Ideology and Social Networks in the U.S. Congress

Transcription

1 Ideology and Social Networks in the U.S. Congress James H. Fowler University of California, San Diego July 11, 2007

2 Improve connectedness scores (Fowler 2005) Cosponsorship is about 2 things: The idea The person behind the idea The goal is to create a Social Space Model that simultaneously estimates ideological and social ideal points. Place legislators and bills in policy space (Poole and Rosenthal 1997; Clinton, Jackman, and Rivers 2004) Place sponsors and cosponsors in social space (Hoff, Raftery, and Handcock 2002) Create alternative Social Utility Model

3 Utility from Policy Social Utility Other factors The Double-Center Operator n legislators choose whether to cosponsor m bills. Each bill j = 1...m is sponsored by legislator s(j) Each legislator i = 1...n chooses between the bill at ζ j and the status quo located at ψ j, both in R d { 1 if legislator i cosponosors bill j y ij = 0 otherwise (1)

4 Utility from Policy Social Utility Other factors The Double-Center Operator Legislator i receives utility for supporting policies close to her own ideal point x i in R d policy space: U bill status quo ij = x i ζ j 2 Uij = x i ψ j 2 U cosponsor ij = x i ζ j 2 + x i ψ j 2 (2)

5 Utility from Policy Social Utility Other factors The Double-Center Operator Legislator i also receives utility for supporting people close to her own social ideal point in R δ social space λ i = legislator i s ideal point in R δ social sender space (or hub space, Kleinberg 1999) αs(j) = the ideal point for bill j s sponsor in R δ social receiver space (or authority space) U social ij = λ i α s(j) 2 U ideological ij = x i x s(j) 2 U personal ij = λ i α s(j) 2 κ x i x s(j) 2 (3)

6 Utility from Policy Social Utility Other factors The Double-Center Operator Utility also affected by bill-specific factors φ j and legislator-specific factors ξ i : U ij = x i ζ j 2 + x i ψ j 2 λ i α s(j) 2 This simplifies to where κ x i x s(j) 2 + ξ i + φ j (4) U ij = 2x i β j + 2λ i α s(j) + η i + θ j (5) β j = ψ j ζ j κx s(j), η i = λ 2 i κx 2 i + ξ i θ j = ψ 2 j ζ 2 j x 2 s(j) κx2 s(j) + φ j

7 Utility from Policy Social Utility Other factors The Double-Center Operator Suppose observed cosponsorship means true utility of cosponsorship is high (1), while not observing one means that the true utility is low (0). True utility is observed cosponsorship decision minus an error term: Substituting, we get: U ij = y ij ν ij (6) y ij = 2x i β j + 2λ i α s(j) + η i + θ j + ν ij (7)

8 Utility from Policy Social Utility Other factors The Double-Center Operator Define the double-center operator D(.) for a matrix Z (Poole 2005; Clinton, Jackman, Rivers 2005): D(z ij ) = (z ij z i. z.j + z.. )/( 2) (8) Suppose ν ij is i.i.d. stable density and dimension-by-dimension means of x, β, λ, and α s(j) equal 0. Apply double-center operator in equation (8) to both sides of equation (7): D(y ij ) = x i β j λ i α s(j) + ɛ ij (9)

9 Utility from Policy Social Utility Other factors The Double-Center Operator By construction, social distance (λ i α s(j) ) is uncorrelated with ideological distance (x i β j ). Therefore, singular value decomposition (SVD) provides the best d dimensional approximation of x i and β j (Frobenius-Perron): D(Y) = XΣB (10)

10 Utility from Policy Social Utility Other factors The Double-Center Operator Suppose each legislator k sponsors n k bills. We can define an n by n matrix such that: µ ik = 1 n k s(j)=k D(y ij ) = x i βk λ i α k + υ ik (11) where β k = 1 n k s(j)=k β j and υ ik is a stable density (but not i.i.d. across sponsors). Rearranging yields another SVD formulation: XΣB M = ΛTA (12)

11 Utility from Policy Social Utility Other factors The Double-Center Operator Standard errors via Metropolis-Hastings! If errors ɛ ij in equation (9) are normally distributed, then: L(x i, β j, λ i, α s(j) y ij ) = ( ) 2 D(yij ) x i β j + λ i α s(j) i,j,(i j) (13)

12 Utility Add Error and Double Center Estimate dyad-specific mean utility transfers γ ik from legislator k to legislator i. These utilities can be thought of as social influence vote-trading specific instances of monetary transfers from one legislator to another (such as PAC contributions to a friend s campaign) To control for legislative activity, assume γ i. = γ.s(j) = γ.. = 0.

