Modeling Complex

1

• Modeling Complex
• Real World Networks
• Or
• Do non-mathematicians do
• better mathematics?
• Dr. Eduardo Mendoza
• Mathematics Department Physics
  Department
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de
• Department of Computer Science
• Munich University of Applied Science

2
Topics

• What are complex real-world networks?
• What have non-mathematicians achieved?
• Can mathematicians contribute and how?
• Some personal remarks

3
Acknowledgement

• Most of the nice slides are from Albert Barabasi
  (University of Notre Dame)
• www.nd.edu/networks

4
Internet-Map
5
Internet
INTERNET BACKBONE
Nodes computers, routers Links physical lines
(Faloutsos, Faloutsos and Faloutsos, 1999)
6
WWW
World Wide Web
Nodes WWW documents Links URL links
800 million documents (S. Lawrence, 1999)
ROBOT collects all URLs found in a document
and follows them recursively
R. Albert, H. Jeong, A-L Barabasi, Nature, 401
130 (1999)
7
Communication networks
The Earth is developing an electronic nervous
system, a network with diverse nodes and links are
-computers -routers -satellites
-phone lines -TV cables -EM waves
Communication networks Many non-identical
components with diverse connections between them.
8
Bacon 1
9
Actors
ACTOR CONNECTIVITIES
Nodes actors Links cast jointly
Days of Thunder (1990) Far and Away (1992)
Eyes Wide Shut (1999)
N 212,250 actors ?k? 28.78
P(k) k-?
?2.3
10
The Erdös Number Project

• Erdös numbers have been a part of the folklore of
  mathematicians throughout the world for many
  years.
• Facts about Erdös Numbers and Collaboration
• Statistical descriptions of Erdös number data, a
  file of the subgraph induced by Erdöss
  coauthors, Erdös number record holders, the
  distribution of Erdös numbers (they range up to
  15, but the average is less than 5, and almost
  everyone with a finite Erdös number has a number
  less than 8)
• facts about collaboration in mathematical
  research and the collaboration graph, including
  some information about publishing habits of
  mathematicians
• 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.
• Famous Paths to Paul Erdös Fields Medalists and
  Nobel Prize winners have small Erdös numbers.
• Compute Your Own Erdös Number It may be smaller
  than you think.
• Related Concepts Six degrees of separation, the
  Kevin Bacon game, Small Worlds, academic
  genealogy, Hank Aaron, graph theory.

11
Citation
SCIENCE CITATION INDEX
Nodes papers Links citations
Witten-Sander PRL 1981
1736 PRL papers (1988)
P(k) k-?
(? 3)
(S. Redner, 1998)
12
Society
Nodes individuals Links social relationship
(family/work/friendship/etc.)
S. Milgram (1967)
Six Degrees of Separation
John Guare
Social networks Many individuals with diverse
social interactions between them.

13
Boehring-Mennheim
14
Metab-movie
Nodes chemicals (substrates) Links bio-chemical
reactions
Metabolic Network
15
Prot Interaction map
Yeast protein network
Nodes proteins Links
physical interactions (binding)
P. Uetz, et al. Nature 403, 623-7 (2000).
16
P53
One way to understand the p53 network is to
compare it to the Internet. The
cell, like the Internet, appears to be a
scale-free network.
17
Bio-Map
18
Food Web
Nodes trophic species Links trophic
interactions
R.J. Williams, N.D. Martinez Nature (2000)
R. Sole (cond-mat/0011195)
19
A few structural invariants for complex networks

• Shortest path
• L(x,y) between two nodes x and y
• Average shortest path Lav of a network (also
  called diameter of a network) this determines
  the effective linear size
• Maximal shortest path Lmax
• Small world property Lav is small, despite
  (large) network size (given by number of nodes)

• Examples
• Lav(Kn) 1 if Kn is complete
• Coauthorship networks
• Math 9.5
• HEP 4
• Neuro 6

20
Degree k, degree distribution P(k)

• Degree total number of connections (edges) from
  a node
• In- and out-degrees for directed graphs
• Average degree ltkgt
• Degree distribution P(k) function expressing
  the probability that a node has degree k
• Log distribution (log P(k) as function of log k)
  is also often used

21
Clustering coefficient C (Watts-Strogatz 98)
22
Clustering coefficient C (2)

• C is also the probability that if a triple of
  nodes of a network is connected by at least two
  edges, then the third edge is also present ?

