Transcript Self-similarity of Complex Social Networks

5th Conference on Applications of Social Network Analysis
ASNA 2008
University of Zurich, 12 September 2008
Self-similarity of Complex
Social Networks – A
Sociological Perspective
Haiko Lietz
University of Duisburg-Essen, Institute of Sociology, Germany
Mittweida University, Department of Mathematics, Physics, and Computer
Science, Germany
[email protected]
Complex Adaptive System (Holland 1995)
http://en.wikipedia.org/wiki/Image:Complex-adaptive-system.jpg
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Introduction
Micro-macro problem: gap between individual and structural
levels in social theory
Individualist theories of social emergence: macro-social
properties and laws can be explained in terms of properties
and laws about individuals and their relations (Homans,
Coleman)
Collectivist theories of social emergence: Emergence is
incompatible with such such reductionist individualism (Blau,
Bhaskar, Archer, Porpora, Kontopoulos, Sawyer)
Emergence has been proposed to mediate structure and
agency, society and the individual (Sawyer 2005)
But Sawyer‘s dedicated account leaves much to be desired
in terms of mechanisms: How do social formations emerge?
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Introduction
Social scaling: Social systems cover 9 orders of magnitude
(if measured in terms of individuals)
Proposition: Emergence happens on all social scales in selfsimilar ways (White 2008)
Self-similarity: A system is self-similar when it has similar
properties as ist components, their components, and so on
Network approach: Components as nodes, relations as
edges (Wasserman & Faust 1994)
Sociological perspective: Input from „new“ science of
complex networks (Barabási 2002, Watts 2004)
Complex network research: At intersection of graph theory
and statistical mechanics (Albert & Barabási 2002, Newman 2003a)
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Content
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2
3
4
5
Introduction
Input from Complexity Science
Identity and Control
Network Analysis
Conclusions
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Input from Statistical Mechanics (Wilson 1979)
T = 2Tc
T = 1.05Tc
T = Tc
A ferromagnet is magnetised
when more than 50% of ist
component spins point in the
same direction (black or white)
Above its critical temperature
Tc = 1044K the system is not
magnetised: clusters of
correlated spins are
characteristically small
Below Tc it is magnetised:
clusters are characteristically
large
At Tc the metal is at its critical
point
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Input from Statistical Mechanics (Newman 2005)
Between order and chaos: At criticality
the system undergoes a phase transition
from order (magnetism) to chaos (no
magnetism)
Scale invariance: At criticality the system
has no characteristic length (it is selfsimilar)
Power laws: At criticality the system is
described by power law probability
distributions with characteristic scaling
exponents
Cluster size: Many small clusters, few
large clusters
Cluster lifetime: Many short-lived clusters,
few long-lived clusters
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Scale-invariant Complex Systems
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Scale-free Complex Networks
Order
Scale-free (Order in Chaos)
Chaos
Scale-free degree distributions: no characteristic degree
Power laws are ubiquitous in biological networks (metabolic, protein
interaction, and neural networks) and technical networks (power grid,
Internet)
They have been found in information networks (WWW, e-mail networks),
large-scale social networks (citation, coauthorship, telephone call, film
actor and musician collaboration networks) and economic networks
(stock and money flow, world trade, production market networks)
(Newman 2003a; Caldarelli 2007)
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Fractal Complex Networks
The discovery of scale invariance in
complex networks sparked a search for
self-similarity as an ordering principle
(Strogatz 2005)
Application of renormalization
procedure: only scale-free networks
with hub (nodes with high degree)
repulsion are fractal (exhibit the selfsimilarity property) (Song et al. 2005)
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Properties of Social, as Opposed to Non-social Networks
Clustering higher (Newman 2003b)
Degree distributions often exhibit exponential cutoffs or no
scale invariance at all (Amaral et al. 2000; Newman 2003a)
Positive degree correlations (no hub repulsion) (Newman 2003b)
As a consequence: no self-similarity property (Song et al. 2006)
How can the concept of self-similarity be applied to
complex social networks in a way that is not purely
structural?
