I’m reading about the clustering coefficient in Networks by Mark Newman, and he defines it as follows:
the fraction of pairs of people with a common friend who are themselves friends, or equivalently as the mean probability that two people with a common friend are themselves friends.
I’m not sure what he means by mean probability here. I found quite a few lecture notes on the subject (the clustering coefficient) and some say “average probability”, some say “expected probability”, while others just use the term “probability”.
Do these just mean the same thing in this context?
Edit:
Newman seems to use the terms interchangeably throughout the book (sometimes even in the same chapter section). In fact, he even refers to the local clustering coefficient as an average probability. In addition, Newman means something different by global clustering coefficient. The notion discussed in the answer below corresponds more closely with the network average clustering coefficient (which is, somewhat confusingly, sometimes referred to by the same name).
The reason behind this is probably that for the clustering coefficient, the way you average makes a difference:
You can consider all cases where two people have a common friend and ask how probable it is that they are themselves friends. This quantity is called transitivity.
You can consider all pairs of people with given common friend and ask how probable it is that they are themselves friends. This would be the (local) clustering coefficient of a node (the given friend). Then you average the local clustering coefficient over all people in the network (as given friends). The result is called the (global) clustering coefficient.
So, for transitivity, you consider only one probability, while for the clustering coefficient, you have one probability per node and you average over this and indeed you have a “mean probability”. In this sense, just writing “probability” would actually change the meaning. Of course, whether anybody can possibly deduce that mean probability means that you first calculate node-wise probabilities and then take their mean is written on another page.
To see that there is a difference between transitivity and the global clustering coefficient, consider a popular person, i.e., somebody with a lot of friends. This person is the common friend for a lot of pairs of people. The clustering coefficient gives these pairs the less weight than pairs where the common friend is less popular. The transitivity gives all pairs the same weight.