What does scale free mean in terms of a scale free graph My understanding of a scale free graph is as follows: Say if we have a large graph $G$ if we were to take random partitions of $G$: $g1, g2,\dots$ 1. Any centrality metric (such as page rank, degree centrality, ...) on $g1, g2,\dots$ 2. would have the same distribution as on $G$ If the above holds for a particular centrality metric, the graph is considered to be scale free vis-a-vis that particular centrality metric. Is this correct?

The random graphs evolves in time. You can divide the time in units almost equals and then combine the vertices and edges created at the same unit of time. What mean by combine is identify those created at the same unit. The larger the unit chosen the fewer vertices you graph will have.

Then a model of random graph is scale-free in time if it generates power law graphs with the same exponent regardless the choice of the unit of time.