Worked SNF Example with Two Omics Blocks

From Two Patient Networks to One Update

Similarity Network Fusion (SNF) combines patient-to-patient information from multiple omics views by letting each network update its relationships using information from the others. In this tutorial, the calculation is deliberately small: three hypothetical patients, two omics blocks, and one diffusion update. The small size makes each matrix operation inspectable, but the same sequence of calculations applies to larger networks.

The goal is not to treat an updated edge as a direct measurement in either block. An updated weight reflects both the other network's transition structure and the local neighborhoods used to diffuse that information. Keeping those ingredients distinct is essential when interpreting the fused result.

The calculation requires the same patients in the same order in every view. Each starting affinity must be nonnegative, and each transition row to be normalized must have positive sum. For the examples below, diagonal affinities are set to zero, there are no isolated patients, and local neighborhoods are selected by largest affinity. We use \(k=1\): each patient retains its single strongest other patient, with ties resolved by a pre-specified rule. A row of \(S^{(v)}\) is normalized over the retained neighbors; because \(k=1\), its retained entry is one.

With two views, one diffusion update for view \(v\) uses the other view's transition matrix. Write \(Q^{(v)}\) for the matrix before row normalization:

Each updated transition matrix \(P_{\mathrm{new}}^{(v)}\) is obtained by dividing each row of \(Q^{(v)}\) by that row's sum. This version of the update makes explicit that one view's local structure surrounds the transition information supplied by the other view. For a final undirected network, we first average the two updated transition matrices and then symmetrize: the fused edge weight for \(i\ne j\) is the mean of the two directed entries in that average. This final symmetrization is a stated reporting convention, not an extra diffusion step.

Worked Calculations Before Diffusion

One Complete SNF Diffusion Update

Why the Update Has This Form

The matrix product can be understood one entry at a time. Before row normalization, the \((i,j)\) entry of the update for view \(v\) is:

where \(P^{(-v)}\) is the other view's transition matrix in a two-view calculation. Each term weights the transition from \(a\) to \(b\) by how strongly patients \(i\) and \(j\) are locally connected to \(a\) and \(b\) in view \(v\). Large contributions arise when the local neighborhoods of \(i\) and \(j\) align with substantial transitions in the other view. This is the sense in which the views exchange information; it does not require that the two blocks have identical edge patterns.

The row normalization has a direct purpose. The matrix product does not generally have rows summing to one, even if its factors are row-stochastic, because the rightmost factor is transposed. Dividing a nonnegative row by its positive sum restores a transition interpretation. In the worked example, the first row sum of \(Q^{(1)}\) is \(1/5\), so its single nonzero value becomes \((1/5)/(1/5)=1\). For its second row, the two nonzero values each become \((1/3)/(2/3)=1/2\).

Diagnostics, Alternatives, and Limitations

Check the arithmetic and the network construction before interpreting a fused edge. Every nonempty row of \(P^{(v)}\), \(S^{(v)}\), and \(P_{\mathrm{new}}^{(v)}\) should sum to one within rounding tolerance. Starting affinities and the pre-normalized products should be nonnegative. Confirm that patient order is identical across views, that diagonal entries were handled consistently, and that the selected neighbors really are the largest affinities in each row. In the worked example, \(S^{(1)}\) is asymmetric by construction; treating it as symmetric would change the update.

A row with zero sum cannot be normalized. This can result from an isolated patient, an empty local neighborhood, or an affinity construction that assigned only zero weights. It must be addressed explicitly—for example, by revisiting the graph construction or applying a stated isolate-handling rule—rather than silently dividing by zero. Ties at the neighborhood boundary can also change \(S^{(v)}\); report the tie rule and examine whether plausible alternatives change the fused edges. Likewise, a small \(k\) can make the result sensitive to individual selected neighbors. Inspect edge rankings across reasonable neighborhood choices and diffusion iterations, and avoid claiming that one edge is robust if its rank changes substantially.

The result depends on the input affinities and their scales, on \(k\), on the number of updates, and on the rule used to create the final undirected network. In particular, row normalization removes overall row magnitude: it retains relative outgoing weights, not the total affinity of a patient to all others. The final symmetrized matrix also need not have rows summing to one, so its entries should be described as fused edge weights, not transition probabilities.

One alternative is to average the starting affinity matrices directly. That is simpler, but it does not use the local-neighborhood diffusion shown here and may be dominated by a block with a larger numerical scale. Another alternative is to concatenate measured features before computing patient similarities; this changes the construction of the network rather than exchanging information between already constructed networks. These approaches answer different construction choices and should not be treated as equivalent to an SNF update.