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04/22/23 1
Data Mining: Concepts and Techniques
— Chapter 9 —9.2. Social Network Analysis
Jiawei Han and Micheline Kamber
Department of Computer Science
University of Illinois at Urbana-Champaign
www.cs.uiuc.edu/~hanj©2006 Jiawei Han and Micheline Kamber. All rights reserved.
Acknowledgements: Based on the slides by Sangkyum Kim and Chen Chen
04/22/23 2
04/22/23 3
Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
Social Networks
Social network: A social structure made of nodes (individuals or organizations) that are related to each other by various interdependencies like friendship, kinship, like, ...
Graphical representation Nodes = members Edges = relationships
Examples of typical social networks on the Web Social bookmarking (Del.icio.us) Friendship networks (Facebook, Myspace, LinkedIn) Blogosphere Media Sharing (Flickr, Youtube) Folksonomies
Web 2.0 Examples Blogs
Blogspot Wordpress
Wikis Wikipedia Wikiversity
Social Networking Sites Facebook Myspace Orkut
Digital media sharing websites Youtube Flickr
Social Tagging Del.icio.us
Others Twitter Yelp
Adapted from H. Liu & N. Agarwal, KDD’08 tutorial
7
Society
Nodes: individuals
Links: social relationship (family/work/friendship/etc.)
S. Milgram (1967)
Social networks: Many individuals with diverse social interactions between them
John Guare Six Degrees of Separation
8
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
9
Complex systemsMade of many non-identical
elements connected by diverse interactions.
NETWORK
10
“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,
… Social network theory and link analysis
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?
11
Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
Networks and Their Representations
A network (or a graph): G = (V, E), where V: vertices (or nodes), and E: edges (or links)
Multi-edge: if more than one edge between the same pair of vertices
Self-edge (self-loop): if an edge connects vertex to itself Simple network/graph if a network has neither self-edges nor
multi-edges Adjacency matrix:
Aij = 1 if there is an edge between vertices i and j; 0
otherwise Weighted networks:
Edges having weight (strength), usually a real number Directed network (directed graph): if each edge has a direction
Aij = 1 if there is an edge from i to j; 0 otherwise
Cocitation and Bibliographic Coupling
Cocitation of vertices i and j: # of vertices having outgoing edges pointing to both i and j
Cocitation of i and j:
Cociation matrix: It is a symmetric matrix
Diagonal matrix (Cii): total # papers citing i Bibliographic coupling of vertices i and j: # of
other vertices to which both point
Bibliographic coupling of i and j:
Cociation matrix: Diagonal matrix (Bii): total # papers cited by i
i j
i
Vertices i and j are co-cited by 3 papers
i j
Vertices i and j cite 3 same papers
Cocitation & Bibliographic Coupling: Comparison
Two measures are affected by the number of incoming and outgoing edges that vertices have
For strong cocitation: must have a lot of incoming edges Must be well-cited (influential) papers, surveys, or books Takes time to accumulate citations
Strong bib-coupling if two papers have similar citations A more uniform indicator of similarity between papers Can be computed as soon as a paper is published Not change over time
Recent analysis algorithms HITS explores both cocitation and bibliographic coupling
SIGMOD
SDM
ICDM
KDD
EDBT
VLDB
ICML
AAAI
Tom
Jim
Lucy
Mike
Jack
Tracy
Cindy
Bob
Mary
Alice
Bipartite Networks
Bipartite Network: two kinds of vertices, and edges linking only vertices of unlike types
Incidence matrix: Bij = 1 if vertex j links to group i
0 otherwise One can create a one-mode project from
the two-mode partite form (but with info loss)
The projection to one-mode can be written in terms of the incidence matrix B is follows
Degree and Network Density
Degree of a vertex i: # of edges m = 1/2 of sum of degrees of all the vertices:
The mean degree c of a vertex in an undirected graph:
Density ρ of a graph: A network is dense if density ρ tends to be a constant as n → ∞ A network is sparse if density ρ → 0 as n → ∞. The fraction of
nonzero element in the adjacency matrix tends to zero Internet, WWW and friendship networks are usually regarded as
sparse16
04/22/23 17
Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
18
Some Interesting Network 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?
