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Overview of Complex NetworksComplex Networks
CSYS/MATH 303, Spring, 2011
Prof. Peter Dodds
Department of Mathematics & StatisticsCenter for Complex Systems
Vermont Advanced Computing CenterUniversity of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
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Outline
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Examples of Complex Networks
Properties of Complex Networks
Modelling Complex Networks
Nutshell
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Class Admin
I Office hours:I 1:00 pm to 3:00 pm, Wednesday;
Farrell Hall, second floor, Trinity Campus.I Appointments by email ([email protected] ).
I Course outlineI ProjectsI Assignments (about 8)I Assignment 1 appears today and involves:
I dolphins
I a Karateclub
I political blogs
I a worm’s brain
I the Internet
I jazzmusicians
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Properties ofComplex Networks
Modelling ComplexNetworks
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Class Admin
I Office hours:I 1:00 pm to 3:00 pm, Wednesday;
Farrell Hall, second floor, Trinity Campus.I Appointments by email ([email protected] ).
I Course outlineI ProjectsI Assignments (about 8)I Assignment 1 appears today and involves:
I dolphins
I a Karateclub
I political blogs
I a worm’s brain
I the Internet
I jazzmusicians
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Class Admin
I Office hours:I 1:00 pm to 3:00 pm, Wednesday;
Farrell Hall, second floor, Trinity Campus.I Appointments by email ([email protected] ).
I Course outlineI ProjectsI Assignments (about 8)I Assignment 1 appears today and involves:
I dolphins
I a Karateclub
I political blogs
I a worm’s brain
I the Internet
I jazzmusicians
Page 6
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Class Admin
I Office hours:I 1:00 pm to 3:00 pm, Wednesday;
Farrell Hall, second floor, Trinity Campus.I Appointments by email ([email protected] ).
I Course outlineI ProjectsI Assignments (about 8)I Assignment 1 appears today and involves:
I dolphins
I a Karateclub
I political blogs
I a worm’s brain
I the Internet
I jazzmusicians
Page 7
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Class Admin
I Office hours:I 1:00 pm to 3:00 pm, Wednesday;
Farrell Hall, second floor, Trinity Campus.I Appointments by email ([email protected] ).
I Course outlineI ProjectsI Assignments (about 8)I Assignment 1 appears today and involves:
I dolphins
I a Karateclub
I political blogs
I a worm’s brain
I the Internet
I jazzmusicians
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Properties ofComplex Networks
Modelling ComplexNetworks
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Properties ofComplex Networks
Modelling ComplexNetworks
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Properties ofComplex Networks
Modelling ComplexNetworks
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Exciting details regarding these slides:
I Three versions (all in pdf):1. Presentation,2. Flat Presentation,3. Handout (2x2).
I Presentation versions are navigable and hyperlinksare clickable.
I Web links look like this (�).I References in slides link to full citation at end. [2]
I Citations contain links to papers in pdf (if available).I Brought to you by a troubling concoction of LATEX,
Beamer, and perl.
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Bonus materials:
Textbooks:I Mark Newman (Physics, Michigan)
“Networks: An Introduction” (�)I David Easley and Jon Kleinberg (Economics and
Computer Science, Cornell)“Networks, Crowds, and Markets: Reasoning About aHighly Connected World” (�)
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Bonus materials:
Review articles:I S. Boccaletti et al.
“Complex networks: structure and dynamics” [5]
Times cited: 1,028 (as of June 7, 2010)
I M. Newman“The structure and function of complex networks” [16]
Times cited: 2,559 (as of June 7, 2010)
I R. Albert and A.-L. Barabási“Statistical mechanics of complex networks” [1]
Times cited: 3,995 (as of June 7, 2010)
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Basic definitions:
Complex: (Latin = with + fold/weave (com + plex))Adjective
I Made up of multiple parts; intricate or detailed.I Not simple or straightforward.
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Modelling ComplexNetworks
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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Basic definitions:
Complex System—Some ingredients:
I Distributed system of many interrelated partsI No centralized controlI Nonlinear relationshipsI Existence of feedback loopsI Complex systems are open (out of equilibrium)I Presence of MemoryI Modular (nested)/multiscale structureI Opaque boundariesI Emergence—‘More is Different’ [2]
I Many phenomena can be complex: social, technical,informational, geophysical, meteorological, fluidic, ...
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net•work |ˈnetˌwərk|noun
1 an arrangement of intersecting horizontal and vertical lines.
• a complex system of roads, railroads, or other transportation routes :
a network of railroads.
2 a group or system of interconnected people or things : a trade network.
• a group of people who exchange information, contacts, and
experience for professional or social purposes : a support network.
• a group of broadcasting stations that connect for the simultaneous
broadcast of a program : the introduction of a second TV network | [as adj. ]
network television.
• a number of interconnected computers, machines, or operations :
specialized computers that manage multiple outside connections to a network | a
local cellular phone network.
• a system of connected electrical conductors.
verb [ trans. ]
connect as or operate with a network : the stock exchanges have proven to be
resourceful in networking these deals.
• link (machines, esp. computers) to operate interactively : [as adj. ] (
networked) networked workstations.
• [ intrans. ] [often as n. ] ( networking) interact with other people to
exchange information and develop contacts, esp. to further one's
career : the skills of networking, bargaining, and negotiation.
DERIVATIVES
net•work•a•ble adjective
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Thesaurus deliciousness:
networknoun
1 a network of arteries WEB, lattice, net, matrix, mesh,
crisscross, grid, reticulum, reticulation; Anatomy plexus.
2 a network of lanes MAZE, labyrinth, warren, tangle.
3 a network of friends SYSTEM, complex, nexus, web,
webwork.
