Network dynamics, cohesion and scaling

Douglas R. WhiteUniversity of California – Irvine

ISCOM Annual MeetingChamps-sur-Marne, France, 5-9 December 2003

Logic of the presentation

• Mathematical models of network scaling– Developed from general to specific

• Hypothesis testing– E.g., cohesive attachment and unit formation– Structure of fields and organizations

• Empirical examples– What do model parameters reflect– Micro-macro linkages

Four aspects of general modeling strategies are presented

• Why and what are power-laws? Integrating scaling coefficients across domains

• Power-law networks with preferential attachment to cohesion

• Stretched exponential modeling with basal units

• Reinterpreting power-law networks with preferential attachment to degree

• Several domains of empirical work are outlined : – Comparison of networks with quasi power-law properties

– studies of industry and organizational fields (biotech)

– instabilities and network diffusion processes (Santa Fe project with Turchin, Chase-Dunn and others) - e.g., the alpha diffusion/epidemic threshold

– studies of archaeological and historical urban dynamics

– network studies of social class (Nord-Pas-de-Calais).

– network dynamics and trade in the 13th century (project with Peter Spufford and Joseph Wehbe)

Why and what are power-laws?• Power-law distributions are easily generated by adding together distributions

– M. Mitzenmacher. 2001. A Brief History of Generative Models for Power Law and Lognormal Distributions.

Internet Mathematics http://www.eecs.harvard.edu/~michaelm/NEWWORK/postscripts/history-revised.pdf • The copying model (e.g., Kumar et al. 2000) generates power-law degree distributions - e.g., web

sites copying others– discussed by Bela Bollobás and Oliver Riordan in Mathematical Results on Scale-free Random Graphs (2002).

• Power-laws are consistent with Boltzmann-Gibbs ergodic theory generalized to Tsallis entropy where events are not independent but network autocorrelated

– Nonextensive Entropy - Interdisciplinary Applications, 2002, eds. M. Gell-Mann and C. Tsallis (Oxford, Oxford University Press)

– http://www.santafe.edu/sfi/publications/Bulletins/bulletinFall00/features/tsallis.html– A. K. Rajagopal, Sumiyoshi Abe. 2002. Statistical mechanical foundations of power-law distributions.

http://arxiv.org/abs/cond-mat/0303064

• It is sufficient to have certain forms of equiprobable copying of canonical distributions of exponential form

• Copying of this sort might describe replication of thermodynamic engines with spin-offs, hydrodynamic disturbances, biological reproduction with mutation, social imitation, cultural and linguistic replication, borrowing, etc.

– Murray Gell-Mann and Constantino Tsallis, Interdisciplinary Applications of Ideas from Nonextensive Statistical Mechanics and Thermodynamics, SFI

A handy interpretation for network theory

• "What Tsallis defined was a simple generalization of Boltzmann entropy that does not add up from system to system and has a parameter q that measures the degree to which the nonextensivity holds," says Seth Lloyd, an External Faculty member at SFI and an associate professor of mechanical engineering at MIT. "Tsallis's is the most simple generalization that you can imagine. And for a variety of systems with long-range interactions—solid-state physics, chaotic dynamics, chemical systems, the list goes on and on—Tsallis entropy is maximized for some value of q. It is mathematically handy."

• In nonextensive situations, correlations between individual constituents in the system do not die off exponentially with distance as they do in extensive cases. Instead, the correlations die off as the distance is raised to some empirically derived or theoretically deduced power, which is called a power law.

• No power-law mystery here but some useful ideas for formulating laws of social scaling

Complexity parameters• A- The parameter q of Tsallis entropy can be considered a

bulking coefficient where u's responses to an interaction with v depend on u-v's common organizational constraints or u's interior organization (e.g., perception, processing of information)

• B- This slows u's response time relative to the u-v interaction time, resulting in the apparent correlation or autocorrelation.

• both A and B have been considered as indices of complexity, and both would arise out of the stacking of interiors relative to exteriors in multilevel processes

Why and how are power-law networks self-organizing?

