A continuity theory for lossless source coding over networks

A continuity theory of lossless source coding over networks is established and its implications are investigated. In the given model, source and side-information random variables X and Y have finite alphabets, and the input sequences are drawn i.i.d. according to a generic distribution P
_X,Y
on (X,Y). We consider traditional source coding, where all demands equal source random variables. We define a family of lossless source coding problems that includes prior example network source coding problems as special cases. We show that the lossless rate region R
_L
(P
_X,Y
) is inner semi-continuous in P
_X,Y
. We further show that for a special type of networks called super-source networks, where there is a super source node v
^*
that has access to (X,Y) and any other node with access to some source random variable X
_i
is directly connected to v
^*
, R
_L
(P
_X,Y
) is also outer semi-continuous in P
_X,Y
. Based on the continuity of super-source networks with respect to P
_X,Y
, we conjecture that R
_L
(P
_X,Y
) is also outer semi-continuous and therefore continuous in P
_X,Y
for general networks.