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Mathematics Illuminated

Connecting with Networks

Virtually everything we experience — in nature as well as human activity — involves a series of connections that link one thing to another. Networks, you might say, make the world go 'round.

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Network Graph

Connections can be physical, as with bridges, or immaterial, as with friendships. Both types of connections can be understood using the same mathematical framework called network theory, or graph theory, which is a way to abstract and quantify the notion of connectivity. This unit looks at how this branch of mathematics provides insights into extremely complicated networks such as ecosystems.


Unit Goals

  • Networks can be represented by graphs, which can be analyzed mathematically.
  • A graph is a set of elements along with another set that defines how the elements are connected.
  • The degree of a node is how many connections it has.
  • A path is a sequence of edges connecting two nodes.
  • A connected component of a graph is a maximal collection of nodes and edges that are mutually connected.
  • Random graphs can undergo “connectivity avalanches” during construction.
  • Distance on a graph is a measure of the fewest number of edges needed to travel between two given nodes.
  • The clustering coefficient is a measure of how many of a node’s neighbors are connected to each other (e.g., the fraction of a given individual’s friends who are also friends with each other).
  • Small-world networks have higher-than-expected clustering coefficients and short mean distances.
  • Scale-free networks follow a power law when describing the distribution of degrees.





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