Network Theory
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Transcript of Network Theory
Network Theory
Some Network Terminology Each case can be thought of as a vertex or node An arc i j = case i cites case j in its majority opinion
(directed or two-mode network) An arc from case i to case j represents
an outward citation for case i an inward citation for case j
A tie i j = nodes are connected to one another (bilateral or symmetric network)
Total arcs/ties leading to and from each vertex is the degree in degree = total inward citations out degree = total outward citations
Clustering Coefficient What is the probability that your friends are
friends with each other? Network level
Count total number of transitive triples in a network and divide by total possible number
Ego level For ego-centered measure, divide total ties between
friends by total possible number
Degree Centrality Degree centrality = number of inward citations
(Proctor and Loomis 1951; Freeman 1979) InfoSynthesis uses this to choose cases for its CD-ROM
containing the 1000 “most important” cases decided by the Supreme Court
However, treats all inward citations the same Suppose case a is authoritative and case z is not Suppose case a i and case z j
Implies i is more important than j
Eigenvector Centrality:An Improvement Eigenvector centrality estimates simultaneously the importance
of all cases in a network (Bonacich 1972) Let A be an n x n adjacency matrix representing all citations in a
network such that aij = 1 if the ith case cites the jth case and 0 otherwise Self-citation is not permitted, so main diagonal contains all zeros
Roe Akron Thornburgh Webster Planned Parenthood
Roe 0 0 0 0 0 Akron 1 0 0 0 0 Thornburgh 1 1 0 0 0 Webster 1 1 1 0 0 Planned Parenethood
1 1 1 1 0
Eigenvector Centrality:An Improvement Let x be a vector of importance measures so that each case’s
importance is the sum of the importance of the cases that cite it:
xi = a1i x1 + a2i x2 + … + ani xn or x = ATx
Probably no nonzero solution, so we assume proportionality instead of equality:
λxi = a1i x1 + a2i x2 + … + ani xn or λx = ATx
Vector of importance scores x can now be computed since it is an eigenvector of the eigenvalue λ
Problems with Eigenvector Centrality Technical
many court cases not cited so importance scores are 0 0 score cases add nothing to importance of cases they cite citation is time dependent, so measure inherently biases
downward importance of recent cases Substantive
assumes only inward citations contain information about importance
some cases cite only important precedents while others cast the net wider, relying on less important decisions
Well-Grounded Cases How well-grounded a case is in past precedent
contains information about the cases it cites Suppose case h is well-grounded in authoritative
precedents and case z is not Suppose case h i and case z j Implies i is more authoritative than j
Hubs and Authorities Recent improvements in internet search engines (Kleinberg
1998) have generated an alternative method
A hub cites many important decisions Helps define which decisions are important
An authority is cited by many well-grounded decisions Helps define which cases are well-grounded in past precedent
Two-way relation well-grounded cases cite influential decisions and influential cases are
cited by decisions that are well-grounded
Hub and Authority Scores Let x be a vector of authority scores and y a vector of hub scores
each case’s inward importance score is proportional to the sum of the outward importance scores of the cases that cite it:
λx xi = a1i y1 + a2i y2 + … + ani yn or x = ATy
each case’s outward importance score is proportional to the sum of the outward impmortance scores of the cases that it cites:
λy yi = ai1 x1 + ai2 x2 + … + ain xn or y = Ax
Equations imply λx x = ATAx and λy y = AATy
Importance scores computed using eigenvectors of principal eigenvalues λx and λy
Closeness Centrality Sabidussi 1966
inverse of the average distance from one legislator to all other legislators
let ij denote the shortest distance from i to j Closeness is()( )121j j j njxnδδδ=−+++L
Closeness Centrality Rep. Cunningham 1.04 Rep. Rogers 3.25
Betweeness Centrality Freeman 1977
identifies individuals critical for passing support/information from one individual to another in the network
let ik represent the number of paths from legislator i to legislator k
let ijk represent the number of paths from legislator i to legislator k that pass through legislator j
Betweenness is ijkjijkikx≠≠=∑
Large Scale Social Networks Sparse
Average degree << size of the network Clustered
High probability that one person’s acquaintances are acquainted with one another (clustering coefficient)
Small world Short average path length
“Six degrees of separation” (Milgram 1967)
Large Scale Social Network Data
----------------Actual---------------- ----Theoretical----
Network
Size
Degree Path
Length
Clustering Path
Length
Clustering Los Alamos National Laboratory
52909 9.7 5.9 0.43 4.79 0.00018
High Energy Physics
56627 173 4 0.726 2.12 0.00300
Mathematics 70975 3.9 9.5 0.59 8.2 0.00005 Neuroscience 209293 11.5 6 0.76 5.01 0.00006 Fortune 1000 Directors
7673 14.4 4.6 0.588 3.8 0.00188
Movie Actors 225226 61 3.65 0.79 2.99 0.00027
Citations in High Energy Physics
Judicial Citations
Number of Cases
1
10
100
1000
1 1 0 1 0 0
1
10
100
1000
1 1 0 1 0 0 Inward Citations Outwa rd Citat ions
Scientific and Judicial Citations Unifying property is the degree distribution
P(k) = probability paper has exactly k citations Degree distributions exhibit power-law tail Common to many large scale networks
Albert and Barabasi 2001 Common to scientific citation networks
Redner 1998; Vazquez 2001 Suggests similar processes
Academics may be as strategic as judges!
