  Quantifying the contributions, or weights, of comparisons or single studies
to the estimates in a network meta-analysis (NMA) is an active area of
research. We extend this to the contributions of paths to NMA estimates. We
present a general framework, based on the path-design matrix, that describes
the problem of finding path contributions as a linear equation. The resulting
solutions may have negative coefficients. We show that two known approaches,
called shortestpath and randomwalk, are special solutions of this equation, and
both meet an optimization criterion, as they minimize the sum of absolute path
contributions. In general, there is an infinite space of solutions, which can
be identified using the generalized inverse (Moore-Penrose pseudoinverse). We
consider two further special approaches. For complex networks we find that
shortestpath is superior with respect to run time and variability, compared to
the other approaches, and is thus recommended in practice. The path-weights
framework also has the potential to answer more general research questions in
network meta-analysis.