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Ancestry:
From Keith Briggs’s excellent etymologicalinvestigation: (�)
I Opus reticulatum:I A Latin origin?
[http://serialconsign.com/2007/11/we-put-net-network]
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
First known use: Geneva Bible, 1560‘And thou shalt make unto it a grate like networke ofbrass (Exodus xxvii 4).’
From the OED via Briggs:
I 1658–: reticulate structures in animalsI 1839–: rivers and canalsI 1869–: railwaysI 1883–: distribution network of electrical cablesI 1914–: wireless broadcasting networks
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Ancestry:
Net and Work are venerable old words:I ‘Net’ first used to mean spider web (King Ælfréd, 888).I ‘Work’ appears to have long meant purposeful action.
I ‘Network’ = something built based on the idea ofnatural, flexible lattice or web.
I c.f., ironwork, stonework, fretwork.
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Ancestry:
Net and Work are venerable old words:I ‘Net’ first used to mean spider web (King Ælfréd, 888).I ‘Work’ appears to have long meant purposeful action.
I ‘Network’ = something built based on the idea ofnatural, flexible lattice or web.
I c.f., ironwork, stonework, fretwork.
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Ancestry:
Net and Work are venerable old words:I ‘Net’ first used to mean spider web (King Ælfréd, 888).I ‘Work’ appears to have long meant purposeful action.
I ‘Network’ = something built based on the idea ofnatural, flexible lattice or web.
I c.f., ironwork, stonework, fretwork.
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
I Piranha physicus
I Hunt in packs.
I Feast on new and interesting ideas(see chaos, cellular automata, ...)
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
I Piranha physicus
I Hunt in packs.
I Feast on new and interesting ideas(see chaos, cellular automata, ...)
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Key Observation:
I Many complex systemscan be viewed as complex networksof physical or abstract interactions.
I Opens door to mathematical and numerical analysis.I Dominant approach of last decade of a
theoretical-physics/stat-mechish flavor.I Mindboggling amount of work published on complex
networks since 1998...I ... largely due to your typical theoretical physicist:
I Piranha physicus
I Hunt in packs.
I Feast on new and interesting ideas(see chaos, cellular automata, ...)
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Popularity (according to ISI)
“Collective dynamics of ‘small-world’ networks” [23]
I Watts and StrogatzNature, 1998
I ≈ 4677 citations (as of January 18, 2011)
I Over 1100 citations in 2008 alone.
“Emergence of scaling in random networks” [3]
I Barabási and AlbertScience, 1999
I ≈ 5270 citations (as of January 18, 2011)
I Over 1100 citations in 2008 alone.
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Popularity according to books:
The Tipping Point: How Little Things canmake a Big Difference—Malcolm Gladwell [11]
Nexus: Small Worlds and the GroundbreakingScience of Networks—Mark Buchanan
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Popularity according to books:
Linked: How Everything Is Connected toEverything Else and What ItMeans—Albert-Laszlo Barabási
Six Degrees: The Science of a ConnectedAge—Duncan Watts [21]
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Numerous others:I Complex Social Networks—F. Vega-Redondo [20]
I Fractal River Basins: Chance and Self-Organization—I.Rodríguez-Iturbe and A. Rinaldo [17]
I Random Graph Dynamics—R. Durette
I Scale-Free Networks—Guido Caldarelli
I Evolution and Structure of the Internet: A StatisticalPhysics Approach—Romu Pastor-Satorras andAlessandro Vespignani
I Complex Graphs and Networks—Fan Chung
I Social Network Analysis—Stanley Wasserman andKathleen Faust
I Handbook of Graphs and Networks—Eds: StefanBornholdt and H. G. Schuster [7]
I Evolution of Networks—S. N. Dorogovtsev and J. F. F.Mendes [10]
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More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
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More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 55
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 56
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 57
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 58
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 59
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 60
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 61
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 62
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
20 of 63
More observations
I But surely networks aren’t new...I Graph theory is well established...I Study of social networks started in the 1930’s...I So why all this ‘new’ research on networks?I Answer: Oodles of Easily Accessible Data.I We can now inform (alas) our theories
with a much more measurable reality.∗
I Real networks occupy a tiny, low entropy part of allnetwork space and require specific attention.
I A worthy goal: establish mechanistic explanations.I What kinds of dynamics lead to these real networks?
∗If this is upsetting, maybe string theory is for you...
Page 63
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
21 of 63
More observations
I Web-scale data sets can be overly exciting.
Witness:I The End of Theory: The Data Deluge Makes the
Scientific Theory Obsolete (Anderson, Wired) (�)I “The Unreasonable Effectiveness of Data,”
Halevy et al. [12]
I c.f. Wigner’s “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” [24]
But:I For scientists, description is only part of the battle.I We still need to understand.
Page 64
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
21 of 63
More observations
I Web-scale data sets can be overly exciting.
Witness:I The End of Theory: The Data Deluge Makes the
Scientific Theory Obsolete (Anderson, Wired) (�)I “The Unreasonable Effectiveness of Data,”
Halevy et al. [12]
I c.f. Wigner’s “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” [24]
But:I For scientists, description is only part of the battle.I We still need to understand.
Page 65
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
21 of 63
More observations
I Web-scale data sets can be overly exciting.
Witness:I The End of Theory: The Data Deluge Makes the
Scientific Theory Obsolete (Anderson, Wired) (�)I “The Unreasonable Effectiveness of Data,”
Halevy et al. [12]
I c.f. Wigner’s “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” [24]
But:I For scientists, description is only part of the battle.I We still need to understand.
Page 66
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
21 of 63
More observations
I Web-scale data sets can be overly exciting.