Discussion:• Theory of the scaling coefficient• Self-organizing in tradeoffs between local and

global inequalities (micro-macro links)• Fields and organizations• Selection acting on non-social networks• Preferential gradientsPreferential gradients allowing more flexible

self-organization also operate in social networks

Theory of the scaling coefficient• with pure preferential attachment as the process

that creates or replaces links in a network, a power-law degree distribution will emerge as n → ∞ with a slope of alpha → 3 from below, as Barabási, Dezsö, Ravasz, Yook and Oltavai (2002) show in their 'scale-free' network model.

• Bollobás and Riordan (2002) proved this result. • Empirical networks with power-law distributions,

however, display a mix of preferential attachment by degree and other processes.

• The only pure 'scale-free' networks are those lacking real-world organizational constraints.

Examples of scale-independent networks and effects on alpha

Proteome yeast alpha=2.4 (Amaral) hierarchical organization, reduces alpha

Greek Gods alpha=3.0 (H&J Newman) with no real organizational constraints, pure 'scale free' alpha (courtesy Briannah Walters)

Biotech alpha=2.0 (Powell, White, Koput, Owen-Smith) cohesive organization, reduces alpha

Power-Law References (next slide)• Dorogovtsev & Mendes 2003 – alpha theory (Fg1)

– A.V. Goltsev, S.N. Dorogovtsev, J.F.F. Mendes. 2003. Critical phenomena in networks. Phys. Rev. E 67, 026123. http://arxiv.org/abs/cond-mat/0204596

• Handcock & Jones 2003 – epidemic threshold (Fg1 and examples) surpassed at alpha=3

• White & Johansen 2003 – theory and review (Fg1,2)• Adamic, Lukose & Huberman 2003 – local/global inequality

(Fg3)• Moody & White 2003 – cohesion follows from degree

correlation ← Newman & Park 2003 • Amaral, Scala, Barthelemy & Stanley 2000 – as cost-per-tie

grows, a cutoff and attenuation of power laws occur and alpha tends towards 1

Theory of scale-free networks power-law coefficient alpha

scale-independence in networks:

Theory of the Scaling Coefficient alpha

Fg1. alpha relative to number of nodes

Fg3. Tradeoff between local inequality (=>

navigability) and global inequality as a function

of alpha

Fg2, as a function of alpha: (1)

Proportion interacting with k others; and (2, inset) Prob. P(k)

of interacting with k or more

others

4. Social networks (not others in this one) have degree correlation and cohesive communities

variance finite for alpha > 3 & alpha ≤ 2

at 2 < alpha < 3 as n → ∞ degree variance →∞

∞ variance

potentially infinite variance

finite variance

as n →∞ alpha → 3 from below in pure

preferential attachment(dotted scatterplot line);

5. Sexual partners distribution may exceed

“no epidemics” threshold;

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Implications of 2 < alpha < 3 for stretched exponential

• (Quasi) Power-law coefficients in the range 2-3 are ubiquitous for degree distributions but as n → ∞ for these networks the variance → ∞. This contradicts finite energy driving connection dynamics and implies broken-scale cutoffs at the tail. Latherer & Sornette (1998) show that Tsallis-consistent stretched exponentials fit the quasi power-law segment of the distribution as well as the drop-off at the tail, and give an estimate of a normed basal unit for the distribution, such as the following.

– Species extinction times e+7 (lifespan years) x 2.2-2.5

– Oilfield formations e+6 (barrels) x 3±1

– Country sizes e+6 (people) x7

– Urban aggregates e+ 3 (people) x 7-20

– Airline connection network e-0 (airports) x 6 (drw) x

– Top physicist citation network e-0 (citations) x ~3

– Temperature, S. Pole 13 (normalized temperature)

– Radio waves from galaxies e-8 (intensities)

– Earthquake sizes e-9 (energy units?)

