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The Duality of Structure and Culture in Social Networks: A Formal Analysis

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We present here an analytic formalism for a formal study of the structure-culture duality.

We present here an analytic formalism for a formal study of the structure-culture duality.

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  • 1. The Duality of Structure and Culture in Social Networks: A Formal Analysis Moses A. Boudourides Department of Mathematics University of Patras Greece Moses.Boudourides@gmail.com Slides: http://nicomedia.math.upatras.gr/sn/dcs_0_slides.pdf Slides on IPPS data: http://nicomedia.math.upatras.gr/sn/IPPScomms_slides.pdf Draft paper on IPPS data: http://nicomedia.math.upatras.gr/sn/cion_0a.pdf Draft paper on the strength of indirect relations: http://nicomedia.math.upatras.gr/sn/dirisn_0.pdf June 4, 2011 Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 2. Prolegomena • Assumed setting: (Social) Structure: Represented by patterns of ties (social relationships) among actors in a social network. Culture: Represented by some germane attributes and attitudes that actors possess and display. • Two approaches for a formal analysis of structure–culture: Exogenous covariate effects on stochastic models of structure, e.g., Snijders, van den Bunt & Steglich (Introduction to stochastic actor-based models for network dynamics, 2010). Duality of structure and culture, e.g., Breiger (A tool kit for practice theory, 2000). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 3. Do Attitudes Matter to Social Networks? Bonnie Erickson (1988): YES, because “attitudes are made, maintained, or modified primarily through interpersonal processes” and, thus: (a) “natural units of analysis for attitudes are not isolated individuals but social networks and (b) viable subjects for explanation are not individual attitudes, but degrees of attitude agreement among individuals in given structural situations.” Doug McAdam (1986): NO, in the context of participation studies, where “attitudinal affinity” is irrelevant, because: • “The argument is that structural availability is more important than attitudinal affinity in accounting for differential involvement in movement activity. Ideological disposition toward participation matters little if the individual lacks the structural contact to ‘pull’ him or her into protest activity.” Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 4. Definition of a Dual Graph System • A bipartite graph H(U, V ) = (U, V , E ) with vertex classes U and V (U ∩ V = ∅) and E a set of connections (or associations or “translations”) between U and V , i.e., E ⊂ U × V .1 • A (simple undirected) graph G (U) = (U, EU ) on the set of vertices U and with a set of edges EU ⊂ U × U. • A (simple undirected) graph G (V ) = (V , EV ) on the set of vertices V and with a set of edges EV ⊂ V × V . • Dual Graph System: G = (U ∪ V , EU ∪ E ∪ EV ) 1 By considering V as a collection of subsets of U (i.e., V as a subset of P(U), the power set of U, that is the set of all subsets of U), the bipartite graph H(U, V ) is the incidence graph that corresponds (in a 1–1 way) to the hypergraph H = (U, V ) (Bollobas, 1998, p. 7). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 5. The Block Image of a Dual Graph System UU UV VU VV Figure: Each block is composed of a number of different vertices connected to each other: the blocks UU and UV contain only vertices of U, while the blocks VU and VV contain only vertices of V . Loops represent internal links (colored blue) and lines represent external links (either among vertices of U or V , colored blue, or among vertices of U and V , colored red, the latter being called translations or traversal links). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 6. An Example of a Dual Graph System 4 E D 3 C 2 B 1 A Figure: A dual graph system composed of two graphs G (U) and G (V ), which are “translated” to each other by a bipartite graph H(U, V ) (with dashed edges), where U = {1, 2, 3, 4} and V = {A, B, C , D, E }. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 7. An Example of a Vertex–Attributed Graph 4 B 3 2 1 A Figure: A vertex–attributed graph as a dual graph system composed of two graphs G (U) and G (V ), which are “translated” to each other by a bipartite graph H(U, V ) (with dashed edges), where U = {1, 2, 3, 4} is the vertex–attributed graph and V = {A, B} the values of the attribute. Note that all vertices of U have traversal degree equal to 1 (as the attribute is exclusionary). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 8. A Vertex–Attributed Graph as a Dual Graph System • Let Gα (W ) = (W , F ) be a graph with set of vertices W and set of edges F ⊂ W × W . • Let all vertices be equipped with an attribute, defined by an assignment mapping α: W → {0, 1}, such that, for any vertex w ∈ W , α(w ) = 1, when the vertex satisfies the attribute, and α(w ) = 0, otherwise. • Setting: • U = {w ∈ W : α(w ) = 1}, • V = {w ∈ W : α(w ) = 0}, • EU = {(wp , wq ) ∈ F: α(wp ) = α(wq ) = 1}, • EV = {(wr , ws ) ∈ F: α(wr ) = α(ws ) = 0}, • E = {(wp , wr ) ∈ F: α(wp ) = 1 and α(wr ) = 0}. • Then Gα (W ) becomes a dual graph system Gα = (U ∪ V , EU ∪ E ∪ EV ). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 9. An Example of a Signed Graph 4 B + − 3 − − 2 + 1 A Figure: A signed graph as a dual graph system composed of two graphs G (U) and G (V ), which are “translated” to each other by a bipartite graph H(U, V ) (with dashed edges), where U = {1, 2, 3, 4} is the signed graph and V = {A, B} is a dipole. Note that all vertices of U have traversal degree equal to 1 (as the 2 poles of the dipole are exclusionary). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 10. A Signed Graph as a Dual Graph System • Let G (U) = (U, EU ) be a graph. • Let G (V ) = ({p, q}, {(p, q)}) be a dipole. • Suppose that there exist “translations” from all vertices of U to one of the two poles of V , i.e., E = {(u, p) ∪ (u, q): for all u ∈ U}. • Define the sign of each edge in G (U) by an assignment mapping σ: EU → {+, −} as follows, for any (ui , uj ) ∈ EU : • σ(ui , uj ) = +, whenever both ui and uj are “translated” to the same pole, and • σ(ui , uj ) = −, otherwise. Then the signed graph Gσ is the dual graph system Gσ = (U ∪ {p, q}, EU ∪ E ∪ {(p, q)}). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 11. An Example of a Time–Dependent Graph 4 C 3 B 2 1 A Figure: A time–dependent graph as a dual graph system composed of two graphs G (U) and G (V ), which are “translated” to each other by a bipartite graph H(U, V ) (with dashed edges), where U = {1, 2, 3, 4} is the signed graph and V = {A, B, C } is the succession of time slots. Note that, now, vertices of U may have any traversal degree (as they could be present or absent at any time slot). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 12. A Time–Dependent Graph as a Dual Graph System • Let Gt (W ) = (W , F ) be a graph parametrized over time t, which might take at least 2 discrete values. • Let all vertices of W (for any time t) be associated with a time assignment mapping τt : W → {0, 1}, such that, for any vertex w ∈ W , τt (w ) = 1, when the vertex w is present at time t, and τt (w ) = 0, otherwise. • Setting, for any t: • Wt = {w ∈ W : τt (w ) = 1}, • Ft = {(wp , wq ) ∈ F: τt (wp ) = τt (wq ) = 1}, • Then Gt (W ) becomes a family of dual graph systems Gt = (Wt , Ft ). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 13. A Time–Translated Graph as a Dual Graph System • Furthermore, setting, for any two times t1 < t2 : • Ut1 = {w ∈ F: τt1 (w ) = 1}, • Vt2 = {w ∈ W : τt2 (w ) = 1}, • EUt1 = {(wp , wq ) ∈ F: τt1 (wp ) = τt1 (wq ) = 1}, • EVt2 = {(wr , ws ) ∈ F: τt2 (wr ) = τt2 (ws ) = 1}, • Et1 ,t2 = {(wp , wr ) ∈ F: either wp = wq ∈ Ut1 ∩ Vt2 or wp ∈ Ut1 Vt2 and wq ∈ Vt2 Ut1 }. • Then the graph Gt1 ,t2 (W ) of time translations from t1 to t2 becomes a dual graph system Gt1 ,t2 = (Ut1 ∪ Vt2 , EUt1 ∪ Et1 ,t2 ∪ EVt2 ). