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  1. A hierarchy of exponential random graph models for the analysis of social networks Pip Pattison University of Melbourne UKSNA, University of Greenwich, June 2013

  2. Acknowledgments Joint work with Garry Robins, Peng Wang and Tom Snijders University of Melbourne Garry Robins, Peng Wang, Galina Daraganova, David Rolls University of Oxford Tom Snijders University of Manchester Johan Koskinen Swinburne University Dean Lusher

  3. Outline • Structure in networks • The ERGM framework for network modelling • Hierarchy of dependence structures for ERGMs • Five networks • Applications

  4. 1. Structure in networks

  5. Cartwright and Harary:Psychological Review, 1956 We expect: • negative ties to be bi-partite in form (or k-partite in generalisations) • positive ties to be potentially clustered

  6. Granovetter: American Journal of Sociology, 1973 We expect: • closed triangles in strong ties • local bridges to be weak

  7. Jackson & Wolinksy: Journal of Economic Theory, 1996 We expect: • disconnected cliques • stars

  8. Watts & Strogatz: Nature, 1998 We expect: • High concentration of triangles • Short paths • Low density • Absence of hubs

  9. Degree effects: degree assortativity and dissassortativity (e.g. Newman, 2003) We expect: • relatively high (or low) rates of connection among high-degree nodes

  10. Burt: American Journal of Sociology, 2004 Robins (2009): We expect to see brokers who are: • embedded in groups • bridging to other groups

  11. Bearman, Moody & Stovel:American Journal of Sociology, 2004 We expect: • An absence of 4-cycles (and 3-cycles)

  12. Jackson, Rodriguez-Barraquer & Tan: American Economic review, 2012 We expect: • m-cliques but not • (m+1)-cycles

  13. An aside Paper Citations (WoS, June 26, 2013) Cartwright & Harary (1956) 534 Granovetter (1973) 5833 Jackson & Wolinsky (1996) 416 Watts & Strogatz (1998) 7572 Newman (2003) 507 Burt (2004) 491 Bearman, Moody & Stovel (2004) 133 Jackson, Rodriguez-Barraquer & Tan (2012) 0 Our fascination with network structure runs deep!

  14. Other regularities in network structure Other hypothesised sources of regularity in network structure include: • Homophily and heterophily effects (e.g. McPherson, Smith-Lovin & Cook, 2001) • Consequences of social foci and other settings (Feld, 1985; Pattison & Robins, 2002) • Embedding in geographical, organisational and sociocultural contexts (e.g. Daraganova et al, 2012; Lomi et al, in press; White, 1992) • Interdependence or embeddedness with other networks (e.g. Granovetter, 1985; Padgett & McLean, 2006)

  15. Harrison White on network ties Notably, almost all of these hypotheses about structural regularity are based on arguments about local interaction in networks: “A social tieexists in, and only in, a relation between actors which catenates, that is entails (some) compound relation through other such ties of those actors. … Thus it is subject to, and known to be subject to, the hegemonic pressures of others engaged in the social construction of that network” (White, 1998)

  16. 2. General modelling framework

  17. Network models Network models should: • reflect known and hypothesised processes for network tie formation (such as those just mentioned) • be dynamic, where possible, and consistent with known or hypothesised dynamics • allow us to test propositions about network structure and process • allow us to understand the consequences of network structure and process For cross-sectional data, the exponential random graph modelling (ERGM) framework is convenient

  18. Exponential random graph models (ERGMs) • We regard the nodes of a network as fixed, and treat potential ties among nodes as variables that are dependent on exogenous attributes of the nodes and potential ties and, potentially, on one another. • The form of assumed dependence among tie variables leads to a general form of a probability model (an exponential random graph model) for the ensemble of tie variables • Additional simplifying assumptions … • The model can be estimated using MCMCMLE from an observation on the network (and relevant node- or dyad-level covariates) - see Snijders (2002)

  19. Exponential random graph model (ERGM) Y(i,j) is a tie variable: Y(i,j) = 1 if node i is tied to node j, 0 otherwise Ensemble of tie variables: Y = [Y(i,j)] tie variables y = [y(i,j)] realisations P(Y=y) = (1/()) exp{ppzp(y)} Frank & Strauss (1986) • zp(y) are network statistics • p are corresponding parameters • ()is a normalising quantity Network effects

