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Complexity Theory in Biological and Social Systems

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Title: Complexity Theory in Biological and Social Systems


1
Complexity Theory in Biological and Social Systems
  • George Kampis
  • Basler Chair, ETSU
  • 2007 Spring

2
Basler lectures, 2007
3
Complexity theory
  • Historical remarks Math, Phys, (PhilSci)
  • Modern theory ABM and Networks Soc/Biol
  • (and then came the Hungarians)
  • Generic properties of complex networks
  • Problems
  • Food webs as an example
  • Future work (limited to our own)

4
(1) Complex systems, math
  • Information theory statistical complexity of
    messages (no content telephone eng.) C. Shannon,
    W. Weaver 1948
  • Kolmogorov (Chaitin) algorithmic complexity 1966
    difficulty of description
  • Both reduce c. to a compression problem, so must
    be (and are) equivalent
  • E.g. (uniform) random numbers, highest entropy,
    highest complexity
  • Random not origin, but math properties, e.g.
    unpredictability, cant win against
  • Kolmogorov version No shorter description (e.g.
    choice sequence)
  • Most sequences are random (complex) in this
    sense!
  • Study of complexity must be fundamental on purely
    formal grounds
  • But does it matter? (randomness vs. true
    complexity, most things are not random in any
    intuitive sense)
  • Uses in theoretical computer science and
    foundations of math

5
A.N. Kolmogorov
C. Shannon
W. Weaver
G. Chaitin
6
Complexity and simplicity
  • Mandelbrot set a very complex object
  • e.g. infinite zoom, infinite details
  • But, a simple algorithm
  • Then, simple or complex?
  • Preservation of complexity from initial conditions

7
LOW-COMPLEXITY ART Jürgen Schmidhuber
www.idsia.ch/juergen
8
(2) Complex Systems, phys
  • Nonlinear dynamics
  • Feedback (activation, inhibition)
  • Chaos, catastrophe theory, bifurcations,
    fractals
  • Itinerancies, long lived transients, strange
    attractors
  • adaptive dynamics (ie learning and other
    restructuring processes)
  • emergent phenomena (e.g. dyamic structures)

9
E. Lorenz A.M. Turing
N. Packard
I. Tsuda K. Kaneko
10
Complexity and simplicity 2.
  • Internet topology a very complex object
  • Finite but excessive details, multilevel zoom
    needed
  • No simple algorithm to generate
  • Yet simple structural measures help (preferential
    attachment, scale-free etc.)

11
Common features of complex systems
  • Unpredictability (but lets be precise about
    this!)
  • Counter-intuitive nature (possible limits of
    understanding?)
  • Complicated behaviours, complex spatio-temporal
    phenomena
  • Non-generalizability of behavior (details matter)
  • Necessity of different approaches (complexity is
    complex)

12
Three Classes of Complexity
  • Warren Weaver 1968
  • Organized simplicity (pendulum, oscillator)
  • Disorganized complexity (statistical systems)
  • Organized complexity
  • Heterogeneity, many components

13
(3) Complex Systems, Biol/Soc
  • Weaver class 3 systems
  • Approaches
  • Local simulation (no analitic model may exist),
    ABM (upshot from 1.)
  • Global network theory (upshot from 2.)

14
Social Networks
e.g. Stanley Wasserman
Narrative theory
social psychology
Financial etc market flows
15
Networks. A Hungarian Phenomenon
Okay, who is this?
Easy, and completely unrelated.
16
Networks. A Hungarian Phenomenon
Okay, who is this?
17
Networks. A Hungarian Phenomenon
Frigyes Karinthy (1887 - 1938, author,
playwright, poet, translator)
18
Networks. A Hungarian Phenomenon
"A mathematician is a device for turning coffee
into theorems."
19
Erdos number project
  • Erdos EN0, co-author EN1, co-author-of-coauthor
    EN2 etc.
  • http//www.oakland.edu/enp/
  • Kevin Bacon numbers http//oracleofbacon.org/
  • In fact, according to the Oracle of Bacon site,
    Paul Erdös himself has an official Bacon number
    of 4, by virtue of the N is a Number (a
    documentary about him), and lots of other
    mathematicians have finite Bacon number through
    this film. (CAVEAT bogus?)
  • Citation networks, friendship networks, sex,

