slides/2014-04-10-Graphs.html

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name: inverse
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---

# Graphs & Networks

---

## Graphs

  + Graphs (or networks) can model a surprising number of domains
  + Modeling with graphs opens up a large number of algorithms
  + Linear Algebra has many connections to graphs

???

## Math

  + Data mining theme: get your problem stated as a math problem, whole slew of
    solutions present themselves
  + Linear Algebra is really useful for running equations of all nodes, or
    simulate moving across network

---

## Vertices & Edges

  + Vertex
    + the interconnected objects, or nodes
  + Edge
    + the lines or curves that connect vertices
  + Graph
    + Collection of vertices and edges ```G = (V,E)```

.white-background[
<img src="img/GraphNodesEdges.png" width=100% />
]

???

## Definitions

  + These are the abstract terms, how do they relate to the real world?

---

## Examples

  + Vertex
    + User, building, router, product
  + Edge
    + Relationship, road, network cable, purchased
  + Graph
    + Social Network, physical infrastructure, Internet, purchasing
    history

???

## Examples

  + Many graphs have assumed edge labels
    + the edges represent something
    consistent
  + Some graphs have multiple types of edges
    + relationship is one of family,
    friend, co-worker, etc.
  + Edge can be anything that ties two things together
    + purchase history, e.g.,
    is not a physical thing connecting, but an idea

---

## Social Networks

  + An edge connects two people
  + If this is just a line, what information are we missing about how the link
    was formed?

<img src="img/jretz-linkedin.png" width=100% />

???

## Symmetric vs. Asymmetric Edges

  + "Just a line" is called *undirected*
  + We're missing information about who invited whom. That would be expressed
    as an arrow and is called *directed*.

---

## Definitions

  + Directed: Connections have a direction. Invitations, water pipes, email.
  + Undirected: Connections have no direction. Friends, walkways on campus,
    physical wires.
  + Cycle: Set of nodes and edges in which you can travel back to a vertex
  + Acyclic: A graph without any cycles

<img src="img/Directed_acyclic_graph.png" width=60% />

???

## Modeling

  + Can always model an undirected graph as a directed graph by having two
    directed edges going in opposite directions in place of each undirected
    edge.

---

## Acyclic?

  + Social network (undirected)
  + Product purchases (directed)
  + Internet links (directed)
  + Class prerequisites (directed)

???

## Answers

  + Social network: cyclic
  + Product purchases: acyclic
  + Internet links: cyclic
  + Class prerequisites: acyclic

---

## Bipartite

.left-column[

  + Graph whose vertices can be divided into two distinct sets
  + Vertices in ```U``` are only connected to those in ```V```, vice versa
  + Product purchases: users ```U```, products ```V```
]

.right-column[
.white-background[
<img src="img/Simple-bipartite-graph.svg.png" width=100% />
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???

## Recommendations

  + Can model recommendations as link following:
  + From a user, follow to products
  + From products, follow back to other users
  + From other users, follow back to products

---

## Measurements

  + Geodesic distance
    + Number of edges to connect two vertices
  + Eccentricity of ```v```
    + Largest geodesic distance from ```v``` to the most distant vertex
  + Radius
    + Minimum eccentricity of any vertex in the graph
  + Diameter
    + Maximum eccentricity of any vertex in the graph
  + Peripheral vertex
    + Vertex with eccentricity == diameter
  + Incoming/Outgoing edge count of a vertex
    + Number of edges coming in/out of a vertex

???

## Data Stats

  + Similar to getting distribution stats from initial datasets, these
    measurements can help you understand graphs as a summary
  + Once you have the incoming/outgoing edge counts, you can use regular stats:
    what is the distribution of counts?

---

## Examples

.white-background[
<img src="img/6n-graf.svg.png" width=100% />
]

???

## Answers

  + Distance 6, 5: 2
  + Eccentricity 2: 3 (disconnected graph is infinity)
  + Radius: 2
  + Diameter: 3
  + Peripheral Verticies: 1, 2, 6

---

## Connections

  + Connected
    + there exists a path from one vertex to another
  + Connectivity
    + minimum number of vertices to remove to disconnect remaining
    vertices
  + Clustering Coefficient
    + Measure of how connected a vertex or group of vertices are

???

## Robustness

  + This is used to understand robustness of a system: if an earthquake
    damaged the Bay Bridge, could we still travel from one point to another?
  + What is the connectedness of Oakland and SF?
  + Closely related to min-cuts, which is discussed in the book
  + Network topology: what happens if a router fails?

---

## Clustering Coefficient

  + How many directed edges are possible between 3 vertices?
  + 4 vertices?
  + ```v*(v-1)```
  + Undirected?
  + ```v*(v-1)/2```
  + Clustering Coefficient: Ratio of actual edges to possible edges amongst neighbors

???

## Reading

  + Used in Reading this week
  + ```v*(v-1)``` connection to every other node but yourself
  + ```/2``` undirected, don't double count connections

---

## Example

.center[
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<img src="img/6n-graf.svg.png" width=90% />
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]

???

## Answer

  + Clustering Coefficient of 1:
    + Neighbors of 1: 5 2
    + 2*(2-1) / 2 = 1
    + Actual links = 1
  + Clustering Coefficient of 4:
    + Neighbors of 4: 3,5,6
    + 3*(3-1) / 2 = 3
    + Actual: 0
    + If 3-5 connected? 1/3

---

## Random Walk

  + Many algorithms based on concept of randomly deciding:
    + Follow link or not
    + Which link to follow
  + Simulate the decision many times
  + What is the probability you will wind up on ```u``` from ```v```?

---

# *Break*

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