1. Social Network Analysis
Social Network Introduction
Statistics and Probability Theory
Models of Social Network Generation
Networks in Biological System
November 21, 2018
Data Mining: Concepts and Techniques

2. Six Degrees of Separation
Society
Nodes: individuals
Links: social relationship (family/work/friendship/etc.)
S. Milgram (1967)
Six Degrees of Separation
John Guare
Social networks: Many individuals with diverse social interactions between them.
November 21, 2018
Data Mining: Concepts and Techniques

3. Communication networks
The Earth is developing an electronic nervous system, a network with diverse nodes and links are
-computers
-routers
-satellites
-phone lines
-TV cables
-EM waves
Communication networks: Many non-identical components with diverse connections between them.
November 21, 2018
Data Mining: Concepts and Techniques

4. New York Times Complex systems NETWORK
Made of many non-identical elements connected by diverse interactions.
NETWORK
November 21, 2018
Data Mining: Concepts and Techniques

5. “Natural” Networks and Universality
Consider many kinds of networks:
social, technological, business, economic, content,…
These networks tend to share certain informal properties:
large scale; continual growth
distributed, organic growth: vertices “decide” who to link to
interaction restricted to links
mixture of local and long-distance connections
abstract notions of distance: geographical, content, social,…
Do natural networks share more quantitative universals?
What would these “universals” be?
How can we make them precise and measure them?
How can we explain their universality?
This is the domain of social network theory
Sometimes also referred to as link analysis
November 21, 2018
Data Mining: Concepts and Techniques

6. Some Interesting Quantities
Connected components:
how many, and how large?
Network diameter:
maximum (worst-case) or average?
exclude infinite distances? (disconnected components)
the small-world phenomenon
Clustering:
to what extent that links tend to cluster “locally”?
what is the balance between local and long-distance connections?
what roles do the two types of links play?
Degree distribution:
what is the typical degree in the network?
what is the overall distribution?
November 21, 2018
Data Mining: Concepts and Techniques

7. A “Canonical” Natural Network has…
Few connected components:
often only 1 or a small number, indep. of network size
Small diameter:
often a constant independent of network size (like 6)
or perhaps growing only logarithmically with network size or even shrink?
typically exclude infinite distances
A high degree of clustering:
considerably more so than for a random network
in tension with small diameter
A heavy-tailed degree distribution:
a small but reliable number of high-degree vertices
often of power law form
MIGHT GIVE REAL EXAMPLES HERE? FROM WATTS?
November 21, 2018
Data Mining: Concepts and Techniques

8. Probabilistic Models of Networks
All of the network generation models we will study are probabilistic or statistical in nature
They can generate networks of any size
They often have various parameters that can be set:
size of network generated
average degree of a vertex
fraction of long-distance connections
The models generate a distribution over networks
Statements are always statistical in nature:
with high probability, diameter is small
on average, degree distribution has heavy tail
Thus, we’re going to need some basic statistics and probability theory
November 21, 2018
Data Mining: Concepts and Techniques

9. Data Mining: Concepts and Techniques
Zipf’s Law
Look at the frequency of English words:
“the” is the most common, followed by “of”, “to”, etc.
claim: frequency of the n-th most common ~ 1/n (power law, α = 1)
General theme:
rank events by their frequency of occurrence
resulting distribution often is a power law!
Other examples:
North America city sizes
personal income
file sizes
genus sizes (number of species)
People seem to dither over exact form of these distributions (e.g. value of α), but not heavy tails
November 21, 2018
Data Mining: Concepts and Techniques

10. Data Mining: Concepts and Techniques
Zipf’s Law
The same data plotted on linear and logarithmic scales. Both plots show a Zipf distribution with 300 datapoints
Linear scales on both axes
Logarithmic scales on both axes
November 21, 2018
Data Mining: Concepts and Techniques

11. Social Network Analysis
Social Network Introduction
Statistics and Probability Theory
Models of Social Network Generation
Networks in Biological System
Summary
November 21, 2018
Data Mining: Concepts and Techniques

12. Some Models of Network Generation
Random graphs (Erdös-Rényi models):
gives few components and small diameter
does not give high clustering and heavy-tailed degree distributions
is the mathematically most well-studied and understood model
Watts-Strogatz models:
give few components, small diameter and high clustering
does not give heavy-tailed degree distributions
Scale-free Networks:
gives few components, small diameter and heavy-tailed distribution
does not give high clustering
Hierarchical networks:
few components, small diameter, high clustering, heavy-tailed
Affiliation networks:
models group-actor formation
November 21, 2018
Data Mining: Concepts and Techniques

13. The Clustering Coefficient of a Network
Let nbr(u) denote the set of neighbors of u in a graph
all vertices v such that the edge (u,v) is in the graph
The clustering coefficient of u:
let k = |nbr(u)| (i.e., number of neighbors of u)
choose(k,2): max possible # of edges between vertices in nbr(u)
c(u) = (actual # of edges between vertices in nbr(u))/choose(k,2)
0 <= c(u) <= 1; measure of cliquishness of u’s neighborhood
Clustering coefficient of a graph:
average of c(u) over all vertices u
k = 4
choose(k,2) = 6
c(u) = 4/6 = 0.666…
November 21, 2018
Data Mining: Concepts and Techniques