Examples C(Kn) 1 if Kn is complete
C(T) 0 if T is a tree, etc
23
Mathematicians are people who turn coffee into
theorems
Erdös-Rényi model (1960)
Pál Erdös (1913-1996)
Also known as random graphs
24
Structural invariants of random graphs

• Diameter/shortest path length
• Lav ln N/ln z1 (z1 average no of nearest
  neighbors)
• random graphs are small worlds
• Clustering coefficient
• Crand p ltkgt/N
• ? random graphs have low clustering

25
Are complex networks random?
26
Highly clustered small worlds
Nature June 4, 1998
August 1999
http//smallworld.sociology.columbia.edu
27
Between1999 2001, researchers found out that
most real world networks have the same internal
structure
Scale-free networks ie P(k) k-r r constant
Why?
What does it mean?
28
Scale-free complex networks
29
Random vs. Scale-free
30
19 degrees
19 degrees of separation The WWW is very big
but not very wide
3
l152 1?2?5 l174 1?3?4?6 ? 7 lt l gt ??
6
1
4
7
5
2
31
Nature July 27, 2000
32
H. Jeong et al Nature, May 3, 2001
33
Sex-web
Nodes people (Females Males) Links sexual
relationships
4781 Swedes 18-74 59 response rate.
Liljeros et al. Nature 2001
34
Halting Viruses in Scale-Free Networks

• Classical Epidemiology epidemic threshold T
  exists, such that transmission probability lt T
  implies disease will die out
• Recent Results
• T 0 in scale free networks (Pastor-Satorras
  Vespigniani 01)
• Network of sexual contacts is scale-free
  (Liljeros et al 01)
• ? spread of AIDS will not be stopped by
  traditional methods
• Solution immunizing hubs (with degree gt k0)
  restores positive T )

35
Public/Press reaction

• General science press
• Nature, News Views Unspinning the web
• National Geographic Herculean Internet and Web
  Have Achilles' Heels
• American Scientist Graph Theory in Practice
  Part I,Part II
• Scientific American The Post-Genome Project
• Science, Science News, Discover, Discovery,
  Microelectronics Tech Alert,.
• Daily press
• New York Times First Cells, Then Species, Now
  the Web
• CNN Scientists spot Achilles heel of the
  Internet
• BBC Unweaving the world wide web
• USA Today Only 19 degrees of Web separation
• Washington Post The Net, the Web, the Catch
• Boston Globe, Seattle Post-Intelligencer, and
  many more
• Le Monde Le diamètre de la Toile est revu à la
  hausse
• articles in Spanish, Italian, Swedish, Dutch,
  German,,.Korean, Japanese, ..

36
Rita Colwell (NSF Director and well-known
microbiologist) in a recent (Oct 11, 2002) speech
Braiding Mathematics and Statistics with Life
Sciences Weaving the Future's Tapestry

• Networks offer a powerful example. From the
  World Wide Web to the citation network of
  scientific papers, and from metabolic networks to
  the complex food chains, mathematics has been
  central to unraveling this ever-changing and
  evolving connectedness.
• Barabasi captures it concisely in his new
  book, Linked The New Science of Networks. He
  writes, "all networks have a deep underlying
  order and operate according to simple but
  powerful rules. This knowledge promises to shed
  light on the spread of fads and viruses, the
  robustness of ecosystems, the vulnerability of
  economies."

37
Traditional modeling Network as a static graph
Given a network with N nodes and L links
?
Create a graph with statistically identical
topology
RESULT model the static network topology
PROBLEM Real networks are dynamical systems!
Evolving networks
OBJECTIVE capture the network dynamics

• identify the processes that contribute to the
  network topology
• develop dynamical models that capture these
  processes

METHOD
?
BONUS get the topology correctly.
38
Origins SF
Modeling SCALE-FREE NETWORKS
(1) The number of nodes (N) is NOT fixed.
Networks continuously expand by the addition of
new nodes
Examples
WWW addition of new documents
Citation publication of new papers
39
BA model
Scale-free model
(1) GROWTH
At every timestep we
add a new node with m edges (connected to the
nodes already present in the system). (2)
PREFERENTIAL ATTACHMENT
The probability ? that a new node will be
connected to node i depends on the connectivity
ki of that node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
40
More models
Other Models

• Non-linear preferential attachment
  ?(k) k? ? P(k) no scaling for ??1
• ? ?lt1 stretch-exponential
• ? ?gt1 no-scaling (?gt2 gelation)
• (Krapivsky et al (2000).)

• Initial attractiveness ?(k) Ak?
• P(k) k-? where ?2 A/m
• (Dorogovtsev et al (2000).)

• Aging each node has a lifetime
  ? node cannot get links
  after retirement. (actor)
• P(k) power-law with exponential cutoff
• (Amaral et al (2000).)