19
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
20
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
21
Probability and Random Variables
A random variable X is simply a variable that probabilistically assumes values in some set
set of possible values sometimes called the sample space S of X sample space may be small and simple or large and complex
S = {Heads, Tails}, X is outcome of a coin flip S = {0,1,…,U.S. population size}, X is number voting democratic S = all networks of size N, X is generated by preferential attachment
Behavior of X determined by its distribution (or density) for each value x in S, specify Pr[X = x] these probabilities sum to exactly 1 (mutually exclusive outcomes) complex sample spaces (such as large networks):
distribution often defined implicitly by simpler components might specify the probability that each edge appears independently this induces a probability distribution over networks may be difficult to compute induced distribution
22
Some Basic Notions and Laws
Independence: let X and Y be random variables independence: for any x and y, Pr[X = x & Y = y] = Pr[X=x]Pr[Y=y] intuition: value of X does not influence value of Y, vice-versa dependence: e.g. X, Y coin flips, but Y is always opposite of X
Expected (mean) value of X: only makes sense for numeric random variables “average” value of X according to its distribution formally, E[X] = S (Pr[X = x] X), sum is over all x in S often denoted by m always true: E[X + Y] = E[X] + E[Y] true only for independent random variables: E[XY] = E[X]E[Y]
Variance of X: Var(X) = E[(X – m)^2]; often denoted by s^2 standard deviation is sqrt(Var(X)) = s
Union bound: for any X, Y, Pr[X=x & Y=y] <= Pr[X=x] + Pr[Y=y]
23
Convergence to Expectations
Let X1, X2,…, Xn be: independent random variables with the same distribution Pr[X=x] expectation m = E[X] and variance s2
independent and identically distributed (i.i.d.) essentially n repeated “trials” of the same experiment natural to examine r.v. Z = (1/n) S Xi, where sum is over i=1,…,n example: number of heads in a sequence of coin flips example: degree of a vertex in the random graph model E[Z] = E[X]; what can we say about the distribution of Z?
Central Limit Theorem: as n becomes large, Z becomes normally distributed
with expectation m and variance s2/n
24
The Normal Distribution
The normal or Gaussian density: applies to continuous, real-valued random
variables characterized by mean m and standard
deviation s density at x is defined as
(1/(s sqrt(2p))) exp(-(x-m)2/2s2) special case m = 0, s = 1: a exp(-x2/b) for
some constants a,b > 0 peaks at x = m, then dies off exponentially
rapidly the classic “bell-shaped curve”
exam scores, human body temperature, remarks:
can control mean and standard deviation independently
can make as “broad” as we like, but always have finite variance
25
The Binomial Distribution
Coin with Pr[heads] = p, flip n
times, probability of getting exactly
k heads:
choose (n, k) = pk(1-p)n-k
For large n and p fixed:
approximated well by a normal
with
m = np, s = sqrt(np(1-p))
s/m 0 as n grows
leads to strong large deviation
bounds
www.professionalgambler.com/ binomial.html
26
The Poisson Distribution
Like binomial, applies to variables taken on integer values > 0
Often used to model counts of events number of phone calls placed in a given
time period number of times a neuron fires in a given
time period Single free parameter l, probability of exactly
x events: exp(-l) lx/x! mean and variance are both l
Binomial distribution with n large, p = l/n (l fixed)
converges to Poisson with mean l
single photoelectron distribution
27
Power Law (or Pareto) Distributions
Heavy-tailed, pareto, or power law distributions:
For variables assuming integer values > 0 probability of value x ~ 1/xa
Typically 0 < a < 2; smaller a gives heavier tail
sometimes also referred to as being scale-free
For binomial, normal, and Poisson distributions the tail probabilities approach 0 exponentially fast
What kind of phenomena does this distribution model?
What kind of process would generate it?
