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

The Duality of Structure and Culturein Social Networks:A Formal Analysis

Moses A. Boudourides

Department of MathematicsUniversity of Patras

Greece

[email protected]

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, 2011Moses A. Boudourides The Duality of Structure and Culture in Social Networks

http://nicomedia.math.upatras.gr/sn/dcs_0_slides.pdf

http://nicomedia.math.upatras.gr/sn/IPPScomms_slides.pdf

http://nicomedia.math.upatras.gr/sn/cion_0a.pdf

http://nicomedia.math.upatras.gr/sn/dirisn_0.pdf

Prolegomena

• Assumed setting:

(Social) Structure: Represented by patterns of ties (socialrelationships) among actors in a social network.

Culture: Represented by some germane attributes andattitudes that actors possess and display.

• Two approaches for a formal analysis of structure–culture:

Exogenous covariate effects on stochastic models ofstructure, e.g.,

Snijders, van den Bunt & Steglich (Introduction tostochastic 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

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 isolatedindividuals but social networks and

(b) viable subjects for explanation are not individual attitudes, butdegrees of attitude agreement among individuals in givenstructural situations.”

Doug McAdam (1986): NO, in the context of participationstudies, where “attitudinal affinity” is irrelevant, because:

• “The argument is that structural availability is more importantthan attitudinal affinity in accounting for differentialinvolvement in movement activity. Ideological dispositiontoward participation matters little if the individual lacks thestructural contact to ‘pull’ him or her into protest activity.”


Definition of a Dual Graph System

• A bipartite graph H(U,V ) = (U,V ,E ) with vertex classes Uand V (U ∩ V = ∅) and E a set of connections (orassociations or “translations”) between U and V , i.e.,E ⊂ U × V .1

• A (simple undirected) graph G (U) = (U,EU) on the set ofvertices U and with a set of edges EU ⊂ U × U.

• A (simple undirected) graph G (V ) = (V ,EV ) on the set ofvertices V and with a set of edges EV ⊂ V × V .

• Dual Graph System: G = (U ∪V ,EU ∪E ∪EV )

1By 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 thehypergraph H = (U, V ) (Bollobas, 1998, p. 7).


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 andUV contain only vertices of U, while the blocks VU and VV contain only vertices of V . Loops represent internallinks (colored blue) and lines represent external links (either among vertices of U or V , colored blue, or amongvertices of U and V , colored red, the latter being called translations or traversal links).


An Example of a Dual Graph System

1

2

3

4

A

B

C

D

E

Figure: A dual graph system composed of two graphs G(U) and G(V ), which are “translated” to each otherby a bipartite graph H(U, V ) (with dashed edges), where U = {1, 2, 3, 4} and V = {A, B, C ,D, E}.


An Example of a Vertex–Attributed Graph

1

2

3

4

A

B

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 thevertex–attributed graph and V = {A, B} the values of the attribute. Note that all vertices of U have traversaldegree equal to 1 (as the attribute is exclusionary).


A Vertex–Attributed Graph as a Dual Graph System

• Let Gα(W ) = (W ,F ) be a graph with set of vertices W and setof edges F ⊂W ×W .

• Let all vertices be equipped with an attribute, defined by anassignment mapping α: W → {0, 1}, such that, for any vertexw ∈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 systemGα = (U ∪ V ,EU ∪ E ∪ EV ).


An Example of a Signed Graph

+

−

−

−

+

1

2

3

4

A

B

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 signedgraph and V = {A, B} is a dipole. Note that all vertices of U have traversal degree equal to 1 (as the 2 poles ofthe dipole are exclusionary).


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 toone 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 mappingσ: EU → {+,−} as follows, for any (ui , uj) ∈ EU :

• σ(ui , uj) = +, whenever both ui and uj are “translated” tothe same pole, and

• σ(ui , uj) = −, otherwise.

Then the signed graph Gσ is the dual graph systemGσ = (U ∪ {p, q},EU ∪ E ∪ {(p, q)}).


