Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
"Network thinking is poised to invade all domains of human activity and most fields of human inquiry. It is more than another useful perspective or tool. Networks are by their very nature the fabric of most complex systems, and nodes and links deeply infuse all strategies aimed at approaching our interlocked universe."
It is a cliché to say that we live in a connected age. Improvements in communication and transportation technologies, starting with the telegraph and locomotive and continuing through the Internet, jumbo jet, and beyond, have brought us increasingly closer together. These technologies enable us to maintain our relationships to one another more easily, and they encourage us to make new connections.
Underlying these connecting technologies is an infrastructure of roads, air routes, power lines, telephone cables, and a variety of electromagnetic wave transmitters and receivers. These systems allow people, electricity, and information to reach even the most remote areas of our country and our world with relative ease. They are vastly complex collections of elements and their connections. Because the elements and connections within a network interact in complicated ways, they exhibit system characteristics that are often unforeseeable when they are viewed simply as a large group of independent network components. Obviously, the way they interrelate makes a huge difference in the overall nature and capacity of the network.
Mathematicians view networks as fundamental objects of study. Networks, as a whole, exhibit behavior that is very difficult, if not impossible, to understand by studying the elements individually. Examples of this abound in the history of our nation's power grid. Small events, such as a single power line coming in contact with an overgrown tree, can set in motion a cascade of events that leads to large-scale power outages many miles away. That such events occur, despite multiple, built-in safety features that are designed to prevent these types of outcomes on a local scale, is a testament to the need to understand network behavior on a broader scale.
Networks are all around us. We are connected to each other, not only through physical links such as power lines, phone lines, and roads, but also through the less-tangible relationships of friendship, family, and business ties. We use a global information network, in the form of the Internet and World Wide Web, almost without thinking. Our connections give us access to information and opportunity.
We can use our understanding of networks to study life itself, on multiple scales. From networks of genes and proteins, to cellular structures, to ecosystems of predators and prey, we realize that living beings are not in any way solitary; they depend heavily on their interactions. Detailed understanding of the functioning of the web of life can help us make better decisions about the future of our planet, and of our species.
If one of the benefits of connectedness is that we are better able to work together, a drawback is that we are more susceptible to small disturbances. As is evident in the example of the failure of a solitary power line causing a huge blackout, small disturbances can rapidly, and unpredictably, grow into real dangers. One computer virus can quickly cripple a business, or an entire nation. A biological virus can spread so rapidly in today's era of broadly affordable airfare that a global pandemic can envelop us before we know it is happening. Terrorists of both the real and cyber-worlds can use the de-centralizing properties of networks not only to mount an attack, but also to evade detection.
Analyzing networks mathematically is a way to understand the complicated world around us. In this unit we will learn a bit about the history and fundamental ideas of the subject. We will start with Euler's early study and his approach to the problem of the Königsberg bridges. Then we will travel through the random networks of Paul Erdös and the small worlds of Duncan Watts and Steve Strogatz. We will then explore the "rich-get-richer" world of the scale-free network. Finally, we will take a look at the emerging study of dynamic networks. Throughout this exploration, we will study both basic ideas and examples of networks in action. By the end, we will have caught a glimpse of some of the networks that are such pervasive influences in our daily lives.