No. Connection requests are free for every member, every time. Your people are the point of Cofelt, so reaching them is never taxed.
What keeps the network calm instead is a daily request limit that stops bulk-firing, plus the verification gauntlet: unverified accounts cannot send requests at all. The recipient can assume the person reaching out is real and chose to connect.
Sparks cover selected premium actions like full compatibility readings and extra messages. They are never required to start a connection.