Spatializing Social Networks

Transcription

1 Spatializing Social Networks Carter T. Butts Department of Sociology and Institute for Mathematical Behavioral Sciences University of California, Irvine

2 Space, Demography, and Social Structure Three areas, three focii Demography: properties of human populations Geography: spatial properties of objects or entities Social Networks: relational properties of social units The intersection: spatial properties of human social macrostructure Today: an introduction to this exciting area!

3 Structure of the Session Part I - Introduction to Network Concepts Part II - Spatial Properties of Network Structure Part III - Application: Extrapolative Simulation

4 Part I - Introduction to Network Concepts

5 Focus on Relations Relationship: an irreducible property of two or more entities Compare with properties of entities alone ( attributes ) Focus: the properties and consequences of relations (rather than individual properties) Entities can be persons, non-human animals, groups, locations, organizations, regions, etc. Relationships can be communication, acquaintanceship, sexual contact, proximity, migration rate, alliance/conflict, etc. Social network analysis: the study of relational data arising from social systems

6 Some Vocabulary Network: a collection of entities, together with a set of relations on those entities 2 3 Entities: nodes, or vertices Relations: edges, or ties Focus on dyadic relations Directed vs. undirected edges May be signed or valued Graph: a set of vertices together with a set of edges Mathematical representation of social structure

7 Sociometric Notation Based on graph theory Graph defined as G=(V,E) Vertex set: V={v,...,vN} Edge set: E Undirected: E {{vi,vj}:vi,vj V} Directed: E {(vi,vj):vi,vj V} Simple iff (vi,vj),(vi,vj) E Simple operations G G2=(V V2,E E2) G G2=(V V2,E E2) { V ={v, v 2, v 3, v 4, v 5 } {v, v 2 }, {v, v 3 }, {v 2, v 3 }, E= {v 2, v 4 }, {v 2, v 5 }, {v 3, v 4 }, {v 3, v 5 }, {v 4, v 5 } V ={v, v 2, v 3, v 4, v 5, v 6 } { v 2, v, v 2, v 4, v 2, v 5, v v, v 3, v 5, v 4, v 2, E= 2, 6 v 4, v 6, v 5, v 6, v 6, v 4, v 6, v 5 } }

8 Subgraphs and Cuts Subgraph G G2 iff V V2, E E2 Selection of vertices, w/all associated edges G[S]=(S,{e E:e S}) G[S,S2]={{s,s2} E: s S,s2 S2} G[S,S2]={(s,s2) E: s S,s2 S2} Edge cut 3 4 Induced Subgraph Selection of vertices and edges from G

9 A Bit More Vocabulary Adjacency 3 Path: a sequence of adjacent vertices (and connecting edges) with no repetitions Walk: like a path, but repetition is allowed Cycle: like a path, but start/endpoints are the same i and j are connected iff an i,j path exists Component: a maximal set of connected vertices Isolate: a component of size Connectedness and components 2 Walks, paths, and cycles i is adjacent to j iff i sends a tie to j

10 Relational Data Relational (network) data concerns connections among entities, rather than attributes of entities Entities can be persons, organizations, concepts, etc. Relations can be interaction, proximity, membership, etc. Two basic types (for now, at least!) One-mode data: connections among one sort of entity Two-mode data: connections among two sorts of entities (Both are useful, though we will deal primarily with one-mode data today...)

11 One-Mode Data Networks with one vertex class Organizations, individuals, concepts, etc. Represented by adjacency matrices Vertices on rows and columns Aij= if i sends a tie to j, else Aij= Can contain edge values, where applicable (Aij is value of i,j edge) Symmetric in undirected case Diagonals represent self-ties Often treated as undefined Mt. Si SAR EMON, Confirmed Ties

12 Two-Mode Data Networks with two vertex classes Different entity types Membership Matching/containment Mixed-class movement matrices Represented by incidence matrices Senders on rows, receivers columns Can be used to obtain dual representations A A 3 B B 4 A B A B

13 Relational Data - Designs Own-tie reports Personal ties elicited from each ego Standard instruments: roster and name generator Pros: Easily implemented, most common design Cons: Vulnerable to reporting error Egocentric network sampling Personal ties elicited from ego, followed by induced ties Standard instrument: name generator followed by roster Pros: Well-suited to large-scale survey sampling; provides information on ego's neighborhood Cons: Vulnerable to reporting error; false positives/negatives on own ties contaminate sampling of neighbors' ties

