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Data Mining for Complex Network Introduction and Background.
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Transcript of Data Mining for Complex Network Introduction and Background.
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Data Mining for Complex Network
Introduction and Background
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Welcome!
Instructor: Ruoming Jin Homepage: www.cs.kent.edu/~jin/ Office: 264 MCS Building Email: [email protected]
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Course overview
The course goal First of all, this is a research course, or a
special topic course. There is no textbook and even no definition what topics are supposed to under the course name
Discussing the state-of-are techniques for mining complex networks
Review & Course Project
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Simple Concepts (Review)
Erdős–Rényi model Random Graph Model
Markov Chain and Random Walk Maximal Likelihood/Model Selection
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Requirement
Each of you will have one presentation Review Paper
Select a topic and review at least four papers Bonus: Develop a new idea on this topic
Course Project One or Two a group Collect data; preprocess data; analyzing the
data;
Final Grade: 30% presentations, 25% review, 35% project, and 10% class participation
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What is a network?
Network: a collection of entities that are interconnected with links. people that are friends computers that are interconnected web pages that point to each other proteins that interact
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Graphs
In mathematics, networks are called graphs, the entities are nodes, and the links are edges
Graph theory starts in the 18th century, with Leonhard Euler The problem of Königsberg bridges Since then graphs have been studied
extensively.
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Networks in the past
Graphs have been used in the past to model existing networks (e.g., networks of highways, social networks) usually these networks were small network can be studied visual inspection
can reveal a lot of information
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Networks now
More and larger networks appear Products of technological advancement
• e.g., Internet, Web
Result of our ability to collect more, better, and more complex data• e.g., gene regulatory networks
Networks of thousands, millions, or billions of nodes impossible to visualize
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The internet map
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Understanding large graphs
What are the statistics of real life networks?
Can we explain how the networks were generated?
What else? A still very young field! (What is the basic principles and
what those principle will mean?)
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Measuring network properties
Around 1999 Watts and Strogatz, Dynamics and
small-world phenomenon Faloutsos3, On power-law relationships
of the Internet Topology Kleinberg et al., The Web as a graph Barabasi and Albert, The emergence of
scaling in real networks
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Real network properties
Most nodes have only a small number of neighbors (degree), but there are some nodes with very high degree (power-law degree distribution) scale-free networks
If a node x is connected to y and z, then y and z are likely to be connected high clustering coefficient
Most nodes are just a few edges away on average. small world networks
Networks from very diverse areas (from internet to biological networks) have similar properties Is it possible that there is a unifying underlying generative
process?
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Generating random graphs
Classic graph theory model (Erdös-Renyi) each edge is generated independently with
probability p
Very well studied model but: most vertices have about the same degree the probability of two nodes being linked is
independent of whether they share a neighbor the average paths are short
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Modeling real networks
Real life networks are not “random” Can we define a model that
generates graphs with statistical properties similar to those in real life? a flurry of models for random graphs
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Processes on networks
Why is it important to understand the structure of networks?
Epidemiology: Viruses propagate much faster in scale-free networks
Vaccination of random nodes does not work, but targeted vaccination is very effective
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The future of networks
Networks seem to be here to stay More and more systems are modeled as
networks Scientists from various disciplines are
working on networks (physicists, computer scientists, mathematicians, biologists, sociologist, economists)
There are many questions to understand.
