Realistically speaking, the Alpha Centauri problem is most likely a plot hole that

just didn't think about. Within the universe, we're obviously left with educated speculation. Subspace communication is definitely out, as everything we've seen in the games suggest that intersystem communications would need to be relayed through the same node that no longer exists. The main options I see as being probable mostly overlap with those Herra mentioned:
1) No one's left in Sol to respond to any attempts at contact, due to Shivan attack through some never-discovered node or some sort of mutually-assured destruction between different factions in Sol. The former would have made for an especially-powerful kick in the nads were it used in the nonexistent sequel.
2) The GTVA has made contact with Sol, but the results of the contact are disturbing enough that the information has been classified at the highest level. Maybe some sort of ultra-militaristic fashion has risen up and seized power. Maybe the Vasudan elements left stranded in Sol have somehow managed to take over the place. Maybe the aforementioned Shivan attack is currently in-progress. Insert your own reason here.
3) The GTVA has attempted to make contact, but some sort of phenomenon existing in the space between Sol and Alpha Centauri has made their efforts futile. Perhaps some sort of interstellar medium that largely absorbs radio wavelengths is positioned smack-dab between the two star systems. Or, like Mobius suggested, maybe the Lucifer's subspace cataclysm created some sort of strange anomaly that deflects certain wavelengths of the EM spectrum.
Out of those three choices, I kind of feel like the third is the most probable reason. Three or four light-years represents a whole crap-load of empty space, and any number of factors contained within it could render getting a coherent message through next-to-impossible. It would certainly explain why the issue was never raised at all in FS2, even in a speculative sense. Still, though, that first option would make for one heck of a campaign...