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Content
1
2
3
4
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Introduction
Input from Complexity Science
Identity and Control
Network Analysis
Conclusions
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Basics of Identity and Control (White 2008)
Identities as actors seek control over turbulent ecology
Stories emerge from interacting control projects as these
build networks
Identities find footing by collectively embedding into higher
level context
As identities couple through stories, a higher level identity
emerges which engages in ist own control projects
Three types of disciplines („social molecules“) serve as
mechanisms of social action that configure identities
Context is constantly shaped by collective dynamics and, at
the same time, feeds back on these
All these concepts are scale invariant (Lietz 2008)
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Order and Chaos (White 2008)
Resulting social structure (space-time) is always borne of
action with constant processes of emergence and feedback
But: „Sociology has to account for chaos and normality
together” (p.1)
Processes in social space-time are shaped by three
stochastic variables:
(1) Contigency: there is a repertoire of possible stories and
story-sets
(2) Ambage: there is uncertainty in social relations
(3) Ambiguity: there is uncertainty in cultural relations
„These [variables] are assessments that may or may not
prove to be measurable like temperature.“ (p.72)
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Style and Self-similarity (White 2008)
Style is a sensibility: It combines interpretations of situations
with sets of practices
Style is settled through continued reenactment: Temporality
emerges from style
Style is self-similar process: Styles reproduce themselves
similarly on short and long length scales (spatial and
temporal)
Style is enacted in social space-time: It presupposes, and is
a means of coping with, stochastic context
On the macro scale „a [power law] size distribution profile is
a surface sign of the likelihood of finding a style.“ (p.149)
There is a similarity in modeling of style and critical
phenomena as processes between order and chaos
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Content
1
2
3
4
5
Introduction
Input from Complexity Science
Identity and Control
Network Analysis
Conclusions
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Data on German Deputies‘ Policy Affiliations
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Modeling Policy as Style
Micro level: Deputies have individual sensibilities and chose
their affiliations to non-parliamentary organizations according
to their style (a)
Meso level: On the level of political parties social contexts
emerge as lasting patterns of sensibility
Macro level: Multiple styles cumulate into a scale invariant
policy profile (b)
1
10
100
1,E+00
1,E-01
1,E-02
1
10
100
1,E+00
a
1,E-01
b
(a) Exponential probability distribution
(logarithmic binning) of number of
affiliations to non-parliamentary
organizations per deputy (R2 = 96.9%)
1,E-02
1,E-03
1,E-03
1,E-04
1,E-04
(b) Power law probability distribution (log.
bin.) of resulting size of nonparliamentary organizations (γ = 2.2; R2 =
99.6%)
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Networks from Affiliations
(Breiger 1974)
Resulting networks are socio-cultural category networks
(catnets)
Trade-off: Ambiguity (cultural uncertainty) is low, ambage
(social uncertainty) is high
Stochastic social context for unfolding of style is provided
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Networks from Affiliations
Only person networks analyzed
Only four political parties considered that were present in all
three legislative (left network)
Categories: black if government coalition partner in 20022005; otherwise white (right network)
9 Networks generated (1-3 periods and 1-3 types of tie) to
study networks at slightly different length scales
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Scaling Analysis of Networks
Exponential fits to
degree distribution
always better than
90% (least squares
method, log. bin.)
Networks not
scale-free, but size
of power law
regime increases
with network size
(as expected)
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Emergent Properties of Networks
Small-world property found (large SW Quotient)
Slightly positive degree correlation
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Methods
Goal is to identify self-similarity using socio-cultural markers
(not purely structural ones)
Two markers:
(1) Proportion government: 0 when all nodes are white; 1
when all are black
(2) E-I Index: -1 when all edges are between nodes of same
color (all ties internal); 1 when all edges are between nodes
of different colors (all ties external) (Krackhardt & Stern 1988)
Two clustering algorithms:
(1) Blockmodeling: identifies structurally equivalent node sets
(akin to reversed renormalization procedure) (White et al. 1976)
(2) Structural Cohesion: finds nested cohesive cores (Moody &
White 2003)
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Blockmodeling Consecutive Networks
7 blocks
after 3
splits
Not shown:
Homophily
as selfsimilarity
(McPherson et
al. 2001)
Prop. Gov.:
trend
towards
less
polarization
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Evolution of Structural Cohesion Among Permanent Deputies
0
k
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A look at deputies
present 20022008
2002
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Evolution of Structural Cohesion Among Permanent Deputies
0
k
20
A look at deputies
present 20022008
Increasing density
2005
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Evolution of Structural Cohesion Among Permanent Deputies
0
k
20
A look at deputies
present 20022008
Increasing density
Evolution of a
cohesive core (k
is level of
structural
cohesion (cf.
Guimera et al. 2005)
2008
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Evolution of Structural Cohesion Among Permanent Deputies
Trends:
(1) Decreasing
polarization towards
cores
(2) Decreasing
polarization in time
(3) Decreasing
proportion government
in core
Self-similarity:
Constant sensibility of
„Moving together“ and
closing gov./opposition
gap
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Content
1
2
3
4
5
Introduction
Input from Complexity Science
Identity and Control
Network Analysis
Conclusions
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Take Home Message
Self-similarity as a principle of social emergence and
feedback offers a lead to the micro-macro problem
Power laws are signatures of scale invariance and selfsimilarity
Purely structural approaches to self-similarity may not be
applicable to complex social networks and don‘t capture the
socio-cultural flesh and blood
Sociological self-similarity analysis does not require large
datasets (although they are recommended to convincingly
show effects over many scales of length)
There seems to be a possibility of modeling certain social
processes as critical phenomena between order and chaos
(Watts 1999; Amaral et al. 2000)
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These slides at
www.haikolietz.de/docs/self-similarity.pdf
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