Witness:I The End of Theory: The Data Deluge Makes the
Scientific Theory Obsolete (Anderson, Wired) (�)I “The Unreasonable Effectiveness of Data,”
Halevy et al. [12]
I c.f. Wigner’s “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” [24]
But:I For scientists, description is only part of the battle.I We still need to understand.
Page 67
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
21 of 63
More observations
I Web-scale data sets can be overly exciting.
Witness:I The End of Theory: The Data Deluge Makes the
Scientific Theory Obsolete (Anderson, Wired) (�)I “The Unreasonable Effectiveness of Data,”
Halevy et al. [12]
I c.f. Wigner’s “The Unreasonable Effectiveness ofMathematics in the Natural Sciences” [24]
But:I For scientists, description is only part of the battle.I We still need to understand.
Page 68
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
22 of 63
Super basic definitions
Nodes = A collection of entities which haveproperties that are somehow related to each other
I e.g., people, forks in rivers, proteins, webpages,organisms,...
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
22 of 63
Super basic definitions
Nodes = A collection of entities which haveproperties that are somehow related to each other
I e.g., people, forks in rivers, proteins, webpages,organisms,...
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
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Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 72
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 73
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 74
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 75
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 76
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 77
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
23 of 63
Basic definitions:
Links = Connections between nodesI links
I may be real and fixed (rivers),I real and dynamic (airline routes),I abstract with physical impact (hyperlinks),I or purely abstract (semantic connections between
concepts).I Links may be directed or undirected.I Links may be binary or weighted.
Page 78
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)
I For undirected networks, connection betweennumber of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 79
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)
I For undirected networks, connection betweennumber of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 80
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)
I For undirected networks, connection betweennumber of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 81
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)
I For undirected networks, connection betweennumber of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 82
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)I For undirected networks, connection between
number of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 83
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)I For undirected networks, connection between
number of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 84
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)I For undirected networks, connection between
number of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 85
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
24 of 63
Basic definitions:
Node degree = Number of links per node
I Notation: Node i ’s degree = ki .I ki = 0,1,2,. . . .I Notation: the average degree of a network = 〈k〉
(and sometimes as z)I For undirected networks, connection between
number of edges m and average degree:
〈k〉 =2mN
I For directed networks,
〈kout〉 = 〈kin〉 =mN
I Defn: Ni = the set of i ’s ki neighbors
Page 86
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
25 of 63
Basic definitions:
Adjacency matrix:
I We represent a graph or network by a matrix A withlink weight aij for nodes i and j in entry (i , j).
I e.g.,
A =
0 1 1 1 00 0 1 0 11 0 0 0 00 1 0 0 10 1 0 1 0
I (n.b., for numerical work, we always use sparse
matrices.)
Page 87
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Basic definitions
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
25 of 63
Basic definitions:
Adjacency matrix:
I We represent a graph or network by a matrix A withlink weight aij for nodes i and j in entry (i , j).
I e.g.,
A =
0 1 1 1 00 0 1 0 11 0 0 0 00 1 0 0 10 1 0 1 0
I (n.b., for numerical work, we always use sparse
matrices.)
Page 88
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
25 of 63
Basic definitions:
Adjacency matrix:
I We represent a graph or network by a matrix A withlink weight aij for nodes i and j in entry (i , j).
I e.g.,
A =
0 1 1 1 00 0 1 0 11 0 0 0 00 1 0 0 10 1 0 1 0
I (n.b., for numerical work, we always use sparse
matrices.)
Page 89
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
26 of 63
Examples
What passes for a complex network?
I Complex networks are large (in node number)I Complex networks are sparse (low edge to node
ratio)I Complex networks are usually dynamic and evolvingI Complex networks can be social, economic, natural,
informational, abstract, ...
Page 90
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
26 of 63
Examples
What passes for a complex network?
I Complex networks are large (in node number)I Complex networks are sparse (low edge to node
ratio)I Complex networks are usually dynamic and evolvingI Complex networks can be social, economic, natural,
informational, abstract, ...
Page 91
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
26 of 63
Examples
What passes for a complex network?
I Complex networks are large (in node number)I Complex networks are sparse (low edge to node
ratio)I Complex networks are usually dynamic and evolvingI Complex networks can be social, economic, natural,
informational, abstract, ...
Page 92
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
26 of 63
Examples
What passes for a complex network?
I Complex networks are large (in node number)I Complex networks are sparse (low edge to node
ratio)I Complex networks are usually dynamic and evolvingI Complex networks can be social, economic, natural,
informational, abstract, ...
Page 93
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
26 of 63
Examples
What passes for a complex network?
I Complex networks are large (in node number)I Complex networks are sparse (low edge to node
ratio)I Complex networks are usually dynamic and evolvingI Complex networks can be social, economic, natural,
informational, abstract, ...
Page 94
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 95
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 96
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 98
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 99
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 100
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 101
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
27 of 63
Examples
Physical networksI River networksI Neural networksI Trees and leavesI Blood networks
I The InternetI Road networksI Power grids
I Distribution (branching) versus redistribution(cyclical)
Page 102
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 103
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 104
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 105
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 106
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 107
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 108
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
Page 109
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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28 of 63
Examples
Interaction networksI The BlogosphereI Biochemical
networksI Gene-protein
networksI Food webs: who
eats whomI The World Wide
Web (?)I Airline networksI Call networks
(AT&T)I The Media
datamining.typepad.com (�)
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Properties ofComplex Networks
Modelling ComplexNetworks
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Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Properties ofComplex Networks
Modelling ComplexNetworks
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References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
29 of 63
Examples
Interaction networks:social networks
I SnoggingI FriendshipsI AcquaintancesI Boards and
directorsI OrganizationsI twitter.com (�)
facebook.com (�), (Bearman et al., 2004)
I ‘Remotely sensed’ by: tweets (open), instantmessaging, Facebook posts, emails, phone logs(*cough*).
