Instead of getting rid of less significant link, may be we could manipulate the opacity of links. For example: If Drama <-> Romance has weight 500 and Drama<->Documentary has weight 40. The opacity of first link is 500/500 and opacity of second one is 40/500. That means the links with larger weight can be more obvious in terms of opacity than the links with less weight.
Instead of getting rid of less significant link, may be we could manipulate the opacity of links. For example: If
Drama <-> Romance
has weight 500 andDrama<->Documentary
has weight 40. The opacity of first link is 500/500 and opacity of second one is 40/500. That means the links with larger weight can be more obvious in terms of opacity than the links with less weight.