GraphZD: Graph-Aware Continuous Zero-Determinant Control in Evolutionary Networks

The emergence and maintenance of cooperation in finite, structured populations remain a fundamental challenge in evolutionary game theory. Zero-determinant (ZD) strategies can enforce linear relations between long-term payoffs, yet fixed extortionate or generous control cannot adapt to heterogeneous neighborhood conditions. This limitation is particularly important in evolutionary networks, where local structure shapes both interaction payoffs and strategy replacement. In this paper, we propose GraphZD, a graph-aware continuous ZD controller for repeated Prisoner's Dilemma games on evolutionary networks. The method instantiates a graph-aware continuous relational-control paradigm that treats the enforceable payoff relation, rather than a predefined strategy label, as the object of adaptation. A graph attention encoder summarizes local behavior, payoff, and structure. An online contextual linear controller evaluates nine control anchors and forms a continuous proposal, which is regulated by an identification phase, exponential moving-average smoothing, and a coexistence gate. An analytically computed common scale then maps the control to a feasible memory-one multiplicative ZD policy without coordinate-wise clipping, preserving the intended Press-Dyson vector direction. We evaluate GraphZD over 5,000 death-birth generations on grid, scale-free, SBM, and small-world networks initialized with 20% adaptive carriers and 80% AllD. In paired experiments, GraphZD improves population share, cooperation, and collective payoff over an otherwise identical controller without graph information. It also outperforms fixed extortionate and generous ZD endpoints: relative to the stronger generous endpoint, GraphZD raises mean population share by 17.8 percentage points and mid-run joint payoff by 0.84. The trajectories support a two-stage interpretation: graph-aware control improves survival under early defector pressure, allowing cooperative groups to persist and generate higher social welfare later. These results connect local structural information with adaptive payoff control in finite evolutionary populations.