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Social Network Analysis - Part I basics Johan Koskinen Workshop: Monday, 29 August 2011 The Social Statistics Discipline Area, School of Social Sciences Mitchell Centre for Network Analysis

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Statistics and networks? Why statistics? -Is the network a unique narrative? -Numbers in lieu of ethnography? Possible answers -Detecting systematic tendencies -Social mechanisms -Why not?

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Outline Statistics for networks Types of analysis Networks ERGM Statistics & S.N. SAOM Non- parameteric Functional dependencies Statistical dependencie s

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Part 1 Social network data?

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to … go to for advice…

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to … consider a friend…

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Social networks mary paul We conceive of a network as a Relation defined on a collection of individuals relates to on off Generally binary Tie present Tie absent

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Social networks mary paul We conceive of a network as a Graph: G(V,E), on relates to on off Generally binary Tie present Tie absent Individuals: V={1,2,…,n} Relation: E {(i,j) : i,j V}

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Social networks We conceive of a network as a Graph: G(V,E), on on off Generally binary Tie present Tie absent Individuals: V={1,2,…,n} Relation: E {(i,j) : i,j V} i (i, j) j

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Social networks We conceive of a network as a Graph: G(V,E), on Individuals: V={i,j,k,l} Relation: E ={(i,j),(i,k),(k,j),(l,j)} john pete mary paul l i j k

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Social networks We conceive of the Graph as a collection of Tie variables: {X ij : i,j V} i (i, j) j

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Social networks We conceive of the Graph as a collection of on off Generally binary x ij = 1 Tie variables: {X ij : i,j V} i (i, j) j x ij = 0

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Social networks We conceive of the Graph as a collection of Tie variables: {X ij : i,j V} john pete mary paul i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x = i j k l =

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Social networks We conceive of the Graph as a collection of Tie variables: {X ij : i,j V} i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x = i j k l = l i j k

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Social networks The Adjacency matrix: The matrix of the collection Tie var. {X ij : i,j V} i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x =

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Social networks The Adjacency matrix: The matrix of the collection Tie var. {X ij : i,j V} i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x = out- degree

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Social networks The Adjacency matrix: The matrix of the collection Tie var. {X ij : i,j V} i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x = In-degree

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Social networks The Adjacency matrix: The matrix of the collection Tie var. {X ij : i,j V} i - x ij x ik x il jx ji -x jk x jl kx ki x kj - x kl lx li x lj x lk - x = In-degree

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Social networks – example in R Lets create an Adjacency matrix: - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - x = x <- matrix(rbinom(100,1,.4),10,10) number of nodes density (#arcs/#possible arcs) number of cells

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Social networks – example in R Lets create an Adjacency matrix: - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - x = x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 number of nodes density (#arcs/#possible arcs) number of cells No diagonal (self-nominations)

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x Print matrix to screen

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) Print matrix to screen To sum third row

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) Print matrix to screen To sum third row

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) To sum all rows

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) To sum all rows

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) To sum all rows Out-degree distribution

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Social networks – example in R Lets create an Adjacency matrix: x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) colSums(x) To sum all columns in-degree distribution

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Social networks To draw the Graph Tie variables: {X ij : i,j V} i (i, j) j

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Social networks To draw the Graph Tie variables: {X ij : i,j V} i (i, j) j ?

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Social networks To draw the Graph load package network x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) colSums(x) library('network')

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Social networks To draw the Graph load package network x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) colSums(x) library('network') myGraph <- as.network(x) Transform the adjacency matrix to a network object

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Social networks To draw the Graph load package network x <- matrix(rbinom(100,1,.4),10,10) diag(x) <- 0 x sum(x[3,]) rowSums(x) colSums(x) library('network') myGraph <- as.network(x) plot(myGraph) plot the new network object

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Part 2 Modes of analysis of Social network data?

