Mechanisms in the Formation of Complex Networks

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Mechanisms in the Formation of Complex Networks

• Luis A. Nunes Amaral
• Northwestern University
• Dept. of Chemical Engineering

Frontiers of Science Pavia, Sept 8-11, 2003
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Complex networks ubiquity in Complex Systems
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From graphs to network theory

• Graphs
• Random graphs
• Small-world networks
• Scale-free networks
• Degree distrs. in small-world networks

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Konigsbergs bridges puzzle
Euler (1736)
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Topology Regular Graphs and Solids

• Tiling of the plane and higher-dimensional spaces
• Relation between number of edges, faces and
  vertices in d-dimensional solids
• Etc

Many people.
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d-dimensional graphsVoronoi lattices

• Voronoi cells tile the plane and
  higher-dimensional spaces
• Properties asymptotically similar to
  d-dimensional lattices

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Random graphs

• Percolation threshold
• Degree distribution, P(k)
• Typical path length, L
• Clustering coefficient, C

Rapaport, Bull Math Biophys 13, 107
(1951) Kochen Pool Erdos Renyi
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Asymptotic behavior
Lattice
Random graph
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Social networksMilgrams experiment
Milgram, Psych Today 2, 60 (1967)
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Six-degrees of separation

• Distance

• Clustering

Milgram, Psych Today 2, 60 (1967)
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Real-world networks

• Percolation threshold
• Degree distribution, P(k)
• Typical path length, L
• Clustering coefficient, C

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Small-world networks
N 1000

• Large clustering coeff.
• Short typical path

Watts Strogatz, Nature 393, 440 (1998)
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Ubiquity of small-world networks
Phys Rev Lett 83, 3180 (1999) Newman Watts,
Phys Lett A 263, 341 (1999) Barrat Weigt, Eur
Phys J B 13, 547 (2000)
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Scale-free networks
Actors Movies
Web-pages Hyper-links
Nodes Links
Trans. stations Power lines
Barabasi Albert, Science 286, 509 (1999)
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Single-scale small-world networks
Proc Nat Acad Sci USA 97, 11149 (2000)
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Scale-free and truncated scale-free networks
Nature 411, 907 (2001) Phys Rev Lett 88, 138701
(2002)
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Classes of small-world networksCosts and
Constraints
Proc Nat Acad Sci USA 97, 11149 (2000)
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Basic mechanisms

• Physical proximity
• d-dimensional ordered or disordered graphs
• Random connections (due to errors?)
• Random graphs with homogeneous nodes that have
  very small diameter and lack local structure
• Preferential attachment (greedy heuristics?)
• Growing networks following preferential
  attachment mechanism become scale-free
  (heterogeneous nodes).
• Costs, constraints, preferences
• Degree distribution of network will depend on
  constraints to establishment of links Some will
  be single-scale (egalitarian) and others will be
  scale-free (aristocratic).

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Characterization of complex networks

• Diameter, clustering coefficient, degree
  distribution
• Betweenness centrality
• Number of short paths going through a node
• Communities
• Can one identify cliques within the network?
• Assortativity
• Do highly-connected nodes preferentially connect
  to other highly-connected nodes?
• Correlations between degree and other quantities
• Local motifs
• What is the structure of the building blocks of
  complex networks?

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Complex networks ubiquity in Complex Systems
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Acknowledgments

• Searle Leadership Fund Award for support.
• Collaborators Alex Arenas, Albert Diaz-Guilera,
  Andre Moreira, Christofer Edling, Fredrik
  Liljeros, Gene Stanley, Marc Barthelemy, Roger
  Guimera, and Stefano Mossa.
• For reprints and details, check
• http//amaral.northwestern.edu

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Epidemiological models Small-world network of
contacts