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Properties ofComplex Networks
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Examples
Relational networksI Consumer purchases
(Wal-Mart: ≈ 2.5 petabyte = 2.5× 1015 bytes)
(�)I Thesauri: Networks of words generated by meaningsI Knowledge/Databases/IdeasI Metadata—Tagging: delicious (�), flickr (�)
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Properties ofComplex Networks
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Examples
Relational networksI Consumer purchases
(Wal-Mart: ≈ 2.5 petabyte = 2.5× 1015 bytes) (�)I Thesauri: Networks of words generated by meaningsI Knowledge/Databases/IdeasI Metadata—Tagging: delicious (�), flickr (�)
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Properties ofComplex Networks
Modelling ComplexNetworks
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Examples
Relational networksI Consumer purchases
(Wal-Mart: ≈ 2.5 petabyte = 2.5× 1015 bytes) (�)I Thesauri: Networks of words generated by meaningsI Knowledge/Databases/IdeasI Metadata—Tagging: delicious (�), flickr (�)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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Examples
Relational networksI Consumer purchases
(Wal-Mart: ≈ 2.5 petabyte = 2.5× 1015 bytes) (�)I Thesauri: Networks of words generated by meaningsI Knowledge/Databases/IdeasI Metadata—Tagging: delicious (�), flickr (�)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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Examples
Relational networksI Consumer purchases
(Wal-Mart: ≈ 2.5 petabyte = 2.5× 1015 bytes) (�)I Thesauri: Networks of words generated by meaningsI Knowledge/Databases/IdeasI Metadata—Tagging: delicious (�), flickr (�)
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Properties ofComplex Networks
Modelling ComplexNetworks
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Clickworthy Science:
Bollen et al. [6]; a higher resolution figure is here (�)
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Properties ofComplex Networks
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
I And even when renderings somehow look good:
“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
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Properties ofComplex Networks
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
I And even when renderings somehow look good:
“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
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Properties ofComplex Networks
Modelling ComplexNetworks
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
⇐ Typical hairball
I number of nodes N = 500
I number of edges m = 1000
I average degree 〈k〉 = 4
I And even when renderings somehow look good:
“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
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Properties ofComplex Networks
Modelling ComplexNetworks
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
⇐ Typical hairball
I number of nodes N = 500
I number of edges m = 1000
I average degree 〈k〉 = 4
I And even when renderings somehow look good:
“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
Page 128
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Properties ofComplex Networks
Modelling ComplexNetworks
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
⇐ Typical hairball
I number of nodes N = 500
I number of edges m = 1000
I average degree 〈k〉 = 4
I And even when renderings somehow look good:“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
Page 129
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
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A notable feature of large-scale networks:
I Graphical renderings are often just a big mess.
⇐ Typical hairball
I number of nodes N = 500
I number of edges m = 1000
I average degree 〈k〉 = 4
I And even when renderings somehow look good:“That is a very graphic analogy which aidsunderstanding wonderfully while being, strictlyspeaking, wrong in every possible way”said Ponder [Stibbons] —Making Money, T. Pratchett.
I We need to extract digestible, meaningful aspects.
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Properties ofComplex Networks
Modelling ComplexNetworks
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Properties
Some key aspects of real complex networks:I degree
distribution∗
I assortativityI homophilyI clusteringI motifsI modularity
I concurrencyI hierarchical
scalingI network distancesI centralityI efficiencyI robustness
I Plus coevolution of network structureand processes on networks.
∗ Degree distribution is the elephant in the room thatwe are now all very aware of...
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Properties
1. degree distribution Pk
I Pk is the probability that a randomly selected nodehas degree k
I k = node degree = number of connectionsI ex 1: Erdos-Rényi random networks:
Pk = e−〈k〉〈k〉k/k !
I Distribution is Poisson
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Properties ofComplex Networks
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Properties
1. degree distribution Pk
I Pk is the probability that a randomly selected nodehas degree k
I k = node degree = number of connectionsI ex 1: Erdos-Rényi random networks:
Pk = e−〈k〉〈k〉k/k !
I Distribution is Poisson
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Properties ofComplex Networks
Modelling ComplexNetworks
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Properties
1. degree distribution Pk
I Pk is the probability that a randomly selected nodehas degree k
I k = node degree = number of connectionsI ex 1: Erdos-Rényi random networks:
Pk = e−〈k〉〈k〉k/k !
I Distribution is Poisson
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Properties ofComplex Networks
Modelling ComplexNetworks
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Properties
1. degree distribution Pk
I Pk is the probability that a randomly selected nodehas degree k
I k = node degree = number of connectionsI ex 1: Erdos-Rényi random networks:
Pk = e−〈k〉〈k〉k/k !
I Distribution is Poisson
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Properties ofComplex Networks
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Properties
1. degree distribution Pk
I Pk is the probability that a randomly selected nodehas degree k
I k = node degree = number of connectionsI ex 1: Erdos-Rényi random networks:
Pk = e−〈k〉〈k〉k/k !
I Distribution is Poisson
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Properties
1. degree distribution Pk
I ex 2: “Scale-free” networks: Pk ∝ k−γ ⇒ ‘hubs’I link cost controls skewI hubs may facilitate or impede contagion
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Properties
1. degree distribution Pk
I ex 2: “Scale-free” networks: Pk ∝ k−γ ⇒ ‘hubs’I link cost controls skewI hubs may facilitate or impede contagion
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Properties ofComplex Networks
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Properties
1. degree distribution Pk
I ex 2: “Scale-free” networks: Pk ∝ k−γ ⇒ ‘hubs’I link cost controls skewI hubs may facilitate or impede contagion
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Properties
Note:I Erdos-Rényi random networks are a mathematical
construct .I ‘Scale-free’ networks are growing networks that form
according to a plausible mechanism.I Randomness is out there, just not to the degree of a
completely random network.
