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Data mining --- mining graphs
CAP5771.1University of South Florida
Xiaoning Qian
Xiaoning Qian ([email protected]) 10/13/11
Today’s Lecture
1. Complex networks
2. Graph representation for networks
3. Markov chain
4. Viral propagation
5. Google’s PageRank
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1. M Faloutsos, P Faloutsos, C Faloutsos, On power-lawrelationships of the Internet topology. Comput. Commun. Rev.29:251-262, 1999
2. JM Kleinberg, Navigation in a small world—it is easier to findshort chains between points in some networks than others. Nature,406:845, 2000
3. AL Barabasi, Linked: The New Science of Networks. Cambridge,MA, 2002
4. DJ Watts, The “new” science of networks. Annu. Rev. Sociol.30:243–70, 2004
5. DL Alderson, Catching the “network science” bug: Insight andopportunity for the operations researcher. Operations Research56(5): 1047-1065, 2008
6. MEJ Newman, Networks: An Introduction. Oxford, 2010
Xiaoning Qian ([email protected]) 10/13/11
“New” Science of Networks
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The first network/graph problem Find a tour crossing every bridge just once, Euler, 1735
Xiaoning Qian ([email protected]) 10/13/11
Network Science
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Bridges of Königsberg
1. S Milgram, The small world problem. Psychol.Today 2:60–67, 1967
“New” Science Unprecedented number of empirical networks Much larger scale networks Visualization does not convey enough information
Computer are much more powerful Highly interdisciplinary
Xiaoning Qian ([email protected]) 10/13/11
Network Science
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Mining networks/graphs
Topology/structure of complex networks Global: degrees, centrality, connectivity, etc.
Scale-free (power-law) networks: 6 degree separation?
Local: clustering (community), network motifs, etc.
Dynamics/behavior of complex networks Global: the topological effect on dynamics
How information, virus, disease, rumors, etc. propagate?
Local: how individual nodes behave
Xiaoning Qian ([email protected]) 10/13/11
Network Science
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Xiaoning Qian ([email protected]) 10/13/11
Mathematics of Networks (Graphs)
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What is a network/graph? A collection of vertices/nodes joined by edges Different types of vertices and edges:
Directed vs. Undirected Weighed vs. Binary Labeled vs. Nonlabeled Bipartite graphs Hypergraphs
Mathematically,G = {V, E}
Xiaoning Qian ([email protected]) 10/13/11
Mathematics of Networks (Graphs)
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Undirected network: Undirected network: <v<vii, , vvjj> > ∈∈ E => < E => < vvjj, v, vii > > ∈∈ E E
Xiaoning Qian ([email protected]) 10/13/11
Mathematics of Networks (Graphs)
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Adjacency matrix L Symmetric for undirected graphs Square matrix for (self-)graphs; rectangular for bipartite
graphsLijij = eijij if <v<vii, , vvjj> > ∈∈ E E
Matrix analysis for graph mining!
Simple graphs, connected graphs, complete graphs, …
Xiaoning Qian ([email protected]) 10/13/11
Mathematics of Networks (Graphs)
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Node degree cii
The number of edges incident with vertex vvii
Neighbor set Input-, output-degrees Degree distributions (power-law) …
Trail (distinct edges), path (distinct nodes), cycle, cut, …
Xiaoning Qian ([email protected]) 10/13/11
What is a Markov chain?
Finite Markov chain -- (Q, P) Q = {q1, q2, …, qs} : a finite set of states P : state transition probability matrix Given a sequence of observations: The probability of the sequence is:
For first-order time-homogeneous Markov chain:
Hence,
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Finite Markov chain -- (Q, P) Q = {B, q1, q2, …, qs} : a finite set of states P : state transition probability matrix
initial state probability:
The probability of a sequence can be expressed with P:
Note: The output are states at each time -- states areobservable!!
Xiaoning Qian ([email protected]) 10/13/11
What is a Markov chain?
