How to cool a graph

We introduce a new graph parameter called the cooling number, inspired by the spread of influence in networks and its predecessor, the burning number. The cooling number measures the speed of a slow-moving contagion in a graph; the lower the cooling number, the faster the contagion spreads. We provide tight bounds on the cooling number via a graph's order and diameter. Using isoperimetric results, we derive the cooling number of Cartesian grids. The cooling number is studied in graphs generated by the Iterated Local Transitivity model for social networks. We conclude with open problems.