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

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

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).


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.”


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).

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).


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 }.


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).

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 ).


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).

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)}).


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).

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 ).


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 ).


Graph Duality to Kemeny’s Hexagon of Social
Choice
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.


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 .


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).


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.


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 .


The Inﬁnite Regress of Potential Ties in Dual Social
Network Systems

Reminiscent of:
• The third man argument (Plato) or
• F.H. Bradley’s regress.


Deﬁnitions of Actual–Direct and Potential–Indirect
Ties

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.


Deﬁnitions of Potential–Virtual Indirect Ties in
Time–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.


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.


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.

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%.


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


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 .


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).


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).


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.


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
War
Church 8 7 4 1 8 6 5 3 1 5 3 2
Anti-Racist 3 6 1 4 2 2 2 7 2 1 3 3
Student 4 6 3 3 3 4 5 1 1 1 4 2
Labor Union – Prof. 1 4 2 3 1 7 1 6 6 4 3 1
Political Party 1 4 2 3 1 7 1 6 6 4 4 1
Women 5 1 1 2 4 5 2 8 2 1 3 3
Sport – Recr. 7 5 3 2 6 6 5 4 1 2 2 2
Environmental 2 6 1 4 5 2 6 9 2 1 3 3
Art, Music & Edu. 7 5 3 2 6 6 5 4 1 2 1 2
Neighborhood 9 3 3 3 9 1 7 2 6 3 3 1
Charitable 6 2 1 1 7 6 3 5 3 6 1 2
Anti-Globalist 3 8 1 4 2 3 4 7 4 1 3 3
Third World 3 8 1 4 2 3 4 7 4 1 1 3
Human Rights 3 8 1 4 2 3 4 7 4 1 1 3
Peace 3 8 1 4 2 3 4 7 4 1 4 3
Other 6 2 1 1 7 5 3 5 5 5 4 2


ITALY: The IPPS Interorganizational Community
Structure

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.


ALL 8 COUNTRIES: The IPPS Interorganizational
Community 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.


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