The Role of Weights¶

Edge weights in DRESS are integral to the equation and have a clean structural interpretation.

How weights enter the equation¶

Every term in the DRESS update has the form \(\bar{w}(u,x) \cdot d_{ux}\). Weights always appear as a multiplicative factor on the similarity value. This coupling means weights and similarity are inseparable: the structural profile of a node is the vector of \(\bar{w} \cdot d\) products over its incident edges.

Interpretation¶

Conductance¶

In the diffusion analogy, weights control how much structural information flows along each edge. A high-weight edge is a strong conductor; a low-weight edge is a weak channel. At steady state, the similarity landscape reflects not just topology but how strongly connected each path is.

Attention¶

When computing the update for edge \((u, v)\), the weight \(\bar{w}(u, x)\) scales how much neighbour \(x\) contributes to \(u\)'s structural profile. Heavy edges amplify a neighbour's vote; light edges dampen it. Weights act as a built-in attention mechanism.

Relative, not absolute¶

Because weights appear identically in the numerator and denominator (both are degree-1 in \(\bar{w} \cdot d\)), uniformly scaling all weights does not change the result. Only the relative weights matter. DRESS captures the pattern of connection strengths, not their absolute magnitude.

Unweighted as a special case¶

When all weights are 1, the equation reduces to pure structural counting. Weights generalise this: they let the graph express "this connection matters more than that one," and DRESS respects it throughout its fixed-point computation without introducing any additional parameters.