20
(No Transcript)
21
High school friendship James Moody Race, school
integration, and friendship segregation in
America, American Journal of Sociology 107,
679-716 (2001).
http//www-personal.umich.edu/mejn/networks/
22
The science of links
  • 1. Properties of random graphs (networks)
  • 2. Six degrees of separation
  • 3. Small worlds the strength of weak ties
  • 4. Hubs and connectors
  • 5. The 80/20 rule
  • 6. Rich get richer preferential attachment
  • 7. Einstein's legacy
  • 8. Achilles' heel
  • 9. Viruses and fads
  • 10. The fragmented Web
  • Albert-Laszlo Barabasi, Linked The New Science
    of Networks (Perseus, 2002)

23
Random network theory
  • P. Erdos and A. Renyi, (1959) "On Random Graphs
    I, Publ. Math. Debrecen 6, p. 290297.
  • One is a threshold one acquaintance per person,
    one link to at least one other neuron for each
    neuron in the brain.
  • As the average number of links per node increases
    beyond the critical one, the number of nodes left
    out the giant cluster decreases exponentially.
  • If the network is large, despite the links'
    completely random placement, almost all nodes
    will have approximately the same number of links.
    (Poisson distribution)
  • Mathematicians call this phenomenon the emergence
    of a giant component, one that includes a large
    fraction of all nodes. Physicists call it
    percolation and we just witness a phase
    transition, similar to the moment in which water
    freezes.

24
Six degrees of separation
  • S. Milgram experiment (1967) proof of small
    world
  • ftp//cs.ucl.ac.uk/genetic/papers/Milgram1967Small
    .pdf
  • D. Watts and S. Strogatz (1998)
  • http//en.wikipedia.org/wiki/Watts_and_Strogatz_mo
    del
  • http//en.wikipedia.org/wiki/Small-world_network
  • http//en.wikipedia.org/wiki/Clustering_coefficien
    t
  • (near-)cliques plus bridges formed by hubs
  • Preferential attachment can generate similar

25
Weak ties
Granovetter, Mark.(1973). "The Strength of Weak
Ties" American Journal of Sociology, Vol. 78,
No. 6., May 1973, pp 1360-1380.
Of those who found jobs through personal contacts
(N54), 16.7 reported seeing their contact
often, 55.6 reported seeing their contact
occasionally, and 27.8 rarely. When asked
whether a friend had told them about their
current job, the most frequent answer was not a
friend, an acquaintance. The conclusion from
this study is that weak ties are an important
resource in occupational mobility. When seen from
a macro point of view, weak ties play a role in
effecting social cohesion.
Granovetter's basic argument is that your
relationship to family members and close friends
("strong ties") will not supply you with as much
diversity of knowledge as your relationship to
acquaintances, distant friends, and the like
("weak ties"). Hence, a person or an organization
may be able to enhance exposure or influence by
creating or maintaining contacts with "weak
ties". In marketing or politics, the weak ties
enable reaching populations and audience that are
not accessible via strong ties.
So this is a bad picture, eh? ?
Weak ties typically not transitive, unlike
strong ties
26
bridges
27
Erdos numbers, contd
  • Perhaps the most famous contemporary
    mathematician, Andrew Wiles, was too old to
    receive a Fields Medal (but was given a Special
    Tribute by the Committee at the 1998 ICM). He has
    an Erdös number of at most 3, via Erdös to ANDREW
    ODLYZKO to Chris M. Skinner.
  • And surely the most famous contemporary "computer
    personality" with a small Erdös number is William
    H. (Bill) Gates, who published with Christos H.
    Papadimitriou in 1979, who published with Xiao
    Tie Deng, who published with Erdös coauthor PAVOL
    HELL, giving Gates Erdös number at most 4.
  • A prolific biologist has an Erdös number of 2,
    through Laszlo A. Szekely, Eugene V. Koonin, at
    the National Center for Biotechnology
    Information. This gives many biologists small
    finite Erdös numbers, as well. Indeed, it is
    probably possible to connect almost everyone who
    has published in the biological sciences to
    Erdös. .
  • Here is a message from another biologist, Bruce
    Kristal, who has Erdös number 2 and lots of
    coauthors, which may provide useful hints for
    other searchers in this area I recently
    published with D Frank Hsu (Erdös number 1), and
    I am writing to briefly point out some potential
    implications of this that Frank and I found very
    interesting. Specifically, I am a biologist who
    works across several areas. Because of this, I
    have published with, among others, major figures
    in research on AIDS, aging, neurologic injury and
    neurodegeneration, and nutritional epidemiology.
    I believe that one of the neuroscientists I have
    published with, M. F. Beal, is among the most
    highly cited in this area. In the last area,
    nutritional epidemiology, I am on one (position)
    paper with many of the world leaders, including
    Walter Willett. Walt has over 1000 publications
    and was recently named as the most highly cited
    biomedical researcher in the last decade.
    Likewise, Frank is a computer scientist with ties
    in both mathematics and information retrieval as
    well as some biology citations. I mention these
    because Frank and I have discussed, among other
    issues, whether I may serve as a weak link of
    sufficient breadth to impact the overall network
    structure both within biology and between biology
    and these other areas of math and computer
    science. Koonin is clearly more prolific than I
    am, but our fields may be sufficiently different
    to complement. Interested people can contact him
    directly.

