2. Outline
1. Basics of Social Networks
2. Real-world problem
3. How to construct graph from real-world problem?
4. What graph theory problem getting from real-world problem?
5. Graph type of Social Networks
6. Special properties in social graph
7. How to find communities and groups in social networks?
(Algorithms)
8. How to interpret graph solution back to real-world problem?

3. 1. Basics of Social Networks

4. Definitions
Social Network:
finite set or sets of actors and the relation or relations defined on them.
Actor = Node = Point = Agent:
social entities such as persons, organizations, cities, etc.
Tie = Link = Edge = Line = Arc:
represents relationships among actors.
Relation:
collection of ties of a specific kind among members of a group.

5. Attributes of Actor (nodes)
- People can be queried
about different
features, like
( age, gender, race,
socioeconomic status,
place of residence,
grade in school, etc. )

6. 2. Real-world Problem

7. Definition : Social Groups and Communities
– “Two or more people , who interact with one another,
share similar characteristics and attributes and
collectively have a sense of unity”
– Actors who have all possible ties among themselves
The real-world problem is:
“Finding Groups And Communities In Social Networks”

8. • Social networks and the social network analysis:
– Is an interdisciplinary academic field
(social psychology, sociology, statistics, and graph theory)
– 1930; first sociograms in the to study interpersonal relationships, by
Jacob
– 1950; sociograms approaches mathematically formalized
– 1980; theories and methods of
social networks became popular
in the social and behavioral sciences
– Social network analysis is now one
of the major paradigms in
contemporary sociology
http://www.cmu.edu/joss/content/articles/volume1/Freeman.html

9. Why to find social groups and communities?
–behavior analysis
–location-based interaction analysis
–recommender systems development
–link prediction
–customer interaction and analysis & marketing
–media use
–Security
–Social studies

10. 3. How to Construct Graph From
Real-world Problem?

11. - Shared Attributes: Actors are grouped based on the shared
attributes among them. i.e. Group of four people (Bob, Carol, Ted,
and Alice)
- Blue for males, red for females
http://faculty.ucr.edu/~hanneman/nettext/C3_Graphs.html
Bob Carol
TedAlice

12. - Attribute 1: "close friends”: who they regarded as close friends in
the group?
A directed graph of friendship ties
Bob Carol
TedAlice
Bob, Carol, and Ted form a "clique" (i.e. each is connected to each of the others)
Alice is a "pendant" (tied to the group by only one connection)

13. - Attribute 2: “Spouse”
A directed graph of spousal ties
Bob Carol
TedAlice

14. 4. What Graph Theory Problem Getting
From Real-world Problem?

15. • Clique problem: refers to any problem to find
particular (complete) subgraphs ("cliques") in
a graph,
• i.e., sets of elements where each pair of
elements is connected.
http://sebastian.doc.gold.ac.uk/

16. • Note: the notion of clique here
dose not necessary refers to a
complete subgraph,
http://sebastian.doc.gold.ac.uk/
Complete Graph: there's an edge between any two node
Dense Graph: number of edges is close to the maximal number of edges
Sparse Graph: when it has only a few edges

17. Dense Graph Definition
• A graph G = (V, E) is said to be dense if for every v ∈ V ,
degree(v) > n/2, where n = |V|
• Density is the ratio between the number of edges |E|
and the number of vertices |V|.
• Density for undirected graphs:
• The maximal density is 1 = complete graphs
• Maximum number of edges ½ |V| (|V|−1)
http://www.cc.gatech.edu/~vigoda/MCMC_Course/Lec7.pdf

18. Complexity of the problem
• Clique problem is NP-Complete problem
– k-clique problem, the input is an undirected graph
and a number k, and the output is a clique of size
k if one exists (or, sometimes, all cliques of size k)

19. 5. Graph Type of Social Networks

20. Small-World Graph = Scale-Free Graph
– most nodes are not neighbors of one another, but most
nodes can be reached from every other by a small number
of hops or steps.
– Specifically, a small-world network is defined to be a
network where the typical distance L between two
randomly chosen nodes (the number of steps required)
grows proportionally to the logarithm of the number of
nodes N in the network, that is:
http://www.lenddo.com/blog/2012/06/facebook-proves-it%E2%80%99s-a-small-world-after-all-we-are-all-connected-by-six-degrees-or-less/
Last time by: Reem

