What are networks (and why should you care)?

What are networks?

• Networks are a mathematical way of representing a set of objects and the relationships among them.
• These "objects" are termed nodes or vertices.
• These relationships are termed edges or, less often, links.
• Networks are also called graphs.

Example networks / applications

Connectome

More connectome

Enron anomaly prediction

Organizational communication

Faculty hiring and other hierarchies

Pandemic spread

Protein-protein interaction

What are NOT networks? - Hypergraphs

• Networks represent dyadic relationships: interactions between two things.
  • Example: an email from me to you
• Polyadic relationships (interactions between more than two things) are common.
  • Example: an email from me to you AND someone else
• We could ignore this:
  • Make an edge from me to you.
  • Make another, separate edge from me to someone else.
• Hypergraphs are a mathematical way of representing general polyadic relationships.

What are NOT networks? - Multigraphs

• Graphs (strictly speaking) usually have at most one edge between node $i$ and node $j$.
• There may be multiple relationships between two nodes in data that we want to model.
  • Example: An email from me to you, and a phone call from me to you.
• Sometimes we can compress this information into at most one edge, and still use a graph.
  • Example: Create an edge if there was an email OR a phone call.
• Multigraphs allow for more than one edge from node $i$ to node $j$.

Every time we represent something in the real world with a network, we're making a modeling choice

What can we do with networks? (i.e. What is this class about?)

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