– Light waves from galaxies e-34 (intensities)

ln p(x)

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Stretched exponential distribution for 1997 U. S. airport connections

• For airlines case see drw working paper "Social Scaling: From scale-free to stretched exponential models for scalar stress, hierarchy, levels and units in human and technological networks and evolution"

References: stretched exponentials• Jean Latherer, D. Sornette. 1998. Stretched exponential distributions in

Nature and Economy: ``Fat tails'' with characteristic scales. Eur.Phys.J. B2: 525-539. http://arxiv.org/abs/cond-mat/9801293

• D. Sornette. 1998. Multiplicative processes and power laws. Phys. Rev. E 57 N4, 4811-4813. http://arxiv.org/abs/cond-mat/9708231

• U. Frisch, D. Sornette 1997. Extreme deviations and applications J. Phys. I France 7, 1155-1171. http://arxiv.org/abs/cond-mat/9705132

• It is notable that Tsallis entropy is a stretched exponential– A. K. Rajagopal, Sumiyoshi Abe. 2002. Statistical mechanical foundations of

power-law distributions. http://arxiv.org/abs/cond-mat/0303064

• Open question:– How is Tsallis q (deviation from exponential) related to power law alpha

(slope of power law) if not through stretched exponential, which gives them a unity relative to the basal organizational bulking unit?

Network copying/diffusion is often advantageous. Not in the case of disease, however, e.g., STDs.

Here the epidemic threshold alpha=3 is surpassed

Swedish MLE estimate alpha previous year =3.26 lifetime =3.125

For Uganda, 4.5 < alpha < 9; for the U.S. 3.1 < alpha < 4.5

Self-organization in scale-free networks:• Tradeoffs between local and global inequalities

– Robustness, resilience, searchability• Feedbacks may be governed by selection

– e.g., STD disease and sexual networks as alpha > 3– navigability in 1.8 region < alpha < 2.3

• Feedbacks may be governed by preference gradients (organizational/ideational diffusion)– e.g., variations in seachability in region 1.8 < alpha < 2.3

• Feedbacks may be governed by external resistance– e.g., opposition and movement breakthrough as alpha < 1, the example being

Bettencourt and Kaiser's Feynman diagram network • Network topologies, then, can be generated by micro-macro links,

hill-climbing optimization, natural selection, preferential gradients, autocorrelation

Fields and Organizations• Fields are networks of interactions; some fields are scale-free

– Among atomisms – May include organizations as well– When organizational constraints lacking and 2 < alpha < 3,

variance of degree → ∞ for n → ∞ • Organizations (not necessarily scale-free) are identified by

– Cohesive boundaries– Hierarchical architecture– Differences in scale, power, social organization– Coupling of external and internal linkages that allow near-equilibrium

homeokinetics• How organizations are linked into fields

– Chutes and ladders– Stacked cohesive cores– Mobility– Study multiple orgs in fields or single orgs with snowball of external links– Snowball samples are ignorable designs (can estimate parameters of field)

Observations

• Organizational networks in the range 1.8 < alpha < 2.3 with external snowball network will have hierarchy, local structure, local inequality and navigability, finite variance, and less global inequality

• Field alphas will range from 2.0 < alpha < 3.0, and be unnavigable where alpha > 2.3, and if large, variance goes to infinity.

Cohesion k ≡ k independent paths

• Field Cohesion:– Occurs in k-cones– Power-law slopes of k-cones– e.g., biotech

• Organizational Cohesion: – Occurs in k-ridges

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Theorem: ≡ no k-1 cutset

Alpha slope =1.8 for biotech

Organizational Cohesion: Validation of the Methodology for Network Research on Social Cohesion

The algorithm for finding social embeddedness in nested cohesive subgroups is applied to high school friendship networks (boundaries of grades are approximate).

Longitudinal Network Studies and Predictive Social Cohesion Theory D.R. WHITE, University of California Irvine, BCS-9978282

Fig 2. Friendship Cohesion in an American high school

8th grade

7th grade

11-12th grade

10th grade

9th

The cohesive groups overlap in k-ridges with components centered on organization by grades.