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3-state Markov chain model for the weather:
Q = {Rain (or snow), Cloudy, Sunny};
P is given in the figure; Initial state probability
RR
CC
SS
Xiaoning Qian ([email protected]) 10/13/11
An example
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Chapman-Kolmogorov equationp(xn) = P(n-1) p(x1)
Limiting distribution (stationary/steady-state distribution) Irreducibility, Periodicity, Ergocity
p = P p
How to solve p? Eigen-decomposition of P Power method
Xiaoning Qian ([email protected]) 10/13/11
Chapman-Kolmogorov Equation
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Random walk on graphs (network diffusion) is a Markovprocess.
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Random walk on graphs
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The algorithm of Google---PageRank
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What’s behind Google?
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Xiaoning Qian ([email protected]) 10/13/11
PageRank
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What is an important Webpage?There are many Webpages pointing to it
Important Webpage point to more important Webpage
Importance diffuses based on links between Webpages
Vertices: Webpages; Edges: hyperlinks;
HITS: JM Kleinberg
Xiaoning Qian ([email protected]) 10/13/11
PageRank
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Diffusion (Random walk) on Web
pi: importance for page i; Lij: link from page j to i;
λ λ
Hence, the problem becomes a Markov chain problem(diffusion process):
λ λ
Xiaoning Qian ([email protected]) 10/13/11
PageRank
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Diffusion (Random walk with restart) on Web
pi: importance for page i; Lij: link from page j to i;
λ λ
Xiaoning Qian ([email protected]) 10/13/11
PageRank
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Diffusion (Random walk with restart) on Web
Matrix form:
λ λ
PseudocountDiffusion factor
λ λ
Xiaoning Qian ([email protected]) 10/13/11
PageRank
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How do we solve this?λ λ
Note that p is simply for ranking and the absolutevalues are not critical! WLOG, we assume
Hence, the problem becomes a Markov chain problem(diffusion process):
λ λ
Xiaoning Qian ([email protected]) 10/13/11
Viral propagation
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How does the virus spread over the network? Will it become an “epidemic” outbreak? How fast the virus will die out or become “epidemic”? How we should design “robust” networks to prevent
cascading failures?
* A Ganesh et. al., The effect of network topology on the spread of epidemics. INFOCOM, 2005
* D Chakrabati, Tools for large graph mining. Ph.D. Thesis, CMU, 2005
Xiaoning Qian ([email protected]) 10/13/11
Mathematical Epidemiology
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SIR (Susceptible-Infective-Recovered) model
SIS (Susceptible-Infective-Susceptible) model Catching the disease from Infective neighbors (birth rate): β
Recover rate: δ
Epidemic threshold: τ
Xiaoning Qian ([email protected]) 10/13/11
SIS model
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????
Sum and Product rules in probability!!
SIS model is again a Markov process!
Xiaoning Qian ([email protected]) 10/13/11
SIS model
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Sum and Product rules in probability!!
Xiaoning Qian ([email protected]) 10/13/11
SIS model
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Sum and Product rules in probability!!
With appropriate approximations, we can derivep(vi
t=susceptible) = p(vit-1=susceptible) ζi + p(vi
t-1=infective) δ
1-p(vit) = [1-p(vi
t-1)] ζi + p(vit-1) δ
and
pt =[ βL + (1-δ)I ] pt-1
Xiaoning Qian ([email protected]) 10/13/11
SIS model
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With appropriate approximations, we can derivept
=[ βL + (1-δ)I ] pt-1
Eigen-decomposition of the matrix S= [ βL + (1-δ)I ]
LL
pp pp 00
LLHence,
Xiaoning Qian ([email protected]) 10/13/11
SIS model
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With appropriate approximations, we can derivept
=[ βL + (1-δ)I ] pt-1
Eigen-decomposition of the matrix S= [ βL + (1-δ)I ]
Epidemic threshold:
LLHence,
LL
Networks/graphs are everywhere and require new toolsto study them efficiently and effectively.
Random walk (Markov chain) on graphs and itsextension can be a useful technique to “mine” complexnetworks/graphs PageRank Viral propagation
Have you learned anything? :)
I am teaching Biological Network Analysis, Spring 2012.
Xiaoning Qian ([email protected]) 10/13/11
Summary
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