[rrajasek]

What they're using is the mathematical definition of randomness - an event whose outcomes cannot be determined prior to its occurence. The assumption is that it's physically impossible to predict with certainty what the true outcome is.

In many cases, what you're describing applies - there are many real world phenomena for which we have very limited information or are too complicated to model exactly. Probability theory is really useful in these situations. Why probability theory is useful is because even unpredictable events have some high level patterns that we can discover.

In the case of the article, they're interested in understanding properties of random graphs. Random graphs are useful because they're useful models of real life graphs (such as for social media websites). This is because the real life graph is constructed in a way that appears "random" (such as to whoever maintains the social media site). A lot of graph properties are proxies for real life social behavior - triangles indicate close knit groups, cycles indicate broad friend circles. If you can estimate some of these properties in a random graph, you can do the same for real life graphs. For e.g. Facebook might be interested in identifying close-knit friend circles to identify potential users who may be friends or recommend groups that users can join based on who their friends are. It will also give them sensible estimates on how many people know each other personally, how many friend circles exist and so on.

However, there are certain real life processes which are considered to be 'truly random' - quantum tunnelling, radioactive decay. It's physically impossible (regardless of what technology we can develop) to predict if an electron will tunnel through a barrier, or how long it takes for an uranium atom to decay. However, there are patterns to this randomness such as the probability that the particle tunnels and the half life for radioactive decay. These are macroscopic descriptions of random phenomena.

EDIT: added more relevant information about random graphs