28
Distinguishing Distributions in Data
All these distributions are idealized models In practice, we do not see distributions, but data Typical procedure to distinguish between Poisson, power law, …
might restrict our attention to a range of values of interest accumulate counts of observed data into equal-sized bins look at counts on a log-log plot
power law: log(Pr[X = x]) = log(1/xa) = -a log(x) linear, slope –a
Normal: log(Pr[X = x]) = log(a exp(-x2/b)) = log(a) – x2/b non-linear, concave near mean
Poisson: log(Pr[X = x]) = log(exp(-l) lx/x!) also non-linear
29
Zipf’s Law
Pareto distribution vs. Zipf’s Law Pareto distributions are continuous probability distributions Zipf's law: a discrete counterpart of the Pareto distribution
Zipf's law: Given some corpus of natural language utterances, the frequency of any
word is inversely proportional to its rank in the frequency table Thus the most frequent word will occur approximately twice as often as
the second most frequent word, which occurs twice as often as the fourth most frequent word, etc.
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)
04/22/23 30
Linear scales on both axes
Logarithmic scales on both axes
The same data plotted on linear and logarithmic scales. Both plots show a Zipf distribution with 300 data points
Zipf Distribution
31
Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
32
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
04/22/23 33
Models of Social Network Generation
Random Graphs (Erdös-Rényi models)
Watts-Strogatz models
Scale-free Networks
Basic Network Measures: Degrees and Clustering
Coefficients Let a network G = (V, E), degree of a vertex
Undirected network: d(vi):
Directed network
In-degree of a vertex din(vi):
Out-degree of a vertex dout(vi):
Clustering coefficients
Let Nv be the set of adjacent vertices of v, kv be the number of
adjacent vertices to node v Local clustering coefficient for directed network
Local clustering coefficient for undirected network
For the whole network: Averaging the local clustering coefficient of all the vertices (Watts & Strogatz):
34
35
The Erdös-Rényi (ER) Model: A Random Graph Model
A random graph is obtained by starting with a set of N vertices and adding edges between them at random
Different random graph models produce different probability distributions on graphs
Most commonly studied is the Erdős–Rényi model, denoted G(N,p), in which every possible edge occurs independently with probability p
Random graphs were first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs”
The usual regime of interest is when p ~ 1/N, N is large e.g., p = 1/2N, p = 1/N, p = 2/N, p=10/N, p = log(N)/N, etc. in expectation, each vertex will have a “small” number of neighbors will then examine what happens when N infinity can thus study properties of large networks with bounded degree Sharply concentrated; not heavy-tailed
36
Erdös-Rényi Model (1959)
- Democratic
- Random
Pál ErdösPál Erdös (1913-1996)
Connect with
probability pp=1/6 N=10
k~1.5 Poisson distribution
37
The -model
The -model has the following parameters or “knobs”: N: size of the network to be generated k: the average degree of a vertex in the network to be generated p: the default probability two vertices are connected : adjustable parameter dictating bias towards local connections
For any vertices u and v: define m(u,v) to be the number of common neighbors (so far)
Key quantity: the propensity R(u,v) of u to connect to v if m(u,v) >= k, R(u,v) = 1 (share too many friends not to connect) if m(u,v) = 0, R(u,v) = p (no mutual friends no bias to connect) else, R(u,v) = p + (m(u,v)/k)^ (1-p)
Generate new edges incrementally using R(u,v) as the edge probability; details omitted
Note: = infinity is “like” Erdos-Renyi (but not exactly)
The Watts and Strogatz Model
Proposed by Duncan J. Watts and Steven Strogatz in their joint 1998 Nature paper
A random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering
The model also became known as the (Watts) beta model after Watts used β to formulate it in his popular science book Six Degrees
The ER graphs fail to explain two important properties observed in real-world networks:
By assuming a constant and independent probability of two nodes being connected, they do not account for local clustering, i.e., having a low clustering coefficient
Do not account for the formation of hubs. Formally, the degree distribution of ER graphs converges to a Poisson distribution, rather than a power law observed in most real-world, scale-free networks
39
C(p) : clustering coeff. L(p) : average path length (Watts and Strogatz, Nature 393, 440 (1998))
Watts-Strogatz Model
40
Small Worlds and Occam’s Razor
For small , should generate large clustering coefficients we “programmed” the model to do so Watts claims that proving precise statements is hard…
But we do not want a new model for every little property Erdos-Renyi small diameter -model high clustering coefficient
In the interests of Occam’s Razor, we would like to find a single, simple model of network generation… … that simultaneously captures many properties
Watt’s small world: small diameter and high clustering
41
Discovered by Examining the Real World…
Watts examines three real networks as case studies: the Kevin Bacon graph the Western states power grid the C. elegans nervous system
For each of these networks, he: computes its size, diameter, and clustering coefficient compares diameter and clustering to best Erdos-Renyi approx. shows that the best -model approximation is better important to be “fair” to each model by finding best fit
Overall, if we care only about diameter and clustering, is better than
p
04/22/23 42
Case 1: Kevin Bacon Graph
Vertices: actors and actresses Edge between u and v if they appeared in a film together
Is Kevin Bacon the most
connected actor?