The Watts-Strogatz (WS) Model(Nature 1998)
Order Chaos
“Real”Social Network
Preferential Attachmentand the Scale Free Model Barabasi and Albert, Science 1999
Add new nodes to a network one by one, allow them to “attach” to existing nodes with a probability proportional to their degree
Yields scale-free degree distribution
Hierarchical Networks Ravasz and Barabasi 2003
Identifying Networks
Turnout in a Small World
Social Logic of Politics 2005, ed. Alan Zuckerman
Why do people vote? How does a single vote affect the outcome of an
election? How does a single turnout decision affect the
turnout decisions of one’s acquaintances?
Pivotal Voting Literature Most models assume independence between voters
Decision-theoretic modelsDowns 1957; Tullock 1967; Riker and Ordeshook 1968; Beck 1974; Ferejohn and Fiorina 1974; Fischer 1999
Empirical modelsGelman, King, Boscardin 1998; Mulligan and Hunter 2001
Game theoretic models imply negative dependence between votersLedyard 1982,1984; Palfrey and Rosenthal 1983, 1985; Meyerson 1998; Sandroni and Feddersen 2006
Social Voting Literature Turnout is positively dependent
between spouses (Glaser 1959; Straits 1990) between friends, family, and co-workers
Lazarsfeld et al 1944; Berelson et al 1954; Campbell et al 1954; Huckfeldt and Sprague 1995; Kenny 1992; Mutz and Mondak 1998; Beck et al 2002
Influence matters many say they vote because their friends and relatives vote (Knack
1992) Mobilization increases turnout
Organizational (Wielhouwer and Lockerbie 1994; Gerber and Green 1999, 2000a, 2000b)
Individual -- 34% try to influence peers (ISLES 1996)
Turnout Cascades
If turnout is positively dependent then changing a single turnout decision may cascade to many voters’ decisions, affecting aggregate turnout
If political preferences are highly correlated between acquaintances, this will affect electoral outcomes
This may affect the incentive to vote Voting to “set an example”
Small World Model of Turnout Assign each citizen an ideological preference
and initial turnout behavior Place citizens in a WS network Randomly choose citizens to interact with
their “neighbors” with a small chance of influence
Hold an election Give one citizen “free will” to measure
cascade
Simplifying Assumptions Social ties are
Equal Bilateral Static
Citizens are Non-strategic Sincere in their discussions
Model Analysis Analytic--to a point:
Create Simulation Analyze Model Using:
A Single Network Tuned to Empirical Data Several Networks for Comparative Analysis
( )( )1 2 1 1
( 1) ( 1) ( 1)1 1
1 1 0 1
!( ( 1))!1 (1/ 2) 1 1 1( )!