Interpretation: 7th-graders- core/periphery; 8th- two cliques, one hyper-solidary, the other marginalized; 9th- central transitional; 10th- hang out on margins of seniors; 11th-12th-

integrated, but more freedom to marginalize

k-ridges in organizations

The usefulness of the measures of cohesion and embeddedness are tested against outcome variables of school attachment in the friendship study and similarity in corporate donations to political parties in the corporate interlock study.

Nearly identical findings are replicated in 12 American high schools for school attachment measures and friendship networks from the AddHealth Study (http://www.cpc.unc.edu/addhealth/), Adolescent Risk and Vulnerability: Concepts and Measurement. Baruch Fischhoff, Elena O. Nightingale, Joah G. Iannotta, Editors, 2002, The National Academy Press.

The cohesion variables outperform other network and attribute variables in predicting the outcome variables using multiple regression.

2003 James Moody and Douglas R. White, Social Cohesion and Embeddedness: A Hierarchical Conception of Social Groups. American Sociological Review 8(1)

Social Cohesion Dynamics in the Field of Biotechnology: Longitudinal Change in the Mix of Attachment Processes, and the emergence of organizational features

Longitudinal Network Studies and Predictive Social Cohesion Theory D.R. WHITE, University of California Irvine, BCS-9978282

Biotech Collaborations

All ties 1989

Four logics of attachment are tested for the development of collaboration among biotech organizations:

accumulative advantage, including preferential attachment to degree homophily, follow-the-trend, and multiconnectivity.

What shapes network evolution, using multi-probability models to estimate dyadic attachments demonstrate how a preference for diversity and multiconnectivity in choice of collaborative partnerships. Cohesion variables outperform scores of other independent variables.

All ties 1989

New ties 1989

And so on to 1999

The process of searching for partners is dynamic and recursive. Preferential attachment to shared and partner cohesion operates with firms moving up a ladder of increasing cohesiveness of their networks. At the lower rungs of the cohesion ladder, there is a preference for expanding diversity by linking to well-connected partners. At the higher rungs of the cohesion ladder, firms may forego cohesion, opting to ally with recent entrants to the field. This relationship is suggestive of a systemic pumping action, with the most connected members pushing out in a diastolic search to pull in newcomers, and those less connected being pulled inward, in a systolic action, to attach to those with more cohesive linkages. The pumping process operates upwardly from the bottom, level by level.

2003 Walter W. Powell, Douglas R. White, Kenneth W. Koput and Jason Owen-Smith. Network Dynamics and Field Evolution: The Growth of Interorganizational Collaboration in the Life Sciences, 1988-99. Forthcoming: American Journal of Sociology. http://eclectic.ss.uci.edu/~drwhite/pub/SFI-WP2003ajs.pdf

Cohesion and diversity are the major biotech network predictors of tie-formation

Test of Multi-connectivity: Odds Ratios from McFadden’s Model 1-mode 2-mode New Repeat New Repeat

Hypothesis Partner Cohesion >1 1.432**

2.589**

1.667**

1.080

Shared Cohesion >1 1.058*

1.103

5.272**

1.905**

Partner Tie Diversity >1 1.025**

1.016

1.037**

1.020**

Prospective Tie Diversity

>1 .972

.738**

.883**

.952**

Partner’s Partner Tie Diversity

>1 1.042

.944**

.986**

.959**

The two-tailed significance levels are *=pvalue<.05; **=pvalue<.01. Controls for accumulative advantage, homophily, and follow-the-trend were also included in these models; odds ratios for these mechanisms are presented in prior tables. All models include fixed effects for firm, year, and type, as well as main effects for form. The complete models are presented in Appendix II.

