Network Science workshop

April 29, 2013 3rd International Business Complexity and Global Leadership Conference 1/37

WARNING!Network Science is extremely

contagious ONCE YOU LEARN IT you . ,START seeing Networks everywhere.

D Zinoviev.


Outline

● What Is Network Science?● Terms and Definitions● Measures● Formation● Complex Behavior● Tools of the Craft● Unusual Applications of Network Science


What is Network Science?

Network science is an interdisciplinary academic field which studies complex networks such as:

telecommunication, transportation, electrical, computer, biological, cognitive and semantic, and social.


What is it based upon?

The field draws on theories and methods including:

Graph theory from mathematics (Erdős, Rényi, Strogatz), Game theory from economics (Jackson), Statistical mechanics from physics (Barabási, Newman, Vespignani,

Watts), Data mining and information visualization from computer science

(Adamic), and Social structure from sociology (Watts).


Terms and definitions● Network = Graph● Nodes (vertexes, actors, members)

represent entities● Nodes have properties (gender,

capacity, political view)● Edges (arcs, links, ties) represent

relationships● Edges have properties (direction,

weight, kind)● Directed vs undirected● Multigraph: graph with parallel

edges● Simple graph: undirected, no loops,

no parallel edges● Connected graphs

Boston SSAlbany

Brunswick

Boston NS

St Albans

ProvidenceHartford

Springfield

New Haven

New York PS

Montreal

Rutland


Adjacency Matrix A

7

5

Boston SSAlbany

Brunswick

Boston NS

St Albans

ProvidenceHartford

Springfield

New Haven

New York PS

Montreal

6 Rutland

9

12

11

4

8

1

3

2

10

A=0 0 1 0 0 0 0 1 0 0 0 00 0 1 0 0 0 0 0 0 0 0 01 1 0 0 0 1 1 0 0 0 0 00 0 0 0 0 0 1 0 0 1 0 00 0 0 0 0 0 1 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 1 1 1 0 0 0 1 0 0 01 0 0 0 0 0 0 0 1 1 0 00 0 0 0 0 0 1 1 0 0 0 00 0 0 1 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 1 0

A

ij=1 if and only if i and j are connected


Incidence Matrix B

B=1 0 1 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 1 0 0 0 00 1 1 0 0 0 0 0 0 0 0 00 0 1 0 0 1 0 0 0 0 0 00 0 1 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 1 0 0 00 0 0 0 0 0 0 1 0 1 0 00 0 0 1 0 0 0 0 0 1 0 00 0 0 1 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 10 0 0 0 1 0 1 0 0 0 0 0

7

5

Boston SSAlbany

Brunswick

Boston NS

St Albans

ProvidenceHartford

Springfield

New Haven

New York PS

Montreal

6 Rutland

9

12

11

4

8

1

3

2

10

A

B

C D

E

F

G

H

I

J

KL

Bij=1 if and only if node i is incident to edge j

edges

node

s

A=B2−2I


PATHS

7

5

Boston SSAlbany

Brunswick

Boston NS

St Albans

ProvidenceHartford

Springfield

New Haven

New York PS

Montreal

6 Rutland

9

12

11

4

8

1

3

2

10

A

B

C D

E

F

G

H

I

J

KL

Path = sequence of connected edges (e.g., B – H – I)

Can be simple (no self-intersections)

Can be a loop (ends where it starts)

Paths have lengths Geodesic = a shortest path (B

– F – G – J is not a geodesic, but B – H – I is)

What if edges are weighted?


Small World

We are on average just 4–6 links (“handshakes”) away from any other living person on Earth (Milgram's experiment)—thence, “six degrees of separation”

Not all networks have the “small world” property

I

Someone I know

Boris Berezovsky

Vladimir Putin

Barak Obama

Wait, how do you know O

bama?


Centrality

● How “central” is a node in the network?

● Possibly affects influence, resilience, susceptibility, etc.

● Several flavors: degree, closeness, betweenness, eigenvalue, etc.


