# Day 7: Network Analysis¶

## Announcements¶

• No class on Monday (2/15)
• Individual (wireless sensor network) project assigned (due 2/29)
• Team project 2 will involve machine learning in the context of financial data (assigned 2/29 and due 3/28)

## Slide feedback¶

Each team will be responsible for providing detailed feedback to one of the other teams. At the start of class on Wednesday, each team will give Johnny their written evaluation so that he can briefly review them before giving the evaluations to the team being evaluated.

Johnny will be reviewing the evaluations to make sure that they are detailed and helpful. We will be incorporating your evaluations in your team’s grade for this project.

## Network Analysis¶

### Examples¶

• Social: Friend networks in the real world as well as online
• Infrastructure: Networks of roads; electrical grids
• Biological: gene-gene networks, networks of nerves cells in the brain

### Graphs¶

Networks are often modeled as graphs. A graph $$G$$ is an ordered pair of (disjoint) sets $$(V, E)$$ where $$E$$ is a subset of $$V \times V$$. We refer to the elements of $$V$$ as nodes or vertices and the elements of $$E$$ as edges. A vertex $$v \in V$$ is incident to an edge $$e \in E$$ if $$v \in e$$. Two nodes incident to the same edge are said to be joined by that edge. We also say that two such nodes are adjacent. Edges can be directed or undirected. In an undirected graphs, an edge between nodes $$u$$ and $$v$$ may be written as $$uv$$ or $$vu$$ (in such cases, we will not make any distinction between the symbols $$uv$$ and $$vu$$). We will only consider graphs without loops (i.e., for all $$v \in V$$, we have $$vv \notin E$$).

The number of nodes $$n = |V|$$ is the order of $$G$$. The size of $$G$$ is the number of edges $$m = |E|$$. The density of G is the proportion of actual edges to possible edges. For undirected graphs, the density is given by

$\frac{2m}{n(n-1)}$

and, for directed graphs, the density is

$\frac{m}{n(n-1)}.$

A path is a non-empty graph $$P = (V, E)$$ where

$\begin{split}V &= (v_0, v_1, \dots, v_k) \text{ and} \\ E &= (v_0v_1, v_1v_2, \dots, v_{k-1}v_k) \\\end{split}$

such that $$i \neq j$$ implies $$v_i \neq v_j$$. The length of a path is the number of edges in it. A graph is connected if there is a path between any two nodes.

For an undirected graph $$G$$, we have the following definitions:

$\begin{split}eccentricity(v) &= \max_u (\text{shortest path }uv) \\ radius(G) &= \min_v eccentricity(v) \\ diameter(G) &= \max_v eccentricity(v) \\ center(G) &= \{v \in G \; | \; eccentricity(v) = radius(G)\} \\ periphery(G) &= \{v \in G \; | \; eccentricity(v) = diameter(G)\}\end{split}$