An Example of a Time–Dependent Graph

1

2

3

4

A

B

C

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 signedgraph and V = {A, B, C} is the succession of time slots. Note that, now, vertices of U may have any traversaldegree (as they could be present or absent at any time slot).


A Time–Dependent Graph as a Dual Graph System

• Let Gt(W ) = (W ,F ) be a graph parametrized over time t, whichmight take at least 2 discrete values.

• Let all vertices of W (for any time t) be associated with a timeassignment mapping τt: W → {0, 1}, such that, for anyvertex w ∈W , τt(w) = 1, when the vertex w is present attime 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 systemsGt = (Wt ,Ft).


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 r Vt2 and wq ∈ Vt2 r Ut1}.• Then the graph Gt1,t2(W ) of time translations from t1 to t2

becomes a dual graph systemGt1,t2 = (Ut1 ∪ Vt2,EUt1

∪ Et1,t2 ∪ EVt2).


Graph Duality to Kemeny’s Hexagon of SocialChoice

zxy z(xy) zyx

(yz)x

yzx

y(xz)

yxz(xy)zxyz

x(yz)

xzy

(xz)y

Figure: This is the target graph of a dual social network system, in which each actor has to rank threealternatives {x, y, z}. The above hexagon represents the 12 possible rankings in the way they are linked togetherwith regards to Kemeny’s distance.


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 bya sequence of vertices a1, a2, . . . , an ∈W such that(aj , aj+1) ∈ F , for all j = 1, 2, . . . , n − 1, where all vertices aredistinct (except possibly the 2 terminal ones).

• A n-path from a1 to an is denoted as (a1, . . . , an).

• If a1 6= 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) =

{(a1, an), when n ≥ 1 and a1 6= an,

{a0}, when n = 0 and a1 = an = a0.


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 thisproblem is #P-complete under polynomial parsimoniousreductions (for any directed or undirected graph).


Examples of Closures

Figure: The actual ties, on the top, are the black colored continuous linesor, in the middle, the dashed lines (translations), while the potential tiesare, at the bottom, colored as follows: red, when induced by a triadicclosure, blue, when induced by a quadruple closuse, and magenta, wheninduced by a quintuple closure.


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–translatedgraph, then (wp,wq) ∈ Et1,t2 if and only if

(wp,wq) = (wp, t1, t2,wq), where

• either wp = wq ∈ Ut1 ∩ Vt2

• or wp ∈ Ut1 r Vt2 and wq ∈ Vt2 r Ut1 .


The Infinite Regress of Potential Ties in Dual SocialNetwork Systems

Reminiscent of:

• The third man argument (Plato) or

• F.H. Bradley’s regress.


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 iscalled traversal or translational.

• A non–traversal dyad in (U × U r EU) ∪ (V × V r EV ) is said toconstitute a potential or indirect tie if it (is not actual butit) forms the closure of an actual traversal path in G ofappropriate length.

• A traversal dyad in U ×V r E is said to constitute a potential orindirect translation if it (is not actual but it) forms theclosure of an actual traversal path in G of appropriate length.


Definitions of Potential–Virtual Indirect Ties inTime–Translated Graph

Let Gt1,t2(W ) = (Ut1 ∪ Vt2,EUt1∪ Et1,t2 ∪ EVt2

) be atime–translated graph between two time instances t1 < t2:

• A non–traversal dyad in Vt2 × Vt2 r EVt2is 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 inGt1,t2(W ) of appropriate length.

• A non–traversal dyad in Vt1 × Vt1 r EVt1is 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 inGt1,t2(W ) of appropriate length.


Typology of Ties in Dual Social Network Systems

• An actual tie is said to actualize (or institutionalize, accordingto Harrison C. White) a potential tie, if the two ties coexistbetween 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 andonly if, ignoring actors with traversal degree ≤ 1, the dualgraph system is bipartite.

• Every potential tie is actualized by a dual quadruple closure ifand only if, ignoring actors with traversal degree = 0, the dualgraph system constitutes a graph isomorphism andtranslations are just permutations of the same vertex set.


