Complex networks

Katarzyna Sznajd-WeronKatedra Fizyki Teoretycznej

Internet Movie DataBase(IMDB)

A Few Good Man

Robert Wagner

Austin Powers: The spy who shagged me

Wild Things

Let’s make it legal

Barry Norton

What Price Glory

Monsieur Verdoux

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Try to play at https://oracleofbacon.org/

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Network science

• A.L. Barabasi, Network Science, Cambridge University Press (2016)

• Online at http://networksciencebook.com/

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Interdisciplinary research

Arkadiusz JędrzejewskiBachelor: PhysicsMaster: Applied MathPhD: Physics (statistical physics)

Several figures from his thesis will be used during this lecture.

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Two basic network parameters

• Number of nodes (𝑵 – the size of the network): the number of components in the system.

• To distinguish the nodes, we label them with 𝑖 = 1, 2, … ,𝑁. • Number of links (𝑳): the total number of interactions

between the nodes. • Links are rarely labeled, as they can be identified through the

pair of nodes they connect: 1,2 , 2,3 , 2,4 , 4,5 , (5,7)

node (vertex)

link (edge)

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Network science and graph theory

Network science Graph theory

network graph

node vertex

link edge

• Two terminologies are used interchangeably!• A subtle distinction between them:

• the terms {network, node, link} often refers to real systems

• the terms {graph, vertex, edge} refers to mathematical representation of networks

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The same graph for different networks

© 2017 Marcin Weron

The social network

The technological network

The corresponding graph

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Adjacency Matrix

A complete description of a network

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• The simplest way: a complete list of the links

– {(1, 2), (1, 3), (2, 3), (2, 4)}

• Hint: for contact processes (opinion dynamics)

– For each node a vector of neighbours

Nn[1] = [2,3]

Nn[2] = [1,3]

Nn[3] = [1,2]

Nn[4] = [2]

Another description of a network

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Examples of networks

Network Nodes Links Dir / Undir N L

Internet Routers Internet connections Undirected 192,244 609,066

WWW Webpages Links Directed 325,729 1,497,134

Mobile-Phone Calls

Subscribers Calls Directed 36,595 91,826

Email Email add Emails Directed 57,194 103,731

Science Collaboration

Scientists Co-authorships Undirected 23,133 93,437

Actor Network Actors Co-acting Undirected 702,388 29,397,908

Citation Network Papers Citations Directed 449,673 4,689,479

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J. J. Potterat et al. (2002): Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs

A social network of sexual contacts

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Dating in a high school

nodes: studentsBlue: boysPink: girlslinks: they were on a date

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Other examples - puzzle

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• Basic properties of a node

– Degree

– Clustering Coefficient

• Basic properties of a network

– Degree distribition

– Average degree

– Average Path Length

– Average Clustering Coefficient

Basic properties

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• Degree of 𝑖 − 𝑡ℎ node (𝑘𝑖): number of links attached (in/out in directed) to the node

• Example: 𝑘1 = 1, 𝑘2 = 3, 𝑘3 = 1, 𝑘4 = 2, 𝑘5 =2, 𝑘6 = 0, 𝑘7 = 1

• The total number of links:

𝐿 =1

2

𝑖=1

𝑁

𝑘𝑖

Basic properties of a node: Degree

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Directed networks

• Incoming degree (𝑘𝑖𝑖𝑛): the number of links that

point to node 𝑖

• Outgoing degree (𝑘𝑖𝑜𝑢𝑡): the number of links that

point from node 𝑖 to other nodes

• A node’s total degree

𝑘𝑖 = 𝑘𝑖𝑖𝑛 + 𝑘𝑖

𝑜𝑢𝑡

• The total number of links in a directed network:

𝐿 =

𝑖=1

𝑁

𝑘𝑖𝑖𝑛 =

𝑖=1

𝑁

𝑘𝑖𝑜𝑢𝑡

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Basic properties of a network: Degreedistribution and average degree

• The probability that a randomly selected node in the network has degree 𝑘

• Average degree:

𝑘 =1

𝑁

𝑖=1

𝑁

𝑘𝑖 =2𝐿

𝑁

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Basic properties: average path length

• A path: a route that runs along the links of the network

• A path’s length: the number of links the path contains

• The shortest path between nodes 𝑖 and 𝑗: the path with the fewest number of links that connect nodes 𝑖 and 𝑗

• Diameter: the longest shortest path in a graph

• Average path length

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Basic properties: clustering coefficient

the number of links between the 𝑘𝑖 neighbors of node 𝑖

𝐶 =1

𝑁

𝑖=1

𝑁

𝐶𝑖

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„Small world” by Stanley Milgram

Stanley Milgram, "The Small World Problem", Psychology Today, vol. 1, no. 1,

May 1967, pp61‐67 (8542 cyt.)

