  Networks determine our social circles and the way we cooperate with others.
We know that topological features like hubs and degree assortativity affect
cooperation, and we know that cooperation is favoured if the benefit of the
altruistic act divided by the cost exceeds the average number of neighbours.
However, a simple rule that would predict cooperation transitions on an
arbitrary network has not yet been presented. Here we show that the unique
sequence of degrees in a network can be used to predict at which game
parameters major shifts in the level of cooperation can be expected, including
phase transitions from absorbing to mixed strategy phases. We use the
evolutionary prisoner's dilemma game on random and scale-free networks to
demonstrate the prediction, as well as its limitations and possible pitfalls.
We observe good agreements between the predictions and the results obtained
with concurrent and Monte Carlo methods for the update of the strategies, thus
providing a simple and fast way to estimate the outcome of evolutionary social
dilemmas on arbitrary networks without the need of actually playing the game.
