How does Elixir handle cluster formation and distributed state?

Cluster formation in Elixir production: LibCluster: auto-forms BEAM clusters using DNS (Kubernetes headless services), Epmd, EC2 tags, or gossip protocols. Nodes join automatically when they discover each other. State distribution strategies: (1) Horde (CRDT-based distributed supervisor and registry): processes can run anywhere in the cluster; Horde ensures a process runs on exactly one node, migrating on node join/leave. (2) Distributed Erlang messaging: route messages to any PID on any node using Node.spawn or distributed :rpc.call. (3) Local state with pub/sub: each node is authoritative for some partition of data; Phoenix PubSub broadcasts changes cluster-wide. (4) Mnesia: replicated database with RAM and disk tables, supporting transactions across nodes. Challenges: network partitions, split-brain scenarios — Elixir/BEAM does not automatically resolve these; design systems to handle partition tolerance explicitly.