Extrapolating from this fact, shouldn't the patterns (and hence the logic) within social networks hold meaning?
Probably, yeah.
What would be the equivalent of running "pulses" through the social graph?
Depends on how you see the analogy. Research on "diffusion of innovations"[1] could relate to this... the idea being that the introduction of a new "thing" (a technology innovation) is your "pulse" and its spread through the network is the analogue of the neural network processing. Also possibly related would be the idea of "information cascades" in networks[2]. Seems like there's something of a parallel between some of the SNA / Network Theory stuff and the Spreading Activation[3] stuff as well.
I don't know about trying to treat the social network as an analogue of a neural network, but the Social Network Analysis[4] folks certainly study the flow of information through a network. You might poke around the journals and forums on that, or the Network Science[5] and/or Complex Adaptive Systems[6] people and see if anybody is doing anything related.
[1]: http://en.wikipedia.org/wiki/Diffusion_of_innovations
[2]: http://en.wikipedia.org/wiki/Information_cascade
[3]: http://en.wikipedia.org/wiki/Spreading_activation
[4]: http://en.wikipedia.org/wiki/Social_network#Social_network_a...
[5]: http://en.wikipedia.org/wiki/Network_theory
[6]: http://en.wikipedia.org/wiki/Complex_adaptive_system
There's an interesting wiki full of links related to network science and complex systems here: http://netwiki.amath.unc.edu/ that might be fun to explore.