Degree Centrality[ ]

7

5

Boston SS (2)Albany (4)

Brunswick (1)

Boston NS (1)

St Albans (1)

Providence (2)

Springfield (4)

New Haven (3)

New York PS (2)

Montreal (1)

6 Rutland (1)

9

12

11

4

8

1

3

2

10Hartford (2)

Just count the neighbors! More neighbors = more

“friends” = more importance Distinguish in-degree, out-

degree, and [total] degree Can be defined in two ways (N

is the total number of nodes, a

ij∈A):

d i=∑ jaij

d i=∑ jaij / N−1


Degree Distribution

Degree [centrality] distribution is an important network measure—it relates to the network formation process

Most common distributions in complex networks: binomial (Poisson for n→∞) and power law (a.k.a. Pareto, Zipf, scale free)

Why it is what it is?


Closeness Centrality

7

5

Boston SS (0.5)

Brunswick (1)

Boston NS (1)

St Albans (0.4)

Providence (0.4)

Springfield (0.6)

New Haven (0.5)

New York PS (0.5)

Montreal (0.4)

6 Rutland (0.4)

9

12

11

4

8

1

3

2

10Hartford (0.5)

Albany (0.6)

Calculate average inverse shortest path to all other nodes

Shorter path = closer “friends” = better connectivity

Can be defined in two ways (N is the total number of nodes, p

ij

is a geodesic path from I to j)

Takes care of disconnected networks!

ci=∑ j1 / pij

ci=∑ j1/ p ij / N−1


Betweenness Centrality

7

5

Boston SS (0.1)

Brunswick (0)

Boston NS (0)

St Albans (0)

Providence (0.04)

Springfield (0.5)

New Haven (0.14)

New York PS (0.13)

Montreal (0)

6 Rutland (0)

9

12

11

4

8

1

3

2

10Hartford (0.06)

Albany (0.5)

Calculate how many shortest paths go through the node

Mores paths = better brokerage opportunities (= more vulnerability)

Can be defined in two ways (N is the total number of nodes, p

ij

is a geodesic path from I to j, n is the number of such paths)

bwi=∑ j≠i≠kn p jik /n p jk

bwi=∑ j≠i≠kn p jik /n p jk / N−1 N−2


Eigenvector Centrality

7

5

Boston SS (0.29)

Brunswick (0)

Boston NS (0)

St Albans (0.19)

Providence (0.25)

Springfield (0.49)

New Haven (0.34)

New York PS (0.31)

Montreal (0.17)

6 Rutland (0.17)

9

12

11

4

8

1

3

2

10Hartford (0.33)

Albany (0.45)

Recursive definition: A node is as important as its neighbors are

ei=1∑ j

aij e j

A− I E=0 E , =eig A


Similarity and Triadic Closure

Connectivity between nodes may imply similarity: A is connected to B A is similar to B (known as homophily in social networks). Two dyads sharing a node become a triad.

A

B

C

A

B

CAlternative interpretation: weak ties become strong ties (Granovetter).

A

B

C A

B

C


Clustering Coefficient

Clustering coefficient of a node with n neighbors:

Ci=0 — star

Ci=1 — clique (1, 4, 5, 6)

C1=6/10

Average clustering coefficient: C=(.6+.67+1+1+1+1)/6=.88

C i=2∑ j , k

aij aik a jk

n n−1

“Birds of a feather flock together...” (William Turner)

1 (.6)2 (.67)

3 (1.)4 (1.)

5 (1.)6 (1.)


Modularity and Components

NSSI (self-cutters) online communities in LiveJournal (blogging social Web site) form six components

If these two components are merged, they form a giant component

Modularity Q∈[-1, 1] measures the density of links inside clusters as compared to links between clusters:

Q=

∑ij [aij−∑iaij∑ j

aij

∑ijaij ]ij

∑ijaij


Assortativity

Assortative networks: nodes connect to nodes with similar degree; high modularity, better community structure

Dissassortative networks: nodes connect to nodes with different degree


Network Formation

● Networks are complex systems composed of interconnected parts that as a whole exhibit properties not obvious from the properties of the individual parts.

● Most networks are not an immediate product of intelligent design.


exponential Networks A.k.a. Erdős–Rényi networks Start with a fixed set of N nodes Randomly connect them with probability p Average degree λ=pN Binomial / Poisson degree distribution

(decays exponentially after max) No small-world property!


