ICPSR - Complex Systems Models in the Social Sciences - Lecture 1 - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 3 - Professor Daniel Martin Katz
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Transcript of ICPSR - Complex Systems Models in the Social Sciences - Lecture 3 - Professor Daniel Martin Katz
Daniel Martin KatzMichigan State University
College of Law
Complex Systems Models in the Social Sciences
(Lecture 3)
Back to Where We Ended Our Last Class
Stanley Milgram’s Other Experiment
Milgram was interested in the structure of society
Including the social distance between individuals
While the term “six degrees” is often attributed to milgram it can be traced to ideas from hungarian author Frigyes Karinthy
What is the average distance between two individuals in society?
Stanley Milgram’s Other Experiment
NE
MA
Six Degrees of Separation?
NE
MA
Target person worked in Boston as a stockbroker
296 senders from Boston and Omaha.
20% of senders reached target.
Average chain length = 6.5.
And So the term ... “Six degrees of Separation”
Six Degrees
Six Degrees is a claim that “average path length” between two individuals in society is ~ 6
The idea of ‘Six Degrees’ Popularized through plays/movies and the kevin bacon game
http://oracleofbacon.org/
Six Degrees of Kevin Bacon
Visualization Source: Duncan J. Watts, Six Degrees
Six Degrees of Kevin Bacon
But What is Wrong with Milgram’s Logic?
150(150) = 22,500
150 3 = 3,375,000
150 4 = 506,250,000
150 5= 75,937,500,000
The Strength of ‘Weak’ Ties
Does Milgram get it right? (Mark Granovetter)
Visualization Source: Early Friendster – MIT Network
www.visualcomplexity.com
Strong and Weak Ties (Clustered
v. Spanning)
Clustering ---- My Friends’ Friends are also likely to be friends
So Was Milgram Correct?
Small Worlds (i.e. Six Degrees) was a theoretical and an empirical Claim
The Theoretical Account Was Incorrect
The Empirical Claim was still intact
Query as to how could real social networks display both small worlds and clustering?
At the Same time, the Strength of Weak Ties was also an Theoretical and Empirical proposition
Watts and Strogatz (1998)
A few random links in an otherwise clustered graph yields the types of small world properties found by Milgram
“Randomness” is key bridge between the small world result and the clustering that is commonly observed in real social networks
Watts and Strogatz (1998)
A Small Amount of Random Rewiring or Something akin to Weak Ties—Allows for Clustering and Small Worlds
Random Graphlocally Clustered
Different Form of Network Representation
1 mode
2 mode
2 mode
Actors and
Movies
Different Forms of Network Representation
1 mode
Actor to Actor
Could be Binary (0,1)
Did they Co-Appear?
Different Forms of Network Representation
Different Forms of Network Representation
1 mode
Actor to Actor
Could also beWeighted
(I.E. Edge Weights by Number of
Co-Appearences)
Features of Networks
Mesoscopic Community StructuresWe will discuss these next week
Macroscopic Graph Level PropertiesWe will discuss these today
Microscopic Node Level Properties We will discuss these Next week
Macroscopic Graph Level Properties
Degree Distributions (Outdegree & Indegree)
Clustering Coefficients
Connected Components
Shortest Paths
Density
Shortest Paths
Shortest Paths
The shortest set of links connecting two nodes
Also, known as the geodesic path
In many graphs, there are multiple shortest paths
Shortest Paths
Shortest Paths
A and C are connected by 2 shortest paths
A – E – B - C
A – E – D - C
Diameter: the largest geodesic distance in the graph
The distance between A and C is the maximum for the graph: 3
Shortest Paths
In the Watts -Strogatz Model Shortest Paths are reduced by increasing levels of random rewiring
Clustering Coefficients
Clustering Coefficients
Measure of the tendency of nodes in a graph to cluster
Both a graph level average for clustering
Also, a local version which is interested in cliqueness of a graph
Density
Density = Of the connections that could exist between n nodes
directed graph: emax = n*(n-1)!(each of the n nodes can connect to (n-1) other nodes)
undirected graph emax = n*(n-1)/2(since edges are undirected, count each one only once)
What Fraction are Present?
DensityWhat fraction are present?density = e / emax
For example, out of 12possible connections.. this graph
this graph has 7, giving it a density of 7/12 = 0.58
A “fully connected graph has a density =1
Connected Components
We are often interested in whether the graph has a single or multiple connected components
Strong Components
Giant Component
Weak Components
“Largest Weakly Connected Component” in the SCOTUS Citation Network
There exist cases that are not in this visual as they are disconnected as of the year 1830
However, by 2009, 99% of SCOTUS Decisions are in the Largest Weakly Connected Component
Connected Components
Open “Giant Component” from the netlogo models Library
Connected Components
Notice the fraction of nodes in the
giant component
Notice the Size of the “Giant
Component”
Model has been
advanced 25+ Ticks
Connected Components
Model has been
advanced 80+ Ticks
Notice the fraction of nodes in the
giant component
Notice the Size of the “Giant
Component”
Connected Components
Model has been
advanced 120+ Ticks
Notice the fraction of nodes in the
giant component
Notice the Size of the “Giant Component”now = “num-nodes”
in the slider
Degree Distributions
outdegreehow many directed edges (arcs) originate at a node
indegreehow many directed edges (arcs) are incident on a node
degree (in or out)number of edges incident on a node
Indegree=3
Outdegree=2
Degree=5
Node Degree from
Matrix Values
Outdegree:
outdegree for node 3 = 2, which we obtain by summing the number of non-zero entries in the 3rd row
Indegree:
indegree for node 3 = 1, which we obtain by summing the number of non-zero entries in the 3rd column
Degree Distributions
These are Degree Count for particular nodes but we are also interested in the distribution of arcs (or edges) across all nodes
These Distributions are called “degree distributions”
Degree distribution: A frequency count of the occurrence of each degree
Degree Distributions
Imagine we have this 8 node network:
In-degree sequence:[2, 2, 2, 1, 1, 1, 1, 0]
Out-degree sequence:[2, 2, 2, 2, 1, 1, 1, 0]
(undirected) degree sequence:[3, 3, 3, 2, 2, 1, 1, 1]
Degree Distributions
Imagine we have this 8 node network:
In-degree distribution:[(2,3) (1,4) (0,1)]
Out-degree distribution:[(2,4) (1,3) (0,1)]
(undirected) distribution:[(3,3) (2,2) (1,3)]
Why are Degree Distributions Useful?
They are the signature of a dynamic process
We will discuss in greater detail tomorrow
Consider several canonical network models
Canonical Network Models
Erdős-Renyi Random Network
Highly Clustered Network
Watts-Strogatz Small World Network
Highly Clustered Highly Clustered
Barabási-Albert Preferential
Attachment Network
Why are Degree Distributions Useful?
Barabási-Albert Preferential
Attachment Network
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Watch the Changing Degree Distribution
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Barabási-Albert Preferential Attachment
Netlogo Models Library --> Networks --> Preferential Attachment
Back to the Milgram
Experiment
The Milgram Experiment
How did the successful subjects actually succeed?
How did they manage to get the envelope from nebraska to boston?
this is a question regarding how individuals conduct searches in their networks
Given most individuals do not know the path to distantly linked individuals
Search in Networks
Most individuals do not know the path to an individual who is many hops away
Must rely on some sort of heuristic rules to determine the possible path
Search in Networks
What information about the problem might the individual attempt to leverage?
visual by duncan watts
dimensional data:
send it to a stockbrokersend it to closet possible city to boston
Follow up to the original Experiment
available at: http://research.yahoo.com/pub/2397
Published in Science in 2003