Sociology and CS
Philip Chan
How close are people connected?
 Are people




closely connected,
not closely connected,
isolated into groups,
…
Degree of Separation
 The number of connections to reach another
person
Milgram’s Experiment
 Stanley Milgram, psychologist
 Experiment in the late 1960’s



Chain letter to gather data
Stockbroker in Boston
160 people in Omaha, Nebraska


Given a packet
Add name and forward it to another person who
might be closer to the stockbroker
 Partial “social network”
Small World
 Six degrees of separation
 Everyone is connected to everyone by a few
people—about 6 on the average.

Obama might be 6 connections away from you
 “Small world” phenomenon
Bacon Number
 Number of connections to reach actor Kevin
Bacon
 http://oracleofbacon.org/
 Is a connection in this network different from
the one in Milgram’s experiment?
Problem Formulation
 Given (input)


People
Connections/links/friendships
 Find (output)

the average number of connections between
two people
 Simplification

don’t care about how long/strong/… the
friendships are
Problem Formulation
 Formulate it into a graph problem
(abstraction)
 Given (input)


People
Connections
 Find (output)

the average number of connections between
two people
Problem Formulation
 Formulate it into a graph problem
(abstraction)
 Given (input)


People -> vertices
Connections -> edges
 Find (output)

the average number of connections between
two people -> ?
Problem Formulation
 Formulate it into a graph problem
(abstraction)
 Given (input)


People -> vertices
Connections -> edges
 Find (output)

the average number of connections between
two people -> average shortest path length
Algorithm
 Shortest Path

Dijkstra’s algorithm
 Limitations?
Algorithm
 Shortest Path

Dijkstra’s algorithm
 This could be an overkill, why?
Algorithm
 Unweighted edges

Each edge has the same weight of 1
 Simpler algorithm?
 Any ideas?
Breadth-First Search
 Start from the origin
 Explore people one link away (dist=1)
 Explore people two links away (dist=2)
…
 Until the destination is reach
Algorithm
 Breadth-first search (BFS)

Single origin, all destinations
Algorithm
 Breadth-first search (BFS)

Single origin, all destinations
 Repeat BFS for each origin

ShortestDistance(x,y) = shortestDistance(y,x)
Algorithm
 Breadth-first search (BFS)

Single origin, all destinations
 Repeat BFS for each origin


ShortestDistance(x,y) = shortestDistance(y,x)
Each subsequent BFS can stop earlier
BFS Algorithm—A Closer Look
 Add origin to the open list
 While open list is not empty
1.
2.
3.
Remove the first person from the open list
Mark the person as visited
Add non-visited neighbors of the person to
the open list
BFS Algorithm—A Closer Look
 Add (origin, 0) to the open list
 While open list is not empty
1.
2.
3.
Remove the first item (person, dist) from the
open list
Mark person as visited
For each non-visited neighbor of person

Add (neighbor, dist + 1) to the open list
Implementation
 Data Structures

What do we need to keep track of?



Whether we have visited some person before
An ordered list of persons to be visited
Implementation


Boolean visited[person]
Open list of to be visited people
 Queue
 First in first out (FIFO)
Queue Operations
 Enqueue

Add an item at the end
 Dequeue

Remove an item from the front
Queue Implementation 1
 Consider using an array
 How to implement the two operations?
 Enqueue

Update tail, add item at tail
 Dequeue
Queue Implementation 1
 Consider using an array
 How to implement the two operations?
 Enqueue

Update tail, add item at tail
 Dequeue


Remove the first item
Shift the rest of the items up, update tail
Additional Operations
 What do we need to check before Enqueue?
 What do we need to check before Dequeue?
Additional Operations
 What do we need to check before Enqueue?
 What do we need to check before Dequeue?
 Enqueue:

Is the queue full?
 Dequeue

Is the queue empty?
Queue Implementation 1(shifting
items)
 Is the queue empty?
Queue Implementation 1(shifting
items)
 Is the queue empty?

When tail is -1
 Is the queue full?
Queue Implementation 1(shifting
items)
 Is the queue empty?

When tail is -1
 Is the queue full?

When tail is length - 1
Number of assignments (Speed of
algorithm)
 N = number of items on the queue
 Count number of assignments

involving items in the queue
Number of assignments
(Implementation 1)
 Enqueue
Number of assignments
(Implementation 1)
 Enqueue

1 (assign new item at tail)
 Dequeue
Number of assignments
(Implementation 1)
 Enqueue

1 (assign new item at tail)
 Dequeue



1 (assign first item to somewhere else)
N-1 shifts (assignments)
Total N
Shifting items in Dequeue
 Quite a bit of work

if we have many items in the queue
 Can we avoid shifting?
 If so, how? Ideas?
Queue Implementation 2
 Consider using an array
 How to implement the two operations?
 Enqueue


Head and tail
Update tail, add at tail
 Dequeue
Queue Implementation 2
 Consider using an array
 How to implement the two operations?
 Enqueue


Head and tail
update tail, add at tail
 Dequeue


Get item from head
Update head
Queue Implementation 2 (head and
tail)
 Is the queue empty?
Queue Implementation 2 (head and
tail)
 Is the queue empty?

When head is larger than tail
 Is the queue full?
Queue Implementation 2 (head and
tail)
 Is the queue empty?

When head is larger than tail
 Is the queue full?

When tail is length - 1
Number of assignments
(Implementation 2)
 Enqueue
Number of assignments
(Implementation 2)
 Enqueue

1 (assign new item at tail)
 Dequeue
Number of assignments
(Implementation 2)
 Enqueue

1 (assign new item at tail)
 Dequeue

1 (assign item at tail to somewhere else)
What if tail reaches the end of
array
 But there is room in the front of the array?
 Any ideas to reuse the space in the front?
Queue Implementation 3
 Circular queue

Wrap around
 If we use head and tail (Implementation 2)

What needs to be changed?
Circular Queue Operations
 Length = length of array
 Enqueue
Circular Queue Operations
 Length = length of array
 Enqueue

If not full

If tail < length
 increment tail

Else
 tail = 0

Add item
 Dequeue
Circular Queue Operations
 Length = length of array
 Enqueue

If not full

If tail < length
 increment tail

Else
 tail = 0

Add item
 Dequeue

Similarly for head
Queue Implementation 3 (circular
queue)
 Is the queue empty?
Queue Implementation 3 (circular
queue)
 Is the queue empty?

When head is larger than tail

With wrap-around adjustment
 Is the queue full?
Queue Implementation 3 (circular
queue)
 Is the queue empty?

When head is larger than tail

With wrap-around adjustment
 Is the queue full?

When head is larger than tail

With wrap-around adjustment
 Same!!

How can we distinguish them?
Queue Implementation 3 (circular
queue)
 Is the queue empty?

When head is larger than tail

With wrap-around adjustment
 Is the queue full?

When head is larger than tail

With wrap-around adjustment
 Same!!

How can we distinguish them?

Keep track of size instead
Keeping track of queue size
 Enqueue: increment size
 Dequeue: decrement size
Summary
 Understand how people are connected
 Degrees of separation

shortest distance between two people
 Average degree of separation (community)
 Sum of shortest distance of all pairs / # of pairs
 Algorithm for shortest distance (distance 1 in all edges)
Breadth-first search (BFS)
 Different origins
 BFS


Open list: people to be explored

Queue
 Circular queue (implementation 3)