NO!
Rank NameAveragedistance
# ofmovies
# oflinks
1 Rod Steiger 2.537527 112 25622 Donald Pleasence 2.542376 180 28743 Martin Sheen 2.551210 136 35014 Christopher Lee 2.552497 201 29935 Robert Mitchum 2.557181 136 29056 Charlton Heston 2.566284 104 25527 Eddie Albert 2.567036 112 33338 Robert Vaughn 2.570193 126 27619 Donald Sutherland 2.577880 107 2865
10 John Gielgud 2.578980 122 294211 Anthony Quinn 2.579750 146 297812 James Earl Jones 2.584440 112 3787…
876 Kevin Bacon 2.786981 46 1811…
876 Kevin Bacon 2.786981 46 1811
Kevin Bacon
No. of movies : 46 No. of actors : 1811 Average separation: 2.79
43
Rod Steiger
Martin Sheen
Donald Pleasence
#1
#2
#3
#876Kevin Bacon
Bacon-
map
44
Case 2: New York State Power Grid
Vertices: generators and substations Edges: high-voltage power transmission lines and transformers Line thickness and color indicate the voltage level
Red 765 kV, 500 kV; brown 345 kV; green 230 kV; grey 138 kV
45
Case 3: C. Elegans Nervous System
Vertices: neurons in the C. elegans worm Edges: axons/synapses between neurons
46
Two More Examples
M. Newman on scientific collaboration networks coauthorship networks in several distinct communities differences in degrees (papers per author) empirical verification of
giant components small diameter (mean distance) high clustering coefficient
Alberich et al. on the Marvel Universe purely fictional social network two characters linked if they appeared together in an issue “empirical” verification of
heavy-tailed distribution of degrees (issues and characters) giant component rather small clustering coefficient
47
Towards Scale-Free Networks
Major limitation of the Watts-Strogatz model It produces graphs that are homogeneous in degree In contrast, real networks are often scale-free networks
inhomogeneous in degree, having hubs and a scale-free degree distribution. Such networks are better described by the preferential attachment family of models, such as the Barabási–Albert (BA) model
The Watts-Strogatz model also implies a fixed number of nodes and thus cannot be used to model network growth
The leads to the proposal of a new model: scale-free network, a network whose degree distribution follows a power law, at least asymptotically
48
Scale-free Networks: World Wide Web
800 million documents (S. Lawrence, 1999)
ROBOT: collects all URL’s found in a document and follows them recursively
Nodes: WWW documents Links: URL links
R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999)
49
k ~ 6
P(k=500) ~ 10-99
NWWW ~ 109
N(k=500)~10-90
Expected Result Real Result
Pout(k) ~ k-out
P(k=500) ~ 10-6
out= 2.45 in = 2.1
Pin(k) ~ k- in
NWWW ~ 109 N(k=500) ~ 103
J. Kleinberg, et. al, Proceedings of the ICCC (1999)
World Wide Web
50
< l
>
Finite size scaling: create a network with N nodes with Pin(k) and Pout(k)
< l > = 0.35 + 2.06 log(N)
l15=2 [125]
l17=4 [1346 7]
… < l > = ??
1
2
3
4
5
6
7
nd.edu
19 degrees of separation
R. Albert et al, Nature (99)
based on 800 million webpages [S. Lawrence et al Nature (99)]
A. Broder et al WWW9 (00)IBM
World Wide Web
51
What Does that Mean?
