- Course ID: CMPLXSYS 530
- Course Title: Computer Modeling of Complex Systems
- Term: Winter 2015
- Schedule: Wednesdays and Friday, 1:00-2:30PM ET
- Location: 120 West Hall (http://www.lsa.umich.edu/cscs/research/computerlab)
- Teachers: Mike Bommarito and Sarah Cherng

In this notebook, we'll explore one of the most popular types of environments - a network. In the cells below, we'll be creating a small-world network and simulating a disease outbreak.

In the import section below, you'll see an important new addition: **networkx**. We'll be using **networkx** as our preferred package for creating and managing network objects; we'll include links to more info and documentation in later cells.

In [98]:

```
%matplotlib inline
# Imports
import networkx as nx
import numpy
import matplotlib.pyplot as plt
import pandas
import seaborn; seaborn.set()
# Import widget methods
from IPython.html.widgets import *
```

In some problems, we can define our own rules to *"grow"* or *"generate"* a network. However, for many problems, we may want to re-use an existing methodology. **networkx** comes with a suite of methods to "sample" random graphs, such as:

- Trees, e.g., balanced trees (networkx.balanced_tree)
- Erdos-Renyi graphs (networkx.erdos_renyi_graph)
- Watts-Strogatz small-world graphs (networkx.watts_strogatz_graph)
Bipartite graphs

For a full list, see this page on Graph Generators in the networkx documentation.

In the sample below, we'll create a Newman-Watts-Strogatz small-world graph and print some basic info about the graph.

In [99]:

```
# Create a random graph
nodes = 30
edges = 2
prob_out = 0.2
g = nx.newman_watts_strogatz_graph(nodes, edges, prob_out)
print((g.number_of_nodes(), g.number_of_edges()))
```

As humans, we are wired for visuals; for networks, visualizations can provide an important first characterization that isn't apparent from simple statistics.

In the cell below, we show how to calculate a "layout" for a network and visualize it using networkx.

For more examples, see the Drawing Graphs section of the NetworkX tutorial.

In [100]:

```
# Draw the random graph
g_layout = nx.spring_layout(g, iterations=1000)
nx.draw_networkx(g, pos=g_layout, node_color='#dddddd')
```

Next, we'll simulate a very simple outbreak in a network. Our disease will start by infecting a random patient, then pass with probability 1 to any connected individuals.

To find the "connected" individuals, we'll need to learn how to find the neighbors of a node. NetworkX makes this easy.

In [101]:

```
# Let's pick a node at random to infect
patient_zero = numpy.random.choice(g.nodes())
patient_zero
healthy_nodes = g.nodes()
healthy_nodes.remove(patient_zero)
```

In [102]:

```
# Now we can visualize the infected node's position
f = plt.figure()
nx.draw_networkx_nodes(g, g_layout,
nodelist=[patient_zero],
node_color='red')
nx.draw_networkx_nodes(g, g_layout,
nodelist=healthy_nodes,
node_color='#dddddd')
nx.draw_networkx_edges(g, g_layout,
width=1.0,
alpha=0.5,
edge_color='#111111')
_ = nx.draw_networkx_labels(g, g_layout,
dict(zip(g.nodes(), g.nodes())),
font_size=10)
```

In our simple disease outbreak model, each step of the model consists of the following:

- for each infected individual, find their neighbors
infect all neighbors

For our first step, we start with a single infected person; in subsequent steps, we need to use a

*for*loop to handle all currently infected persons.

In [103]:

```
# Find patient zero's neighbors
neighbors = g.neighbors(patient_zero)
neighbors
# Let's infect all of his neighbors!
infected_patients = [patient_zero]
infected_patients.extend(neighbors)
infected_patients
# Remove the infected from healthy nodes
healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]
```

In [104]:

```
# Now we can visualize the infected node's position
f = plt.figure()
nx.draw_networkx_nodes(g, g_layout,
nodelist=infected_patients,
node_color='red')
nx.draw_networkx_nodes(g, g_layout,
nodelist=healthy_nodes,
node_color='#dddddd')
nx.draw_networkx_edges(g, g_layout,
width=1.0,
alpha=0.5,
edge_color='#111111')
_ = nx.draw_networkx_labels(g, g_layout,
dict(zip(g.nodes(), g.nodes())),
font_size=10)
```

As mentioned above, we now need to use a *for* loop over all individuals.

In [105]:

```
# Now let's infect the neighbors of all infected patients!
newly_infected = []
for infected_patient in infected_patients:
# Find patient zero's neighbors
neighbors = [neighbor for neighbor in g.neighbors(infected_patient) if neighbor not in infected_patients]
newly_infected.extend(neighbors)
newly_infected
```

Out[105]:

In [106]:

```
# Update infected and healthy
infected_patients.extend(newly_infected)
# Remove the infected from healthy nodes
healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]
```

In [107]:

```
# Now we can visualize the infected node's position
f = plt.figure()
nx.draw_networkx_nodes(g, g_layout,
nodelist=infected_patients,
node_color='red')
nx.draw_networkx_nodes(g, g_layout,
nodelist=healthy_nodes,
node_color='#dddddd')
nx.draw_networkx_edges(g, g_layout,
width=1.0,
alpha=0.5,
edge_color='#111111')
_ = nx.draw_networkx_labels(g, g_layout,
dict(zip(g.nodes(), g.nodes())),
font_size=10)
```

Let's wrap our model step method in yet another *for* loop, allowing us to simulate multiple steps of the model at once.

In [108]:

```
# Now let's infect the neighbors of all infected patients!
model_steps = 5
# Iterate over steps
for i in range(model_steps):
# Iterate over infected
newly_infected = []
for infected_patient in infected_patients:
# Find patient neighbors and infect
neighbors = [neighbor for neighbor in g.neighbors(infected_patient) if neighbor not in infected_patients]
newly_infected.extend(neighbors)
# Remove the infected from healthy nodes
infected_patients.extend(newly_infected)
healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]
# Now we can visualize the infected node's position
f = plt.figure()
nx.draw_networkx_nodes(g, g_layout,
nodelist=infected_patients,
node_color='red')
nx.draw_networkx_nodes(g, g_layout,
nodelist=healthy_nodes,
node_color='#dddddd')
nx.draw_networkx_edges(g, g_layout,
width=1.0,
alpha=0.5,
edge_color='#111111')
_ = nx.draw_networkx_labels(g, g_layout,
dict(zip(g.nodes(), g.nodes())),
font_size=10)
```