Small World Networks A.k.a. Watts–Strogatz networks Start with a fixed set of N nodes Connect each node to its m neighbors Rewire the connections with probability β Degree distribution: δ-function for β→0, binomial/Poisson

for β→1 (unrealistic) Small-world—but no clustering!

β=0

0<β<1

β=1


Scale Free Networks A.k.a. Barabási–Albert networks Start with few nodes Attach a new node X to m existing nodes

Yi with probability proportional to the

degrees of Yi (preferential attachment)

Power law degree distribution Small-world, community structure No meaningful average degree (scale-

free) Fat tail


Strategic Network formation Formed on purpose Start with a fixed set of N nodes Add links to maximize utility: either

globally or pairwise Topology depends on the costs and

benefits

Link cost c Benefit from direct

connection δ Benefits from indirect

connections δ2, δ3, δ4, etc.

3δ-3c

3δ-3c

3δ-3c

3δ-3c

δ+2δ2-c3δ-3c

δ+2δ2-cδ+2δ2-c

0

0

0

0

δ vs c

“cheap” li

nks

“expensive” links


Complex Behaviors

● Simple contagion: epidemics, rumor propagation

● Complex contagion: collective action, political views, fashion

● Information diffusion: effect of feedback

● Resilience


Simple Contagion

Susceptible – Infectious – Susceptible (SIS): At each step, a “healthy” (but susceptible) node gets infected by an infected neighbor with probability p, and an infected node recovers with probability r

Susceptible – Infectious – Recovered (SIR): same as in SIS, but a node cannot be reinfected

Spreads fast in power-law networks


Collective Action

A node becomes infected with probability p when either a certain number M or a certain fraction m of its neighbors is infectious ✔ “I will wear red pants if at least 50% of my friends wear red

pants.”✔ “I will use protocol X if at least 10 of my partners support

protocol X.”✔ “I will go to protest tax hikes if all my friends go with me.”✔ “I will feel happy if people around me are happy.”

Supported by community structure:✔ Structural trapping (few external links)✔ Social reinforcement (many internal links)✔ Homophily (“connected” means “similar”)

Success depends on the point of origin


Information Diffusion

A network of senders and receivers Each actor has knowledge, credibility,

and popularity Options for sender (speaker):

To send (gain popularity, gain or lose credibility)

Not to send (lose popularity) Options for receiver (listener):

Listen silently (gain knowledge, lose popularity)

Listen and provide feedback (gain knowledge, gain popularity, gain or lose credibility)

Action based on Nash equilibrium


Resilience

Random attacks: Fail

random nodes

Targeted attacks: Attack

selected nodes

Exponential random networks

No difference: The network gracefully degrades

Scale-free networks (robust yet fragile)

The giant component survives.

The giant component rapidly falls

apart.


Tools of the Craft

● Gephi—graph visualization● Pajek—network algorithms and some

visualization● NetLogo—simple simulation environment (good

for small-scale experiments)● CFinder—community finder● NodeXL—network visualization plugin for Excel● networkx—Python library for network

algorithms


Gephi Network

Science “Paintbrush”

Analysis and visualization of large networks

Windows, Linux, MacOS

Developed by Gephi consortium

Free and open source


Pajek “Spider” in

Slovene Analysis and

visualization of large networks

Windows (run on Linux in wine)

Developed by Batagelj and Mrvar

Free, but not open source


Unusual applications

Reminder:If all you know is Network Science

everything looks like a Network.


Unusual networks

● Networks of recipes and cooking ingredients (Adamic)

● Product space networks (Hidalgo)● Human disease networks (Barabási)● Flavor networks (Ahn)● Soccer player networks (Onody / de Castro)● And more!..


Semantic networks Two words are similar if they

are used by similar people (But two people are similar if

they use similar words!) Zinoviev, Stefanescu,

Swenson, and Fireman, “Semantic Networks of Interests in Online NSSI Communities,” Proc. of Workshop “Words and Networks,” Evanston, IL, June 2012


Textual Networks Co-occurrence of actors in

the New Testament A node is an actor, an

edge is introduced if two actors are mentioned in the same chapter of a book at least once

Bigger nodes—more mentioning

Zinoviev, research in progress, unpublished


Thank you!

Network Science workshop

Technology

definitions network

graph theory

node i

graph nodes vertexes

paralleledges simple

parallel edges

properties gender

political view edges