  Network meta-analysis (NMA) is a central tool for evidence synthesis in
clinical research. The results of an NMA depend critically on the quality of
evidence being pooled. In assessing the validity of an NMA, it is therefore
important to know the proportion contributions of each direct treatment
comparison to each network treatment effect. The construction of proportion
contributions is based on the observation that each row of the hat matrix
represents a so-called 'evidence flow network' for each treatment comparison.
However, the existing algorithm used to calculate these values is associated
with ambiguity according to the selection of paths. In this work we present a
novel analogy between NMA and random walks. We use this analogy to derive
closed-form expressions for the proportion contributions. A random walk on a
graph is a stochastic process that describes a succession of random 'hops'
between vertices which are connected by an edge. The weight of an edge relates
to the probability that the walker moves along that edge. We use the graph
representation of NMA to construct the transition matrix for a random walk on
the network of evidence. We show that the net number of times a walker crosses
each edge of the network is related to the evidence flow network. By then
defining a random walk on the directed evidence flow network, we derive
analytically the matrix of proportion contributions. The random-walk approach,
in addition to being computationally more efficient, has none of the associated
ambiguity of the existing algorithm.
