SOST71032 Social Network AnalysisRandom Graph ModelsDr Termeh ShafieDepartment of Social StatisticsSchool of Social SciencesThe University of ManchesterDay 4SOST71032 Social Network Analysis Random Graph Models Day 4 1 / 31parametric vs. non-parametricI parametric methods– fitting a model– parametric tests based on the theoretical distribution of the summarystatistics (only available for some models)– the data follow some sort of theoretical probability distribution– from simple to more complicated models that incorporatedependencies among tie variablesI non- parametric methods– statistical tests for model fit– when we don’t think that assumptions of parametric tests are satisfied(distribution free methods)– evaluates H0 against H1without assuming any parametric model– p-values have the same structure of interpretation:probability of seeing such extreme data given the null hypothesis is true– tests: shuling edges while fixing an observed summary measurelast lecturethis lectureSOST71032 Social Network Analysis Random Graph Models Day 4 2 / 31example.Knecht (2008): Friendship Selection and Friends’ Influencefriendship networks at 4 time points for 25 pupils at a schoolrunning hypotheses:H1 pupils chose friends with the same genderH2 pupils reciprocate friendshipH3 the friend of a friend is a friendH4 pupils chose friends with similar delinquency behaviourH5 pupils adopt delinquent behaviour from their friendsSOST71032 Social Network Analysis Random Graph Models Day 4 3 / 31example.Knecht (2008): Friendship Selection and Friends’ Influencefriendship networks at 4 time points for 25 pupils at a schoolrunning hypotheses:H1 pupils chose friends with the same genderH2 pupils reciprocate friendshipH3 the friend of a friend is a friendH4 pupils chose friends with similar delinquency behaviourH5 pupils adopt delinquent behaviour from their friendsSOST71032 Social Network Analysis Random Graph Models Day 4 3 / 31example.Knecht (2008): Friendship Selection and Friends’ InfluenceSOST71032 Social Network Analysis Random Graph Models Day 4 4 / 31example.Knecht (2008): Friendship Selection and Friends’ InfluenceSOST71032 Social Network Analysis Random Graph Models Day 4 5 / 31example.first test: social selection by genderhypothesis: pupils chose friends with the same gendermore precisely:the probability of friendship between pupils with same gender is highermethod: divide pairs of pupils (dyads) into two categoriesD1 = f(A, B); gender(A) = gender(B)gD2 = f(A, B); gender(A) 6= gender(B)gcompare the ratio of friendship ties in the two groups# ties in D1# dyads in D1 vs.# ties in D2# dyads in D2results:105312 = 0.337 vs. 288 31 = 0.108SOST71032 Social Network Analysis Random Graph Models Day 4 6 / 31example.significance of observed dierence 0.11 probability for friendship betweendierent gender0.34 probability for friendship between same gendercould this dierence be just accidental?if we divided pupils into two meaningless groups,the tie probability would also not be equalthe non-parametric approach:repeat the analysis 1000 times with random gender assignment=) average dierence is 0.035; maximum is 0.142maybe friendship is only seemingly influenced by gender equality; the“true” explanatory variable might beI behaviourI other ties in the networkwe need a model that control for the influence of other variablesSOST71032 Social Network Analysis Random Graph Models Day 4 7 / 31logistic regressionclassic starting point: why not treat edges as independent,with log-odds as a linear function of covariates?=) modelling the occurrence of ties with logistic regressionrandom variable Yuv for tie from node u to node vYuv = (1 with probability 0 with probability 1 Puv – PuvPuv = FunctionOf(parameters,statistics)– statistics quantify characteristics of dyad (u, v) in observed network– parameters quantify influence of those variables on tie probability:I a positive (negative) parameter means:the higher (lower) the statistic the higher (lower) the probabilityI a zero parameter means:the statistic has no influence on the tie-probabilityparameters are estimated from the observed networkSOST71032 Social Network Analysis Random Graph Models Day 4 8 / 31logistic regressionprobability puv of a tie from u to v is specified aspuv = logit-1(q · s) = exp(q · s)1 + exp(q · s)where s = (s1, . . . , sk) ⊆ Rk statisticsq = (q1, . . . , qk) ⊆ Rk parametersq · s =k∑i=1qi · siThe statistics si = si(u, v; y) are functions of the observed dataThe parameters are estimated to maximize the probability ofthe observed network y:P(Y = y) = ∏u6=vpyuvuv (1 – puv)1-yuvSOST71032 Social Network Analysis Random Graph Models Day 4 9 / 31example.results from logistic regressiongender model: friendship ties explained by gender equalitypuv = logit-1(q0 + q1 · SameGender(u, v))output:statistic parameter st. error Pr(> jzj)(intercept) -2.1151 0.1901 jzj)(intercept) -4.3664 0.3915