14 Designs, Cont. Link-tracing Personal ties elicited from ego; new ego(s) chosen from alters; process is iterated (possibly many times) Standard instruments: multiwave own-report, RDS Pros: Allows estimation of network properties for large and/or hard to reach populations; highly scalable; can be robust to poor seed sampling Cons: Vulnerable to reporting error; reporting errors can contaminate design (but may be less damaging than ego net case); often difficult to execute Arc-sampling Reports on third-party ties elicited from ego; multiple egos may be sampled for each third-party tie Archival/observer data is a special case Standard instrument: CSS Pros: Very robust to reporting error (via modeling); can be very robust to missing data Cons: Can impose large burden on respondents; can be difficult to execute

15 Related Issue: Network Boundary Problem Important problem: whence the vertex set? If misspecified, theoretically relevant ties may be missed Typical cases Exogenously defined Relationally defined Isolated social unit (component), locally dense Design defined Physical region, group/cohort membership Alters named in ego net, link trace; sampled egos Major distinction: local versus global properties Different sampling methods needed for each

16 Comment: Interpersonal vs Other Networks As noted, networks come in many flavors Focus for the remainder of the talk: interpersonal networks Vertices, edges chosen based on substantive interest Immediate interest to population researchers Related to migration, cultural diffusion, epidemiology Provides a link between social structure, population demography These are not the only interesting networks with spatial properties, however! Migration, international relations, capital flows, trade, transportation, etc. Feel free to ask about other kinds of networks, too!

17 Part I Summary Emphasis on relational structure Graph theoretic/sociometric terms and notation Relational data: one mode, or two? Typical designs for relational data collection Boundary issues

18 Part II: Spatial Properties of Network Structure (Faust et al., 999)

19 Early Thoughts If we ever get to the point of charting a whole city or a whole nation, we would have an intricate maze of psychological reactions which would present a picture of a vast solar system of intangible structures, powerfully influencing conduct, as gravitation does bodies in space. Such an invisible structure underlies society and has its influence in determining the conduct of society as a whole... Jacob Moreno, NYT, April 3 933

20 Spatial Embeddings Simple idea: assign vertices to spatial locations Spatial embedding of G=(V,E) Location function ℓ:V S, where S is an abstract space To simplify, will use notation ℓi=ℓ(vi) Properties of S May or may not be continuous May or not be metric May contain social dimensions ( Blau space) as well as physical ones

21 Homophily and Propinquity Let d be a distance on S Homophily: a phenomenon in which Pr({vi,vj} E) is decreasing in d(ℓi,ℓj) Propinquity: homophily based on physical distance or travel time Stylized facts Homophily is routinely observed for most interpersonal relations Many mechanisms can lead to it Differential formation/survival of ties by distance Alter-directed migration Exposure to shared focii (Data from Freeman et al., 988)

22 Aside: Scale in Structural Research Microinteraction Group Behavior Classical Network Analysis Social Macrostructure

23 Quantifying Macrostructure Basic concept: spatially embedded populations Closed curves in space define sets of actors Regions are synonymous with populations Standard network analysis as a microfoundation Conception of macrostructure as a fabric of relations among populations of spatially embedded actors Features of the fabric - not ties or configurations - are the focus Individual nodes are not visible, but underlying networks are assumed Measures of macrostructure are derived from aggregate networks Measures must be defined in [primarily] graph theoretic terms Measures must lend themselves to large-population study Interregional relations should emerge from low-level network features

24 Spatial Subgraph Measures Based on the properties of spatially induced subgraphs Ex: Internal Tie Volume (Vi) Equal to total number of internal ties within a region Formally: V i A = E G [ A ] Related to social density and (indirectly) to connectivity and mean path length Easily sampled using own-tie report data

25 Spatial Cut Measures Based on properties of spatially induced (edge) cut sets Ex: External Tie Volume (Ve) Equal to the total number of ties entering/leaving a region Formally: V e A = G [ A, A ] Related to group exposure and influence May also be sampled from own-tie reports

26 (Directional) Tie Flow Across Cuts Based on the net direction of edges within spatially induced cuts Treats all edges as outwardly directed vectors Ex: Directed External Tie Volume Net directional outward tie flow Formally: e G [ A, A ] V e A = v e ve

27 Interregional Relations Reflect aggregate properties of interpersonal networks spanning two or more distinct spatial regions Communication between states Contacts between cities, regions Form classical macro-level networks Vertices are regions Edges are interregional relations (generally valued) Traffic/migration analysis Disease/cultural diffusion Urban planning Numerous uses