  20. 3. Dependence structures

  21. Characterising the proximity of potential network ties Under what circumstances is the tie linking node a and node b conditionally dependent on the tie linking node c and node d? a b When each of actors a and b is already linked to both actors c and d, and conversely? c d Strict inclusion

  22. Characterising the proximity of potential network ties Under what circumstances is the tie linking node a and node b conditionally dependent on the tie linking node c and node d? a b When each of actors a and b is already linked to at least one of actors c and d, and conversely? c d Inclusion

  23. Characterising the proximity of potential network ties Under what circumstances is the tie linking node a and node b conditionally dependent on the tie linking node c and node d? a b When at least one of actors a and b is already linked to both actors c and d? c d Partial inclusion

  24. Characterising the proximity of potential network ties Under what circumstances is the tie linking node a and node b conditionally dependent on the tie linking node c and node d? a b When at least one of actors a and b is already linked to at least one of actors c and d, and conversely? c d Distance criterion

  25. a b a b d d c c a. Strict p-inclusion SIp(p>0) b. p-inclusion Ip a b a b d d c c c. Partial p-inclusion PIp d. p-distance criterion Dp A second dimension: varying path length Key: Red lines indicate existing paths of length p or less (p 0) Blue dashed lines indicate potential ties, Yaband Ycd

  26. I0 = PI0 SI1 D0 I1 PI1 SI2 D1 I2 PI2 D2 The dependence hierarchy Pattison & Snijders, 2013)

  27. Associated model configurations Each configuration is a subgraph of diameter p (p-club, Mokken, 1979) For p = 1: cohesive subsets SIp: Strict p-inclusion a b c d

  28. Associated model configurations Each configuration has the property that every pair of edges lies on a cyclic walk of length  (2p+2) For p = 1: closure Ip: p-inclusion a b c d

  29. Associated model configurations Each configuration has the property that every pair of edges lies on a cyclic walk of length  (2p+2) or on a cyclic walk of length  (2p+1) with an edge incident to a node on the cycle For p = 1: brokerage PIp: Partial p-inclusion a b c d

  30. Associated model configurations Each configuration has the property that every pair of edges lies on a path of length  p+2 For p = 1: connectivity Dp: p-distance a b c d

  31. Model configurations for the case of p = 0 SI0: not defined I0: each configuration is an edge PI0: each configuration is an edge D0: each configuration is such that every pair of edges lies on a path of length  2 Bernoulli or Erdös-Rényi model: edges are independent Markov model (Frank & Strauss, 1986)

  32. I0 = PI0 (Bernoulli) SI1 (clique) D0 (Markov) I1 (social circuit) PI1 (edge-triangle) SIp (p-club) D1 (3-path) Ip (cyclic walk of length  2p+2) PIp ((r+1)-path-(2(p-r)+1)-cyclic walk, 0 rp-1) Dp (path of length p) The dependence hierarchy Pattison & Snijders, 2013) Cohesion Closure Brokerage Connectivity

  33. Other assumptions • Homogeneity: isomorphic configurations have equal parameters (Frank & Strauss, 1986) • Related effects: a single statistic for a family of related configurations, such as: • m-stars • m-triangles, • m-2-paths • m-edge-triangles • … (Snijders et al, 2006; Hunter & Handcock, 2006)

  34. Resulting model effects often include: • Edge: Propensity for edge to occur • Alternating star: (Endogenous) propensity for edges to attach to nodes with edges (progressively discounted for additional edges) – hence level of dispersion of degree distribution • Alternating 2-path: Propensity for presence of shared partners (progressively discounted for additional shared partners) • Alternating triangle: Propensity for an association between an edge linking nodes and their propensity for shared partners (progressively discounted for additional shared partners) (closure) • Alternating edge-triangle: Propensity for an association between degree and closure (progressively discounted for higher degrees)

  35. 4. Five networks

  36. Gift-giving (taro exchange) among households in a Papuan village* (n = 22) Hage P. and Harary F. (1983). Structural models in anthropology. Cambridge: Cambridge University Press. Schwimmer E. (1973). Exchange in the social structure of the Orokaiva. New York: St Martins.