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network
52
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
53
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
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
54
Case1: Internet Backbone
(Faloutsos, Faloutsos and Faloutsos, 1999)
Nodes: computers, routers Links: physical lines
04/22/23 55
Internet-Map
56
Case2: Actor Connectivity
Nodes: actors Links: cast jointly
N = 212,250 actors k = 28.78
P(k) ~k-
Days of Thunder (1990) Far and Away
(1992) Eyes Wide Shut (1999)
=2.3
04/22/23 57
Case 3: Science Citation Index
( = 3)
Nodes: papers Links: citations
(S. Redner, 1998)
P(k) ~k-
2212
25
1736 PRL papers (1988)
Witten-SanderPRL 1981
04/22/23 58
Nodes: scientist (authors) Links: write paper together
(Newman, 2000, H. Jeong et al 2001)
Case 4: Science Coauthorship
59
Case 5: Food Web
Nodes: trophic species Links: trophic interactions
R.J. Williams, N.D. Martinez Nature (2000)R. Sole (cond-mat/0011195)
04/22/23 60
Robustness of Random vs. Scale-Free Networks
The accidental failure of a number of nodes in a random network can fracture the system into non-communicating islands.
Scale-free networks are more robust in the face of such failures
Scale-free networks are highly vulnerable to a coordinated attack against their hubs
04/22/23 61
protein-gene interactions
protein-protein interactions
PROTEOME
GENOME
Citrate Cycle
METABOLISM
Bio-chemical reactions
Bio-Map
04/22/23 62
Boehring-Mennheim
63
Nodes: proteins
Links: physical interactions (binding)
P. Uetz, et al., Nature 403, 623-7 (2000)
Prot Interaction Map: Yeast Protein Network
64
Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
65
Information on Social Network
Heterogeneous, multi-relational data represented as a graph or network Nodes are objects
May have different kinds of objects Objects have attributes Objects may have labels or classes
Edges are links May have different kinds of links Links may have attributes Links may be directed, are not required to be binary
Links represent relationships and interactions between objects - rich content for mining
Metrics (Measures) in Social Network Analysis (I)
Betweenness: The extent to which a node lies between other nodes in the network. This measure takes into account the connectivity of the node's neighbors, giving a higher value for nodes which bridge clusters. The measure reflects the number of people who a person is connecting indirectly through their direct links
Bridge: An edge is a bridge if deleting it would cause its endpoints to lie in different components of a graph.
Centrality: This measure gives a rough indication of the social power of a node based on how well they "connect" the network. "Betweenness", "Closeness", and "Degree" are all measures of centrality.
Centralization: The difference between the number of links for each node divided by maximum possible sum of differences. A centralized network will have many of its links dispersed around one or a few nodes, while a decentralized network is one in which there is little variation between the number of links each node possesses.
66
Metrics (Measures) in Social Network Analysis (II)
Closeness: The degree an individual is near all other individuals in a network (directly or indirectly). It reflects the ability to access information through the "grapevine" of network members. Thus, closeness is the inverse of the sum of the shortest distances between each individual and every other person in the network
Clustering coefficient: A measure of the likelihood that two associates of a node are associates themselves. A higher clustering coefficient indicates a greater 'cliquishness'.
Cohesion: The degree to which actors are connected directly to each other by cohesive bonds. Groups are identified as ‘cliques’ if every individual is directly tied to every other individual, ‘social circles’ if there is less stringency of direct contact, which is imprecise, or as structurally cohesive blocks if precision is wanted.
Degree (or geodesic distance): The count of the number of ties to other actors in the network.
67
Metrics (Measures) in Social Network Analysis (III)
(Individual-level) Density: The degree a respondent's ties know one another/ proportion of ties among an individual's nominees. Network or global-level density is the proportion of ties in a network relative to the total number possible (sparse versus dense networks).
Flow betweenness centrality: The degree that a node contributes to sum of maximum flow between all pairs of nodes (not that node).
Eigenvector centrality: A measure of the importance of a node in a network. It assigns relative scores to all nodes in the network based on the principle that connections to nodes having a high score contribute more to the score of the node in question.
Local Bridge: An edge is a local bridge if its endpoints share no common neighbors. Unlike a bridge, a local bridge is contained in a cycle.
Path Length: The distances between pairs of nodes in the network. Average path-length is the average of these distances between all pairs of nodes.
68
Metrics (Measures) in Social Network Analysis (IV)
Prestige: In a directed graph prestige is the term used to describe a node's centrality. "Degree Prestige", "Proximity Prestige", and "Status Prestige" are all measures of Prestige.