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Properties
Note:I Erdos-Rényi random networks are a mathematical
construct .I ‘Scale-free’ networks are growing networks that form
according to a plausible mechanism.I Randomness is out there, just not to the degree of a
completely random network.
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Properties
Note:I Erdos-Rényi random networks are a mathematical
construct .I ‘Scale-free’ networks are growing networks that form
according to a plausible mechanism.I Randomness is out there, just not to the degree of a
completely random network.
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Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.
Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.
Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
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Properties ofComplex Networks
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Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.
Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.
Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
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Properties ofComplex Networks
Modelling ComplexNetworks
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Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.
Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.
Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.
Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.
Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
Page 146
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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37 of 63
Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.
Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
Page 147
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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37 of 63
Properties
2. Assortativity/3. Homophily:
I Social networks: Homophily (�) = birds of a featherI e.g., degree is standard property for sorting:
measure degree-degree correlations.I Assortative network: [15] similar degree nodes
connecting to each other.Often social: company directors, coauthors, actors.
I Disassortative network: high degree nodesconnecting to low degree nodes.Often techological or biological: Internet, WWW,protein interactions, neural networks, food webs.
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Properties ofComplex Networks
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Local socialness:
4. Clustering:
I Your friends tend to knoweach other.
I Two measures (explained onfollowing slides):
1. Watts & Strogatz [23]
C1 =
⟨∑j1 j2∈Ni
aj1 j2
ki(ki − 1)/2
⟩i
2. Newman [16]
C2 =3×#triangles
#triples
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Properties ofComplex Networks
Modelling ComplexNetworks
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38 of 63
Local socialness:
4. Clustering:
I Your friends tend to knoweach other.
I Two measures (explained onfollowing slides):
1. Watts & Strogatz [23]
C1 =
⟨∑j1 j2∈Ni
aj1 j2
ki(ki − 1)/2
⟩i
2. Newman [16]
C2 =3×#triangles
#triples
Page 150
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
38 of 63
Local socialness:
4. Clustering:
I Your friends tend to knoweach other.
I Two measures (explained onfollowing slides):
1. Watts & Strogatz [23]
C1 =
⟨∑j1 j2∈Ni
aj1 j2
ki(ki − 1)/2
⟩i
2. Newman [16]
C2 =3×#triangles
#triples
Page 151
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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First clustering measure:
Example network:
Calculation of C1:
I C1 is the average fraction ofpairs of neighbors who areconnected.
I Fraction of pairs of neighborswho are connected is∑
j1j2∈Niaj1j2
ki(ki − 1)/2
where ki is node i ’s degree,and Ni is the set of i ’sneighbors.
I Averaging over all nodes, wehave:C1 = 1
n∑n
i=1
Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
=⟨Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
⟩i
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Properties ofComplex Networks
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39 of 63
First clustering measure:
Example network:
Calculation of C1:
I C1 is the average fraction ofpairs of neighbors who areconnected.
I Fraction of pairs of neighborswho are connected is∑
j1j2∈Niaj1j2
ki(ki − 1)/2
where ki is node i ’s degree,and Ni is the set of i ’sneighbors.
I Averaging over all nodes, wehave:C1 = 1
n∑n
i=1
Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
=⟨Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
⟩i
Page 153
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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References
39 of 63
First clustering measure:
Example network:
Calculation of C1:
I C1 is the average fraction ofpairs of neighbors who areconnected.
I Fraction of pairs of neighborswho are connected is∑
j1j2∈Niaj1j2
ki(ki − 1)/2
where ki is node i ’s degree,and Ni is the set of i ’sneighbors.
I Averaging over all nodes, wehave:C1 = 1
n∑n
i=1
Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
=⟨Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
⟩i
Page 154
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
39 of 63
First clustering measure:
Example network:
Calculation of C1:
I C1 is the average fraction ofpairs of neighbors who areconnected.
I Fraction of pairs of neighborswho are connected is∑
j1j2∈Niaj1j2
ki(ki − 1)/2
where ki is node i ’s degree,and Ni is the set of i ’sneighbors.
I Averaging over all nodes, wehave:C1 = 1
n∑n
i=1
Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
=⟨Pj1 j2∈Ni
aj1 j2ki (ki−1)/2
⟩i
Page 155
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
39 of 63
First clustering measure:
Example network:
Calculation of C1:
I C1 is the average fraction ofpairs of neighbors who areconnected.
I Fraction of pairs of neighborswho are connected is∑
j1j2∈Niaj1j2
ki(ki − 1)/2
where ki is node i ’s degree,and Ni is the set of i ’sneighbors.
I Averaging over all nodes, wehave:C1 = 1
n∑n
i=1
Pj1 j2∈Ni
aj1 j2ki (ki−1)/2 =⟨P
j1 j2∈Niaj1 j2
ki (ki−1)/2
⟩i
Page 156
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Properties ofComplex Networks
Modelling ComplexNetworks
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Triples and triangles
Example network:
Triangles:
Triples:
I Nodes i1, i2, and i3 form a triplearound i1 if i1 is connected to i2and i3.
I Nodes i1, i2, and i3 form atriangle if each pair of nodes isconnected
I The definition C2 = 3×#triangles#triples
measures the fraction of closedtriples
I The ‘3’ appears because foreach triangle, we have 3 closedtriples.
I Social Network Analysis (SNA):fraction of transitive triples.
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Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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40 of 63
Triples and triangles
Example network:
Triangles:
Triples:
I Nodes i1, i2, and i3 form a triplearound i1 if i1 is connected to i2and i3.
