  As an extended companion paper to [1], we elaborate in detail how the tensor
network construction of a p-adic CFT encodes geometric information of a dual
geometry even as we deform the CFT away from the fixed point by finding a way
to assign distances to the tensor network. In fact we demonstrate that a unique
(up to normalizations) emergent graph Einstein equation is satisfied by the
geometric data encoded in the tensor network, and the graph Einstein tensor
automatically recovers the known proposal in the mathematics literature, at
least perturbatively order by order in the deformation away from the pure
Bruhat-Tits Tree geometry dual to pure CFTs. Once the dust settles, it becomes
apparent that the assigned distance indeed corresponds to some Fisher metric
between quantum states encoding expectation values of bulk fields in one higher
dimension. This is perhaps a first quantitative demonstration that a concrete
Einstein equation can be extracted directly from the tensor network, albeit in
the simplified setting of the p-adic AdS/CFT.
