Lead with liking ("you matter") over utility ("you're useful to me"); use their name, ask about what they care about, and listen.
The idea
Dale Carnegie nailed this in 1936: showing someone you value them and showing them you like them land very differently. "I value your expertise" says you're useful to me. Genuine interest in them as a person says you matter. Robert Cialdini put it plainly: "The number one rule for salespeople is to show customers that you genuinely like them." The counterintuitive first move is to prove you don't need their help. An agenda written all over your face triggers their defenses. Curiosity makes you the person who sees them.
How to do it
Remember their name and use it.
Do two minutes of homework. Find something they posted that wasn't about work, and ask about that.
Listen 75% of the time, without interrupting.
Find the shared experience: you both have teenagers, you both hate networking events. Connection happens in the overlap.
Between meetings, keep showing it in the simplest way (see Script).
Example
I once watched someone at the next table torpedo a first meeting by opening with "I really value what you do in the defense space." The other person gave that polite, dead-behind-the-eyes smile we all know, and the meeting looked like it limped along for 20 awkward minutes before mercifully ending. (I was supposed to be reviewing a legal document. I needed the distraction.)
Script
"Hey, just thought of you. Hope you're doing well!"
Watch out for
Openers that sound like LinkedIn wrote them. Start over.
Make One Friend: Your one goal at a networking event is to make a friend, because referrals come from knowing, trusting and liking each other.
Connection Over Value: Choose who to stay close to by who you genuinely like, not who seems important, because liking is what makes you actually reach out.
Remembering Names: Let four seconds pass, use the name later, build a weird association, and when it's gone anyway, just admit it and ask.
Know, Like and Trust: People work with people they know, like and trust. Trust comes from shared context, previous engagement and endorsements from people already trusted.