Strength of Actualized or Emergent Ties

Let a scalar–valued utility function δ be defined over anyactualized or emergent tie (ui , uj) as follows:

δ(ui , uj) =cij

1 + νij,

νij is the traversal geodesic distance between ui and uj and cij is anormalization constant such that cij > 1, for actualized ties, andcij = 1, for emergent ties.

• The tie (ui , uj) is stronger than the tie (uk , ul) (or (uk , ul) isweaker 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 pathsestablishing the closure of the corresponding terminal actors.


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 thatperiod) war on Iraq took place throughout the world.

• More than seven million people in more than 300 cities all over theworld had participated.

• The largest peace protests since the Vietnam War on one single day.

• An international team of social movement scholars set up the IPPSProject Survey (2003-4), coordinated by Stefaan Walgrave, tostudy this international protest event.

• Over 10,000 questionnaires were distributed in 8 countries during thedemonstrations: in the UK, Italy, the Netherlands, Switzerland,USA, Spain, Germany and Belgium.

• About 6,000 completed questionnaires have been sent back, with asuccessful response rate of well above 50%.


http://webh01.ua.ac.be/m2p/index.php?page=projects&page2=pproject&id=11

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:

1

2

3

4

5

6

7

8

9

10

11

12

In the IPPS data:

• B (blue nodes) is the populationof respondents (varies in each country),

• A (red nodes) is a set of 16 (types of)organizations, to which respondentsdeclared affiliation,

• C (green nodes) is a set of 10 attitudeswith regard to the meaning of the war,about which respondents expressed theirpositions, opinions etc.


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 ofsubsets C = C(V ) = {C1,C2, . . . ,Cp}, called communities, suchthat C maximizes the following benefit function Q, calledmodularity, 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 thebasis of a random graph, which preserves the same degreedistribution with the examined graph G .


Thus, the exact expression of modularity becomes:

Q =c∑

k=1

[ lkm−( dk

2m

)2],

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).


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, inthe sense that it would be decomposed to subgraphs with veryfew internal connections and many external connectionsbetween them).


Modularity Maximization

• If maxQ > 0, over all possible partitions, the graph has acommunity structure, in the sense that most of the graphconnections fall within the communities (of the optimalpartition) than what would have been expected by chance(under the null model). This community structure is strongerthe more Q approaches to 1.

• However, this optimization problem has been proven to beNP-complete (Brandeis et al., 2008) and, thus, onlyapproximate optimization techniques, such as greedyalgorithms, simulated annealing, extremal optimization,expectation maximization, spectral methods etc. can bepractically useful.


ITALY: Membership of Organizations to Community ids for different War Attitudes

(N = 972)

Noatti-tudes

USACru-sadeagainstIslam

Anti-Dicta-torialRegimeWar

UNSecu-rityCoun-cilAu-tho-rizedWar

Warfor Oil

RacistWar

IraqiThreattoWorldPeace

AlwaysWrongWar

WartoOver-throwtheIraqiRegime

FeelingsagainstNe-olib-eralGlob-aliza-tion

Govern-mentalDis-satis-fac-tion

Allatti-tudes

Church 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


ITALY: The IPPS Interorganizational CommunityStructure

Church

Anti−Racist

Student

Labor Union & Prof.

Political Party

Women

Sport & Recr.

Environmental

Art, Music & Edu.

NeighborhoodCharitable

Anti−Globalist

Third WorldHuman Rights

Peace

Other

Figure: Ignoring activists’ attitudes.

ChurchAnti−Racist

Student

Labor Union & Prof.

Political Party

Women

Sport & Recr.

Environmental

Art, Music & Edu.

Neighborhood

Charitable

Anti−Globalist

Third World

Human Rights

Peace

Other

Figure: Taking into account allactivists’ attitudes.


ALL 8 COUNTRIES: The IPPS InterorganizationalCommunity Structure

Church

Anti−Racist

Student

Labor Union & Prof.

Political Party

Women

Sport & Recr. Environmental

Art, Music & Edu.

Neighborhood

Charitable

Anti−Globalist

Third World

Human Rights

Peace

Other

Figure: The meta–community interorganizational network in the eight countries of the IPPS survey, takinginto account the attitudes of all activists.


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