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 models

Downs 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)11

1 1 0 1

!( ( 1))!1 (1/ 2) 1 1 1

( )!

bi j

bbi 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 q

Dk

→

→

−

− − −−−

= = = = = =

⎛ ⎞−= + − − −⎜ ⎟⎜ ⎟

⎝ ⎠∑∑ ∑ ∑ ∑ ∏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”

Respondent

Discussant

Discussant

Discussant

Discussant’s Discussant

Discussant’s Discussant

Discussant’s Discussant

Discussant’s Discussant

Discussant’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

C

FG

D E

BA

C

FG

D E

BA

C

FG

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 contacts

Wielhouwer 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)