Statistical Analysis of Network Configurations in a Communication Network

1. MIS 644Social Newtork Analysis2017/2018 Spring Chapter 7 C Statistical Analysis of Networks: ERGM Examples

2. Outline

3. Random Graphs • Some configurations may occur by chance even for random graphs • ties are fromed independently of one another • no dependence • Fig 4.1a random graph • n=38, m=146 • Communication network Fig 4.1b • Selected network statistics Table 4.1

6. Distributions of Graphs • whether there are significantly more network structures than expected by chance • simuate a large sample of random directed nets • on 38 edges and 146 arcs • sample of graphs from a graph distribution • U|L: uniform condition on L: # of arcs • uniform - each graphs of 146 arcs has equal probability • 1,000 simple random graphs Fig 4.2 • distribution of reciprocated arcs

8. ranges 0 – 15 • on average 7.8 reciprocated arcs – by chance • unusual to see more than 15 • RT=44 • 1- the network is not a random graph • 2- there is dependence among network ties • 3- there are reciprocity processes active in the network • prob of observing 44 ties in a RG is alomst 0 • if :%5 less than 

9. null hypthesis: graph is random U|L model • # of RA is significantly greater than what is expected from U|L • there is significant reciprocation in the network • Similar conculusion about transitive traids • from 1,000 graphs mean TT:55 with std: 7.8 • observed 212 significantly greater

10. Two limitations • 1- comparison distribtion to a random graph • another distribution • 146 ties and 44 RA • # TT observed data may not be extream • 2- consider each effect one at a time • ignore nesting of configurations • There are significantly more TTs • do not know • increased number of stars or • triangulation process

11. saying that signficant number of RAs and TTs • rejecting the null: observed graph from U|L • not saying anything about the alternative • wheter reciprocation or transitivity are significant processes

12. ERGM tries to do • model the effects of interest • e.g., reciprocated ties or transitive traids • in relation to an observed netowrk • to find a distribution of graphs • the observed data – at the centered • fit the model – by esitmating parameters

13. for the communication net – ERGM • 38 nodes, • average of 146 ties, 44 RA, 212 TT • 312 in-2 start, 283 out-2 strat for nested effects • which configurations are importnat • which effects – independent explanatory value • whch can be expalined by other effects

14. ERGM – compair the observed net with other possible ways it could be arranged • saample space – tie can be arranged • model estimation - assign probabiilities • observed net is central not an exteam • in terms of the effects being modeled

15. ERGM examine competing theories - formation of network ties – within single analysis • test one network theoretical concept against others • homophily and/or reciporcity

16. Applied Example • communication in the Corporation • in entertainment industry • n=38 executives • communication network - directed • tie – response of an actor to a survay item about another executive with whome it was important to communicate to coplete a work • Fig 4.1b, Table 4.1 • understand structure of informal communication ties

17. information abour executives • # of projects they involved • level of seniority • office membership • Information about advice relations within the organization – binary directed network • ties pointing executives form whom adive is recieved • Download data and PNet models from • http://www.sna.unimelb.edu.au/

18. network self-organization – communication ties • Fig 5.1 - # traids • triadic structure • some regions densly clustered • nodes 24 and 34 – hubs • Fig 5.2 – mutual ties undirected reciprocated ties • communication - partially explained by reciprocity • actor attributes – network structure • Fig 5.3: node size  # projects completed • suggest – homophily effect (experience)

22. Fig 5.4: seniority • Fig 5.5: branch office • may be homophily from seniority • brach office – not clear • communication ties – affected from adsvice relationships • Fig 5.6. ties align (entrained) between communbication and advice networks • 24 ties • whether by chance or entainment effect significant

26. many possible compating explanations for network structure • if examine one process at a time • may overestimate its worth • ERGM infer • whether – independent tendencise for a configuration to occur • or whether – presence of condiguration more parsimoniously explained by combined presene of other feects

27. ERGM model and interpretation • parameter estimates –strength and direction of patterns Table 5.1 • a positive (negative) more (less) of the configuration then expected – given the ogther effects in the model • magnitutes not be comparable – scaling is different

