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## Network Theory

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**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**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 xnorx = ATx • Probably no nonzero solution, so we assume proportionality instead of equality:λxi = a1i x1 + a2i x2 + … + ani xnor 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 ynorx = 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 = ai1x1 + ai2x2 + … + ain xnory = Ax • Equations imply λx x = ATAxand λy y = AATy • Importance scores computed using eigenvectors of principal eigenvalues λx andλy**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**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**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)**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**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 Preferential Attachmentand the Scale Free Model**Ravasz and Barabasi 2003**Hierarchical 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 thenchanging 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**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” Discussant’s Discussant Discussant Discussant’s Discussant Respondent Discussant Discussant’s Discussant Discussant’s Discussant 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**Net Favorable Change in Turnout in a Social Network Like the**ISLES**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 Intention toInfluence and Turnout Individual NetworkCharacteristics TurnoutCascades**Prediction: How Individual-Level Clustering Affects**Simulated 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 A B A B A B E E E D D D C C C F G F F G G**Results: How Individual Clustering Affects Intention to**Influence**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**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)