41
Other Models (continued)

• Saturation each node has maximum link number.
• node cannot get links after finite of links
• P(k) power-law with exponential cutoff
• (Amaral et al (2000).)

42
Can Latecomers Make It?
Fitness Model SF model k(t)t ½
(first mover advantage)Real systems
nodes compete for links -- fitnessFitness
Model fitness (h )
k(h,t)tb(h)
where
b(h) h/C G. Bianconi and A.-L.
Barabási, Europhyics Letters. 54, 436 (2001).
43
Bose-Einstein Condensation in Evolving Networks
G. Bianconi and A.-L. Barabási, Physical Review
Letters 2001 cond-mat/0011029
44
2 Excellent Reviews

• R. Albert, A. Barabasi
• Statistical Mechanics of Complex Networks,
  Reviews of Modern Physics 74 (2002)
• avail www.arXiv.org cond-mat/0106096
• S.N. Dorogovtsev, J.F.F.Mendes
• Evolution of networks, Advances in Physics 51
  (2002)
• avail www.arXiv.org cond-mat/0106144

45
Interpretation of the clustering coefficient as
geometric curvature (J.Eckmann, E. Moses PNAS
April 02)
Key developments 2002 (1)

Jost-Joy Aug 02
46
Map of C. Elegans brain
47
Key developments 2002 (2)

• A step towards an encompassing model (Jost-Joy,
  Evolving networks with distance preferences
  preprint Aug 02, www.santafe.edu )
• Distance preference a new node xn forms first
  link randomly, then according to distance
  preference function p(d(xn,x)) up to a fixed
  number m
• Special cases preferredshortest, longest, equal
• Advantage over previous models (eg scale-free
  model) need only to evaluate local information

48
Key developments in 2002 (3)Hierarchical
structures

• Problem scale free model did not explain recent
  discovery of Dorogovtsev et al (in the
  deterministic scale-free case, 12/01) that
• C(k) k -1
• A new hierarchical model in recent papers by
  Ravascz, Barabasi et al (Science Sep 02, Phys Rev
  E in press) integrates modularity and
  scale-freedom

49
Hierarchical growth
50
Hierarchical vs. classical scale-free models
51
Measurements for some networks
52
4. A major step forward in effective treatment
for HIV/AIDSby C Kamp and S BornholdtOctober 7
2002 issue of Proc Royal Soc London B )
Using computer simulations to map the
'predator-prey' dynamics between viruses and the
immune system, the research offers a better
understanding of the conditions that inhibit the
progression of the virus into AIDS and the points
at which HIV may be most effectively
defeated.Commenting on the research, Christel
Kamp, from the Institute of Theoretical Physics,
University of Kiel, says "Our research findings
are a step ahead in understanding conditions that
promote a non-progression to AIDS, suggesting
vaccination and receptor blocking/fusion
inhibition as efficient ways of overcoming an HIV
infection. Receptor blocking is a process by
which the HIV strains are blocked from attaching
to markers on the T-helper cells which are
necessary to start the process of cell membrane
fusion. The first clinical trials are also
showing these strategies to be very promising."
53
Mathematics, yes. But mathematicians?..

• All recent models (with one exception) from
  non-mathematicians (mostly physicists)
• Few efforts (up to now) by mathematicians
• P. Erdös, A. Renyi (1960,)
• Papers by Juergen Jost (well-known expert on
  dynamical systems), 2002
• Reka Albert (one of the pioneer physicists) hired
  by U Minnesota Mathematics Department (2002)
• Dont really understand why
• Too real? (have to analyze data)
• not elegant enough? (I find the simple
  underlying principles discovered till now quite
  elegant)
• Involves more than one research area (how about
  collaborating)
• Too shy to compete with physicists? (how about
  collaborating)

54
How can mathematicians contribute?
Wuchty-Stadler, preprint 2003

• Apply extensive graph theory knowledge to find
  new structural invariants
• Example 1 Spectral properties of scale-free
  networks (Farkas et al 2001). Extensive numerical
  analysis and computational tools discovered new
  phenomena (s. next slide)

55
Spectral density of scale-free networks

• Open Problems
• Find a simple closed expression for small-world
  and scale-free networks (or sub-classes) as in
  the case of random graphs
• What other results of Spectral Graph Theory can
  be applied?

56
Example 2 Centers of Complex Networks

• preprint by S. Wuchty (Theoretical Biochemistry)
  and P. Stadler (Bioinformatics) on
  www.santafe.edu
• Main notion of centrality used till now vertex
  degree
• Authors introduce and investigate the following
  graph-theoretic concepts
• Essentiality (? center )
• Status (? median)
• Centroid value (? centroid)

57
Further examples?

• Try to apply theory of infinite graphs
  (considering complex networks as sequences of
  finite graphs appropriately embedded in
  /related to an infinite graph)
• Is there a way to factorize certain classes of
  scale-free networks in order to see the
  functional module structure (eg in
  intra-cellular networks)?