28 Interregional Tie Volume Total flow of ties between two regions Equal to cardinality of the interregional edge cut Formally: V A, B = G [ A, B ] Related to raw interregional diffusion rates Basic building block for interregional relations

29 Structural Influence Proportion of alter s total tie volume given by the interregional tie volume from ego to alter Equal to the interregional tie volume divided by alter s total tie volume Formally: V A, B I A, B = V i B V e B Sample application: cultural diffusion

30 Structural Absorption Extent to which ego s ties to alter outnumber alter s ties to itself Equal to ratio of interregional tie volume to alter s internal tie volume Formally: V A, B A A, B = V i B Sample applications: solidarity, regional identification, annexation

31 Part II Summary Spatial embeddings Homophily and propinquity Macrostructural measures Interregional relations

32 Part III - Application: Extrapolative Simulation

33 Tie Probability as a Function of Distance Intuition: Problem: How can we find such a function (if one even exists)? Solution: If we can find a functional form linking distance to tie probability, we can predict large-scale structure from population distribution Find data with numbers of possible and realized ties at multiple distances Infer the functional relationship from observed incidence So, what kinds of functions are plausible candidates?

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35 Bayesian Inference/Model Selection for Tie Models General strategy Bayesian inference for tie model parameters Model selection using Bayes factors Computation of posterior quantities and BFs using MCMC sampling and MC integration Overall model properties Single hierarchical level Tie population taken as known Distance taken as known Edge realizations independent conditional on the model Parameters are independent in their prior distributions

36 Three Data Sets Festinger et al. (95) - Social Friendship Collected in post WW-II housing project during Subjects asked to provide three people you most see socially Hägerstrand (967) - Weak Contacts Collected in rural Sweden during the summer of 95 Telephone calls between exchanges monitored Realized calls, distances, and subscribers reported for a variety of regions Freeman et al. (988) - Face-to-Face Interaction Collected on a southern California beach 54 subjects observed for 6 hours over a 3 day period Mean distance between actors and minutes interacting reported

37 Extrapolative Simulation of Social Macrostructure Model Tie model based on posterior mode for Festinger et al. data: Pr { v i, v j } G d ij = pb d ij p b.52, 29.87, 2.83 Relational properties Fairly sparse Non-saturated for small distances Should account for nearly all structure for scales >km (see paper)

38 Simulated Regional Relations Using spatial models, we can simulate regional relations Simple power law model, posterior mode parameters (Festinger) Expected values of relations assessed via MC integration (antithetic variates employed for variance reduction) Effect of population geometry on center periphery relations Population concentrated in apx 2x2km region Apx 5, individuals total Four population configurations Single (Gaussian) center Center with subsidiary differentiation Binary system (double Gaussian) Weak center with high-density ring Effect of central 3x3km region on surroundings in xkm grid cells

39 Drawing Moreno s Map With global data population data sets, it becomes possible to make macrostructural predictions worldwide Macrostructural variables may be evaluated across cities, states, regions, countries, etc. Instantaneous versions of macrostructural measures can be computed across space to create structural maps Crossovers with geography, spatial demography, and urban sociology Effects of transportation systems, political boundaries, and physical barriers on social networks can be determined Network influences on migration, population distribution, and regional economic systems can be accounted for

40 Simulation Strategy Model Posterior mode, Festinger et al. data Data Population data from Tobler et al.'s (995) gridded world population set Smoothed data, 5'x5' cells Expected tie volumes estimated using MC quadrature Uniform within-cell population density Point-mass approximation where cross-cell edge probabilities vary by less than.

41 Part III Summary Strong, power-law propinquity effects on several relations Extrapolative simulation of interregional relations Generation of social maps using extrapolative simulation

42 The Take-Away Spatializing social networks yields new tools for relational analysis Strong homophily/propinquity effects Macrostructural measures, interregional ties from interpersonal relationships Tie probability falls quickly with distance Population+Distance=Macrostructure Extrapolations from distance models allow us to predict large-scale structure By comparing structures across hypothetical population geometries, we can anticipate effects of demographic shifts Estimation of structural geography possible for large regions

43 Lastly: Roads Not Taken (Here) Other spatial networks Transportation, trade, migration,... (you know the litany!) Connections between spatial, network models Network/spatial autocorrelation Exponential family ( Gibbs ) models for networks Spatial processes on networks Models for flow, movement, congestion Spatially generated network dynamics And many more try coming up with some!