  We introduce the controllable graph generation problem, formulated as
controlling graph attributes during the generative process to produce desired
graphs with understandable structures. Using a transparent and straightforward
Markov model to guide this generative process, practitioners can shape and
understand the generated graphs. We propose ${\rm S{\small HADOW}C{\small
AST}}$, a generative model capable of controlling graph generation while
retaining the original graph's intrinsic properties. The proposed model is
based on a conditional generative adversarial network. Given an observed graph
and some user-specified Markov model parameters, ${\rm S{\small HADOW}C{\small
AST}}$ controls the conditions to generate desired graphs. Comprehensive
experiments on three real-world network datasets demonstrate our model's
competitive performance in the graph generation task. Furthermore, we show its
effective controllability by directing ${\rm S{\small HADOW}C{\small AST}}$ to
generate hypothetical scenarios with different graph structures.