14. The Clustering Coefficient of a Network
Clustering: My friends will likely know each other!
Probability to be connected C » p
# of links between 1,2,…n neighbors
C =
n(n-1)/2
Networks are clustered [large C(p)] but have a small characteristic path length [small L(p)].
November 21, 2018
Data Mining: Concepts and Techniques

15. Erdos-Renyi: Clustering Coefficient
Generate a network G according to G(N,p)
Examine a “typical” vertex u in G
choose u at random among all vertices in G
what do we expect c(u) to be?
Answer: exactly p!
In G(N,m), expect c(u) to be 2m/N(N-1)
Both cases: c(u) entirely determined by overall density
Baseline for comparison with “more clustered” models
Erdos-Renyi has no bias towards clustered or local edges
November 21, 2018
Data Mining: Concepts and Techniques

16. Data Mining: Concepts and Techniques
Scale-free Networks
The number of nodes (N) is not fixed
Networks continuously expand by additional new nodes
WWW: addition of new nodes
Citation: publication of new papers
The attachment is not uniform
A node is linked with higher probability to a node that already has a large number of links
WWW: new documents link to well known sites (CNN, Yahoo, Google)
Citation: Well cited papers are more likely to be cited again
November 21, 2018
Data Mining: Concepts and Techniques

17. Data Mining: Concepts and Techniques
Scale-Free Networks
Start with (say) two vertices connected by an edge
For i = 3 to N:
for each 1 <= j < i, d(j) = degree of vertex j so far
let Z = S d(j) (sum of all degrees so far)
add new vertex i with k edges back to {1, …, i-1}:
i is connected back to j with probability d(j)/Z
Vertices j with high degree are likely to get more links!
“Rich get richer”
Natural model for many processes:
hyperlinks on the web
new business and social contacts
transportation networks
Generates a power law distribution of degrees
exponent depends on value of k
November 21, 2018
Data Mining: Concepts and Techniques

18. Data Mining: Concepts and Techniques
Scale-Free Networks
Preferential attachment explains
heavy-tailed degree distributions
small diameter (~log(N), via “hubs”)
Will not generate high clustering coefficient
no bias towards local connectivity, but towards hubs
November 21, 2018
Data Mining: Concepts and Techniques

19. Social Network Analysis
Social Network Introduction
Statistics and Probability Theory
Models of Social Network Generation
Networks in Biological System
Mining on Social Network
Summary
November 21, 2018
Data Mining: Concepts and Techniques

20. Bio-Map GENOME protein-gene interactions PROTEOME
protein-protein interactions
PROTEOME
GENOME
Citrate Cycle
METABOLISM
Bio-chemical reactions
November 21, 2018
Data Mining: Concepts and Techniques

21. Bio-chemical reactions
Metabolic Network
Bio-chemical reactions
METABOLISM
Citrate Cycle
November 21, 2018
Data Mining: Concepts and Techniques

22. Data Mining: Concepts and Techniques
Boehring-Mennheim
November 21, 2018
Data Mining: Concepts and Techniques

23. Data Mining: Concepts and Techniques
Metab-movie
Metabolic Network
Nodes: chemicals (substrates)
Links: bio-chemical reactions
November 21, 2018
Data Mining: Concepts and Techniques

24. Meta-P(k) Metabolic Network
Archaea
Bacteria
Eukaryotes
Organisms from all three domains of life are scale-free networks!
H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, (2000)
November 21, 2018
Data Mining: Concepts and Techniques

25. Bio-Map GENOME protein-gene interactions PROTEOME
protein-protein interactions
PROTEOME
GENOME
Citrate Cycle
METABOLISM
Bio-chemical reactions
November 21, 2018
Data Mining: Concepts and Techniques

26. protein-protein interactions
Protein Network
PROTEOME
protein-protein interactions
November 21, 2018
Data Mining: Concepts and Techniques

27. Data Mining: Concepts and Techniques
Prot Interaction map
Yeast Protein Network
Nodes: proteins
Links: physical interactions (binding)
P. Uetz, et al. Nature 403, (2000).
November 21, 2018
Data Mining: Concepts and Techniques

28. Topology of the Protein Network
Prot P(k)
Topology of the Protein Network
H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, (2001)
November 21, 2018
Data Mining: Concepts and Techniques

29. Data Mining: Concepts and Techniques
p53 Network
Nature (2000) …
“One way to understand the p53 network is to compare it to the Internet The cell, like the Internet, appears to be a ‘scale-free network’.”
November 21, 2018
Data Mining: Concepts and Techniques

30. Data Mining: Concepts and Techniques
P53 P(k)
p53 Network (mammals)
November 21, 2018
Data Mining: Concepts and Techniques