The most notable trade-off between EEG and fMRI is that with EEG you have much better time resolution, but awful spatial resolution (and the spatial resolution you do get is "relative"); conversely, fMRI has much better spatial resolution (and you can map it to absolute brain structures with an anatomical scan) but less temporal resolution.

I should also mention that EEG is orders of magnitude cheaper and simpler to set up.

MEG is good compromise since the MEG measures both tangential and axial magnetic gradients.

I agree with Alberto Sepulveda Rodriguez: It is basically a trade-off between temporal and spatial resolution. (f)MRI is much more expensive than EEG. However, as a researcher conducting an (f)MRI study one might leave the participants in the scanner a couple of minutes longer to add a resting-state "task". This way you might be able to record data for three studies within one session:

- structural data for a VBM- and/or DTI-analysis

- functional data during resting state

- functional data during the a (cognitive) task

While I find resting state-analysis fascinating, there are a couple of "might as well evaluate resting state now that my participant is in the very expensive scanner anyway but I wont put too much effort in the study..."-studies. fMRI being more expensive than EEG might actually lead towards more resting-state-connectivity studies as researchers try to get the most out of their sessions with each participants.

Personally, I do not share your impression that both techniques have low signal-to-noise ratio. When we have to repeat the same stimulus 20 times in order to have signal emerging from noise... I do not think it is low.

Trying to answer your questions and in agreement with the other contributions here, EEG spatial resolution is far from optimal... and networks are mainly a matter of... space.

Although there has been some progress on EEG signal source localization, in fact, the signal measured at the electrode site is the sum of contributions from several sources in disparate places in the brain, which means that you may have contributions from different places to the signal in one electrode, which, in turn, increases the complexity of the networking approaches to brain functioning.

Typically, in EEG, you work with one electrode independently of the others, which is not the best way to address networks. Therefore, in almost one century of EEG researchers have been using mostly univariate approaches and this still is the status quo. Multivariate approaches are coming, for example, microstates. Take a look at this site, which may inform you better:

http://www.scholarpedia.org/article/EEG_microstates

In any case, if you were looking for a gap in Science for your PhD thesis... you are welcome! You have a lot to do and many eager people waiting for your results ;-)