I recently saw a documentary (sorry I forget the name) where some astronomers were using long range telescopes, on Earth and in space, to see if they could determine whether or not there is in fact a large black hole at the center of our galaxy keeping everything together.
By observing a specific area for a long time they were able to determine the orbits of several stars/solar systems, and they found that the stars, all in different, erratic orbits, were in fact circling something central; what they now presume is that supermassive black hole.
What's my point? Stars move through space the same way that planets move through space in orbit around stars etc. etc.
A Dyson Sphere (if it were even possible to build) would orbit right along with the star that its around, but a bridge between two Dyson Spheres would be pulled apart because the stars are moving in two separate orbits. The bridge would snap! Your web of bridges would be even worse. It would break apart into millions of pieces. And I think this would all happen fast too. As a matter of fact this movement between these astral bodies would prohibit you from being able to build such a thing in the first place.
Imagine a spiders web where every point that the threads meet/connect is constantly moving in all sorts of different directions. In the case of the web it would become a tangled mess (because the strands are malleable and its all a relatively small structure).
But in the case of your web of massive, intergalactic, hard, solid structures, it would be catastrophic chaos! I don't think I'm even exaggerating.
Not to mention, building such a thing in the first place... The only substance that is abundant enough to be used as the building material might be dark matter and we cant even observe the stuff (outside of its gravitational pull)!
So, that brings up an idea: Maybe your web should be made out of energy or dark matter. That's all hand wave of course, but its better than physical bridges across galaxies imo.
p.s. since I brought up black holes, can you imagine what those and any large gravitational pulls would do to specific spots of your bridge/web?! The warping of space/time around such large structures?! Chaos!
Good luck with you story/idea!