  37. Interaction network in a university karate club (n = 34) Zachary W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33, 452-473.

  38. Kapferer’s tailor shop in Zambia, sociational (friendship and socioemotional) ties, time 2* (n = 39) *Kapferer B. (1972). Strategy and transaction in an African factory. Manchester: Manchester University Press.

  39. An Australian government organisation (n=60): ‘important’ ties

  40. A dolphin community near Doubtful Sound, NZ* (n = 62) *D. Lusseau, K. Schneider, O. J. Boisseau, P. Haase, E. Slooten, and S. M. Dawson, The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations, Behavioral Ecology and Sociobiology 54, 396-405 (2003).

  41. 5. Applications

  42. Gift-giving (taro exchange) among households in a Papuan village* (n = 22) Hage P. and Harary F. (1983). Structural models in anthropology. Cambridge: Cambridge University Press. Schwimmer E. (1973). Exchange in the social structure of the Orokaiva. New York: St Martins.

  43. Heuristic goodness of fit: degree statistics The t statistic locates the observed value of each statistic in the distribution of statistics associated with the ergm simulated using model parameters: if t   2, the observed statistic is within the envelope expected by the model For example: For the Bernoulli model: edge effect = -1.59(est se = .17) statistic observed simulated mean (sd) t triangles 10 7.481 (4.151) 0.607

  44. Taro exchange: Bernoulli effects estimates stderr Edge -1.590 0.174 effects observed mean stddev t-ratio 2-star 109 132.5 39.0 -0.604 3-star 80 141.6 67.4 -0.913 triangles 10 7.481 4.151 0.607 SD degrees 0.963 1.658 0.261 -2.663 Skew degrees 1.254 0.236 0.405 2.515 GCC* 0.275 0.160 0.057 2.017 Mean LCC* 0.339 0.151 0.066 2.851 VarLCC* 0.045 0.044 0.028 0.044 *GCC is the global clustering coefficient, LCC is the local clustering coefficient

  45. Taro exchange: edge-triangle models Model 2 effects estimates stderr edge -1.180 0.524 * AT(2.00) 2.296 0.602 * AET(2.00) -1.147 0.385 * Model 3 effects estimates stderr edge 1.472 2.169 2-star -0.369 0.401 triangle 4.618 1.511 * edge-triangle -0.588 0.283 * • Both models suggest: • Triadic closure • A negative association between participation in closed triads and degree

  46. Comparison of Models 2 and 3 Model 2Model 3 effects obsmean SDt-ratiomean SD t-ratio 2-star 109 125.0 27.4 -0.6108.3 15.7 0.1 3-star 80 127.5 48.1 -1.075.3 19.9 0.2 Triangles 10 9.9 1.9 0.110.0 2.8 -0.0 SD_deg 0.96 1.5 0.2 -2.40.9 0.154 0.2 Skew _deg 1.25 0.59 0.5 1.30.042 0.428 2.8 GCC 0.27 0.24 0.1 0.60.286 0.098 -0.1 Mean LCC 0.34 0.39 0.10 -0.60.346 0.115 -0.1 Var LCC 0.04 0.11 0.03 -2.10.077 0.033 -1.0 Model 3 appears to be more closely centred on the data

  47. Taro exchange simulated from Model 3

  48. The edge-triangle model for Taro exchange effect estimates stderr edge 1.472 2.169 2-star -0.369 0.401 triangle 4.618 1.511 * ET -0.588 0.283 * A triadicclosureeffect, accompanied by a negative association betweentriadicclosure and tie formation

  49. Interaction network in a university karate club (n = 34) Zachary W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33, 452-473.

  50. Zachary’s karate club effectestimatestderr edge -1.930 1.553 AS(2.00) -0.523 0.459 AT(2.00) 0.624 0.191 * A2P(2.00) 0.130 0.022 * Goodness of fit is good except for: effect observed mean stddev t-ratio 5-clique 2 0.080 0.325 5.905 Positive tendencies for closure in both 3- and 4-cycles