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I am a bachelor student trying to comprehend the subject of my bachelor thesis.

I wanted to ask if there is a rigorous definition of random graphs, like there is one for random variables or function, or is that not possible?

I have seen that most of the literature (like Bollobas,Newman) looks at the Erdos-Renyi and Gilbert model of random graphs. Some of the authors say that a random graph is its distribution on the family of possible graphs, which is the closest of what I could find of a definition. Now if I am correct there have been more models of random graphs floating around in the recent, like the exchangeable random graph model induced by the graphon.

Does it still make sense to find a rigorous definition of THE general random graph or are the random graphs just treated and defined in a case by case basis, depending on which model is used (erdos-renyi, gilbert, graphon,etc)?

Any insight would be appreciated.

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Bollobás defines in his book "Modern Graph Theory" that "rather trivially every probability space whose points are graphs gives us a notion of a random graph". I guess thinking about the notion of random graph is like thinking about the notion of a random variable, while thinking about a specific model of a random graph then would be equivalent of thinking about a specific probability distribution of your random variable.