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DYNAMICS OF COMPLEX SYSTEMS Self-similar
phenomena and Networks
Guido Caldarelli CNR-INFM Istituto dei Sistemi
Complessi Guido.Caldarelli_at_roma1.infn.it
6/6
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• STRUCTURE OF THE COURSE

1. SELF-SIMILARITY (ORIGIN AND NATURE OF POWER-LAWS)
2. GRAPH THEORY AND DATA
3. SOCIAL AND FINANCIAL NETWORKS
4. MODELS
5. INFORMATION TECHNOLOGY
6. BIOLOGY

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• STRUCTURE OF THE FIRST LECTURE

6.1) PROTEIN INTERACTION NETWORKS 6.2) FOOD
WEBS 6.3) FOOD WEBSOptimisation 6.4) PLANT
TAXONOMIES
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• 6.1 PROTEIN INTERACTION NETWORK

Network of Interaction for the protein of Bakers
Yeast (Saccharomyces Cerevisiae)
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• 6.1 PROTEIN INTERACTION NETWORK

How do growth and preferential attachmentapply
to protein networks?

• Growth genes (that encode proteins) can be,
  sometimes, duplicated mutations
  change some of the interactions
  with respect to the parent protein

• Preferential attachment the probability that a
  protein acquires a new
  connection is related to the
  probability that one of its neighbors is
  duplicated proportional to its
  connectivity

A. Vazquez et al., ComPlexUs 1, 38-44 (2003)
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• 6.1 PROTEIN INTERACTION NETWORK

The two hybrid method way of detecting protein
interactions
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• 6.1 PROTEIN INTERACTION NETWORK

With the solvation free energies taken from an
exponential probability distribution p(f) e-f,
we obtain
P(k) k-2

• The real network is random
• The detection method sees only pairs with large
  enough binding constants
• The binding constant is related to the
  solubilities of the two proteins
• Solubilities are given according to some
  distribution

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• 6.2 FOOD WEBS

sequence of predation relations among different
living species sharing the same physical space
(Elton, 1927)
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• 6.2 FOOD WEBS

Set of interconnected food chains resulting in a
much more complex topology
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• 6.2 FOOD WEBS The topology

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• 6.2 FOOD WEBS The topology

? Pamlico Estuary (North Carolina) 14 species
? Aggregated Food Web of Little Rock Lake
(Wisconsin) 182 species ? 93 trophic species
How to characterize the topology of Food Webs?
See Neo Martinez Group at http//userwww.sfsu.ed
u/webhead/lrl.html
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• 6.2 FOOD WEBS The degree

irregular or scale-free? P(k)? k-?
R.V. Solé, J.M. Montoya Proc. Royal Society
Series B 268 2039 (2001) J.M. Montoya, R.V. Solé,
Journal of Theor. Biology 214 405 (2002)
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• 6.2 FOOD WEBS The spanning tree

A spanning tree of a connected directed graph is
any of its connected directed subtrees with the
same number of vertices.
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• 6.3 FOOD WEBS The Optimisation

1
1
1
1
1
1
3
1
1
5
2
3
11
5
1
8
22
1
10
33
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• 6.3 FOOD WEBS The Optimisation

West, G. B., Brown, J. H. Enquist, B. J. Science 284, 1677-1679 (1999) Banavar, J. R., Maritan, A. Rinaldo, A. Nature 399, 130-132 (1999).  
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• 6.3 FOOD WEBS The Optimisation

AX drained area of point X
Hacks Law
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• 6.3 FOOD WEBS The Optimisation

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• 6.3 FOOD WEBS The Optimisation

(D.Garlaschelli, G. Caldarelli, L. Pietronero
Nature 423 165 (2003))
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• 6.3 FOOD WEBS The Optimisation

Original Webs
Aggregated Webs
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• 6.3 FOOD WEBS The Optimisation

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• 6.4 PLANT TAXONOMIES

Lazio
Utah
Amazonia
Iran
Peruvian and Atacama Desert
Argentina
Ecosystem
Set of all living organisms and environmental
properties of a restricted geographic area
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• 6.4 PLANT TAXONOMIES

