@Suvrit: Thank you for the comment. I agree with you, indeed I think that the thing is not that easy to prove due to the fact that the limit could be singular. Relying on numerical simulations, it seems that if the iteration admits a positive-definite fixed point (which is not always true) then this fixed point is unique and globally attractive. In any case, I think that the $\varepsilon$-perturbation argument might be used to prove (at least) local attractivity around a positive-definite fixed point...I will think about it in the next days...

@Suvrit: I had a look at (3.10), although I haven’t gone through the proof yet. However, it seems to me a bit different to my iteration (due to the fact that in (3.10) $\Gamma_p$ only appears in the “denominator”). I found (3.8) more similar to it. My guess is that starting from an “$\varepsilon I$ version” of (3.8), by using a kind of “continuity” argument for $\varepsilon\to 0$ and the additional assumption that my iteration always possesses a positive-definite fixed point, it can be proved that the iteration has a unique and globally attractive (positive-definite) fixed point.

@Suvrit: I've read your paper and I found it very interesting! Moreover, the iteration (3.8) seems to be the “regularized” version of mine! Hence, I wonder if I can use an argument similar to that used in Sec 3.1 to prove convergence of my iteration for $p=1/2$ (perhaps, replacing $I$ by $\varepsilon I$ and letting $\varepsilon\to 0$?). I think that one issue is that Lemma 3 doesn't hold anymore and this implies that my iteration can tend to a singular matrix (rank-1 projection, actually). However, I know that my iteration always admits a positive-definite fixed point and maybe this can help.

Yes, I mean directed graphs.

@Suvrit -- Thanks a lot! Weeks ago I tried to google for references in which the analysis of nonlinear mappings of the type you mentioned is addressed. Unfortunately, I didn't manage to find anything useful. I found this weird because I think it is a quite appealing and interesting form for a matrix equation.

@Suvrit -- I would like to ask you if you eventually managed to prove that your proposed nonlinear mapping $\mathcal{G}$ is contractive w.r.t. a suitable metric. I ask you this because I found a kind of similarity with another nonlinear matrix mapping (see here) which I have been struggling to prove convergence for quite a long time. Thanks in advance for your help.