I doubt #shiny is the bottleneck, but if you want to put that to the test, you could export the #networkd3 plot to an HTML file/s using the saveNetwork() function and test if that works better... you might be able to at least eliminate shiny as the culprit.
More likely, I would guess the performance is related to D3's SVG and DOM manipulation, and then on top of that, simulating animation by updating every element multiple times per second. You might be able to gain some insight into that by using the browser's built-in debugging tools, but practically speaking, you won't be able to change or fix anything you learn from that without hacking the JavaScript code packaged inside of networkD3, and if you're able and willing to do that, you probably don't need to use networkD3 in the first place (and if you can/are, please consider contributing to networkD3 at https://github.com/christophergandrud/networkD3).