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Modes of Analysis SNA Graphical Descriptive Statistical

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Modes of Analysis SNA Graphical john pete mary paul A social network of tertiary students – Kalish (2003)

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Modes of Analysis SNA Graphical john pete mary paul Yellow: JewishBlue: Arab A social network of tertiary students – Kalish (2003)

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Modes of Analysis SNA Descriptive john pete mary paul Centrality index Density arabjew arabmediumlow jewhigh

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Modes of Analysis SNA Statistical nonparametric john pete mary paul Centrality index Density arabjew arabmediumlow jewhigh Differences in centrality may be explained by chance Differences in densities unlikely if classes assumed equal

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Modes of Analysis SNA Statistical model based john pete mary paul Bernoulli arabjew arabmediumlow jewhigh Ties are distributed independently with parameter The network may be described by an - a priori BBM - social selection ERGM with separate effects for clustering and homophily on race

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Part 2 Background: statistical analysis

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Statistical analysis – why statistics? Why statistics? Statistics assessing whether observed measured quantities are big reject chance or not six in 50 out of 51: balanced dice? Networks not as easy - Good model for chance in SNA? - Model to capture systematic patterns (the typical)

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Cant we simply do t-tests? I give advice… Consider testing:

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Cant we simply do t-tests? … to people I consider my friends Consider testing:

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Cant we simply do t-tests? Consider testing: advice friendship Association?

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jk y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jk y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jk y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j - x ij x ik x il x ji -x jk x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jk y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? Consider testing: advice friendship Correlate advice x with friendship y ? i j i j Here we get r = 0.21 Large?

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Cant we simply do t-tests? Using standard statistical techniques Is r = 0.21 big? Standard* statistical approach: Reject H 0 (no correlation) if *though careless is greater than 2 Here t = 6.44 (df 868) 2-sided p-value: 2x10 -10

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Cant we simply do t-tests? Does this – p-value of 2x – mean that advice friendship advice x and friendship y ? are truly associated? i j i j I give advice… … to my friends

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Cant we simply do t-tests? Does this – p-value of 2x – mean that advice friendship advice x and friendship y ? are truly associated? i j i j I give advice… … to my friends N O !

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Cant we simply do t-tests? Here I generated advice friendship friendship y independently of advice x i j i j

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Cant we simply do t-tests? Friendship and advice ties are independent but There may be dependence on actors Some people: I give advice to everyone

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Cant we simply do t-tests? Friendship and advice ties are independent but There may be dependence on actors Some people: …and everyone is my friend

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Cant we simply do t-tests? Friendship and advice ties are independent but There may be dependence on actors other people: I dont really give advice …and no one is my friend

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Cant we simply do t-tests? Sends ties to 0% others - x ij x ik x il x ji -x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? Sends ties to 70% others - x ij x ik x il x ji -x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jl y ki y kj - y kl y li y lj y lk -

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Cant we simply do t-tests? - x ij x ik x il x ji -x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jl y ki y kj - y kl y li y lj y lk - inbetween 70% 0%

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Cant we simply do t-tests? Mostly ones - x ij x ik x il x ji -x jl x ki x kj - x kl x li x lj x lk - - y ij y ik y il y ji -y jl y ki y kj - y kl y li y lj y lk - inbetween 70% 0% Mostly zeros

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Cant we simply do t-tests? x ij x ik x il x ji … x kl x li x lj x lk inbetween 70% 0% y ij y ik y il y ji … y kl y li y lj y lk

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Cant we simply do t-tests? x ji … x kl inbetween 70% 0% y ji … y kl 1 0 1

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Cant we simply do t-tests? x ji … x kl inbetween 70% 0% y ji … y kl correlations assuming no association 0.21

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History: non-parametric approaches From late 1930s the first generation of research dealt with the distribution of various network statistics, under a variety of null models (Wasserman and Pattison, 1996) Observed network Summary measure (e.g. centralization) Distribution of measure under null distribution

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friendship Non-parametric: 2 relations Conformity of 2 sociometric measures ( Katz and Powell, 1953 ) If no association between A and B, for each pair: B: advice network A: friendship network mary paul heads tails paul friendship mary advice friendship mary paul concordant discordant Distribution of #concordant under null distribution obs #concordant # pairs: or

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Compare?

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on expected density U |E( X ++ ) : Bernoulli XX XX X XX XX XX X XXX E(X ++ ) X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ X XX X XXX X XX XX XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX XX XX XXX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX XX XX XXX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond.