Radiality Degree: an individual’s network reaches out into the network and provides novel information and influence.
Reach: The degree any member of a network can reach other members of the network.
Structural cohesion: The minimum number of members who, if removed from a group, would disconnect the group
Structural equivalence: Refers to the extent to which nodes have a common set of linkages to other nodes in the system. The nodes don’t need to have any ties to each other to be structurally equivalent.
Structural hole: Static holes that can be strategically filled by connecting one or more links to link together other points. Linked to ideas of social capital: if you link to two people who are not linked you can control their communication
69
70
A Taxonomy of Common Link Mining Tasks
Object-Related Tasks Link-based object ranking Link-based object classification Object clustering (group detection) Object identification (entity resolution)
Link-Related Tasks Link prediction
Graph-Related Tasks Subgraph discovery Graph classification Generative model for graphs
71
Link-Based Object Ranking (LBR)
Exploit the link structure of a graph to order or prioritize the set of objects within the graph Focused on graphs with single object type and single link type
A primary focus of link analysis community Web information analysis
PageRank and Hits are typical LBR approaches In social network analysis (SNA), LBR is a core analysis task
Objective: rank individuals in terms of “centrality” Degree centrality vs. eigen vector/power centrality Rank objects relative to one or more relevant objects in the
graph vs. ranks object over time in dynamic graphs
72
PageRank: Capturing Page Popularity (Brin & Page’98)
Intuitions Links are like citations in literature A page that is cited often can be expected to be more
useful in general PageRank is essentially “citation counting”, but improves over
simple counting Consider “indirect citations” (being cited by a highly cited
paper counts a lot…) Smoothing of citations (every page is assumed to have a
non-zero citation count) PageRank can also be interpreted as random surfing (thus
capturing popularity)
73
The PageRank Algorithm (Brin & Page’98)
1( )
0 0 1/ 2 1/ 2
1 0 0 0
0 1 0 0
1/ 2 1/ 2 0 0
1( ) (1 ) ( ) ( )
1( ) [ (1 ) ] ( )
( (1 ) )
j i
t i ji t j t kd IN d k
i ki kk
T
M
p d m p d p dN
p d m p dN
p I M p
d1
d2
d4
“Transition matrix”
d3
Iterate until converge Essentially an eigenvector problem….
Same as/N (why?)
Stationary (“stable”) distribution, so we
ignore time
Random surfing model: At any page,
With prob. , randomly jumping to a pageWith prob. (1 – ), randomly picking a link to follow
Iij = 1/N
Initial value p(d)=1/N
74
HITS: Capturing Authorities & Hubs (Kleinberg’98)
Intuitions Pages that are widely cited are good authorities Pages that cite many other pages are good hubs
The key idea of HITS Good authorities are cited by good hubs Good hubs point to good authorities Iterative reinforcement …
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The HITS Algorithm (Kleinberg 98)
d1
d2
d4( )
( )
0 0 1 1
1 0 0 0
0 1 0 0
1 1 0 0
( ) ( )
( ) ( )
;
;
j i
j i
i jd OUT d
i jd IN d
T
T T
A
h d a d
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“Adjacency matrix”
d3
Again eigenvector problems…
Initial values: a=h=1
Iterate
Normalize: 2 2
( ) ( ) 1i ii i
a d h d
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Block-level Link Analysis (Cai et al. 04)
Most of the existing link analysis algorithms, e.g. PageRank and HITS, treat a web page as a single node in the web graph
However, in most cases, a web page contains multiple semantics and hence it might not be considered as an atomic and homogeneous node
Web page is partitioned into blocks using the vision-based page segmentation algorithm
extract page-to-block, block-to-page relationships Block-level PageRank and Block-level HITS
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Link-Based Object Classification (LBC)
Predicting the category of an object based on its attributes, its links and the attributes of linked objects
Web: Predict the category of a web page, based on words that occur on the page, links between pages, anchor text, html tags, etc.
