algorithms and applications in social networks · algorithms and applications in social networks...
TRANSCRIPT
![Page 1: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/1.jpg)
Algorithms and Applications in Social Networks
2019/2020, Semester B
Slava Novgorodov 1
![Page 2: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/2.jpg)
Lesson #2
• Random network models
• Centrality measures
2
![Page 3: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/3.jpg)
Random Graphs
3
![Page 4: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/4.jpg)
Erdős–Rényi model
• Two variants of the model:– G(n, m) – a graph is chosen uniformly from a set of
graphs with n nodes and m edges
– G(n, p) – a graph is constructed on n nodes, with probability of edge equals to p
• We will focus on the second variant
• Expected number of edges and average degree:
4
![Page 5: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/5.jpg)
Erdős–Rényi model
• Probability of node i having a degree k:
• Binomial distribution, which becomes Poisson when n 🡪 infinity
• Phase transition at pc (critical point) = 1/n
5
![Page 6: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/6.jpg)
Erdős–Rényi model
• Example – 40 nodes, different p
p = 0.01, 9 edges p = 0.025, 19 edges p = 0.1, 69 edges
Avg. degree = 0.45 Avg. degree = 0.95 Avg. degree = 3.45
6
![Page 7: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/7.jpg)
Erdős–Rényi model
• Clustering coefficient = p
For a node with k neighbors:
#links between neighbors/#max links between neighbors =
= [ p*(k(k-1)/2) ] / [k(k-1)/2] = p
7
![Page 8: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/8.jpg)
Erdős–Rényi model
• Example – degree distribution for G(1000, 0.1)
• Small diameter, but: (1) Non “power-law” distribution. (2) Low clustering coefficient. 🡪 Not a “small-world”
8
![Page 9: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/9.jpg)
“Small-world” model
• Properties:– Small diameter (proportional to log N)
– High clustering coefficient
• A class of random graphs by Watts and Strogatz
9
![Page 10: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/10.jpg)
Scale-free networks
• A network whose degree distribution follows power law.
10
![Page 11: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/11.jpg)
Random vs Scale-free networks
11
![Page 12: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/12.jpg)
“Small-world” model
• Small-world examples:– Co-authors in the same domain
– Colleagues
– Classmates
• Non small-world examples:– “went-to-same-school” people
12
![Page 13: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/13.jpg)
Watts-Strogatz model
• Input: N nodes, with average degree K and probability p of “recreating” the edge.
Step 1:
Create N nodes, connect
each node to K/2 neighbors
on the left and right (by IDs)Result: High clustering coefficient,
but also big diameter 13
![Page 14: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/14.jpg)
Watts-Strogatz model
Step 2:
For each edge (i, j), decide if it should be
recreated with probability p
Result: High clustering coefficient,
and smaller diameter
14
![Page 15: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/15.jpg)
Watts-Strogatz model
15
![Page 16: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/16.jpg)
Summary
16
![Page 17: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/17.jpg)
Real World examples
• Erdős number – collaboration distance to Erdős
• Kevin Bacon number:– Kevin Bacon himself has a Bacon number of 0.– Those actors who have worked directly with Kevin Bacon
have a Bacon number of 1.– If the lowest Bacon number of any actor with whom X has
appeared in any movie is N, X's Bacon number is N+1.
17
![Page 18: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/18.jpg)
Erdős–Bacon number
• Paul Erdős has Erdős–Bacon number 3– Erdős number 0
– Bacon number 3
Ronald Graham Dave Johnson18
![Page 19: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/19.jpg)
Erdős–Bacon number
• Natalie Portman has Erdős–Bacon number 7– Erdős number 5
– Bacon number 2
19
![Page 20: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/20.jpg)
Centrality Measures
20
![Page 21: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/21.jpg)
Centrality
• Identify the most important vertices in a graph
• Applications:– Most influential people
– Key infrastructure nodes
– Information spread points
• The measure we chose is often depends on the application
21
![Page 22: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/22.jpg)
Preliminaries
• Eccentricity (of node v) – maximal distance between v and any other node.
• Diameter – maximum eccentricity in graph
• Radius – minimum eccentricity in graph
22
![Page 23: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/23.jpg)
Preliminaries
• Central point – node with eccentricity = radius
• Graph center – set of central points
• Periphery – set of nodes with eccentricity = diameter
23
![Page 24: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/24.jpg)
Types of Centrality
• There are many types of centrality measures:– Degree Centrality
– Closeness Centrality
– Betweenness Centrality
– Eigenvector Centrality
• To demonstrate, we use 3 types of graphs:Star graph, Circle graph, Line graph
24
![Page 25: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/25.jpg)
Things to measure
• Degree Centrality:– Connectedness
• Closeness Centrality:– Ease of reaching other nodes
• Betweenness Centrality:– Role as an intermediary, connector
• Eigenvector Centrality– “Whom you know…”
25
![Page 26: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/26.jpg)
Degree Centrality
• How “connected” is a node?
