1. CS 590 Term Project Epidemic model on Facebook
ChoungRyeol LEE, Shubham Agrawal, Ashwin Jiwane

2. Facebook (partial) Network
Source: Facebook ego network, Stanford Network Analysis Project

3. Data Limitation and Processing
It is infeasible for us to access (and handle) the complete Facebook data
Analysis is done on partial dataset obtained from Stanford Network Analysis Project
The original data is the directed ego-network (without ego) of 10 nodes which we had to reconstruct, i.e. make it undirected and add the ego-edges
Ego-network:

4. Source: Slides by Giorgos Cheliotis, National University of Singapore
What is Ego Network?
Source: Slides by Giorgos Cheliotis, National University of Singapore

5. Network Characteristics
Value
Number of nodes (n)
3963
Number of edges (m)
88156
Number of cluster (c) 1
Minimum degree (dmin) 2
Maximum degree (dmax)
1034
Average degree (d)
22.245
Average path length (l)
3.776
Diameter (D) 8
Global clustering coefficient (cc)
0.5212
Maximum clique size 57

6. Centrality Measures Weight Centrality Measures Node Weighted
Eigenvector
2160
Pagerank
1641
Closeness
100
Betweenness
Degree
Non-Weighted
1868
3381

7. Facebook Interpretation
Basic Analysis
Centrality Measure
Facebook Interpretation
Pagerank
It is very likely to visit his profile in random surfing starting from anyone else’s profile
Eigenvector
This person has ‘influential’ (or social) friends
Betweenness
This person is an important connection between different people
Closeness
This person uses minimum amount of ‘mutual friends’ link to connect to anyone else
Degree
This person has maximum number of friends
Observations: The graph follows the “Small World Phenomenon” as the average path length is but it is not a “Scale-Free” network since it doesn’t follow Power-Law

8. Power-Law

9. Friendship Strength In FB, possible ways to measures friendship:
Mutual friends
Common biography (location, education, etc)
Mutual interests (pages, likes, etc)
Common social groups
Due to limitation of data, we considered only Mutual Friends as the weighing measure

10. Cosine Similarity
Cosine similarity measures the normalized number of common friends
Basic principle is to take the cosine of the vectors (rows) from adjacency matrix
In study network:
Maximum cosine value =
Minimum cosine value =

11. Epidemic Models SI and SIR Model: SI Model: SIR Model:
A node is susceptible to infected node with certain probability
You repost/share from friends
SI Model:
Once a node is infected, it remains infected
Post remains active on the wall
SIR Model:
Once a node is infected, it remains infected for certain time period
Post gets inactive after certain time period

12. Model Simulation Simulated epidemic model on the graph
Pre-infected a particular node
Compared the results with different nodes of importance
Checked for the time steps required for complete cascade in SI model
Checked for the time steps required to reach stable condition in SIR model
Stable condition means no more node is getting infected due to ‘died’ nodes

13. Model Simulation Model assumptions:
Probability of infection
Discrete time intervals
Assumed two scenarios of probability:
Function of weight
Similar to ‘Top News’ posts
Independent of weight
Similar to ‘Most Recent’ posts

14. SIR Model Results Function W 0.2 Model SI Importance Eigen Page Degree
Node
2160
1641
100
1868
3381
TimeStep
Freq 1 2
167
172
219
168
120
233 3
254
476
693
491
276
679 4
269
133
403
463
225
669 5 41
338
789
872
464
925 6 46
598
723
482
771
507 7
435
352
427
663
766
309 8
629
702
249
412
906
227 9
549
631
164 96
257
114 10
633
169
109 78 38 16 11
273 93 55
117 15 12
213
110 30 13
123 14
116 21 28 17 18 19 20

15. SIR Model Results

16. SIR Interpretation Unweighted graph (p=0.2): Weighted graph(p=0.2):
Degree: steepest curve, infects less people,
EigenVector: steep curve, infects most people
Pagerank: grows slowest, infects more people
Weighted graph(p=0.2):
Degree: steepest curve, infects less people
EigenVector: grows slowest, infects most people
Pagerank: grows slow, infects more people, better than eigenvector due to weights

17. SI Model Results
2160 – Weig, 1641 – Wpage, 100 – Wdegree; 1868 – Eig, 3381 – Page, Degree

18. Currently working on.. Quarantine Strategy: Vaccination Strategy:
Choose the nodes to quarantine at a certain time interval such that they don’t affect others
Account blocked (reported as spam)
Vaccination Strategy:
Choose the nodes to vaccinate, i.e. make them safe from certain viral, such that epidemic doesn’t flow through them
Spam filter
Objective is to minimize the cost of prevention and/or precaution with the aim of ‘curing’ epidemic
Quarantine – We are picking up people based on the time they have been infected and their importance in the network Vaccination – We are saving people based on the number of infected neighbors and their importance in the network

19. Communities in Facebook network
Held together by some common interests and ideas of a large group of people in Facebook
Any one person may be part of many communities which are overlapping and nested structure
Groups within social networks might highly correspond to social units or communities in reality
A subset of Facebook users within the graph such that connections between the users are denser than connections with the rest of the network.
One person has only one community

20. Reviews of Community Detection
Two methods for discovering groups in networks
Graph partitioning
Pre-fixed number of parts by minimizing “cut edge”
Computation load(NP-Hard)
Community structure detection
Suitable for the structure of large-scale network data
Provides information on topology of the network
Two approaches really want to address the same question with somewhat different means.

21. Community Detection using iGraph
Algorithms in Igraph
Optimal communities
Basic framework
Infomap
92 communities
Compressing the description of information flows on networks. 
Leading Eigenvector
18 communities
Calculation of Leading non-negative eigenvector of the modularity matrix of the graph and distributions of vertices by the sign of eigenvector
Label Propagation
57 communities
Labeling with unique labels and updating by majority voting in the neighbors of the vertex
Multilevel
17 communities
Contribution to modularity with sequential changes of assignment of nodes

22. Thank You