bi jb
bi j
D D
LD k D k D k DN P L a
j b a a a a a b
D D kT q qDk
→
→
−
− − −− −
= = = = = =
⎛ ⎞−= + − − −⎜ ⎟⎜ ⎟⎝ ⎠∑∑ ∑ ∑ ∑ ∏L
Political Discussion Network Data 1986 South Bend Election Study (SBES)
1996 Indianapolis-St. Louis Election Study (ISLES)(Huckfeldt and Sprague)
“Snowball survey” of “respondents” and “discussants”
RespondentDiscussant
Discussant
Discussant
Discussant’s DiscussantDiscussant’s DiscussantDiscussant’s DiscussantDiscussant’s DiscussantDiscussant’s Discussant
Features of a Political Discussion Network Like the ISLES Size: 186 million, but limited to 100,000-1 million
Degree: 3.15 (but truncated sample)
Clustering: 0.47 for “talk” 0.61 for “know”
Interactions: 20 (3/week, 1/3 political, 20 weeks in campaign)
Influence Rate: 0.05 (consistent w/ 1st,2nd order turnout corr.) Preference Correlation: 0.66 for lib/cons, 0.47 for Dem/Rep
Results: Total Change in Turnout in a Social Network Like the ISLES
0%
5%
10%
15%
20%
0 5 10 15 20 25
Total Change in Turnout
Frequency
Net Favorable Change in Turnout in a Social Network Like the ISLES
0%
5%
10%
15%
20%
-10 -5 0 5 10 15 20
Net Favorable Change
Frequency
Turnout CascadesMagnify the Effect of a Single Vote A single turnout decision
changes the turnout decision of at least 3 other people increases the vote margin of one’s favorite candidate by at
least 2 to 3 votes Turnout cascades increase the incentive to vote by
increasing the pivotal motivation (Downs 1957) signaling motivation (Fowler & Smirnov 2007) duty motivation (Riker & Ordeshook 1967)
Consistent with people who say they vote to “set an example”
Do Turnout Cascades Exist?
Cascades increase with number of discussants But this correlates strongly with interest
How does individual-level clustering affect the size of turnout cascades? Social capital literature suggests monotonic and increasing
Individual NetworkCharacteristics
TurnoutCascades
Intention toInfluence and Turnout
Prediction: How Individual-Level Clustering Affects Simulated Turnout
-0.6-0.4-0.2
00.20.40.60.8
1
0 0.5 1
Probability Acquaintances Know One Another (C )
Net Favorable Change in
Turnout
What’s Going On? Clustering increases the number of paths of influence both
within and beyond the group
With a fixed number of acquaintances, clustering decreases the number of connections to the rest of the network
BA
CFG
D EBA
CFG
D EBA
CFG
D E
Results: How Individual Clustering Affects Intention to Influence
-10%
0%
10%
20%
0 0.2 0.4 0.6 0.8 1
Probability Acquaintances Know One Another (C )
Change in Influence
Probability
How Individual Clustering Affects Intention to Vote
-2%-1%0%1%2%3%4%
0 0.2 0.4 0.6 0.8 1
Probability Acquaintances Know One Another (C )
Change in Turnout
Probability
The Strength of Mixed Ties “Weak” ties may be more influential than
“strong” ties because they permit influence between cliques (Grannovetter 1973)
Evidence here suggests that a mixture of strong and weak ties maximizes the individual incentive to set an example by participating
Stylized Facts for Aggregate Turnout Turnout increases in:
Number of contactsWielhouwer and Lockerbie 1994; Ansolabehere and Snyder 2000; Gerber and Green 1999, 2000
Clustering of social tiesCox, Rosenbluth, and Thies 1998; Monroe 1977
Concentration of shared interestsBusch and Reinhardt 2000; Brown, Jackson, and Wright 1999; Gray and Caul 2000; Radcliff 2001
Number of Contacts
Clustering of Social Ties
Concentration of Shared Interests
Implications Turnout Cascades & Rational Voting
Turnout cascades magnify the incentive to vote by a factor of 3-10
Even so, not sufficient Explaining the Civic Duty Norm
Establishing a norm of voting with one’s acquaintances can influence them to go to the polls
People who do not assert such a duty miss a chance to influence people who share similar views, leading to worse outcomes for their favorite candidates
Implications Over-Reporting Turnout
Strategic people may tell others they vote to increase the margin for their favorite candidates
It is rational to do this without knowing anything about the candidates in the election! May explain over-reporting of turnout
(Granberg and Holmberg 1991) Paradox: why would people ever say they don’t vote?
Social Capital Bowling together is better for participation than bowling alone (Putnam
2000) BUT, who we bowl with is also important
People concerned about participation should be careful to encourage a mix of strong and weak ties (Granovetter 1973)