58
How to contribute (2)

• 2. Extend the models to other interesting
  variants
• Directed networks
• Weighted networks, optimization, allometric
  scaling
• Specific questions on the Internet and WWW
• .

59
How to contribute (3)

• Evolving complex networks are dynamical systems
  (as seen from the models)
• Which results from (discrete) dynamical systems
  can be applied appropriately?
• Which questions?
• Example preprint from J. Jost M.F. Joy on
  Evolving networks with distance preferences on
  www.santafe.edu
• Same for stochastic dynamical systems!

60
How to contribute (4)

• 4. Expand on geometric concepts and introduce
  other relevant ideas

Eckmann-Moses Apr 02
Jost-Joy, Aug 02
61
How to contribute (5)

• 5. Work on joint projects with experimental
  scientists on specific complex networks or
  systems
• For biological systems, we have started an
  interdisciplinary Mathematical and
  Computational Biology Intiative cf.
  www.engg.upd.edu.ph/compbio
• Could also be on novel social networks (perhaps
  together with the Institute for Communications
  Research on specific Filipino phenomena like
  texting (or SMS comms)
• There is an active complex systems modeling group
  at NIP (some projects model social behavior)
• cf www.nip.upd.edu.ph/ipl

62
Continued high level of (public) interest
Spring 02
Fall 02
63
Nature Immunology Editorial (Oct 2002)

• Examination of the interrelationships among the
  elements of the immune system, in the manner
  described in Linked could provide us with
  another approach. Taking such a broad view may
  shape the information into previously
  unrecognized hubs, clusters of interactions or
  regulatory circuits.
• Historically, immunologists rarely turn to models
  for answers or inspiration, as the complexity of
  the immune system invalidates and thwarts most of
  them. But as Barabasis book indicates, it is now
  becoming a more approachable endeavor, with
  massive amounts of data collected and centralized
  into public databases . Although we may not like
  it, we can little afford to ignore a structure as
  all pervasive as scale-free networks.

64
An emerging fieldwith lots of opportunities for
impact

• Barabasi et al
• Scale-free and hierarchical structures in complex
  networks
• Preprint 2003 (avail www.nd.edu/networks)

65
The New Generation?Fan Chung Graham
(professional name Fan Chung) is the Akamai
Professor in Internet Mathematics at UC San
Diego.

• Her main research interests lie in spectral graph
  theory and extremal graph theory. These topics
  provide powerful methods in dealing with problems
  arising in a wide range of areas. She has about
  200 papers, in areas ranging from pure
  mathematics ( e.g., differential geometry, number
  theory) to the applied (e.g., optimization,
  computational geometry, telecommunications and
  Internet computing). She has about 100 coauthors
  including many mathematicians, computer
  scientists, statisticians and chemists.
• She is the author of two books -- Erdös on Graphs
  and Spectral Graph Theory. She is the
  Editor-in-Chief of a new journal Internet
  Mathematics and she is also a Co-Editor-in-Chief
  of Advances in Applied Mathematics. In addition,
  she serves on the editorial boards of a dozen or
  so journals. She was awarded the Allendoerfer
  Award by Mathematical Association of America in
  1990. Since 1998, she has been a fellow of
  American Academy of Arts and Sciences.

66
Some personal remarks

• My conviction great mathematics comes from
  working on great problems (challenges). Real
  world phenomena like complex evolving networks
  are a great source of such challenges.
• My personal focus complex biological networks
  (this area of research is also part of systems
  biology).

67
Caveat on mathematicians involvement

• Active in the area of biological networks
• Mikhael L. Gromov
• and biological systems (in general)
• David Mumford (math of perception)
• Sergi Novikov (topology of condensed matter,
  incl biomolecules)

68
Rita Colwell Quote (Oct 02)

• The very scale of scientific quests - cures
  for disease and cancer, environmental
  preservation, and scientific literacy, to name a
  few-beckons the interweaving of science itself...
  
• The progression of mathematical modeling,
  statistical methods, and computational algorithms
  has lifted many fields of science to new levels
  that brim with promise. The biological sciences
  are not immune. Complex challenges call for novel
  mathematical, statistical, and computational
  approaches. ..

69
Lots of opportunities to make a difference...
(Nature, Jan 23 2003)
70
My version of Erdös

• Mathematicians are people
• who turn coffee
• into theorems for a better world.
• also tee.
• Thanks for your attention!