I'm building a graph in neo4j using the free version (I'm using the software I downloaded to my computer). I'm building quite a large graph there (at least a hundred thousand nodes) on the desktop computer I work with. I also have a laptop where I wrote different code that I want to use to analyze the graph that exists on the desktop computer. My question is: How can I connect to the data through the laptop?
Yes, that's possible.
You need to tell the local server to listen on ip address 0.0.0.0 (which means all local network adapters access from local and remote), and allow the BOLT protocoll (port (7687) through your local firewall.
You can find a description here: Connect to a remote DBMS - Neo4j Desktop