I Nodes i1, i2, and i3 form atriangle if each pair of nodes isconnected
I The definition C2 = 3×#triangles#triples
measures the fraction of closedtriples
I The ‘3’ appears because foreach triangle, we have 3 closedtriples.
I Social Network Analysis (SNA):fraction of transitive triples.
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Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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40 of 63
Triples and triangles
Example network:
Triangles:
Triples:
I Nodes i1, i2, and i3 form a triplearound i1 if i1 is connected to i2and i3.
I Nodes i1, i2, and i3 form atriangle if each pair of nodes isconnected
I The definition C2 = 3×#triangles#triples
measures the fraction of closedtriples
I The ‘3’ appears because foreach triangle, we have 3 closedtriples.
I Social Network Analysis (SNA):fraction of transitive triples.
Page 159
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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40 of 63
Triples and triangles
Example network:
Triangles:
Triples:
I Nodes i1, i2, and i3 form a triplearound i1 if i1 is connected to i2and i3.
I Nodes i1, i2, and i3 form atriangle if each pair of nodes isconnected
I The definition C2 = 3×#triangles#triples
measures the fraction of closedtriples
I The ‘3’ appears because foreach triangle, we have 3 closedtriples.
I Social Network Analysis (SNA):fraction of transitive triples.
Page 160
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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40 of 63
Triples and triangles
Example network:
Triangles:
Triples:
I Nodes i1, i2, and i3 form a triplearound i1 if i1 is connected to i2and i3.
I Nodes i1, i2, and i3 form atriangle if each pair of nodes isconnected
I The definition C2 = 3×#triangles#triples
measures the fraction of closedtriples
I The ‘3’ appears because foreach triangle, we have 3 closedtriples.
I Social Network Analysis (SNA):fraction of transitive triples.
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Properties
I For sparse networks, C1 tends to discount highlyconnected nodes.
I C2 is a useful and often preferred variantI In general, C1 6= C2.I C1 is a global average of a local ratio.I C2 is a ratio of two global quantities.
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
41 of 63
Properties
I For sparse networks, C1 tends to discount highlyconnected nodes.
I C2 is a useful and often preferred variantI In general, C1 6= C2.I C1 is a global average of a local ratio.I C2 is a ratio of two global quantities.
Page 163
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
41 of 63
Properties
I For sparse networks, C1 tends to discount highlyconnected nodes.
I C2 is a useful and often preferred variantI In general, C1 6= C2.I C1 is a global average of a local ratio.I C2 is a ratio of two global quantities.
Page 164
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Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
41 of 63
Properties
I For sparse networks, C1 tends to discount highlyconnected nodes.
I C2 is a useful and often preferred variantI In general, C1 6= C2.I C1 is a global average of a local ratio.I C2 is a ratio of two global quantities.
Page 165
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
41 of 63
Properties
I For sparse networks, C1 tends to discount highlyconnected nodes.
I C2 is a useful and often preferred variantI In general, C1 6= C2.I C1 is a global average of a local ratio.I C2 is a ratio of two global quantities.
Page 166
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
42 of 63
Properties
5. motifs:I small, recurring functional subnetworksI e.g., Feed Forward Loop:
Shen-Orr, Uri Alon, et al. [18]
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
42 of 63
Properties
5. motifs:I small, recurring functional subnetworksI e.g., Feed Forward Loop:
Shen-Orr, Uri Alon, et al. [18]
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
42 of 63
Properties
5. motifs:I small, recurring functional subnetworksI e.g., Feed Forward Loop:
Shen-Orr, Uri Alon, et al. [18]
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
43 of 63
Properties
6. modularity and structure/community detection:
Clauset et al., 2006 [9]: NCAA football
Page 170
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 171
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 172
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 173
Overview
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 174
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 175
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 176
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
44 of 63
Properties
7. concurrency:
I transmission of a contagious element only occursduring contact
I rather obvious but easily missed in a simple modelI dynamic property—static networks are not enoughI knowledge of previous contacts crucialI beware cumulated network dataI Kretzschmar and Morris, 1996 [14]
Page 177
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
45 of 63
Properties
8. Horton-Strahler ratios:I Metrics for branching networks:
I Method for ordering streams hierarchicallyI Number: Rn = Nω/Nω+1I Segment length: Rl = 〈lω+1〉/〈lω〉I Area/Volume: Ra = 〈aω+1〉/〈aω〉
(a) (b) (c)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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45 of 63
Properties
8. Horton-Strahler ratios:I Metrics for branching networks:
I Method for ordering streams hierarchicallyI Number: Rn = Nω/Nω+1I Segment length: Rl = 〈lω+1〉/〈lω〉I Area/Volume: Ra = 〈aω+1〉/〈aω〉
(a) (b) (c)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
45 of 63
Properties
8. Horton-Strahler ratios:I Metrics for branching networks:
I Method for ordering streams hierarchicallyI Number: Rn = Nω/Nω+1I Segment length: Rl = 〈lω+1〉/〈lω〉I Area/Volume: Ra = 〈aω+1〉/〈aω〉
(a) (b) (c)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
45 of 63
Properties
8. Horton-Strahler ratios:I Metrics for branching networks:
I Method for ordering streams hierarchicallyI Number: Rn = Nω/Nω+1I Segment length: Rl = 〈lω+1〉/〈lω〉I Area/Volume: Ra = 〈aω+1〉/〈aω〉
(a) (b) (c)
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
45 of 63
Properties
8. Horton-Strahler ratios:I Metrics for branching networks:
I Method for ordering streams hierarchicallyI Number: Rn = Nω/Nω+1I Segment length: Rl = 〈lω+1〉/〈lω〉I Area/Volume: Ra = 〈aω+1〉/〈aω〉
(a) (b) (c)
Page 182
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 183
Overview
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 184
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 185
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 186
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 187
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 188
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 189
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
46 of 63
Properties
9. network distances:
(a) shortest path length dij :
I Fewest number of steps between nodes i and j .I (Also called the chemical distance between i and j .)
