5th Conference on Applications of Social Network Analysis ASNA 2008 University of Zurich, 12 September 2008 Self-similarity of Complex Social Networks.

5th Conference on Applications of Social Network Analysis

ASNA 2008University of Zurich, 12 September 2008

Self-similarity of Complex Social Networks – A

Sociological Perspective

Haiko LietzUniversity of Duisburg-Essen, Institute of Sociology, Germany

Mittweida University, Department of Mathematics, Physics, and Computer Science, Germany

[email protected]

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Complex Adaptive System (Holland 1995)

http://en.wikipedia.org/wiki/Image:Complex-adaptive-system.jpg

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 self-similar ways (White 2008)

Self-similarity: A system is self-similar when it has similar properties as ist components, their components, and so onNetwork 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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Introduction

1 Introduction

2 Input from Complexity Science

3 Identity and Control

4 Network Analysis

5 Conclusions

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Content

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 (Wilson 1979)

T = 2Tc

T = 1.05Tc

T = Tc

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 self-similar)Power laws: At criticality the system is described by power law probability distributions with characteristic scaling exponentsCluster size: Many small clusters, few large clustersCluster lifetime: Many short-lived clusters, few long-lived clusters

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Input from Statistical Mechanics (Newman 2005)

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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)

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 self-similarity property) (Song et al. 2005)

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Fractal Complex 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)

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Properties of Social, as Opposed to Non-social Networks

How can the concept of self-similarity be applied to complex social networks in a way that is not purely structural?

1 Introduction

2 Input from Complexity Science

3 Identity and Control

4 Network Analysis

5 Conclusions

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Content

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 theseAll these concepts are scale invariant (Lietz 2008)

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Basics of Identity and Control (White 2008)

Resulting social structure (space-time) is always borne of action with constant processes of emergence and feedbackBut: „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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Order and Chaos (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 contextOn the macro scale „a [power law] size distribution profile is a surface sign of the likelihood of finding a style.“ (p.149)

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Style and Self-similarity (White 2008)

There is a similarity in modeling of style and critical phenomena as processes between order and chaos

1 Introduction

2 Input from Complexity Science

3 Identity and Control

4 Network Analysis

5 Conclusions

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Content

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Data on German Deputies‘ Policy Affiliations

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)

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Modeling Policy as Style

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1 10 100

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1 10 100 (a) Exponential probability distribution (logarithmic binning) of number of affiliations to non-parliamentary organizations per deputy (R2 = 96.9%)

(b) Power law probability distribution (log. bin.) of resulting size of non-parliamentary organizations (γ = 2.2; R2 = 99.6%)

a b

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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 2002-2005; otherwise white (right network)

9 Networks generated (1-3 periods and 1-3 types of tie) to study networks at slightly different length scales

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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Scaling Analysis of Networks

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Emergent Properties of Networks

Small-world property found (large SW Quotient)

Slightly positive degree correlation

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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Methods

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Blockmodeling Consecutive Networks

7 blocks after 3 splitsNot shown: Homophily as self-similarity (McPherson et al. 2001)

Prop. Gov.: trend towards less polarization

A look at deputies present 2002-2008

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Evolution of Structural Cohesion Among Permanent Deputies

2002

0 k 20

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Evolution of Structural Cohesion Among Permanent Deputies

2005

0 k 20A look at deputies present 2002-2008

Increasing density

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Evolution of Structural Cohesion Among Permanent Deputies

2008

0 k 20A look at deputies present 2002-2008

Increasing densityEvolution of a cohesive core (k is level of structural cohesion (cf. Guimera et al. 2005)

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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

1 Introduction

2 Input from Complexity Science

3 Identity and Control

4 Network Analysis

5 Conclusions

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Content

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 self-similarity

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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Take Home Message

These slides at

www.haikolietz.de/docs/self-similarity.pdf

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