Prof. Götz Pfeiffer

School of Mathematics, Statistics and Applied Mathematics

NUI Galway

Provide answers to the problems in the boxes provided. Marks will be awarded for participation and engagement.

When finished, print this notebook into a **pdf** file and submit this to
**blackboard**.

**Deadline** is next Monday at 5pm.

This is a `jupyter`

notebook. Find an environment that allows you to work with it. You can either
install `jupyter`

as a python packag on your own laptop or PC. Or you can use a suitable website
on the internet, such as nbviewer and `binder`

.

The following packages need to be loaded. In order to execute the code in a box, use the mouse or arrow keys to highlight the box and then press SHIFT-RETURN.

Should it ever happen that the notebook becomes unusable, start again with a fresh copy.

In [ ]:

```
import networkx as nx
import matplotlib.pyplot as plt
```

The purpose of this task is to get you used to working with the `networkx`

package
in the `jupyter`

notebook environment.

- Define a new (simple) graph
`G`

on the vertex set $X = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ with edges $0-1$, $1-2$, $2-3$, $3-4$, $4-5$, $5-6$, $6-7$, $7-8$, $8-9$, and $9-0$. Draw the graph. Hence or otherwise determine its**order**(the number of nodes) and its**size**(the number of links).

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- Find the
**adjacency matrix**`A`

of the graph`G`

. Then compute its square, $A^2$, and draw the graph`G2`

that has $A^2$ as its adjacency matrix. What are the connected components of`G2`

?

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```

For the affilliation network below, with six people labelled $A$ to $F$, and three foci labelled $X$, $Y$ and $Z$, draw the projection on (just) the people, in which two people are joined by an edge if they have a common focus.

(Of course, one can do this easily by hand. It would be nice to get
`networkx`

to do it for you.)

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The **social graph** of a node $x$ in a (social) network
is the **induced subgraph** on the set of friends of $x$
(that is the graph which has (only) the friends of $x$
as its vertices, and between them all the edges from the
original network).
The **clustering coefficient** of $x$ is the density
$m / \binom{n}{2}$
of the social graph of $x$, the proportion its number of edges,
$m$, and its potential number of edges, $\binom{n}{2} = \frac12 n(n-1)$,
where $n$ is its number of vertices.

MathSciNet describes the social network of mathematical researchers defined by collaboration.

**Pick** a (local) mathematician with at least $10$
friends (i.e., co-authors), determine their social graph
and hence compute their clustering coefficient.

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Define a graph `I`

on the $32$ counties of Ireland by
joining two counties whenever they have a common border.
(A list of county names, suitable for cut-and-paste,
can be found on the internet)

What is the order and the size of the resulting graph?

In terms of centrality measures, what are the $3$ most central counties, for

- degree centrality?
- eigenvector centrality?
- closeness centrality?
- betweenness centrality?

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```