(b) average path length 〈dij〉:
I Average shortest path length in whole network.I Good algorithms exist for calculation.I Weighted links can be accommodated.
Page 190
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
47 of 63
Properties
9. network distances:I network diameter dmax:
Maximum shortest path length between any twonodes.
I closeness dcl = [∑
ij d −1ij /
(n2
)]−1:
Average ‘distance’ between any two nodes.I Closeness handles disconnected networks (dij = ∞)I dcl = ∞ only when all nodes are isolated.I Closeness perhaps compresses too much into one
number
Page 191
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
47 of 63
Properties
9. network distances:I network diameter dmax:
Maximum shortest path length between any twonodes.
I closeness dcl = [∑
ij d −1ij /
(n2
)]−1:
Average ‘distance’ between any two nodes.I Closeness handles disconnected networks (dij = ∞)I dcl = ∞ only when all nodes are isolated.I Closeness perhaps compresses too much into one
number
Page 192
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
47 of 63
Properties
9. network distances:I network diameter dmax:
Maximum shortest path length between any twonodes.
I closeness dcl = [∑
ij d −1ij /
(n2
)]−1:
Average ‘distance’ between any two nodes.I Closeness handles disconnected networks (dij = ∞)I dcl = ∞ only when all nodes are isolated.I Closeness perhaps compresses too much into one
number
Page 193
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
47 of 63
Properties
9. network distances:I network diameter dmax:
Maximum shortest path length between any twonodes.
I closeness dcl = [∑
ij d −1ij /
(n2
)]−1:
Average ‘distance’ between any two nodes.I Closeness handles disconnected networks (dij = ∞)I dcl = ∞ only when all nodes are isolated.I Closeness perhaps compresses too much into one
number
Page 194
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 195
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 196
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 197
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 198
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 199
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
48 of 63
Properties
10. centrality:
I Many such measures of a node’s ‘importance.’I ex 1: Degree centrality: ki .I ex 2: Node i ’s betweenness
= fraction of shortest paths that pass through i .I ex 3: Edge `’s betweenness
= fraction of shortest paths that travel along `.I ex 4: Recursive centrality: Hubs and Authorities (Jon
Kleinberg [13])
Page 200
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 201
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 202
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 203
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 204
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 205
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
49 of 63
Models
Some important models:
1. generalized random networks (touched on in 300)2. scale-free networks (�) (covered in 300)3. small-world networks (�) (covered in 300)4. statistical generative models (p∗)5. generalized affiliation networks (partly covered in
300)
Page 206
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 207
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 208
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 209
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 210
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 211
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
50 of 63
Models
1. generalized random networks:
I Arbitrary degree distribution Pk .I Wire nodes together randomly.I Create ensemble to test deviations from
randomness.I Interesting, applicable, rich mathematically.I We will have fun with these guys...
Page 212
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 213
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 214
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 215
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 216
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 217
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
51 of 63
Models
2. ‘scale-free networks’:
γ = 2.5〈k〉 = 1.8N = 150
I Introduced by Barabasi andAlbert [3]
I Generative modelI Preferential attachment
model with growth:I P[attachment to node i] ∝ kα
i .I Produces Pk ∼ k−γ when
α = 1.I Trickiness: other models
generate skewed degreedistributions.
Page 218
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
52 of 63
Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:
I local regularity (an individual’s friends know eachother)
I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
Page 219
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
52 of 63
Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
Page 220
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
52 of 63
Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
Page 221
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
52 of 63
Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
Page 222
Overview
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Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
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Properties ofComplex Networks
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Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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References
52 of 63
Models
3. small-world networksI Introduced by Watts and Strogatz [23]
Two scales:I local regularity (an individual’s friends know each
other)I global randomness (shortcuts).
I Shortcuts allow disease to jumpI Number of infectives increases
exponentially in timeI Facilitates synchronization
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Properties ofComplex Networks
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Models5. generalized affiliation networks
c d ea b
2 3 41
a
b
c
d
e
contexts
individuals
unipartitenetwork
Bipartite affiliation networks: boards and directors,movies and actors.
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Models
5. generalized affiliation networks
eca
high schoolteacher
occupation
health careeducation
nurse doctorteacherkindergarten
db
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Models
5. generalized affiliation networks
100
eca b d
geography occupation age
0
I Blau & Schwartz [4], Simmel [19], Breiger [8], Watts etal. [22]
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Nutshell:
Overview Key Points:
I The field of complex networks came into existence inthe late 1990s.
I Explosion of papers and interest since 1998/99.I Hardened up much thinking about complex systems.I Specific focus on networks that are large-scale,
sparse, natural or man-made, evolving and dynamic,and (crucially) measurable.
I Three main (blurred) categories:1. Physical (e.g., river networks),2. Interactional (e.g., social networks),3. Abstract (e.g., thesauri).
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Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
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56 of 63
Nutshell:
Overview Key Points:
I The field of complex networks came into existence inthe late 1990s.
I Explosion of papers and interest since 1998/99.I Hardened up much thinking about complex systems.I Specific focus on networks that are large-scale,
sparse, natural or man-made, evolving and dynamic,and (crucially) measurable.
I Three main (blurred) categories:1. Physical (e.g., river networks),2. Interactional (e.g., social networks),3. Abstract (e.g., thesauri).
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
56 of 63
Nutshell:
Overview Key Points:
I The field of complex networks came into existence inthe late 1990s.
