### Summary

Students learn about complex networks and how to use graphs to represent them. They also learn that graph theory is a useful part of mathematics for studying complex networks in diverse applications of science and engineering, including neural networks in the brain, biochemical reaction networks in cells, communication networks, such as the internet, and social networks. Students are also introduced to random processes on networks. An illustrative example shows how a random process can be used to represent the spread of an infectious disease, such as the flu, on a social network of students, and demonstrates how scientists and engineers use mathematics and computers to model and simulate random processes on complex networks for the purposes of learning more about our world and creating solutions to improve our health, happiness and safety.### Engineering Connection

Investigators from diverse disciplines of science and engineering use graphs and random processes on graphs to effectively describe complex networked systems, such as the human brain or the internet. Networks composed of biochemical reactions in cells that keep humans alive (or cause diseases when they malfunction, such as cancer) are also studied by biomolecular engineers using graphs. The same is true for networks of interacting individuals in a social environment, such as Facebook friends, who are studied by software engineers. It turns out that understanding the mathematics behind graph theory can lead biomedical engineers to develop better cancer treatments, and electrical and computer engineers to create faster and more reliable communication networks among electronic devices. Moreover, it can help bioengineers and neuroscientists discover the secrets of how human consciousness arises from the complex interactions of billions of neurons in the human brain, help social engineers figure out how ideas are formed from social interactions, and public health scientists understand how contagious diseases, such as the flu, spread by social contact.

### More Curriculum Like This

**Processes on Complex Networks**

Building on their understanding of graphs, students are introduced to random processes on networks. They walk through an illustrative example to see how a random process can be used to represent the spread of an infectious disease, such as the flu, on a social network of students.

**Sets-Nodes-Edges: Representing Complex Networks in Graph Theory**

Students learn about complex networks and how to represent them using graphs. They also learn that graph theory is a useful mathematical tool for studying complex networks in diverse applications of science and engineering, such as neural networks in the brain, biochemical reaction networks in cells...

**Making the Connection**

Students learn and apply concepts and methods of graph theory to analyze data for different relationships such as friendships and physical proximity. They are asked about relationships between people and how those relationships can be illustrated.

**Curb the Epidemic!**

Using a website simulation tool, students build on their understanding of random processes on networks to interact with the graph of a social network of individuals and simulate the spread of a disease. They decide which two individuals on the network are the best to vaccinate in an attempt to minim...

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Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

###### Common Core State Standards - Math

- Represent data with plots on the real number line (dot plots, histograms, and box plots). (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### International Technology and Engineering Educators Association - Technology

- Systems, which are the building blocks of technology, are embedded within larger technological, social, and environmental systems. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### State Standards

###### Florida - Science

- Explain how scientific knowledge and reasoning provide an empirically-based perspective to inform society's decision making. (Grades 9 - 12 ) Details... View more aligned curriculum... Do you agree with this alignment?

###### National Council of Teachers of Mathematics - Math

- use representations to model and interpret physical, social, and mathematical phenomena (Grades Pre-K - 12 ) Details... View more aligned curriculum... Do you agree with this alignment?

### Unit Overview

This unit is composed of two lessons, each with an associated activity, and requires four 45-minute class periods to conduct.

**Lesson 1 **introduces the topics of complex networks and graph theory by presenting information on set theory and graphs and by defining the degree of a node and the degree distribution of a graph. Students then interact with their classmates and document the interactions to create their own example of a social network. Then they apply what they learned about networks and graphs and make calculations to analyze the social network.

**Lesson 2** introduces the topics of random processes on networks and modeling the spread of an infecting disease, using the SIR (susceptible, infectious, resistant) model of epidemiology. Students then use a free online simulation tool to interact with the graph of a social network and simulate the spread of a flu virus. They run multiple simulations, and then analyze the effectiveness of their vaccination strategies.

### Unit Schedule

- Day 1: Sets-Nodes-Edges: Representing Complex Networks in Graph Theory lesson
- Day 2: Start Networking! activity
- Day 3: Processes on Complex Networks lesson
- Day 4: Curb the Epidemic! activity

### Contributors

Garrett Jenkinson and John Goutsias, The Johns Hopkins University, Baltimore, MD; Debbie Jenkinson and Susan Frennesson, The Pine School, Stuart, FL### Copyright

© 2013 by Regents of the University of Colorado; original © 2012 The Johns Hopkins University### Supporting Program

Complex Systems Science Laboratory, Whitaker Biomedical Engineering Institute, The Johns Hopkins University### Acknowledgements

The generous support of the National Science Foundation, Directorate for Computer and Information Science and Engineering (CISE), Division of Computing and Communication Foundations (CCF), is gratefully acknowledged.

Last modified: March 17, 2018

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