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Basic Mathematical Tools
Graph theory Probability theory Linear Algebra
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Graph Theory
Graph G=(V,E) V = set of vertices E = set of edges
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undirected graphE={(1,2),(1,3),(2,3),(3,4),(4,5)}
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Graph Theory
Graph G=(V,E) V = set of vertices E = set of edges
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directed graphE={‹1,2›, ‹2,1› ‹1,3›, ‹3,2›, ‹3,4›, ‹4,5›}
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Undirected graph
1
2
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degree d(i) of node i number of edges
incident on node i
degree sequence [d(1),d(2),d(3),d(4),d(5)] [2,2,2,1,1]
degree distribution [(1,2),(2,3)]
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Directed Graph
1
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in-degree din(i) of node i number of edges pointing
to node i
out-degree dout(i) of node i number of edges leaving
node i in-degree sequence
[1,2,1,1,1]
out-degree sequence [2,1,2,1,0]
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Paths
Path from node i to node j: a sequence of edges (directed or undirected from node i to node j) path length: number of edges on the path nodes i and j are connected cycle: a path that starts and ends at the same node
1
2
3
45
1
2
3
45
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Shortest Paths
Shortest Path from node i to node j also known as BFS path, or geodesic
path
1
2
3
45
1
2
3
45
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Diameter
The longest shortest path in the graph
1
2
3
45
1
2
3
45
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Undirected graph
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Connected graph: a graph where there every pair of nodes is connected
Disconnected graph: a graph that is not connected
Connected Components: subsets of vertices that are connected
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Fully Connected Graph
Clique Kn
A graph that has all possible n(n-1)/2 edges
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2
3
45
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Directed Graph
1
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Strongly connected graph: there exists a path from every i to every j
Weakly connected graph: If edges are made to be undirected the graph is connected
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Subgraphs
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Subgraph: Given V’ V, and E’ E, the graph G’=(V’,E’) is a subgraph of G.
Induced subgraph: Given V’ V, let E’ E is the set of all edges between the nodes in V’. The graph G’=(V’,E’), is an induced subgraph of G
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Trees
Connected Undirected graphs without cycles
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2
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45
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Bipartite graphs
Graphs where the set V can be partitioned into two sets L and R, such that all edges are between nodes in L and R, and there is no edge within L or R
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Linear Algebra
Adjacency Matrix symmetric matrix for undirected graphs
1
2
3
45
01000
10100
01011
00101
00110
A
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Linear Algebra
Adjacency Matrix unsymmetric matrix for undirected
graphs
00000
10000
01010
00001
00110
A 1
2
3
45
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Random Walks
Start from a node, and follow links uniformly at random.
Stationary distribution: The fraction of times that you visit node i, as the number of steps of the random walk approaches infinity if the graph is strongly connected, the
stationary distribution converges to a unique vector.
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Random Walks
stationary distribution: principal left eigenvector of the normalized adjacency matrix x = xP for undirected graphs, the degree distribution
1
2
3
45
00001
10000
0210210
00001
0021210
P
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Eigenvalues and Eigenvectors
The value λ is an eigenvalue of matrix A if there exists a non-zero vector x, such that Ax=λx. Vector x is an eigenvector of matrix A The largest eigenvalue is called the
principal eigenvalue The corresponding eigenvector is the
principal eigenvector Corresponds to the direction of
maximum change
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Types of networks
Social networks Knowledge (Information) networks Technology networks Biological networks
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Social Networks
Links denote a social interaction Networks of acquaintances actor networks co-authorship networks director networks phone-call networks e-mail networks IM networks
• Microsoft buddy network Bluetooth networks sexual networks home page networks
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Knowledge (Information) Networks
Nodes store information, links associate information Citation network (directed acyclic) The Web (directed) Peer-to-Peer networks Word networks Networks of Trust Bluetooth networks
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Technological networks
Networks built for distribution of commodity The Internet
• router level, AS level
Power Grids Airline networks Telephone networks Transportation Networks
• roads, railways, pedestrian traffic
Software graphs
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Biological networks
Biological systems represented as networks Protein-Protein Interaction Networks Gene regulation networks Metabolic pathways The Food Web Neural Networks
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Now what?
