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.