Difference between Barabási-Albert model and Erdos-Renyi model

Question

I am trying to do some network science and have constructed a Barabasi-Albert network (BA) and a Erdos-Renyi network (ER) with the same amount of nodes and same p.

From the degree distribution (k) plots I see a clear difference as the ER don't have nodes which are connected to many nodes as in the BA (pictures are below).

BA:

ER:

However what other differences should I notice and what is the reason for these differences? I think that the difference is something with that the BA is a scale-free network?

Answer 1

So fundamentally, they are created in different ways:

In the Erdos-Renyi network, we assign N nodes, and then connect each pair with probability p. This means that no one node will have much higher degree than any other.

In the B-A network, we assign N nodes, but to create them, we first start with a small set of connected nodes. Then we add nodes one at a time till we get N nodes. When we add a node, we connect it to a small number of existing nodes with probability proportional to the degree of the existing node. As a result, nodes with higher degree (the earlier ones) tend to get even higher degree.

Think of it like the current American economy. If you're born rich, you're practically handed money, but if you are born poor you have extra fees. This is known as the matthew effect, from the book of Matthew: "For to everyone who has, more will be given, and to those who have nothing, even that will be taken away".

Anyways, the result is that the network ends up with a powerlaw distribution. This also tends to affect things like the typical distance between nodes in the network, and various centrality measures. Because there are a few very high degree nodes, they seem to be at the center of everything, while the low degree nodes connect to the rest of the population through their high degree neighbors.

In the Erdos-Renyi network it's much more equal.