28
Hubs are connectors
29
Scale free (i.e. power law)
Pareto 80-20 rule, e.g. 80 of profit is produced
by 20 of firms
wherever you are, the ratio is invariant e.g.
n times fewer people have k times more friends,
Pn/k is constant across x, the number of friends
considered
30
Simulation library of basics
e.g. in NetLogo
Erdos-Rényi Barabási
Watts-Strogatz
31
Topics in network theory
  • Fault tolerance and resilience
  • Topological transitions (e.g. scale free - star)
  • Modularity versus globality
  • Evolvability
  • Network optimization (a combination of these)
  • Self-healing etc.

32
Scale free universal
  • Paretos Law
  • Zipfs Law
  • Levy flight http//en.wikipedia.org/wiki/Levy_flig
    ht Earthquakes http//www.iop.org/EJ/article/0295-
    5075/65/4/581/epl8017.html
  • Gutenberg-Richter Law
  • Rain (Noe effect)
  • Internet web, emails, site visits..
  • .
  • A compilation http//www.insna.org/INSNA/Hot/scal
    e_free.htm

http//www.hpl.hp.com/research/idl/papers/ranking/
ranking.html
33
Power and weakness of scale free
  • Scale-free networks are extremely tolerant of
    random failures. In a random network, a small
    number of random failures can collapse the
    network. A scale-free network can absorb random
    failures up to 80 of its nodes before it
    collapses. The reason for this is the
    inhomogeneity of the nodes on the network --
    failures are much more likely to occur on
    relatively small nodes.
  • Scale-free networks are extremely vulnerable to
    attacks on their hubs.
  • Scale-free networks are extremely vulnerable to
    epidemics. Same is true for purely random
    networks (Erdos-Rényi networks)

34
Structure and tie strengths in mobile
communication networks J.-P. Onnela, J.
Saramäki, J. Hyvönen, G. Szabó, D. Lazer, K.
Kaski, J. Kertész, A.-L. Barabási (2006)
http//arxiv.org/abs/physics/0610104
Distribution of degrees and link strength
Giant component size vs removal, and percolation
(connectivity)
Strong and weak links in real and random case
Confirms Granovetter. Exercise question is this
result nevertheless trivial?
35
Scale free universal?
  • (1) Subnetworks of scale free nets are not scale
    free!
  • (2) Drosophila PIM is not...
  • (3) Food webs are not

