. This measures linear similarity of the centered edge patterns; it is not a test of equality and does not establish that one view caused the fused pattern.

Also compare pairs directly. Identify which pairs are most similar in each view, whether the same pairs remain highly connected after fusion, and whether a view supplies most of the weighted value for particular fused edges. A view can dominate some patient pairs but not others. A single correlation can conceal this, especially when many pairs are tied or when a few extreme similarities control the result.

Worked Example: Checking Which View Drives Particular Pairs

For three hypothetical patients, suppose the pair order is 1–2, 1–3, 2–3. View A has similarities $0.90, 0.70, 0.50$, while view B has $0.10, 0.20, 0.10$. Using equal-weight additive fusion gives $F=0.5K^{(A)}+0.5K^{(B)}$.

For pair 1–2, the weighted terms are $0.5(0.90)=0.45$ from A and $0.5(0.10)=0.05$ from B, giving $F_{12}=0.50$. For pair 1–3 they are $0.35$ and $0.10$, giving $F_{13}=0.45$. For pair 2–3 they are $0.25$ and $0.05$, giving $F_{23}=0.30$.

A supplies the larger weighted term for every pair, and the fused ordering $1\text{–}2 > 1\text{–}3 > 2\text{–}3$ matches A’s ordering. Under both this edgewise contribution check and this rank check, A dominates this small fused network. The conclusion is descriptive and specific to the given weights, similarities, and three pairs; it is not an inference that A is the more biologically meaningful view.

SNF and Kernel Fusion Require Different Diagnostics

In additive kernel fusion, the weighted components sum entry by entry, so edge-mass shares and pairwise weighted terms have a direct algebraic interpretation. Similarity Network Fusion (SNF), by contrast, iteratively updates networks using information from the other views. Its final network is not generally a weighted sum of the original view matrices. Consequently, assigning each final edge a unique additive share from each input view is not justified merely because the inputs were combined.

For an SNF result, compare the input networks and the fused network using their off-diagonal scales and patient-pair patterns, while keeping the interpretation descriptive. Ask whether the fused network’s strongest pairs resemble those in one input, whether its overall edge distribution is closer to one view, and whether another view contributes visibly different pair relationships. The comparison of SNF and kernel fusion is useful here: the fusion rule determines which contribution claims are algebraically valid.

For either method, keep the same patients, pair order, and edge definition across views. If kernel normalization or network construction differs, note the difference before comparing. A diagnostic on differently scaled inputs may flag a construction choice rather than a substantive dominance phenomenon.

Diagnostics, Pitfalls, and Limits

Several warning signs help identify an unreliable dominance conclusion. A much larger off-diagonal mean or RMS suggests a scale imbalance; a high mean combined with very low variation suggests that the view may contribute a broad baseline rather than useful pair distinctions. Large differences between raw and normalized summaries indicate that scale choices matter. A fused pattern that resembles one view in correlation but differs on a few high-interest pairs calls for pair-level inspection.

Confusing weight with contribution. Equal coefficients do not imply equal contributions when the input matrices have different scales.
Confusing scale with pattern. A view can have larger values but the same or a less informative ordering of patient pairs.
Treating a descriptive share as a biological fraction. Edge mass is a property of the chosen numerical representation and formula, not a percentage of disease biology explained.
Forcing additive attribution onto SNF. An iterative nonlinear fusion does not in general provide unique per-view edge shares.
Ignoring preprocessing. Normalization, centering, or different similarity definitions can alter scale and sometimes the apparent pattern.

These checks diagnose the fused object; they do not by themselves determine whether the views contain real cross-view signal or whether a difference is larger than would occur by chance. That requires a separate inferential design. Dominance summaries are also sensitive to the patients and pairs included: a small set of patients can produce unstable summaries, and many pair entries share patients, so the entries are not independent observations. Do not treat a pairwise correlation as if it were based on independent replicates.

Key takeaway: Diagnose dominance using more than one lens: compare off-diagonal scales, state an appropriate contribution measure for the fusion rule, and inspect patient-pair patterns. Report what each diagnostic supports, not a broader claim than the calculation allows.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two nonnegative input kernels have equal fusion coefficients, but one has four times the other’s off-diagonal mean. What can and cannot be concluded about dominance?
Why do the Frobenius norms of two additive kernel components not generally sum to the Frobenius norm of their fused matrix?
In an additive fusion with total off-diagonal mass 2.4 from view A and 0.6 from view B, calculate each view’s mass share.
Why is an exact edge-mass attribution to each input generally unjustified for an SNF result?

Solutions. 1. The larger-mean kernel has a higher overall similarity scale and may supply more edge mass under the stated fusion, but the mean alone does not show that it dominates the patient-pair pattern or contains more biological signal. 2. The squared norm of a sum includes a cross-product term, $2\langle C^{(1)},C^{(2)}\rangle_F$, so overlap or cancellation affects the fused norm. 3. The total mass is $2.4+0.6=3.0$; the shares are $2.4/3.0=0.800$ (80.0%) for A and $0.6/3.0=0.200$ (20.0%) for B. 4. SNF iteratively updates networks using the other views, so the final edges are not generally a simple sum of independently weighted input edges; a unique additive share is not supplied by the procedure.