# Sociology and CS. Small World. Sociology Problems. Degree of Separation. Milgram s Experiment. How close are people connected? (Problem Understanding)

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1 Sociology Problems Sociology and CS Problem 1 How close are people connected? Small World Philip Chan Problem 2 Connector How close are people connected? (Problem Understanding) Small World Are people closely connected, not closely connected, isolated into groups, Problem 1 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 1

2 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 Is a connection in this network different from the one in Milgram s experiment? Problem Formulation Given (input) Find (output) Simplification don t care about 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 ->? 2

3 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 Ideas? Algorithm Shortest Path Dijkstra s algorithm Algorithm Shortest Path Dijkstra s algorithm Limitations? Limitations? Single-source All-pair Shortest Path Floyd s algorithm This could be an overkill, why? Algorithm Unweighted edges Each edge has the same weight of 1 Algorithm Breadth-first search (BFS) Simpler algorithm? 3

4 Algorithm Breadth-first search (BFS) Algorithm Breadth-first search (BFS) Data structure to remember visited vertices Data structure to remember visited vertices Single source; repeat for each vertex to start Algorithm Breadth-first search (BFS) Data structure to remember visited vertices Implementation Which data structure to represent a graph (vertices and edges)? Single source; repeat for each vertex to start ShortestPath(x,y) = shortestpath(y,x) Implementation Which data structure to represent a graph (vertices and edges)? Adjacency matrix Adjacency list Tradeoffs? Implementation Which data structure to represent a graph (vertices and edges)? Adjacency matrix Adjacency list Tradeoffs? Time Space 4

5 Adjacency Matrix vs List Time Speed of what? Adjacency Matrix vs List Time Speed of key operations in the algorithm Algorithm: Key operation: Adjacency Matrix vs List Time Speed of key operations in the algorithm Algorithm: BFS Key operation: identifying children Adjacency Matrix vs List Time Speed of key operations in the algorithm Algorithm: BFS Key operation: identifying children Space Amount of data in the problem Adjacency Matrix vs List Time Speed of key operations in the algorithm Algorithm: BFS Key operation: identifying children Connector Space Amount of data in the problem Number of people/vertices Number of friends/edges each person has Problem 2 5

6 Revolutionary War Spreading the word that the British is going to attack Paul Revere vs William Dawes Revere was more successful than Dawes History books remember Revere more (Problem understanding) The person with the most friends? The person with the most friends? Phone book experiment 250 random surnames Number of friends with those surnames The person with the most friends? Phone book experiment 250 random surnames Number of friends with those surnames The person with the most friends? How to formulate it into a graph problem? Number of friends have a wide range Random sample: Conference in Princeton:

7 The person with the most friends? How to formulate it into a graph problem? Output: the vertex with the highest degree The person with the most friends? Are all friends equal? The person with the most friends? Are all friends equal? You have 100 friends Michelle Obama has only one friend: Barack Obama, who has a lot of friends Not just how many, but who you know Milgram s Experiment 24 letters get to the stockbroker at home 16 from Mr. Jacobs The rest get to the stockbroker at work Majority from Mr. Brown and Mr. Jones Overall, half of the letters came through the three people But Milgram started from a random set of people What does this suggest? Milgram s Experiment Average degree of separation is six, but: A small number of special people connect to many people in a few steps Small degree of separation The rest of us are connected to those special people Called Connectors by Gladwell Getting a Job experiment Mark Granovetter, sociologist Experiment in %: formal means advertisements, headhunters 20%: apply directly 56%: personal connection 7

8 Getting a Job experiment Personal connection 17%: see often 56%: see occasionally 28%: see rarely What does this suggest? Getting a Job experiment Personal connection 17%: see often friends 56%: see occasionally acquaintances 28%: see rarely almost strangers? What does this suggest? Getting jobs via acquaintances Why? Getting a Job experiment Personal connection 17%: see often friends 56%: see occasionally acquaintances 28%: see rarely almost strangers? What does this suggest? Getting jobs via acquaintances connect you to a different world might have a lot connections The Strength of Weak Ties Connector How many friends does one have? What kind of friends does one have? How do you find Connectors? How do you formulate it into a graph problem? Problem Formulation Given (input) People -> vertices Connections -> edges Find (output) Person with the best Connector score Part of the algorithm is to define the Connector score Simplification Don t care about how strong/long/ the friendships/connections are Algorithm 1: Connector Score Friends who are closer have higher scores Friends of distance 1, score =? distance 2, score =? distance 3, score =? distance d, score =? 8

9 Algorithm 1: Connector Score Friends who are closer have higher scores Algorithm 1: Adding the scores How to enumerate the people so that we can add the scores? Friends of distance 1, score =? distance 2, score =? distance 3, score =? distance d, score = 1/d, 1/d 2, Algorithm 1: Adding the scores How to enumerate the people so that we can add the scores? BFS Algorithm 1: Adding the scores How to enumerate the people so that we can add the scores? BFS Is score(x, y) the same as score(y, x)? Algorithm 2: Connector Score Degree of separation (number of connections) to other people is small Algorithm 2: Connector Score Degree of separation (number of connections) to other people is small Connector score: Ideas? Connector score: Average degree of separation from a person to every other person 9

10 Algorithm 2: Adding the scores How to enumerate the people so that we can add the scores? Algorithm 1 vs 2 How do you compare the two algorithms? Algorithm 1 vs 2 How do you compare the two algorithms? Algorithm 1 vs 2 How do you compare the two algorithms? Changing Algorithm 1 slightly will yield Algorithm 2, how? Changing Algorithm 1 slightly will yield Algorithm 2, how? Algorithm 1 is more flexible, why? But? Algorithm 3: Connector Score bridge If the person is not there, it takes a longer path for two people to connect Algorithm 3: Connector Score bridge If the person is not there, it takes a longer path for two people to connect Connector Score: Ideas? Connector Score ( betweenness ): Number of times the person appears on the shortest path between all pairs For one pair, what if multiple shortest paths of the same length (ties)? 10

11 Algorithm 3: Connector Score bridge If the person is not there, it takes a longer path for two people to connect Connector Score ( betweenness ): Number of times the person appears on the shortest path between all pairs Algorithm 3: Connector Score bridge If the person is not there, it takes a longer path for two people to connect Connector Score ( betweenness ): Number of times the person appears on the shortest path between all pairs For one pair, what if multiple shortest paths of the same length exist (ties)? Fractional score for each person/vertex Summary Problem 1: Degree of Separation how close are we from each other? Can Sociology Help CS? Problem 2: Connector who is the most connected? Algorithm 1: score = 1/d Algorithm 2: score = degree of separation Length (not vertices) of shortest path is needed Algorithm 3: score = betweenness Vertices (not length) on the shortest path are needed Search Engines How do they rank pages? Search Engines How do they rank pages? Content: words Most search engines in 1990 s 11

14 Reading Assignment Handout on Representation of Spatial Objects P. Rigaux, M. Scholl & A. Voisard Spatial Databases with Application to GIS Morgan Kaufmann, 2002 How and why does the Dijkstra s shortest path algorithm work? 14

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