How can the number of connections be Gaussian if it cannot be negative? I am analyzing social networks (not virtual) and I am observing the connections between people. If a person would choose another person to connect with randomly, the number of connections within a group of people would be distributed normally - at least according to the book I am currently reading.
How can we know the distribution is Gaussian (normal)? There are other distributions such as Poisson, Rice, Rayliegh, etc. The problem with the Gaussian distribution in theory is that the values go from $-\infty$ to $+\infty$ (although the probabilities go toward zero) and the number of connections cannot be negative.
Does anyone know which distribution can be expected in case each person independently (randomly) picks-up another person to connect with?
 A: When there are $n$ people and the number of connections made by person $i, 1 \le i \le n,$ is $X_i$, then the total number of connections is $S_n = \sum_{i=1}^n{X_i} / 2$.  Now if we take the $X_i$ to be random variables, assume they are independent and their variances are not "too unequal" as more and more people are added to the mix, then the Lindeberg-Levy Central Limit Theorem applies.  It asserts that the cumulative distribution function of the standardized sum converges to the cdf of the normal distribution.  That means roughly that a histogram of the sum will look more and more like a Gaussian (a "bell curve") as $n$ grows large.
Let's review what this does not say:


*

*It does not assert that the distribution of $S_n$ is ever exactly normal.  It can't be, for the reasons you point out.

*It does not imply the expected number of connections converges.  In fact, it must diverge (go to infinity).  The standardization is a recentering and rescaling of the distribution; the amount of rescaling is growing without limit.

*It says nothing when the $X_i$ are not independent or when their variances change too much as $n$ grows.  (However, there are generalizations of the CLT for "slightly" dependent series of variables.)
A: The answer is dependent on the assumptions that you are willing to make. A social network constantly evolves over time and hence is not a static entity. Therefore, you need to make some assumptions about how the network evolves over time.
The trivial answer under the stated conditions is: If the network size is $n$ then as  asymptotically (in the sense of 'as time goes to infinity')
$Prob(\mbox{No of connections for any individual} = n-1) =1$.
If a person selects another person at random to connect to then eventually everyone will be connected. 
However, real life networks do not behave this way. People differ in several aspects.


*

*At any time a person has a fixed network size and the probability of another connection being made is a function of his/her network size (as people introduce other people etc).

*A person has his/her own intrinsic tendency to form a connection (as some are introvert/exterovert etc).
These probabilities change over time, context etc. I am not sure there is a straightforward answer unless we make some assumptions about the structure of the network (e.g., density of the network, how people behave etc).
