We say that a pair (K, m) of a knot K in the 3-sphere and
an integer m is a Seifert surgery if the result (K; m) of
the m-surgery on K is a Seifert fiber space;
here we allow a degenerate Seifert fiber space,
meaning (K; m) has a fiber of index zero.
All the known Seifert surgeries (K, m) have a knot c
disjoint from K such that c is unknotted in the 3-sphere and becomes
a Seifert fiber in the resulting Seifert fiber space (K; m).
We call such a knot as c a ``seiferter'' for (K, m).
As we have previously shown,
if an r-surgery on K yielding a Seifert fiber space for some rational number
r has a seiferter, then r is integral,
except when K is a torus knot or a cable of a torus knot.
Let us look each Seifert surgery as a ``vertex'',
and connect two vertices by an ``edge'' if they are related by
a single twisting along a seiferter.
This leads us to build a ``network'' of Seifert surgeries in which
each Seifert surgery is a vertex.
As we will observe,
many Seifert surgeries are connected to those on torus knots and
we expect that the network gives us a global picture of Seifert surgeries.
In this context, if we have a path in the network from a Seifert surgery (K, m)
to a known Seifert surgery (K_0, m_0), say K_0 being a torus knot,
then we can understand inductively (along the path)
how to obtain the given Seifert surgery.
We will establish some fundamental properties of the network.
(1) Most seiferters for Seifert surgeries on hyperbolic knots
which become exceptional fibers
are shortest geodesics in the knot complements.
(2) The Berge's lens surgeries are ``close'' to
Seifert surgeries on torus knots.
From the networking viewpoint we show also:
(3) The network contains an infinite family of
Seifert fibered surgeries on hyperbolic knots which cannot be embedded
in genus 2 Heegaard surface of the 3-sphere.