Citation: Predict the topic of a paper, based on word occurrence, citations, co-citations
Epidemics: Predict disease type based on characteristics of the patients infected by the disease
Communication: Predict whether a communication contact is by email, phone call or mail
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Challenges in Link-Based Classification
Labels of related objects tend to be correlated Collective classification: Explore such correlations and jointly
infer the categorical values associated with the objects in the graph
Ex: Classify related news items in Reuter data sets (Chak’98) Simply incorp. words from neighboring documents: not
helpful Multi-relational classification is another solution for link-based
classification
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Group Detection
Cluster the nodes in the graph into groups that share common characteristics Web: identifying communities Citation: identifying research communities
Methods Hierarchical clustering Blockmodeling of SNA Spectral graph partitioning Stochastic blockmodeling Multi-relational clustering
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Entity Resolution
Predicting when two objects are the same, based on their attributes and their links
Also known as: deduplication, reference reconciliation, co-reference resolution, object consolidation
Applications Web: predict when two sites are mirrors of each other Citation: predicting when two citations are referring to the
same paper Epidemics: predicting when two disease strains are the
same Biology: learning when two names refer to the same
protein
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Entity Resolution Methods
Earlier viewed as pair-wise resolution problem: resolved based on the similarity of their attributes
Importance at considering links Coauthor links in bib data, hierarchical links between spatial
references, co-occurrence links between name references in documents
Use of links in resolution Collective entity resolution: one resolution decision affects
another if they are linked Propagating evidence over links in a depen. graph
Probabilistic models interact with different entity recognition decisions
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Link Prediction
Predict whether a link exists between two entities, based on attributes and other observed links
Applications Web: predict if there will be a link between two pages Citation: predicting if a paper will cite another paper Epidemics: predicting who a patient’s contacts are
Methods Often viewed as a binary classification problem Local conditional probability model, based on structural and
attribute features Difficulty: sparseness of existing links Collective prediction, e.g., Markov random field model
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Link Cardinality Estimation
Predicting the number of links to an object Web: predict the authority of a page based on the number
of in-links; identifying hubs based on the number of out-links Citation: predicting the impact of a paper based on the
number of citations Epidemics: predicting the number of people that will be
infected based on the infectiousness of a disease Predicting the number of objects reached along a path from an
object Web: predicting number of pages retrieved by crawling a
site Citation: predicting the number of citations of a particular
author in a specific journal
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Subgraph Discovery
Find characteristic subgraphs Focus of graph-based data mining
Applications Biology: protein structure discovery Communications: legitimate vs. illegitimate groups Chemistry: chemical substructure discovery
Methods Subgraph pattern mining
Graph classification Classification based on subgraph pattern analysis
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Metadata Mining
Schema mapping, schema discovery, schema reformulation
cite – matching between two bibliographic sources web - discovering schema from unstructured or semi-
structured data bio – mapping between two medical ontologies
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Link Mining Challenges
Logical vs. statistical dependencies Feature construction: Aggregation vs. selection Instances vs. classes Collective classification and collective consolidation Effective use of labeled & unlabeled data Link prediction Closed vs. open world
Challenges common to any link-based statistical model (Bayesian Logic Programs, Conditional Random Fields, Probabilistic Relational Models, Relational Markov Networks, Relational Probability Trees, Stochastic Logic Programming to name a few)
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Social Network Analysis
Social Networks: An Introduction
Primitives for Network Analysis
Different Network Distributions
Models of Social Network Generation
Mining on Social Network
Summary
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Ref: Mining on Social Networks
D. Liben-Nowell and J. Kleinberg. The Link Prediction Problem for Social Networks. CIKM’03
P. Domingos and M. Richardson, Mining the Network Value of Customers. KDD’01
M. Richardson and P. Domingos, Mining Knowledge-Sharing Sites for Viral Marketing. KDD’02
D. Kempe, J. Kleinberg, and E. Tardos, Maximizing the Spread of Influence through a Social Network. KDD’03.
P. Domingos, Mining Social Networks for Viral Marketing. IEEE Intelligent Systems, 20(1), 80-82, 2005.
S. Brin and L. Page, The anatomy of a large scale hypertextual Web search engine. WWW7.
S. Chakrabarti, B. Dom, D. Gibson, J. Kleinberg, S.R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins, Mining the link structure of the World Wide Web. IEEE Computer’99
D. Cai, X. He, J. Wen, and W. Ma, Block-level Link Analysis. SIGIR'2004. Lecture notes from Lise Getoor’s website: www.cs.umd.edu/~getoor/
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