Social networks, social space, social structure

Transcription

1 Social networks, social space, social structure Pip Pattison Department of Psychology University of Melbourne Sunbelt XXII International Social Network Conference New Orleans, February 22

2 Social space, social structure, social networks: quantitative reflections Acknowledgments Garry Robins and also Ron Breiger David Gibson Emmanuel Lazega Alessandro Lomi Laura Koehly Ann Mische Tom Snijders Stanley Wasserman Harrison White Jodie Woolcock

3 Outline Conceptual issues social space as relational, multi-mode, multi-layered complex social structure as patterned regularities in relational forms A quantitative imperative and a quantitative framework neighbours, neighbourhoods and neighbourhood processes Some illustrative applications structure and randomness in network models exploring a hierarchy of network models setting-dependent network models Issues and next steps

4 Why? social networks provide information about the relational substratum of people s actual lives (Padgett & Ansell) networks as wiring (or plumbing ) for sociocultural processes structure of opportunities and constraints an interest in the interactivity of social processes (eg Durlauf, 21) invites an interest in networks: interactions are neither random nor homogeneous networks create proximities and hence channel opportunities for interactivity nature of local interactive micro-processes affects global system properties social networks are implicated in what we might term social space, but can and should we distinguish the terms?

5 Intuitions about social space 1. Geography: GIS locations, regions, neighbourhoods, settings e.g., spatial epidemiology, urban sociology 2. Sociocultural/psychosocial profiles shared sociocultural resources, Blau space 3. Networks: connectedness through interpersonal relations friendship, kinship, work, advice, acquaintance, sexual contact 4. Affiliation structures: connectedness through memberships clubs, groups, organisations, societies, activities

6 So how can we can conceptualise social space? Social space is a complex of all of these things: it involves multiple types of entities people, groups, organisations, cultural symbols, beliefs, geographical locations it is primarily relational it is made up of links among these entities it is multi-layered relational entities are themselves linked Social structure: regularities in multi-mode relational forms within and across levels

7 A personal note (version I): how I became interested in networks Key: Lorrain & White Michael R read reading-induced conversation weak tie person Friedell me article

8 A (less personal) personal note (version 2) Oscar Oeser Frank Harary Warren Bartlett me Key: PhD student of collaborator

9 A simplified multi-layered framework Social units individuals groups... Ties among social units Settings person-to-person person-to-group... geographical sociocultural... For example: Interactions between social units depend on proximity through ties Interactions between ties depend on proximity through settings

10 How can we represent proximity in social space? Two steps: methodological: choose a notion of proximity that it is convenient from a modelling point of view: proximity interactivity define two variable entities to be neighbours if they are conditionally dependent, given the values of all other entities Hammersley-Clifford theorem then leads to a general model form substantive: what are appropriate assumptions about proximity in this sense?

11 Some assumptions about proximity: actor level Two actors are neighbours if: they share a network tie influence models they are at adjacent spatial locations spatial models

12 Some assumptions about proximity: tie level Two ties are neighbours if: they share a dyad dyad-independent model they share an actor Markov model they share a connection with the same tie realisation-dependent model etc.

13 Some assumptions about proximity: setting level Two settings are neighbours if: they share an actor they share a tie etc.

14 To summarise: Social units individuals groups... Relations on social units Settings person-to-person person-to-group... geographical sociocultural... entities at each level are potentially stochastic and subject to interactive processes proximities at one level lead to interactivity at other levels (interdependence within and across levels) assumptions about proximity (neighbourhood structure) need to be made explicit such complex and interactive systems require modelling

15 Variables attribute variables for social units: Y = [Y i ] eg decisions, beliefs, knowledge, Y i = 1 if i has some attribute otherwise tie variables for pairs of social units: X = [X ij ] X ij = 1 if i is related to j otherwise setting variables for pairs of tie variables: S = [S ij,kl ] S ij,kl = 1 if X ij, X kl share a setting otherwise realisations of Y, X and S are denoted y, x and s, respectively

16 Models for interactive systems of variables Two variables are neighbours if they are conditionally dependent given the observed values of all other variables A neighbourhood is a set of mutually neighbouring variables A model for a system of variables has a form determined by its neighbourhoods This general approach leads to: Pr(X = x) exponential random graph model Frank & Strauss, 1986 Extension to directed dependence assumptions: Pr(Y = y X = x) social influence models Robins et al, 21 Pr(X = x Y = y) Pr(X = x S = s) social selection models setting-dependent and setting-constrained network models

