katarzyna sznajd-weron katedra fizyki teoretycznejkatarzynaweron/students/intro... · 6 two basic...
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Complex networks
Katarzyna Sznajd-WeronKatedra Fizyki Teoretycznej
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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.)
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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)