Francisco, if you have access to real networks, just do the analysis. I think that not all information is public. For example, set of phone or internet connections for a group of people.  I have seen presentations of such type during one conference.

You might know Duncan Watt's book "Six degrees of separation". In general, some networks are of small world type, others are free scale. People study how the number of links is distributed and check what law fits it better. So plenty of interesting work if you have data.

Why connectivity in a network may be practically important? Here we can set plenty of models starting from propagation of AIDS (network of sexual contacts) to the effect of popularity of a certain persons in arts, where the information is spread via private contacts over network.

The problem is that there are several clusters of researchers dealing with networks from different sides and having low connectivity between themselves. Like in the story about blind people describing an elephant. Physicists publish a lot about power laws in distribution of number of links between nodes. Mathematicians find how longest connected sets are distributed. Social scientists very often do not understand physicists and mathematicians but know that network interaction is important for social processes. Economists study games on a fixed network with simple topology.

Hi

Your question is fundamental and I argue is something we do all the time when applying statistics. Suppose you say that someone is tall or talented. What you actually do is compare this person to a random person drawn from the population. A tall person is someone who is taller than many such attempts to sample at random from the population. An efficient way of doing so is to use the fact that the distribution of heights in the population is Normal and therefore fully characterized by its mean and variance.

Similarly, if you have a real life network you can learn a lot by comparing it to a random network - but a relevant random network, namely one that has the same mean connectivity for example (i.e. each node has the same number of neighbours on average). For example in a social network the number of triangles is consistently higher than in a random network with the same number of links per site. This corresponds to the fact that if you have two friends it is likely that they will be friends too. This insight is not huge but it captures something important regarding social networks, that your exposure to new people is mediated by your existing social links and not created at random - you get to know new people usually because someone you know already knows them. This is just an example - but I hope you see where it is going.

The main issue is matching your real world network with a relevant random graph, technically speaking, with a proper null-model. A quite general framework is provided by the so-called "the configuration model", namely an ensemble of random graphs that realize a certain degree distribution. In other words - I think that if you compare your real-life network with a random network with the same degree sequence for a certain property you can learn how probably certain features are.

@Eytan thanks you for your informative response to the question of real-world network comparison with randon graphs. Indeed, I'm familiar with some of the methods that you mention, such as degree, communities, or eigenvector benchmarking. However, being new to Network analysis and working with real-world economic networks, makes me think once and again that conclusions of the type "this or that property of a real-life network is, or is not, randomly distributed" might be nice as a theoretical exercise, but maybe are too poor for real practical consequences.

I understand better your position with this now.

So let me just argue again that this is like any other statistical test everyone is doing like the z-test - you compare your data to a random distribution which encodes what we know about the real world (like the mean and variance) but which is otherwise random.

The analog of this with network is what you describe. However unlike the z-test where the Normal distribution is fixed and decided, choosing the right family of random networks is a non trivial issue and is the core question

Eytan

Dear Francisco, random graphs are used not because all the properties of networks are described by its parameters, like distribution of links, length of the longest connected set, etc - but because the number of possible networks is enormous and because we need to cluster them in types. For each type (here random network with certain properties) we can simulate some social interaction. A big question still remains: what part of interaction can be described by random network parameters and what part by its particular realization. Statistics will just tell us that the rest is random noise. But the difference from other statistical applications is that origin of network is not essentially random. For every social application (propagation of diseases, propagation of gossips, etc) the separation of social outcome into explained and random component may differ. But still this is the best we can do. In fact, we often cannot measure all topological structure of real network and have to derive result based on approximate values and topology.