17 Models at the tie level: random graph models for networks Pr (X = x) = (1/c) exp{ Q λ Q z Q (x)} normalizing quantity parameter network statistic the summation is over all neighbourhoods Q z Q (x) = Π Xij Q x ij signifies whether all ties in Q are observed in x c = x exp{ Q λ Q z Q (x)}

18 What is a neighbourhood here? A neighbourhood is a subset of tie variables each pair are neighbours (conditionally dependent, given the values of all other variables) Each neighbourhood corresponds to a network configuration: for example: {X 12, X 13, X 14 } corresponds to the 2 configuration of tie variables: 1 3 {X 12, X 13, X 23 } corresponds to:

19 Neighbourhoods depend on proximity assumptions Assumptions: two ties are neighbours: Configurations for neighbourhoods if they share a dyad dyad-independence edge if they share an actor Markov + 2-star 3-star 4-star... triangle if they share a connection with the same tie realisation-dependent path 4-cycle coathanger

20 Homogeneous network models Pr (X = x) = (1/c) exp{ Q* λ Q* z Q* (x)}} If we assume that parameters for isomorphic neighbourhood configurations are the same: edges 2-stars 3-stars triangles... Then there is one parameter for each class Q* of isomorphic configurations and the corresponding statistic is a count of such observed configurations in x

21 An example for a bivariate network Two types of tie: A, B on n = 3 actors With a Markov assumption, neighbourhoods correspond to configuration classes:single ties, multiplex ties, triangles, 2-stars, 3-stars, etc. Consider the following classes: Key: A B A and B

22 Model 1a:neighbourhood configurations and parameter values Q λ Q Q λ Q Q λ Q

23 Simulating the model Using the Metropolis algorithm: random starting graph 1,, iterations sample every 1th graph (yielding 1 sampled graphs) The algorithm sets up a Markov Chain on the space of all possible bivariate networks (on 3 nodes) that has Pr(X=x) as its stationary distribution at each step, we consider changing the value of the (i,j) tie (from 1 to or to 1) for a randomly selected pair (and relation): x x ij the change is made with probability min[1,exp({ Q λ Q (z Q (x)-z Q (x ij )})] for diagnostic purposes, we count the number of changes made for each pair in the last 5, iterations

24 Network statistics for sampled graphs (as a function of Metropolis step) 2 Model 1a BBB 3 A_AND_B 1 BB 2 ABB B AB 1 AAB -1-2 A AA edges 2-stars triangles Random graph AAA BBB 1 A_AND_B 4 BB 1 ABB B 3 AB AAB A AA AAA

25 Checking on the simulations Distribution of number of value changes across ties during simulation Model 1a COUNT 5 4 N = REL Random graph COUNT 1 N = REL

26 Models 1b and 1c Multiply parameters for model 1 by: 1/T = 2 (Model 1b) 1/T = 4 (Model 1c) think of: T as temperature, mutliplication by 1/T for T < 1 as chilling mutliplication by 1/T for T << 1 as freezing (eg Grenander, 1993) attempt simulation, as before

27 Models 1b and 1c: network statistics Model 1a edges 2-stars triangles Model 1b A_AND_B B A BB AB AA BBB ABB AAB AAA A_AND_B B A BB AB AA BBB ABB AAB AAA

28 Model 1c: a typical graph Key: A B

29 Checking on the simulations Model 1b Model 1c N = REL COUNT N = REL COUNT

30 The results As T decreases: the model freezes to become a regular pattern - in this case the familiar balance model (Cartwright & Harary, Heider) mathematically, there are several elegant characterisations of balance: algebraic: A B=, A 2 = A = B 2, AB=BA=B graph theoretical: if A and B ties have values of 1 and -1, the product of tie values in any cycle is positive different runs yield different graphs at the millionth step, but the graph is balanced with high probability (and remains for many steps) As T increases: the model becomes a more random manifestation of the underlying regularities (structure)

31 Freezing: structure and randomness As T decreases, the model tends to one in which probability is uniformly distributed over minimum energy networks x (those for which - Q λ Q z Q (x) is a minimum) and is zero elsewhere (eg Grendander, 1993) As T increases, the model tends to a Bernoulli model In other words: low T increasing regularity high T increasing randomness