36
(No Transcript)
37
Drosophila PIM
  • Originally published in Science Express on 6
    November 2003 Science 5 December 2003 Vol. 302.
    no. 5651, pp. 1727 - 1736 DOI 10.1126/science.109
    0289
  • RESEARCH ARTICLES
  • A Protein Interaction Map of Drosophila
    melanogaster
  • L. Giot,1 J. S. Bader,1 C. Brouwer,1 A.
    Chaudhuri,1 B. Kuang,1 Y. Li,1 Y. L. Hao,1 C. E.
    Ooi,1 B. Godwin,1 E. Vitols,1 G. Vijayadamodar,1
    P. Pochart,1 H. Machineni,1 M. Welsh,1 Y. Kong,1
    B. Zerhusen,1 R. Malcolm,1 Z. Varrone,1 A.
    Collis,1 M. Minto,1 S. Burgess,1 L. McDaniel,1 E.
    Stimpson,1 F. Spriggs,1 J. Williams,1 K.
    Neurath,1 N. Ioime,1 M. Agee,1 E. Voss,1 K.
    Furtak,1 R. Renzulli,1 N. Aanensen,1 S.
    Carrolla,1 E. Bickelhaupt,1 Y. Lazovatsky,1 A.
    DaSilva,1 J. Zhong,2 C. A. Stanyon,2 R. L.
    Finley, Jr.,2 K. P. White,3 M. Braverman,1 T.
    Jarvie,1 S. Gold,1 M. Leach,1 J. Knight,1 R. A.
    Shimkets,1 M. P. McKenna,1 J. Chant,1 J. M.
    Rothberg1
  • Drosophila melanogaster is a proven model system
    for many aspects of human biology. Here we
    present a two-hybridbased protein-interaction
    map of the fly proteome. A total of 10,623
    predicted transcripts were isolated and screened
    against standard and normalized complementary DNA
    libraries to produce a draft map of 7048 proteins
    and 20,405 interactions.

38
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39
http//www.biocomp.unibo.it/school/html2004/ABSTRA
CT/Caldarelli.pdf
40
Food webs
D. Lavigne The North-Atlantic Food Web
41
Neo Martinez http//www.foodwebs.org/index.html
42
Food Webs as Networks
Williams et al 2002 Two degrees of separation in
complex food webs, PNAS 99, 12913-6. But then
43
A keystone species is a species that has a
disproportionate effect on its environment
relative to its abundance. Such an organism plays
a role in its ecosystem that is analogous to the
role of a keystone in an arch. While the keystone
feels the least pressure of any of the stones in
an arch, the arch still collapses without it.
Similarly, an ecosystem may experience a dramatic
shift if a keystone species is removed, even
though that species was a small part of the
ecosystem by measures of biomass or productivity.
It has become a very popular concept in
conservation biology.
Are keystone species weak links?
Sea stars eat mussels to make room for other
species, grizzlys import sea nutrients
44
R. Albert, H. Jeong, A.-L. Barabási Error and
attack tolerance of complex networks, Nature 406,
378-482 (2000).
Sometimes weak links are hubs (Barabasi),
sometimes they link up hubs (Csermely), sometimes
keystones are weak links, sometimes not
Unresolved the relation bw hubs, keystones,
weak links
Jordán, F., Liu, W.-C. and Davis, A.J. 2006,
Oikos, 112535-546, Topological keystone species
measures of positional importance in food webs.
45
Connectivity/stability
  • Translates as a diversity/stability problem in
    ecology
  • May-Wigner theorem (1971) low connectivity
    stabilizes
  • McCann (2000) high diversity/generalist species
    stabilize
  • A mixing of methodologies ABM study of the
    evolution of foodwebs modeled as phenotype
    interaction networks

46
(Work in progress)
with W. de Back at Collegium Budapest
Question are there generic emergent properties
in the toplogy of trophic interaction nets? How
do they depend on biological parameters (agent
properties, external perturbations
etc.) Obviously, a selective (self-simplifying)
process. Is it systematic or contingent? If the
former (or latter), how does this relate to real
ecosystems? The study of such questions has just
began (and not only for our team)
47
Summary
  • Complexity not (just) Math and Phys
  • ABM and networks provide two typical Biol/Soc
    paradigms
  • Networks have universal properties
  • which are not
  • Study of real and real (i.e. ABM) networks
  • A final word networks (and/or ABM) are fun!

48
THANK YOU
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