I Explosion of papers and interest since 1998/99.I Hardened up much thinking about complex systems.I Specific focus on networks that are large-scale,
sparse, natural or man-made, evolving and dynamic,and (crucially) measurable.
I Three main (blurred) categories:1. Physical (e.g., river networks),2. Interactional (e.g., social networks),3. Abstract (e.g., thesauri).
Page 231
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
56 of 63
Nutshell:
Overview Key Points:
I The field of complex networks came into existence inthe late 1990s.
I Explosion of papers and interest since 1998/99.I Hardened up much thinking about complex systems.I Specific focus on networks that are large-scale,
sparse, natural or man-made, evolving and dynamic,and (crucially) measurable.
I Three main (blurred) categories:1. Physical (e.g., river networks),2. Interactional (e.g., social networks),3. Abstract (e.g., thesauri).
Page 232
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
56 of 63
Nutshell:
Overview Key Points:
I The field of complex networks came into existence inthe late 1990s.
I Explosion of papers and interest since 1998/99.I Hardened up much thinking about complex systems.I Specific focus on networks that are large-scale,
sparse, natural or man-made, evolving and dynamic,and (crucially) measurable.
I Three main (blurred) categories:1. Physical (e.g., river networks),2. Interactional (e.g., social networks),3. Abstract (e.g., thesauri).
Page 233
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
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Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
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Properties ofComplex Networks
Modelling ComplexNetworks
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57 of 63
Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
Page 235
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
57 of 63
Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
Page 236
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Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
57 of 63
Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
Page 237
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Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
57 of 63
Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
Page 238
Overview
Class admin
Basic definitions
Popularity
Examples ofComplex Networks
Properties ofComplex Networks
Modelling ComplexNetworks
Nutshell
References
57 of 63
Nutshell:
Overview Key Points (cont.):
I Obvious connections with the vast extant field ofgraph theory.
I But focus on dynamics is more of aphysics/stat-mech/comp-sci flavor.
I Two main areas of focus:1. Description: Characterizing very large networks2. Explanation: Micro story ⇒ Macro features
I Some essential structural aspects are understood:degree distribution, clustering, assortativity, groupstructure, overall structure,...
I Still much work to be done, especially with respect todynamics... exciting!
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Properties ofComplex Networks
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References I
[1] R. Albert and A.-L. Barabási.Statistical mechanics of complex networks.Rev. Mod. Phys., 74:47–97, 2002. pdf (�)
[2] P. W. Anderson.More is different.Science, 177(4047):393–396, 1972. pdf (�)
[3] A.-L. Barabási and R. Albert.Emergence of scaling in random networks.Science, 286:509–511, 1999. pdf (�)
[4] P. M. Blau and J. E. Schwartz.Crosscutting Social Circles.Academic Press, Orlando, FL, 1984.
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References II
[5] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, andD.-U. Hwang.Complex networks: Structure and dynamics.Physics Reports, 424:175–308, 2006. pdf (�)
[6] J. Bollen, H. Van de Sompel, A. Hagberg,L. Bettencourt, R. Chute, M. A. Rodriguez, andB. Lyudmila.Clickstream data yields high-resolution maps ofscience.PLoS ONE, 4:e4803, 2009. pdf (�)
[7] S. Bornholdt and H. G. Schuster, editors.Handbook of Graphs and Networks.Wiley-VCH, Berlin, 2003.
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References III[8] R. L. Breiger.
The duality of persons and groups.Social Forces, 53(2):181–190, 1974. pdf (�)
[9] A. Clauset, C. Moore, and M. E. J. Newman.Structural inference of hierarchies in networks, 2006.pdf (�)
[10] S. N. Dorogovtsev and J. F. F. Mendes.Evolution of Networks.Oxford University Press, Oxford, UK, 2003.
[11] M. Gladwell.The Tipping Point.Little, Brown and Company, New York, 2000.
[12] A. Halevy, P. Norvig, and F. Pereira.The unreasonable effectiveness of data.IEEE Intelligent Systems, 24:8–12, 2009. pdf (�)
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References IV
[13] J. M. Kleinberg.Authoritative sources in a hyperlinked environment.Proc. 9th ACM-SIAM Symposium on DiscreteAlgorithms, 1998. pdf (�)
[14] M. Kretzschmar and M. Morris.Measures of concurrency in networks and thespread of infectious disease.Math. Biosci., 133:165–95, 1996. pdf (�)
[15] M. Newman.Assortative mixing in networks.Phys. Rev. Lett., 89:208701, 2002. pdf (�)
[16] M. E. J. Newman.The structure and function of complex networks.SIAM Review, 45(2):167–256, 2003. pdf (�)
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References V
[17] I. Rodríguez-Iturbe and A. Rinaldo.Fractal River Basins: Chance and Self-Organization.
Cambridge University Press, Cambrigde, UK, 1997.
[18] S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon.Network motifs in the transcriptional regulationnetwork of Escherichia coli.Nature Genetics, pages 64–68, 2002. pdf (�)
[19] G. Simmel.The number of members as determining thesociological form of the group. I.American Journal of Sociology, 8:1–46, 1902.
[20] F. Vega-Redondo.Complex Social Networks.Cambridge University Press, 2007.
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References VI
[21] D. J. Watts.Six Degrees.Norton, New York, 2003.
[22] D. J. Watts, P. S. Dodds, and M. E. J. Newman.Identity and search in social networks.Science, 296:1302–1305, 2002. pdf (�)
[23] D. J. Watts and S. J. Strogatz.Collective dynamics of ‘small-world’ networks.Nature, 393:440–442, 1998. pdf (�)
[24] E. Wigner.The unreasonable effectivenss of mathematics in thenatural sciences.Communications on Pure and Applied Mathematics,13:1–14, 1960. pdf (�)