• Normalized: Divide by (n-1)
• High centrality – direct contact with many others
• Low centrality – not active
26
![Page 27: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/27.jpg)
Degree Centrality
CD(A) = 6
CD(B) = 1
CD(C) = 1
CD(D) = 1
CD(E) = 1
CD(F) = 1
CD(G) = 1
27
![Page 28: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/28.jpg)
Degree Centrality
CD(A) = 2
CD(B) = 2
CD(C) = 2
CD(D) = 2
CD(E) = 2
CD(F) = 2
CD(G) = 2
28
![Page 29: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/29.jpg)
Degree Centrality
CD(A) = 1
CD(B) = 2
CD(C) = 2
CD(D) = 2
CD(E) = 2
CD(F) = 2
CD(G) = 1
29
![Page 30: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/30.jpg)
Closeness Centrality
• How close the node to other nodes in a graph
• Normalized: Multiply by (n-1)
• High centrality – quick interaction with others, short communication path, low number of steps
30
![Page 31: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/31.jpg)
Closeness Centrality
Cc(A) = 1/6
Cc(B) = 1/11
Cc(C) = 1/11
Cc(D) = 1/11
Cc(E) = 1/11
Cc(F) = 1/11
Cc(G) = 1/11
31
![Page 32: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/32.jpg)
Closeness Centrality
Cc(A) = 1/12
Cc(B) = 1/12
Cc(C) = 1/12
Cc(D) = 1/12
Cc(E) = 1/12
Cc(F) = 1/12
Cc(G) = 1/12
32
![Page 33: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/33.jpg)
Closeness Centrality
Cc(A) = 1/21
Cc(B) = 1/16
Cc(C) = 1/13
Cc(D) = 1/12
Cc(E) = 1/13
Cc(F) = 1/16
Cc(G) = 1/21
33
![Page 34: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/34.jpg)
Betweenness Centrality
• Number of shortest pathes going through i
• Normalized: Divide by (n-1)(n-2)/2
• High centrality – probability of communication between s and t going through i
34
![Page 35: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/35.jpg)
Betweenness Centrality
CB(A) = 15
CB(B) = 0
CB(C) = 0
CB(D) = 0
CB(E) = 0
CB(F) = 0
CB(G) = 0
35
![Page 36: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/36.jpg)
Betweenness Centrality
CB(A) = 3
CB(B) = 3
CB(C) = 3
CB(D) = 3
CB(E) = 3
CB(F) = 3
CB(G) = 3
36
![Page 37: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/37.jpg)
Betweenness Centrality
CB(A) = 0
CB(B) = 5
CB(C) = 8
CB(D) = 9
CB(E) = 8
CB(F) = 5
CB(G) = 0
37
![Page 38: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/38.jpg)
Centralities
A) BetweennessB) ClosenessC) EigenvectorD) Degree
38
![Page 39: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/39.jpg)
HW question example
Compute and explain 3 types of centrality
39
![Page 40: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/40.jpg)
Families of Florence
• Marriage and relationships of 16 families in Florence in middle ages
• Very interesting, “classic” network to analyze
• The rise of Medici family (https://www2.bc.edu/candace-jones/mb851/Mar12/PadgettAnsell_AJS_1993.pdf)
40
![Page 41: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/41.jpg)
Families of Florence
41
![Page 42: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/42.jpg)
Families of Florence
From: https://simonfink.wordpress.com/2016/05/11/the-medici-marriage-network-in-florence/42
![Page 43: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/43.jpg)
Degree CentralityMedici – 6Strozzi – 4Guadagni - 4
43
![Page 44: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/44.jpg)
Closeness CentralityMedici – 14/25Ridolfi – 14/28
44
![Page 45: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/45.jpg)
Betweeness CentralityMedici – 0.522Guadagni – 0.255
45
![Page 46: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/46.jpg)
Families of Florence
46
![Page 47: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/47.jpg)
Another usecase
• Sister found her “lost” brother by analyzing his (online)social network connections
47
![Page 48: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/48.jpg)
Another usecase
• Sister found her “lost” brother by analyzing his (online)social network connections
48
![Page 49: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/49.jpg)
Another usecase
Happy End – brother was found through the connection E!
49
![Page 50: Algorithms and Applications in Social Networks · Algorithms and Applications in Social Networks 2019/2020, Semester B Slava Novgorodov 1. ... Erdős–Rényi model •Two variants](https://reader034.vdocuments.site/reader034/viewer/2022042920/5f645be476c05f563a70d994/html5/thumbnails/50.jpg)
Thank you!Questions?
50