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Section 8 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvardPowerPoint Presentation

Section 8 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard

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Section 8 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard

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### Section 8 – Ec1818Jeremy Barofskyjbarofsk@hsph.harvard.edu

March 31st and April 1st, 2010

Section 8 Outline (lectures 15, 16)

- Social Network Introduction
- Types of Networks / Graphs
- Random
- Regular
- Small-world

- Erdos / Bacon Numbers
- Review Questions?
- Evaluations
- Office Hours - Thursday, 4/1/10 10-11am, outside 320 CGIS North.

Small World Phenomenon - Milgram

- Question: Probability that two randomly selected people know each other?
- Design: In 1967, Milgram sent packages to 160 random people living in Omaha NE asking them to send the package to a friend or acquaintance they thought might know or be connected to the final individual – a stock broker in Boston. (Postcards also sent back to Harvard to track progress).
- Results:
1) Of those letters that found destination, average path length 5.5-6.

2) Significant selection bias – in one experiment 232 of 296 were not sent on.

3) Most of cards given to target through a few people. Experiment with 160 packages sent, 24 reached target at his home and 16 of these were given to target by one person (nodes in network).

-Reasons for under-estimate or over-estimate of avg. path length?

Social networks

- A graph G consists of a set V(G) of vertices (or nodes) together with a set of edges E(G) (or links) that connect vertices.
- Degree: number of edges connected to a given vertex.
- Order: the number of vertices V(G) in graph G represent its order.
- Size: the number of edges E(G) in G represents its size.
- Directed graph / undirected graph: graph is directed if all its edges are directional, ie- the network tells us not just whether people are friends but whether each person considers the other a friend. If none of edges are directional, then graph G is undirected.

Social Networks Metrics

- Characteristic path length L(G, p): measures average distance between vertices. By distance we mean the shortest path that connects vertices v and v’.
- Clustering coefficient C(G, p): Measures a vertex / person’s level of cliquishness within its neighborhood. Answers – are the friends of my friends, my friends also?
- Formally C(G, p)= actual edges in network within its neighborhood / maximum possible edges in that neighborhood.
- Maximum number of graph edges / number of connections in social network: n(n-1)/2 where n = number of vertices.

Types of Graphs

- Regular Network: each vertex is connected to same number k of their nearest neighbors only. All vertices have the same degree. Long characteristic path length because takes a long time to get from one vertex to another, large clustering coefficient because vertices connected to all other nearby vertices.
- Random Network: Edges between vertices occur randomly with prob. = 1/V(G). Full connectedness occurs non-linearly when Pr(connection) = 1/V(G). Small characteristic path length and clustering coefficient.
- Adjacency matrix: Way to represent network data with each row/ column representing whether those vertices have a connection.

Regular -> Small World -> Random Graphs

- Rewire: Start with a regular graph with vertices in a circle and each connected to 4 closest neighbors. Rewire each edge at random with probability p.
- Changing p means tunes graph such that p = 0 defines a regular graph, p = 1 random.
- Watts and Strogratz define small-world networks with two characteristics:
- Large Clustering Coefficient C(G, p) – most of my neighbors are friends and friends with me too.
- Small Characteristic Path Length L(G, p) – Presence of random, long-distance connections mean that moving from one part of the graph to the other can be done quickly.

Characteristic path length L(p) and clustering coefficient C(p) for rewired graphs as p varies. (Watts and Strogatz, 1998)

Empirical Examples (Watts and C(p) for rewired graphs as p varies. (Watts and Strogatz, 1998)Strogatz, 1998)

Power Laws Again? Are you serious? C(p) for rewired graphs as p varies. (Watts and Strogatz, 1998)(Random means normal distribution and small world means power law)

Erdos and Bacon Numbers C(p) for rewired graphs as p varies. (Watts and Strogatz, 1998)

- Small World Networks exhibit strong connections between neighbors (cliques) but information can still travel quickly because of random connections to other highly connected groups of vertices.
- Erdos/ Bacon numbers: Level of connection in peer-reviewed journal articles or movie credits. Bacon number of 1 means individual acted in same movie as Kevin Bacon. Nearly all actors connected in this way – exhibits characteristics of small world networks.

Is Bacon Best? C(p) for rewired graphs as p varies. (Watts and Strogatz, 1998)

- “By processing all of the 1.6 million people in the Internet Movie Database I discovered that there are currently 506 people who are better centers than Kevin Bacon!” –Oracle of Bacon website.
- Compute average Bacon number and compare to others.

Midterm Questions? IMD.