  We introduce models and algorithmic foundations for graph watermarking. Our
frameworks include security definitions and proofs, as well as
characterizations when graph watermarking is algorithmically feasible, in spite
of the fact that the general problem is NP-complete by simple reductions from
the subgraph isomorphism or graph edit distance problems. In the digital
watermarking of many types of files, an implicit step in the recovery of a
watermark is the mapping of individual pieces of data, such as image pixels or
movie frames, from one object to another. In graphs, this step corresponds to
approximately matching vertices of one graph to another based on graph
invariants such as vertex degree. Our approach is based on characterizing the
feasibility of graph watermarking in terms of keygen, marking, and
identification functions defined over graph families with known distributions.
We demonstrate the strength of this approach with exemplary watermarking
schemes for two random graph models, the classic Erd\H{o}s-R\'{e}nyi model and
a random power-law graph model, both of which are used to model real-world
networks.
