Explain how networks have contributed to the perception that we live in a  "small world" and summarize the findings of small-world researchers.

Networks have contributed to the perception that we live in a "small world" by connecting people and information across vast distances, making it easier to communicate and interact with individuals from different parts of the globe. Social media platforms, email, and video conferencing have made it possible to stay connected with friends, family, and colleagues regardless of geographical location. This interconnectedness has led to the perception that the world is smaller and more accessible, as people can easily reach out to others and share experiences in real-time.

Small-world researchers have found that networks exhibit a "small-world phenomenon," where most nodes can be reached from every other node by a small number of steps. This concept is often illustrated by the "six degrees of separation" idea, which suggests that any two people in the world can be connected through a chain of acquaintances with no more than six intermediaries. This idea has been supported by studies in social networks, where researchers have found that individuals are typically connected by a small number of intermediate connections, leading to the perception of a "small world." Additionally, small-world networks have been observed in various other systems, such as biological neural networks and the internet, further highlighting the interconnectedness and accessibility of the world. Overall, small-world researchers have demonstrated that networks play a significant role in shaping our perception of the world as a smaller and more connected place.