Relationships certainly help in identity formation.
Our most important relationships are those with our immediate family. These relationships give us our most basic and lasting role identities: son, daughter, brother, sister, mother, father, etc.
Growing into adulthood, relationships continue to help define a person's "identity". Being friends with doctors and lawyers can shape the way you think of yourself just as being friends with actors and musicians can.
The fact that our relationships help to define us is clear, I think, in the construction of our values, our language, our social and political perspectives etc.
However, if we had to ask the question "What percentage of a person's identity is defined by relations, interactions, and connections to others in a social network?" the number is not going to be 100%. The percentage may, however, be very high.