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Non-parametric: conditional uniform null distributions Different null distributions for directed graphs Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ XX XX X XX XX XX X XXX X ++ X 1+ Xi+Xi+ Xn+Xn+ X +1 X+iX+i X+nX+n Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ Permute|cond. For a systematic statistical approach to successive conditioning see Pattison et al., 2000

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Non-parametric: conditional uniform null distributions Condition on density: U | X ++ Condition on expected density U |E( X ++ ) : Bernoulli Condition on activity/out-degrees: U | X+ Condition on popularity/in-degrees: U | X + Condition on both in-degrees and out-degrees : U | X +, X+ library(help=sna) # e.g.: rgnm Try and identify these distributions in sna:

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Investigating the triad census conditional on the dyad census ( Holland & Leinhardt 1970 ) Different directed triangles Distribution of #030T given observed MAN Types of dyads: M (mutual):A (asymetric):N (null): Observed network U | MAN : uniform graphs with same MAN as observed obs #030T

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Investigating the triad census conditional on the dyad census Interpretation Distribution of #030T given observed MAN Given that weve accounted for different types of reciprocation M A N obs #030T What triads occur more (less) freq. than chance? Alt.: What triads occur more (less) freq. than what is explained by density and reciprocation alone?

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Triad census in R Load the data set coleman that comes with the package sna ?coleman data(coleman) # loads data set colenet <- as.network(coleman[1,,]) # create network obj colenet # check properties plot(colenet) # plot dyad.census(colenet) ObsTriad <- triad.census(colenet)

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Triad census in R Generate a null-distribution of NumReplics graphs with the same MAN as colenet NumReplics <- 500 g<- rguman(NumReplics,73,mut=62,asym=119,null=2447,method = "exact") TriadRes <- matrix(c(0),NumReplics,16) for (i in 1:NumReplics) { TriadRes[i,] <- triad.census(g[i,,]) } Calculate the triad census for each simulated graph

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Triad census in R Generate a null-distribution of NumReplics graphs with the same MAN as colenet plot the simulated triad census against the observed par( mfrow = c( 4, 4 ) ) for (k in 1:16) { hist(TriadRes[,k],xlim = c(min(ObsTriad[k],TriadRes[,k]),max(ObsTriad[k],TriadRes[,k] ) ),xlab=dimnames(ObsTriad)[[2]][k],main="") lines(ObsTriad[k],0,type="o", col="red") }

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX X XX XX XX X XXX XX XX XXX X X X XX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX X XX XX XX X XXX XX XX XXX X X X XX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX XXX X X X XX X XXX XX XX X XX XX XX X XXX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX XXX X X X XX X XXX X XX XX XX X XXX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX XXX X X X XX X XXX X XX XX XX X XXX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX XXX X X X XX XX XX X XX XX X XXX XX

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Quadratic Assignment Procedure (QAP) (Krackhardt, 1987) B: advice network A: friendship network XX XX XXX X X X XX XX XX X XX XX X XXX XX How unusual is the observed number of concordant pairs compared to the permutation distribution?

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QAP in R Load the data set coleman that comes with the package sna ?coleman data(coleman) # loads data set q.12<-qaptest(coleman,gcor,g1=1,g2=2)# qap test summary(q.12)# summary of test plot(q.12)# plot of null distribution

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Drawbacks of non model based statistical analysis Weak (uninteresting) null hypotheses – what is it we are rejecting? Test: Testing centralization using conditioning on density: U | X ++ Interpretation: network more centralised than expected by chance, but also, network not generated by randomly distributing edges Test: Testing association between relations using QAP Interpretation: relations are not unrelated, but also, ties are more concordant than if identities of vertices did not matter (sic) john pete mary john petemary

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Drawbacks of non model based statistical analysis We have no model for what we are interested in – are significant effects artifacts of other effects ? Test: Testing structural effects using U | MAN Limit in interpretation: what if we are interested in both reciprocity and triangulation?

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Models for networks Models allow us to model features of the data that we are interested in If we are able to fit a model we (may) have adequately described the data (c.p. only holds true for non-parametric analysis when null hypothesis not rejected) Common critique: (a) only one observation (b) not inferring to population (c) where does chance come from? chance = uncertainty; possible process rather than sample (c.p. time series analysis)

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Models for networks Stochastic block models (e.g. Nowicki and Snijders, 2001) Latent class/ clustering models (e.g. Schweinberger, and Snijders, 2003; Handcock et al., 2007) Regressing variables on networks and covariates - the influence model (Robins et al., 2001) - the network effects and network autocorrelation models (Marsden and Friedkin, 1994) Models for longitudinal social network data (e.g. Snijders et al., 2007)

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