Linnean Tree hierarchical structure organized
on different
levels, called taxonomic levels, representing

• classification and identification of different
  plants

• history of the evolution of different species

A Linnean tree already has the topological
structure of a tree graph

• each node in the graph represents a different
  taxa
• (specie, genus, family, and so on). All nodes
  are
• organized on levels representing the taxonomic
  one

• all link are up-down directed and each one
• represents the belonging of a taxon to the
  relative
• upper level taxon

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• 6.4 PLANT TAXONOMIES

Degree distribution
P(k)
k
The best results for the exponent value are given
by ecosystems with greater number of species. For
smaller networks its value can increase reaching
? 2.8 - 2.9.
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• 6.4 PLANT TAXONOMIES

Colosseo
(Terza Università, Rome)

• 6 historical periods
• (1643 - 2001)
• historical events
• climatic changes

Valmalenco (Bernina)
(University of Pavia) (G. Rossi, M. Gandini)

• 3 historical periods
• (1949 - 2003)
• climatic changes
• (Global Warming)

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• 6.4 TAXONOMY Real Subsets

P(k)
P(k)
k
k
2.6 ? 2.8
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• 6.4 TAXONOMY Random Subsets

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• 6.4 TAXONOMY Random Subsets

P(kf, kg) that a genus with degree kg belongs to
a family with degree kf
? kg? ?g kg P(kf,kg)
P(kf,kg)? kg -?
? 2.2 ? 0.2
? kf? ?f kf P(ko,kf)
P(ko,kf)? kf -?
? 1.8 ? 0.2
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• 6.4 TAXONOMY Random Subsets

1) create N species to build up an ecosystem
2) Group the different species in genus, the
genus in families, then families in
orders and so on realizing a Linnean tree
- Each species is represented by a string with 40
characters representing 40 properties which
identify the single species (genes) - Each
character is chosen between 94 possibilities all
the characters and symbols that in the ASCII
code are associated to numbers from 33 to 126
Two species are grouped in the same genus
according to the extended Hamming distance dWH
c1i character of species 1 with
i1,.,40 c2i character of species
2 with i1,.,40
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• 6.4 TAXONOMY Random Subsets

dEH C
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• 6.4 TAXONOMY Random Subsets

No correlation species randomly created with
no relationship between them
Genetic correlation species are no more
independent but
descend from the same ancestor

• ecosystems of 3000 species
• each character of each string is chosen
• at random
• quite big distance between two different
• species

• No correlation

?dEH? 20
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• 6.4 TAXONOMY Random Subsets

• single species ancestor of all species in the
  ecosystem
• at each time step t a new species appear
• - chose (randomly) one species already
  present in the ecosystem
• - change one of its character
• 3000 time steps

• Environment average of all species present in
  the
• the ecosystem at each
  time step t.
• At each time step t we calculate the distance
  between
• the environment and each species

dEH lt Csel
dEH gt Csel

• small distance between different species

?dEH? 0.5
P(k) k -? ? 2.8 ? 0.2
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• 6.4 TAXONOMY Random Subsets

Correlated
Not Correlated
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• CONCLUSIONS

Results

• networks (SCALE-FREE OR NOT) allow to detect
  universality
• (same statistical properties) for FOOD WEBS and
  TAXONOMY.
• Regardless the different number of species and
  environment

• STATIC AND DYNAMICAL NETWORK PROPERTIES other
  than
• the degree distribution allow to validate
  models.
• NEITHER RANDOM GRAPH NOR BARABASI-ALBERT WORK

Future

• models can be improved with
• particular attention to environment and
  natural selection
• FOR FOOD WEBS AND TAXONOMY

• new data

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COSIN COevolution and Self-organisation In
dynamical Networks
RTD Shared Cost Contract IST-2001-33555
http//www.cosin.org

• Nodes 6 in 5 countries
• Period of Activity April 2002-April 2005
• Budget 1.256 M
• Persons financed 8-10 researchers
• Human resources 371.5 Persons/months

EU countries
Non EU countries
EU COSIN participant
Non EU COSIN participant