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 14. Graph Duality to Kemeny’s Hexagon of SocialChoice xyz (xy)z yxz x(yz) y(xz) xzy yzx (xz)y (yz)x zxy z(xy) zyx Figure: This is the target graph of a dual social network system, in which each actor has to rank three alternatives {x, y , z}. The above hexagon represents the 12 possible rankings in the way they are linked together with regards to Kemeny’s distance. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 15. Paths and Closures • Let G = (W , F ) (undirected) graph. • A path of length n (or n-path) in G , from a1 to an , is formed by a sequence of vertices a1 , a2 , . . . , an ∈ W such that (aj , aj+1 ) ∈ F , for all j = 1, 2, . . . , n − 1, where all vertices are distinct (except possibly the 2 terminal ones). • A n-path from a1 to an is denoted as (a1 , . . . , an ). • If a1 = an , the path (a1 , . . . , an ) is open. • If a1 = an , the path (a1 , . . . , an−1 , a1 ) is closed and it forms a (n − 1)-cycle. • For n = 0, a 0-path reduces to a vertex. • The (transitive) closure of a path (a1 , . . . , an ), denoted as (a1 , . . . , an ), is defined as follows: (a1 , an ), when n ≥ 1 and a1 = an , (a1 , . . . , an ) = {a0 }, when n = 0 and a1 = an = a0 . Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 16. The Complexity of Computing Paths • Powers of adjacency matrices yield walks not paths. • This is the problem of “self-avoiding walks” (Hayes, 1998). • Remarkably, Leslie G. Valiant (1979) has shown that this problem is #P-complete under polynomial parsimonious reductions (for any directed or undirected graph). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 17. Examples of Closures Figure: The actual ties, on the top, are the black colored continuous lines or, in the middle, the dashed lines (translations), while the potential ties are, at the bottom, colored as follows: red, when induced by a triadic closure, blue, when induced by a quadruple closuse, and magenta, when induced by a quintuple closure. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 18. Closures in Signed and Time–Translated Graphs • If Gσ = (U ∪ {p, q}, EU ∪ E ∪ {(p, q)}) is a signed graph, then, for all (ui , uj ) ∈ U, • σ(ui , uj ) = + if and only if (ui , uj ) = (ui , p, uj ) = (ui , q, uj ) and • σ(ui , uj ) = − if and only if (ui , uj ) = (ui , p, q, uj ) = (ui , q, p, uj ). • If Gt1 ,t2 = (Ut1 ∪ Vt2 , EUt1 ∪ Et1 ,t2 ∪ EVt2 ) is a time–translated graph, then (wp , wq ) ∈ Et1 ,t2 if and only if (wp , wq ) = (wp , t1 , t2 , wq ), where • either wp = wq ∈ Ut1 ∩ Vt2 • or wp ∈ Ut1 Vt2 and wq ∈ Vt2 Ut1 . Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 19. The Infinite Regress of Potential Ties in Dual SocialNetwork Systems Reminiscent of: • The third man argument (Plato) or • F.H. Bradley’s regress. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 20. Definitions of Actual–Direct and Potential–IndirectTies Given a dual social network system G = (U ∪ V , EU ∪ E ∪ EV ): • Any tie in EU ∪ E ∪ EV is called actual or direct. • Any actual tie in E connecting the dual graph components is called traversal or translational. • A non–traversal dyad in (U × U EU ) ∪ (V × V EV ) is said to constitute a potential or indirect tie if it (is not actual but it) forms the closure of an actual traversal path in G of appropriate length. • A traversal dyad in U × V E is said to constitute a potential or indirect translation if it (is not actual but it) forms the closure of an actual traversal path in G of appropriate length. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 21. Definitions of Potential–Virtual Indirect Ties inTime–Translated Graph Let Gt1 ,t2 (W ) = (Ut1 ∪ Vt2 , EUt1 ∪ Et1 ,t2 ∪ EVt2 ) be a time–translated graph between two time instances t1 < t2 : • A non–traversal dyad in Vt2 × Vt2 EVt2 is said to constitute a potential or past–indirect tie if it (is not actual at time t2 but it) forms the closure of an actual time–traversal path in Gt1 ,t2 (W ) of appropriate length. • A non–traversal dyad in Vt1 × Vt1 EVt1 is said to constitute a virtual or future–indirect tie if it (is not actual at time t1 but it) forms the closure of an actual time–traversal path in Gt1 ,t2 (W ) of appropriate length. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 22. Typology of Ties in Dual Social Network Systems • An actual tie is said to actualize (or institutionalize, according to Harrison C. White) a potential tie, if the two ties coexist between the same pair of actors. • A non–actualized potential tie is said to be emergent. • Every potential tie is emergent by a dual triadic closure if and only if, ignoring actors with traversal degree ≤ 1, the dual graph system is bipartite. • Every potential tie is actualized by a dual quadruple closure if and only if, ignoring actors with traversal degree = 0, the dual graph system constitutes a graph isomorphism and translations are just permutations of the same vertex set. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 23. Strength of Actualized or Emergent Ties Let a scalar–valued utility function δ be defined over any actualized or emergent tie (ui , uj ) as follows: cij δ(ui , uj ) = , 1 + νij νij is the traversal geodesic distance between ui and uj and cij is a normalization constant such that cij > 1, for actualized ties, and cij = 1, for emergent ties. • The tie (ui , uj ) is stronger than the tie (uk , ul ) (or (uk , ul ) is weaker than (ui , uj )) whenever δ(ui , uj ) > δ(uk , ul ). • If δ(ui , uj ) = δ(uk , ul ), (ui , uj ) is stronger than (uk , ul ) (or (uk , ul ) is weaker than (ui , uj )) whenever ων (ui , uj ) > ων (uk , ul ), where the weight ων is the number of traversal paths establishing the closure of the corresponding terminal actors. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 24. The International Peace Protest Survey (IPPS)http://webh01.ua.ac.be/m2p/index.php?page=projects&page2=pproject&id=11 • On February 15, 2003, mass protests against the imminent (at that period) war on Iraq took place throughout the world. • More than seven million people in more than 300 cities all over the world had participated. • The largest peace protests since the Vietnam War on one single day. • An international team of social movement scholars set up the IPPS Project Survey (2003-4), coordinated by Stefaan Walgrave, to study this international protest event. • Over 10,000 questionnaires were distributed in 8 countries during the demonstrations: in the UK, Italy, the Netherlands, Switzerland, USA, Spain, Germany and Belgium. • About 6,000 completed questionnaires have been sent back, with a successful response rate of well above 50%. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 25. Social Network Analysis of the IPPS Data Here, we have decoded the survey data so that, for each of the 8 countries, we obtain a partial tripartite graph G (A, B, C ) of the following form: 5 In the IPPS data: 1 • B (blue nodes) is the population 6 10 of respondents (varies in each country), 2 • A (red nodes) is a set of 16 (types of) organizations, to which respondents 7 11 declared affiliation, 3 • C (green nodes) is a set of 10 attitudes 8 12 with regard to the meaning of the war, about which respondents expressed their 4 positions, opinions etc. 9 Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 26. Communities in Graphs Let G be a graph on a set of vertices V . A community structure in G is a partition of V in a family of subsets C = C(V ) = {C1 , C2 , . . . , Cp }, called communities, such that C maximizes the following benefit function Q, called modularity, which is defined (Newman & Girvan, 2004) as: Q = (fraction of connections within communities) - (expected fraction of such connections). In the null model, the expected fraction above is calculated on the basis of a random graph, which preserves the same degree distribution with the examined graph G . Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 27. Thus, the exact expression of modularity becomes: c lk dk 2 Q= − , m 2m k=1 where• c is the total number of communities in G ,• m is the total number of connections in G ,• lk is the total number of edges inside community Ck and• dk is the sum of degrees of all vertices in Ck (in both latter cases, counting multiplicity of edges, when the graph is weighted). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 28. Properties of Q:• By normalization in definition, −1 ≤ Q ≤ 1.• Q = 0 if and only if the whole graph is a single community (i.e., |C| = 1).• If every vertex of the graph is a community–singleton (i.e., |C| = |V |), then Q ≤ 0.• If Q ≤ 0, for every partition, then G has no community structure (in fact, such a graph would be strongly multipartite-like, in the sense that it would be decomposed to subgraphs with very few internal connections and many external connections between them). Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 29. Modularity Maximization • If max Q > 0, over all possible partitions, the graph has a community structure, in the sense that most of the graph connections fall within the communities (of the optimal partition) than what would have been expected by chance (under the null model). This community structure is stronger the more Q approaches to 1. • However, this optimization problem has been proven to be NP-complete (Brandeis et al., 2008) and, thus, only approximate optimization techniques, such as greedy algorithms, simulated annealing, extremal optimization, expectation maximization, spectral methods etc. can be practically useful. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 30. ITALY: Membership of Organizations to Community ids for different War Attitudes (N = 972) No USA Anti- UN War Racist Iraqi Always War Feelings Govern- All atti- Cru- Dicta- Secu- for Oil War Threat Wrong to against mental atti- tudes sade torial rity to Over- Dis- tudes against Regime War throw Ne- satis- Coun- World olib- Islam War cil Peace the fac- Iraqi eral tion Au- Glob- tho- Regime aliza- rized tion WarChurch 8 7 4 1 8 6 5 3 1 5 3 2Anti-Racist 3 6 1 4 2 2 2 7 2 1 3 3Student 4 6 3 3 3 4 5 1 1 1 4 2Labor Union – Prof. 1 4 2 3 1 7 1 6 6 4 3 1Political Party 1 4 2 3 1 7 1 6 6 4 4 1Women 5 1 1 2 4 5 2 8 2 1 3 3Sport – Recr. 7 5 3 2 6 6 5 4 1 2 2 2Environmental 2 6 1 4 5 2 6 9 2 1 3 3Art, Music & Edu. 7 5 3 2 6 6 5 4 1 2 1 2Neighborhood 9 3 3 3 9 1 7 2 6 3 3 1Charitable 6 2 1 1 7 6 3 5 3 6 1 2Anti-Globalist 3 8 1 4 2 3 4 7 4 1 3 3Third World 3 8 1 4 2 3 4 7 4 1 1 3Human Rights 3 8 1 4 2 3 4 7 4 1 1 3Peace 3 8 1 4 2 3 4 7 4 1 4 3Other 6 2 1 1 7 5 3 5 5 5 4 2 Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 31. ITALY: The IPPS Interorganizational CommunityStructure Art, Music & Edu. Political Party Charitable Neighborhood Labor Union & Prof. Student Sport & Recr. Neighborhood Environmental Church Environmental Anti−Globalist Third World Student Human Rights Peace Other Church Anti−Globalist Anti−Racist Political Party Women Peace Women Sport & Recr. Human Rights Other Charitable Third World Anti−Racist Labor Union & Prof. Art, Music & Edu. Figure: Taking into account all Figure: Ignoring activists’ attitudes. activists’ attitudes. Moses A. Boudourides The Duality of Structure and Culture in Social Networks
  • 32. ALL 8 COUNTRIES: The IPPS InterorganizationalCommunity Structure Anti−Globalist Neighborhood Anti−Racist Peace Women Human Rights Political Party Other Sport & Recr. Environmental Labor Union & Prof. Third World Art, Music & Edu. Student Church Charitable Figure: The meta–community interorganizational network in the eight countries of the IPPS survey, taking into account the attitudes of all activists. Moses A. Boudourides The Duality of Structure and Culture in Social Networks

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