  It has been shown by Voiculescu and Biane that the analytic subordination
property holds for free additive and multiplicative convolutions. In this
paper, we present an operatorial approach to subordination for free
multiplicative convolutions. This study is based on the concepts of `freeness
with subordination', or `s-free independence', and `orthogonal independence',
introduced recently in the context of free additive convolutions. In
particular, we introduce and study the associated multiplicative convolutions
and construct related operators, called `subordination operators' and
`subordination branches'. Using orthogonal independence, we derive
decompositions of subordination branches and related decompositions of s-free
and free multiplicative convolutions. The operatorial methods lead to several
new types of graph products, called `loop products', associated with different
notions of independence (monotone, boolean, orthogonal, s-free). We also prove
that the enumeration of rooted `alternating double return walks' on the loop
products of graphs and on the free product of graphs gives the moments of the
corresponding multiplicative convolutions.