The world is full with networks. What do we do with them? understand their topology and measure
their properties study their evolution and dynamics create realistic models create algorithms that make use of the
network structure
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Measuring Networks
Degree distributions Small world phenomena Clustering Coefficient Mixing patterns Degree correlations Communities and clusters
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Degree distributions
Problem: find the probability distribution that best fits the observed data
degree
frequency
k
fk
fk = fraction of nodes with degree k = probability of a randomly selected node to have degree k
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Power-law distributions
The degree distributions of most real-life networks follow a power law
Right-skewed/Heavy-tail distribution there is a non-negligible fraction of nodes that has very
high degree (hubs) scale-free: no characteristic scale, average is not
informative
In stark contrast with the random graph model! highly concentrated around the mean the probability of very high degree nodes is
exponentially small
p(k) = Ck-α
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Power-law signature
Power-law distribution gives a line in the log-log plot
α : power-law exponent (typically 2 ≤ α ≤ 3)
degree
frequency
log degree
log frequency α
log p(k) = -α logk + logC
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Examples
Taken from [Newman 2003]
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A random graph example
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Maximum degree
For random graphs, the maximum degree is highly concentrated around the average degree z
For power law graphs
Rough argument: solve nP[X≥k]=1
1)1/(αmax nk
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Exponential distribution
Observed in some technological or collaboration networks
Identified by a line in the log-linear plot
p(k) = λe-λk
log p(k) = - λk + log λ
degree
log frequency λ
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Collective Statistics (M. Newman 2003)
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Clustering (Transitivity) coefficient
Measures the density of triangles (local clusters) in the graph
Two different ways to measure it:
The ratio of the means
i
i(1)
i nodeat centered triples
i nodeat centered trianglesC
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Example
1
2
3
4
583
6113
C(1)
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Clustering (Transitivity) coefficient
Clustering coefficient for node i
The mean of the ratios
i nodeat centered triplesi nodeat centered triangles
Ci
i(2) C
n1
C
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Example
The two clustering coefficients give different measures
C(2) increases with nodes with low degree
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4
5
3013
611151
C(2)
83
C(1)
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Collective Statistics (M. Newman 2003)
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Clustering coefficient for random graphs
The probability of two of your neighbors also being neighbors is p, independent of local structure clustering coefficient C = p when z is fixed C = z/n =O(1/n)
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Small world phenomena
Small worlds: networks with short paths
Obedience to authority (1963)
Small world experiment (1967)
Stanley Milgram (1933-1984): “The man who shocked the world”
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Small world experiment
Letters were handed out to people in Nebraska to be sent to a target in Boston
People were instructed to pass on the letters to someone they knew on first-name basis
The letters that reached the destination followed paths of length around 6
Six degrees of separation: (play of John Guare)
Also: The Kevin Bacon game The Erdös number
Small world project: http://smallworld.columbia.edu/index.html
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Measuring the small world phenomenon
dij = shortest path between i and j Diameter:
Characteristic path length:
Harmonic mean
ijji,
dmaxd
ji
ijd1)/2-n(n
1
ji
1-ij
d1)/2-n(n
11
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Collective Statistics (M. Newman 2003)
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Mixing patterns
Assume that we have various types of nodes. What is the probability that two nodes of different type are linked? assortative mixing (homophily)
E : mixing matrix
p(i,j) = mixing probability
ji,
j)E(i,j)E(i,
j)p(i,
p(j | i) = conditional mixing probability
j
j)E(i,j)E(i,
i)|p(j
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Mixing coefficient
Gupta, Anderson, May 1989
Advantages: Q=1 if the matrix is diagonal Q=0 if the matrix is uniform
Disadvantages sensitive to transposition does not weight the entries
1N
1-i)|p(iQ i
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Mixing coefficient
Newman 2003
Advantages: r = 1 for diagonal matrix , r = 0 for uniform matrix not sensitive to transposition, accounts for
weighting
i
i i
a(i)b(i)1
a(i)b(i)-i)|p(ir
j
j)p(i,a(i)
j
i)p(j,b(i)
(row marginal)
(column marginal)
r=0.621 Q=0.528
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Degree correlations
Do high degree nodes tend to link to high degree nodes?
Pastor Satoras et al. plot the mean degree of the neighbors
as a function of the degree Newman
compute the correlation coefficient of the degrees of the two endpoints of an edge
assortative/disassortative
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Collective Statistics (M. Newman 2003)
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Communities and Clusters
Use the graph structure to discover communities of nodes essentially clustering and classification
on graphs
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Other measures
Frequent (or interesting) motifs bipartite cliques in the web graph patterns in biological and software
graphs
Use graphlets to compare models [Przulj,Corneil,Jurisica 2004]
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Other measures
Network resilience against random or
targeted node deletions
Graph eigenvalues
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Other measures
The giant component
Other?
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References
M. E. J. Newman, The structure and function of complex networks, SIAM Reviews, 45(2): 167-256, 2003
M. E. J. Newman, Random graphs as models of networks in Handbook of Graphs and Networks, S. Bornholdt and H. G. Schuster (eds.), Wiley-VCH, Berlin (2003).
N. Alon J. Spencer, The Probabilistic Method