We demonstrate that protein-protein interaction networks in several eukaryotic organisms contain significantly more self-interacting proteins than expected if such homodimers randomly appeared in the course of the evolution. We also show that on average homodimers have twice as many interaction partners than non-self-interacting proteins. More specifically, the likelihood of a protein to physically interact with itself was found to be proportional to the total number of its binding partners. These properties of dimers are in agreement with a phenomenological model, in which individual proteins differ from each other by the degree of their 'stickiness' or general propensity toward interaction with other proteins including oneself. A duplication of self-interacting proteins creates a pair of paralogous proteins interacting with each other. We show that such pairs occur more frequently than could be explained by pure chance alone. Similar to homodimers, proteins involved in heterodimers with their paralogs on average have twice as many interacting partners than the rest of the network. The likelihood of a pair of paralogous proteins to interact with each other was also shown to decrease with their sequence similarity. This points to the conclusion that most of interactions between paralogs are inherited from ancestral homodimeric proteins, rather than established de novo after duplication. We finally discuss possible implications of our empirical observations from functional and evolutionary standpoints.