When we build a random graph model, we start from an empty graph of $n$ nodes and add $m$ edges or generate a graph with edge probability $p$.
What happens if we start from an arbitrary graph instead of an empty graph?
Consider a connected graph with $n_0$ nodes and $m_0$ edges. Will adding a $m$ random edges with probability $p$ make the graph "random" in any sense? Or more appropriately when do the properties of random graphs emerge such as connectivity property etc.? What is this concept called?
What happens if we start from an arbitrary graph instead of an empty graph?
It's difficult to give a rigorous answer to such a broad question. In general, the answer will be: "probably not much".
Will adding a $m$ random edges with probability $p$ make the graph "random" in any sense?
First of all, random is not a property of a graph in the same way as, say, planar or symmetric. Thus you can't simply take a graph and make it "random". When we study random graphs, we actually investigate probability distributions of graph properties. We do that by studying random processes that generate graphs and not by examining the specific instances. So "random" always implies context of which random process are we talking about.
When do the properties of random graphs emerge?
I guess, you can say, when the sample size is large enough.
To be fair, there are other models besides Erdős–Rényi where the initial state can make a huge difference. As somewhat trivial example, consider some random tree generation procedure $T(G)$ that adds a random node and connects it to a node in $G$. The properties of random graphs will be quite different depending on whether it was initialised with a single node or a cycle graph.
Of course, you can also get very technical and examine for a given random process $R(G)$, initial state $G_0$, and a property $P$, what will be the probability of $P_{R^n(G_0)}$ for a certain $n$ if you want some definitive (as far, as probabilities go) answers to when the property arises.