13 Utility Add Error and Double Center Personal utility from equation (3) becomes: U personal ij = γ is(j) κ x i x s(j) 2 (14) The full utility equation from (4) becomes: which simplifies to: U ij = x i ζ j 2 + x i ψ j 2 + γ is(j) κ x i x s(j) 2 + ξ i + φ j (15) U ij = 2x i β j + γ is(j) + η i + θ j (16)

14 Utility Add Error and Double Center Establishing a relationship between cosponsorship and the utility it provides yields an analogue to equation (7): y ij = 2x i β j + γ is(j) + η i + θ j + ν ij (17) Double-centering both sides yields: D(y ij ) = x i β j γ is(j) 2 + ɛ ij (18)

15 Utility Add Error and Double Center The SVD provides the best d dimensional approximation of x i and β j : D(Y) = XΣB (19) Suppose each legislator k sponsors n k bills. We can define an n by n matrix such that: µ ik = 1 n k s(j)=k D(y ij ) = x i βk γ ik 2 Rearranging yields the social utility matrix: (20) Γ = 2(XΣB M) (21)

16 Utility Add Error and Double Center Standard errors via Metropolis-Hastings! L(x i, β j y ij ) = i,j (D(y ij ) x i β j ) 2 (22)

17 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Generate true data with independent random draw from a uniform distribution for each legislator (x) bill (ζ) status quo (ψ) and either: legislator s sender space (λ) and receiver space (α) for Social Space Model legislator dyad s specific social utility (γ) for Social Utility Model

18 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Parameterize social weight: U total ij = ωu social ij + (1 ω)u ideological ij (23) Normally-disributed error term Choose a constant threshold U to generate specific rate of cosponsorship such that { 1 if Uij > U y ij = 0 otherwise (24) Option to change the utilities from quadratic u 2 to gaussian σφ(u).

19 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Baseline configuration: 100 legislators, each sponsors on average 10 bills (for 1000 total bills) rate of cosponsorship is set to 0.1 (Fowler 2005) the effect of social utility is approximately equal to ideological utility (ω = 0.5) no error in the utility utilities are quadratic Use Spearman rank correlation (doesn t matter).

20 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Legislator Ideal Point Bill Parameter Ideological Utility Social Utility ρ ρ ρ NOM ρ ρ ρ NOM ρ ρ ρ NOM ρ λ α Baseline , , , , , , , Bills per Legislator 0.966, , , , , , , Bill per Legislator 0.935, , , , , , , Legislators (1 Bill Each) 0.977, , , , , , , Social Weight 0.992, , , , , , , Social Weight 0.997, , , , , , ,0.022

21 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Legislator Ideal Point Bill Parameter Ideological Utility Social Utility ρ ρ ρ NOM ρ ρ ρ NOM ρ ρ ρ NOM ρ λ α Baseline , , , , , , , Rate of Cosponsorship 0.974, , , , , , , Rate of Cosponsorship 0.965, , , , , , , Standard Deviation Error 0.972, , , , , , , Standard Deviation Error 0.968, , , , , , ,0.843 Gaussian Utility 0.871, , , , , , ,0.829

22 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Legislator Ideal Point Bill Parameter Ideological Utility Social Utility ρ ρ ρ NOM ρ ρ ρ NOM ρ ρ ρ NOM ρ γ Baseline , , , , , , , Bills per Legislator 0.987, , , , , , , Bill per Legislator 0.947, , , , , , , Legislators (1 Bill Each) 0.991, , , , , , , Social Weight 0.997, , , , , , , Social Weight 0.997, , , , , , ,0.020

23 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Legislator Ideal Point Bill Parameter Ideological Utility Social Utility ρ ρ ρ NOM ρ ρ ρ NOM ρ ρ ρ NOM ρ γ Baseline , , , , , , , Rate of Cosponsorship 0.990, , , , , , , Rate of Cosponsorship 0.993, , , , , , , Standard Deviation Error 0.991, , , , , , , Standard Deviation Error 0.990, , , , , , ,0.721 Gaussian Utility 0.910, , , , , , ,0.732

24 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Social Space Model Social Utility Model Congress Years ρ x ρ β ρ λ α ρ x ρ β ρ γ 93rd , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , ,0.933

25 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Social Space Model Social Utility Model Congress Years ρ x ρ β ρ λ α ρ x ρ β ρ γ 101st , , , , , , nd , , , , , , rd , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , , th , , , , , ,0.925

26 The Process Data Generation Baseline Configuration and Evlauation Social Space Model Results Social Utility Model Results U.S. Senate Results Social Space Model Social Utility Model Real Data, 99th Senate Singular Value Singular Value Singular Value Eigenvalue rank Eigenvalue rank Eigenvalue rank is Superior for Cosponsorship Outlier values in left panel indicate two structured dimensions in data generated by the Social Space Model Center panel shows lack of structure in data generated by the Social Utility Model Right panel shows a lack of structure in data taken from the 99th Senate

27 We can recover social utilities without ideology! SVD procedure (surprisingly) works better than W-NOMINATE Social Utility Model works best on real data Can generalize to simultaneous modeling of any bipartite network with a unipartite projection.