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Results

Stanley Milgram, "The Small World Problem", Psychology Today, vol. 1, no. 1,

May 1967, pp61‐67 (8542 cyt.)

23Sociometry, Vol. 32, No. 4 (Dec., 1969), pp. 425-443

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Results

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Small world of facebook

The New York Times, 21.11.2011: 4.74The world is shrinking!

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It began in 1998? Not really ...

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• 𝐺(𝑁, 𝐿): 𝑁 labeled nodes are connected with 𝐿 randomly placed links[Erdős & Rényi, 1959-1968]

• 𝑮(𝑵, 𝒑): Each pair of 𝑁 labeled nodes is connected with probability 𝑝 [Gilbert, 1959]

Erdős–Rényi random graphs, 1960

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1) Start with N isolated nodes

2) Select a node pair and generate 𝑟 ~𝑈 0,1

3) If 𝑝 < 𝑟 then connect the selected pair

4) Otherwise leave them disconnected

5) Repeat step 2) for each of the 𝑁 𝑁−1

2node

pairs

Random graph algorithm 𝑮(𝑵, 𝒑)

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1) Start with N isolated nodes

2) Select a node pair and generate 𝑟 ~𝑈 0,1

3) If 𝑝 < 𝑟 then connect the selected pair

4) Otherwise leave them disconnected

5) Repeat step 2) for each of the 𝑁 𝑁−1

2node

pairs

Random graph algorithm 𝑮(𝑵, 𝒑)

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Erdős–Rényi random graphs, 1960

(c) 2016, Arkadiusz Jędrzejewski

𝑘 = 𝑝 𝑁 − 1

𝑝𝑘 =𝑁 − 1𝑘

𝑝𝑘 1 − 𝑝 𝑁−1−𝑘

Most real networks are sparse, i.e. 𝑘 ≪ 𝑁:

𝑝𝑘 = exp(− 𝑘 )𝑘 𝑘

𝑘!Poisson Distribution

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• Small diameter

• High clustering coefficient

Properties of a real social network

Source: Wolfram|Alpha Personal Analytics for Facebook

Ego graph: Kasia is not present in the graph

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Watts-Strogatz model: small-world network

Watts, D. J.; Strogatz, S. H. (1998)"Collective dynamics of 'small-world' networks„ Nature 393: 440–442.

(c) 2016, Arkadiusz Jędrzejewski

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1/2

1/2

2/4

1/4

1/4

1/6

3/6

1/6

1/6

1/8

3/8

2/8

1/8

1/8

...

Barabasi-Albert network

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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Barabási, Albert-László; Albert, Réka (1999) "Emergence of scaling in random networks" Science 286, 509–512.

Scale-free networks

hub

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Network robustness

• The Impact of Node Removal1. Attacks 2. Errors

• How to measure robustness?

• Percolation theory

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The Impact of Node Removal

How you define an error and how you define an attack?

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Think about two types of networks

carbon nanotubessource: https://arstechnica.com

Internet

• The Impact of Node Removal1. Attacks 2. Errors

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The Impact of Node Removal

Diameter of the network

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Network fragmentation under failures and attacks

the average size of the isolated clusters

the relative size of the largest clusters S

the fraction of removed nodes f

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• Spreading phenomena on networks

• Temporal networks

• Multiplex

Petter Holme, Jari Saramäki

Physics Reports

Vol 519, Pages 97–125 (2012)

There is more …

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Multiplex (Anna Chmiel)CS-AARHUS

The multiplex social network consists of five kinds of online and offline relationships

(Facebook, Leisure, Work, Co-authorship, Lunch) between the employees of Computer Science department at Aarhus

5 layers Multiplex

Nodes: 61

Edges: 620

Matteo Magnani, Barbora Micenkova, Luca Rossi - Combinatorial Analysis of Multiple Networks. arXiv:1303.4986 (2013)

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More on http://networksciencebook.com/