21. 6. Special Properties
in Social Graphs

22. • Community Structure: Real-world social graphs are found to exhibit a
modular structure; with nodes forming groups, and possibly groups within
groups
– In a modular graph, the nodes form communities where groups of nodes in
the same community are tighter connected to each other than to those nodes
outside the community
• Heavy-tailed Degree Distribution:
– few “hubs”,
– most nodes have few neighbors
- The degree distribution has a power law (functional relationship)
- many low degree nodes - only a few high degree nodes in real graphs
• Small Diameter: also known as the ‘small-world phenomenon’ or the ‘six
degrees of separation’
M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69:026113, 2004.

23. 7. How to Find Communities nnd Groups
in Social Networks? (Algorithms)

24. Taxonomy of Community Criteria
- Community detection methods categories:
• Node-Centric Community Detection
– Each node in a group satisfies certain properties
• Group-Centric Community Detection
– Consider the connections within a group as a whole. The group has
to satisfy certain properties without zooming into node-level
• Network-Centric Community Detection
– Partition the whole network into several disjoint sets
• Hierarchy-Centric Community Detection
– Construct a hierarchical structure of communities

25. A classification of
community
detection and graph
clustering methods

26. A classification of
community
detection and graph
clustering methods

27. Clique Percolation Method (CPM)
• Clique is a very strict definition, unstable
• Normally use cliques as a core to find larger communities
• CPM is such a method to find overlapping communities
– Input
• A parameter k, and a network
– Procedure
1. Find out all cliques of size k in a given network
2. Construct a clique graph. Two cliques are adjacent if
they share k-1 nodes
3. Each connected components in the clique graph
form a community 27

28. Example: Clique Percolation Method
{1, 2, 3}, {1, 3, 4}, {4, 5, 6}, {5, 6, 7}, {5, 6, 8}, {5, 7, 8},
{6, 7, 8}
28
Step 1: Find all Cliques of size 3

29. 29
Step 2: Construct Clique Graph
{1, 2, 3}, {1, 3, 4}, {4, 5, 6},
{5, 6, 7}, {5, 6, 8}, {5, 7, 8},
{6, 7, 8}

30. 30
Step 3: Finding Communities
Two cliques are adjacent if
they share k-1 nodes (i.e. k-1=2)
Communities:
{1, 2, 3, 4}
{4, 5, 6, 7, 8}
{1, 2, 3}, {1, 3, 4}, {4, 5, 6},
{5, 6, 7}, {5, 6, 8}, {5, 7, 8},
{6, 7, 8}

31. 8. How to Interpret Graph Solution
Back to Real-life Problem?

32. - Finding Cliques in the Social Graph of the Social
Network leads to the communities and groups
inside the Social Networks, based on the
attributes and characteristics of actors in the
communities

33. References
• Community detection in Social Media, (2012), Symeon Papadopoulos, Yiannis Kompatsiaris,
Athena Vakali, Ploutarchos Spyridonos, Data Mining and Knowledge Discovery May 2012,
Volume 24, Issue 3, pp 515-554
• Community Detection in Graphs, (2010), Santo Fortunato, Complex Networks and Systems
Lagrange Laboratory, ISI Foundation, Viale S. Severo 65, 10133, Torino,I-ITALY.
• A Comparison of Community Detection Algorithms on Artificial Networks, (2009), Günce
Keziban Orman1,2 and Vincent Labatut , Discovery Science Lecture Notes in Computer
Science Volume 5808, pp 242-256
• Social Network Analysis. Methods and Applications, (2008), Wasserman, Stanley, Faust,
Katherine, Cambridge, University Press
• Computing Communities in Large Networks Using Random Walks, (2005), Pascal Pons and
Matthieu Latapy, Computer and Information Sciences – ISCIS, Lecture Notes in Computer
Science Volume 3733, 2005, pp 284-293
• Introduction to social network methods, (2005) Robert A. Hanneman and Mark Riddle,
University of California,

35. http://www.lifelearn.com/2013/11/5-easy-creative-ways-thank-social-media-followers/