29. Arc: - negagtive arc effect • like intercept in regression • baseline propensity fo the occurene of ties • Reciprocity: employies likely to reciprocat3ed communbication

30. Popularity and actgivity (in and out degree effects): tendencies for centrilization in and out degree distrubution • in-degree param. (-)insignificant – no distinctgively poular employee - net of other effects • out-degree param. (-) significant • absence of centralization in network activity • people tend to be uniform in the number of choice of comunication patterns

31. simple 2-pats: not significant • neither more nor less 3-paths than expected given other params • no evidence – people send more ties also receive them • if (+) and significant – • actors most popular also most active • positive correlation between in and out degree distributions

32. Multiple 2-paths: depth of local connectivity between pairs of nodes • 2-path – center of the two path • explaining - connectivity between pairs of nodes at the end of the path • not significant – local connectivity • neither stronger not weaker than expected

33. Transitivity: transitive path closure of multiple 2-paths • (+) significant • tendency for hierarchical path closure • not use single triangles – problematic • not coherent models - not converge • triangles tend to occur together in denser regions of the network

34. multiple 2-paths – nested in multiple triangles • combination of the two param estimate – • when mulitple 2-paths occur – tend to be closed in transive form

35. Cyclic closure: cyclic closure of multiple 2-paths • (-) negative • nonhierarchical (generalized exchange) network closure • associated with mulitpe 2-patghs - together • when multipe 2-paths occur • tend to be closed in transitive form • and not to be closed in cyclic form

36. onces these two closure processes takein into accout together • no other evident tendencies for multple 2-paths • present or absent • local connectivity explained by • tendency for transitive closure • and tendency against cyclic closure

37. Sender effects: degree to chich actors with a specific attribute send more ties compared to others • (-) not quite significant for binary attribute seniority • managers not senior (seniority = 0) - tendency to send ties • number of project – continuous variable • no sender effect – experience in projects effect tendency to communicate • for varible “office” no sender and receiver effects

38. Receiver effefct: degree to chich actors with a specific attribute has the propensity to receive more ties compared to others • no significant effect for variales seniority or number of projects

39. Homophily: (+) significant • for seniority and number of projects • tend to communicatre wtih others with the same seniority and similar experience • the param is – for number of projects • absolute differences • no significant effect for office

40. Covariae advice network: • exogenous to the communication network • fixed in the model • advice may have an effect on communication • but communication has no effect on advice • advice ties may effect communication ties • (+) significant: • advice and communication ties occur together

41. Multiple explanations for network structure • there are significant effects for • purly structural • actor relation • covaraite network effects • each subtype effect – independent explanatory capacity

42. Illustrations: Simulations, Estimations and Goodness of Fit • how to simulate ERGMs • as a heuristic method for GOF • to see wheter a model can fit to many emprical features of networks • Short examples of simulation and GOF • Corporation Communication Network

43. Goals • simulation – effects of parameters • how they can work together • model interpretation • suggestions for model specification • with a GOF example • show how GOF may guide model selection • so effects may be added to improve fit • can be replicated with PNet

44. Simulation • simulations about – • parameters and their interpretations • once burn-in • take sample of graphs • mean and SD of statiscits • n=30, burn-in 100,000 • 1,000,000 interatioons, 1,000 samples at intervals of 1,000 • last graphs is visualization • fixed density of 0.1 (43 ties) – undirected nets

45. when on effects other than density 0.1 • random graph distribution U|L=43 • Fig 13.1 – illustative exaple • baseline compariosn

48. Triangulation • effect of triangulation parameters T(on network closure Fig 13.2 • left-column: with different values of Markov TP • right-column: AT parameters with =2 • MTP of 0.5 or 1.0 – not differnt from RG • from 1.0 to 1.5 – sudden jump of triangulation to a cliquelike structure • for fixed # of edge – clique – maximum # of triangles

49. phase transmition – • why MRG are not good • sample of graphs from RG • mean # triangles = 3.6, SD = 1.8 • a weak MTP increases samohow • when MTP=1.0 # triangles=12.1 SD=6 • t test significantly different from RG • when MTP=1.5 # triangles=102.5, SD=3.9