32 Model 2a:strong and weak ties? Q λ Q Q λ Q Q λ Q

33 Model 2a simulations Model 2a edges 2-stars triangles distribution of moves made A_AND_B B A BB AB AA BBB ABB AAB AAA N = REL COUNT

34 Model 2b (T=.5) Model 2b edges 2-stars triangles distribution of moves made A_AND_B B A BB AB AA BBB ABB AAB AAA N = REL COUNT

35 What have we learnt? This formulation provides a natural means for building models with local network regularities and varying amounts of randomness Some regular structures look more interesting than others; all raise theoretical questions It is easy to model balance (but balance is not too common) Our model for strong and weak ties is probably not very good, yet strong and weak tie patterns abound. Settings may be important The challenge: theoretical accounts model specification model properties empirical evaluation

36 Specific quantitative challenges model identification: we need families of theoretically-principled models within which to frame the model identification problem eg realisation- and setting-dependent models (Pattison & Robins, in press) parameter estimation and model properties (see Snijders, 22): the development of practical approaches depends on model specification and model behaviour model evaluation: do identified regularities fit theoretical conceptualisations? does the model account for important network characteristics (eg global connectivity properties, as in Watts, 1999)?

37 Example 3: network ties among elite families in Renaissance Florence elite families of 15 th century Florence (Kent; Padgett & Ansell) Tie level: Multivariate interfamily network X of nondirected ties: business ties marriage ties Setting level: neighbourhood locations (gonfalone, city quarter level) We model: (a) business tie network as realisation-dependent (n=116 families) (b) multivariate business and marriage networks as setting-dependent (n=87 families)

38 Example 3a: Realisation-dependent model for Business ties Configuration PLE Configuration PLE Configuration PLE edge star 2-star star 3-path 4-cycle star triangle coathanger

39 Example 3b: Markov model for business and marriage ties configuration λ Q (PLE) configuration λ Q (PLE) Key: business marriage both

40 Example 3c: a setting-dependent Markov model for marriage and business ties Suppose (as in Model 3b) that two ties are neighbours if they share an actor: But for any neighbourhood configuration Q, distinguish: configurations in which all ties lie in a common setting e.g. configurations in which not all ties lie in a common setting e.g. parameter: λ QS λ Q

41 Example 3c: PLEs for setting-dependent bivariate Markov model configuration λ Q λ QS configuration λ Q λ QS Key: business marriage both

42 Example 4: Spatial setting structures Define spatial settings as follows: suppose that actors are located according to a lattice-like grid, and that the distance between actors is computed by a city-block metric A subset of actors has diameter r if the largest distance between any pair is r A setting is the set of all ties among any subset of diameter r or less the grid indicates which actors are at adjacent spatial locations some settings of diameter 2 And consider a setting-dependent Markov graph model, as before

43 Example 4: Setting-dependent Markov models for a 6 6 grid For parameters chosen as follows: configuration λ Q λ QS diameter We simulate the model using settings of different maximum diameters and compute average configuration counts and most triangles are within-settings

44 To summarise: Social units individuals groups... Relations on social units Settings person-to-person person-to-group... geographical sociocultural... Social space consists of variable social entities that are of different types and at different levels Proximities in space also come in different kinds and lead to interactivity within and across levels Social space and social process are difficult to disentangle, although we can make simplifying assertions in particular cases Forget the Cartesian model!

45 Some key issues Measurement: what should we observe if we are interested in modelling processes in social space? Generalised networks and setting information? Estimation methods and model properties: Much promise (Snijders, 22), but there are some important caveats concerning model specification, model behaviour, and data constraints Model evaluation: empirical and theoretical agenda Can we model substantively important network properties? Can we further articulate the links between theoretically-based model specification and identified structural regularities? Interdependence across levels: dynamic formulations

46 Why are dynamic formulations important? Because social processes are both interactive and dynamic: We require ways of investigating complex spatial interdependence, and of making this spatial interdependence more and more temporally structured, till we arrive at the description and measurement of interactional fields (Abbott, 1997) We need to understand the ways in which networks evolve over time through cumulative processes of tie creation and dissolution as they are embedded in a changing community of multiplex relations spawned by multiple organizational affiliations (McPherson, Smith-Lovin & Cook, 21)

47 Co-evolution of action, networks, settings