My method for connecting to people starts with the request message, don’t make this too sales orientated, I personally hate receiving a message from people saying can we connect so I can extend my network, very little relevance.
If they connect with you, then that initial message has worked well, if they don’t then they’re not intrested. The next message is key, gotta draw them in without again too much sales, I utilise different messages to see what usually works well.
In my opinion, if they’ve connected and won’t speak they’re not the right people to be speaking with if you tune your initial message and they connect then you’ve opened the door to a good connection and then should be able to open communication with them.
key is polish the process that you connect with people and also find the right people to speak with.