pdodds/teaching/courses/2011-01UVM-303/docs/20… · Overview Class admin Basic deﬁnitions Popularity Examples of Complex Networks Properties of Complex Networks Modelling Complex

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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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http://www.uvm.edu/~vacc/

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Outline

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Modelling Complex Networks

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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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Class Admin




I dolphins

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Class Admin




I dolphins

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Class Admin




I dolphins

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Class Admin




I dolphins

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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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http://www.amazon.com/Networks-Introduction-Mark-Newman/dp/0199206651

http://www.cs.cornell.edu/home/kleinber/networks-book/

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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]


I R. Albert and A.-L. Barabási“Statistical mechanics of complex networks” [1]


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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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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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http://keithbriggs.info/network.html

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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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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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I Piranha physicus

I Hunt in packs.

I Feast on new and interesting ideas(see chaos, cellular automata, ...)

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I Piranha physicus

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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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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.

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More observations




Halevy et al. [12]



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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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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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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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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

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(and sometimes as z)I For undirected networks, connection between

number of edges m and average degree:

〈k〉 =2mN




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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.)

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Basic definitions:

Adjacency matrix:


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


matrices.)

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Basic definitions:

Adjacency matrix:


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


matrices.)

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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, ...

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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

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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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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http://www.facebook.com

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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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http://www.economist.com/node/15557443

http://www.delicious.com

http://www.flickr.com

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Examples


(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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Clickworthy Science:

Bollen et al. [6]; a higher resolution figure is here (�)

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http://www.plosone.org/article/slideshow.action?uri=info:doi/10.1371/journal.pone.0004803&imageURI=info:doi/10.1371/journal.pone.0004803.g005

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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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⇐ Typical hairball

I number of nodes N = 500

I number of edges m = 1000

I average degree 〈k〉 = 4




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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.


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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.


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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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Pk = e−〈k〉〈k〉k/k !


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Pk = e−〈k〉〈k〉k/k !


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Pk = e−〈k〉〈k〉k/k !


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Pk = e−〈k〉〈k〉k/k !


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Properties


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

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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http://en.wikipedia.org/wiki/Homophily

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connecting to each other.Often social: company directors, coauthors, actors.



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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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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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Local socialness:

4. Clustering:




C1 =

⟨∑j1 j2∈Ni

aj1 j2

ki(ki − 1)/2

⟩i

2. Newman [16]

C2 =3×#triangles

#triples

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Local socialness:

4. Clustering:




C1 =

⟨∑j1 j2∈Ni

aj1 j2

ki(ki − 1)/2

⟩i

2. Newman [16]

C2 =3×#triangles

#triples

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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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Example network:

Calculation of C1:



j1j2∈Niaj1j2

ki(ki − 1)/2



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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Example network:

Calculation of C1:



j1j2∈Niaj1j2

ki(ki − 1)/2



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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Example network:

Calculation of C1:



j1j2∈Niaj1j2

ki(ki − 1)/2



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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Example network:

Calculation of C1:



j1j2∈Niaj1j2

ki(ki − 1)/2



n∑n

i=1

Pj1 j2∈Ni

aj1 j2ki (ki−1)/2 =⟨P

j1 j2∈Niaj1 j2

ki (ki−1)/2

⟩i

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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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Example network:

Triangles:

Triples:







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Example network:

Triangles:

Triples:







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Example network:

Triangles:

Triples:







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Example network:

Triangles:

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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Properties

5. motifs:I small, recurring functional subnetworksI e.g., Feed Forward Loop:

Shen-Orr, Uri Alon, et al. [18]

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Properties

6. modularity and structure/community detection:

Clauset et al., 2006 [9]: NCAA football

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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]

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7. concurrency:



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7. concurrency:



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7. concurrency:



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7. concurrency:



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7. concurrency:



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7. concurrency:



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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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(a) (b) (c)

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(a) (b) (c)

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(a) (b) (c)

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(a) (b) (c)

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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.

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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

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ij d −1ij /

(n2

)]−1:


number

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ij d −1ij /

(n2

)]−1:


number

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ij d −1ij /

(n2

)]−1:


number

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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])

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Properties

10. centrality:




Kleinberg [13])

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Properties

10. centrality:




Kleinberg [13])

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Properties

10. centrality:




Kleinberg [13])

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Properties

10. centrality:




Kleinberg [13])

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Properties

10. centrality:




Kleinberg [13])

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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)

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http://en.wikipedia.org/wiki/Scale-free_network

http://en.wikipedia.org/wiki/Small-world_network

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Models



300)

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300)

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300)

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300)

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300)

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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...

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Models




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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.

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Models


γ = 2.5〈k〉 = 1.8N = 150







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γ = 2.5〈k〉 = 1.8N = 150







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γ = 2.5〈k〉 = 1.8N = 150







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γ = 2.5〈k〉 = 1.8N = 150







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γ = 2.5〈k〉 = 1.8N = 150







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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 eachother)

I global randomness (shortcuts).

I Shortcuts allow disease to jumpI Number of infectives increases

exponentially in timeI Facilitates synchronization

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Models


Two scales:I local regularity (an individual’s friends know each

other)I global randomness (shortcuts).



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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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5. generalized affiliation networks

eca

high schoolteacher

occupation

health careeducation

nurse doctorteacherkindergarten

db

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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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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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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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http://www.uvm.edu/~pdodds/research/papers/others/2002/albert2002a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1972/anderson1972a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1999/barabasi1999a.pdf

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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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http://www.uvm.edu/~pdodds/research/papers/others/2006/boccaletti2006a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2009/bollen2009a.pdf

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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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http://www.uvm.edu/~pdodds/research/papers/others/1974/breiger1974a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2006/clauset2006a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2009/halevy2009a.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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http://www.uvm.edu/~pdodds/research/papers/others/1998/kleinberg1998a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1996/kretzschmar1996a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2002/newman2002a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/2003/newman2003a.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 (�)

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http://www.uvm.edu/~pdodds/research/papers/others/2002/watts2002b.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1998/watts1998a.pdf

http://www.uvm.edu/~pdodds/research/papers/others/1960/wigner1960a.pdf

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pdodds/teaching/courses/2011-01UVM-303/docs/20… · Overview Class admin Basic deﬁnitions Popularity Examples of Complex Networks Properties of Complex Networks Modelling Complex