Method: time-increment prediction of new and repeat links

1-mode is biotech-to-biotech;

2-mode is biotech-to-other partner network

the Schumpeterian pump of innovation

• Diastolic (in)

• All ties, e.g., 1997

1997 All Ties, Main Component, Nodes Colored by Cohesion

Node KeySize scaled to cohesionRed = five componentGreen = 3 or 4 componentBlue = 1 or 2 component

pumping action: cohesive core reaching out

• Systolic (out)

• New ties, 1998

1998 New Ties, Main Component, Colored by Cohesion

Key

Size scaled to 1997 cohesionTriangles = New Entrants (e.g. 0 component in 1997)

Figure 3. Degree Distributions by Type of Partner

Biotech degree distributions appear power-law

When summed over time

But…

…shift from power law to exponential Temporal Shift in 1-mode biotech Degree Distributions, 1988-1999, from Power Law

to Exponential, Contra the Barabási scale-free network model α=1.75

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2 clicks

Ring Cohesion• Cohesion is an important predictor of network attachment,

demonstrated in schools (AdHealth), industry (e.g. biotech), kinship, social class, and other fields and organizations. Ring cohesion theory focuses on preferential cohesive-linking mechanisms and how they are constructed.

• As a form of autocorrelation ring cohesion is consistent with Tsallis-theory of how power laws are generated

• Programming for finding sufficient nonisomorphic sets of ring fragments to account for total cohesion now completed (Wehbe), and algorithm counting all such fragments within model graphs implemented in Pajek.

• Ring cohesion analysis has now been completed for biotech and numerous kinship examples (work underway with Wehbe, Houseman)

Complexity and Small-World Phenomena in Kinship Networks

For kinship networks to operate as self-

organizing systems, they must possess small-world characteristics (clustering and low network distances relative to size of the network), including that of navigability -- the capability of finding others in the network through a path of known links – that requires in turn scalability of link-frequency with distance.

All these features are present for the Turkish nomads, and are posited for segmented lineage systems in general. Navigable small world networks of strong kinship ties of trust entail serve as a network structure that supports economic exchange & political recruitment.

Fig 8 shows a power law for preferential attachments of Turkish nomads marriages to closer types of blood marriage.

Fig 9 shows that the frequencies of blood marriages follow a power-law distribution while frequencies of affinal relinking follow exponential decay.

The Turkish Nomads as a prototype of self-

organization in segmented lineage

systems 2003 Douglas R. White and Michael

Houseman The Navigability of Strong Ties: Small Worlds, Tie Strength and Network Topology, in Networks and Complexity, January Special Issue, Complexity 8(1).

Applying Morphogenesis Methodology to Ring Cohesion in Kinship Networks

Longitudinal Network Studies and Predictive Social Cohesion Theory D.R. WHITE, University of California Irvine, BCS-9978282

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turk relink2

Log. (turk relink2)

Power (turk cons)

Fig 8 shows the decay of marriage frequencies with kinship distance

Ranking of Types

# of Couples

Fig 9 ranks types rather than #, with axes reversed, and shows that #s of blood marriages follow a power-law while affinal relinking frequencies follow an exponential

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(power law preferential curve)

Ring Cohesion:

• from the nomad analysis emerged

• a general theory of kinship complexity

e.g., are ring distributions drawn from the same universe?

Generalizing a Complexity Theory of Morphogenesis in Kinship Networks

Longitudinal Network Studies and Predictive Social Cohesion Theory D.R. WHITE, University of California Irvine, BCS-9978282

Frequency distributions that are power law for blood marriages (as in Fig 8) and exponential for affinal relinking are common in societies where blood marriages are frequent, like Parakana and Ticunya (Figs. 9- 10).

In societies where blood marriages are rare, like the Tory Islanders, Wilcania or Nyungar, frequency distributions are power-law for affinal relinking, as in Figs. 11-13. This also indicates a preferentially self-organized kinship network, based on multifamily relinking.

Number of Couples

Fig 9 Parakana Fig 10 Ticunya

BloodMarriage:Power law

Affinal Relinking: Exponential Decay Affinal

Relinking: Power law

Further applications of ring cohesion

• Nord-Pas-de-Calais study: spatial dimensions of ring cohesion (joint scaling model; with Hervé Le Bras)

• Networks of the previous world-system (13th century trade and monetary linkages; with Peter Spufford)

• Networks of the first world-system (Jemdet Nasr; Henry Wright)

tutto

http://jung.sourceforge.net/

is the platform independent java network analysis and visualization programming language

http://vlado.fmf.uni-lj.si/pub/networks/pajek/

is the program for large network analysis