One of the most counterintuitive examples involving randomness is the birthday problem. From 23 persons onwards, the probability of finding at least two people in a group with the same birthday is above 50%. Leonhard Euler’s solution of the Koenigsberg bridge problem heralded the start of the fascinating field of graphs and networks with applications to numerous applied problems across many disciplines. In 1929 the Hungarian writer Frigyes Karinthy highlighted the world’s smallness through his wonderful story “Chains” where he introduced the by now well-known “separation by six” idiom. Starting from these examples, we discuss some risks due to network effects present on the World Wide Web and social media. We present the reader with a glimpse of the fascinating world of coincidences. For instance, the law of truly large numbers states that, with a large enough sample, any outrageous thing is likely to happen. Real-life examples highlight the meaning of this law.

To describe coauthorship networks, we begin with the Erdös number, which links mathematicians to their famously prolific colleague through the papers they have collaborated on. Coauthorship networks help us capture collaborative patterns and identify important features that characterize them. We can also use them to predict how many collaborators a scientist will have in the future based on her coauthorship history. We find that collaboration networks are scale-free, following a power-law distribution. As a consequence of the Matthew effect, frequent collaborators are more likely to collaborate, becoming hubs in their networks. We then explore the small-world phenomenon evidenced in coauthorship networks, which is sometimes referred to as “six degrees of separation.” To understand how a network’s small-worldliness impacts creativity and success, we look to teams of artists collaborating on Broadway musicals, finding that teams perform best when the network they inhabit is neither too big or too small. We end by discussing how connected components within networks provide evidence for the “invisible college.”