To keep things decentralized, control over each identity must be ambiently verifiable. This means that verifying control of the identity can be done anywhere, anytime, without a need to contact a central authority (or ledger!)
Technically this is implemented using Public Key Cryptography. Each identity is named by its associated public key. Each identity declares which other identities it is linked to. A link between two identities is only valid if both identities link to each