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Topics in Applied Graph Theory – Modeling and Searching Networks Lecture 2 - Complex Networks
and their Properties Dr. Anthony Bonato
Ryerson University
Hamilton Institute - NUI May 2013
Networks - Bonato 2
Complex Networks • web graph, social networks, biological networks, internet
networks, …
What is a complex network? • no precise definition • however, there is general consensus on the
following observed properties 1. large scale 2. evolving over time 3. power law degree distributions 4. small world properties
• other properties depend on the kind of network being discussed
3
Examples of complex networks • technological/informational: web graph, router
graph, AS graph, call graph, e-mail graph
• social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph
• biological networks: protein interaction networks, gene regulatory networks, food networks
4
Networks - Bonato 5
Example: the web graph
• nodes: web pages • edges: links • one of the first
complex networks to be analyzed
• viewed as directed or undirected
Anthony Bonato - The web graph 6
Example: On-line Social Networks (OSNs)
• nodes: users on some OSN
• edges: friendship (or following) links
• maybe directed or undirected
Example: Co-author graph
7
• nodes: mathematicians and scientists
• edges: co-authorship
• undirected
Heirarchical social networks
9
• social networks which are oriented from top to bottom • information flows
one way • examples: Twitter,
executives in a company, terrorist networks
Introducing the Web Graph - Anthony Bonato
10
Example: protein interaction networks
• nodes: proteins in a
living cell • edges: biochemical
interaction
• undirected
Properties of complex networks
1. Large scale: relative to order and size
• web graph: order > trillion – some sense infinite: number of strings entered into
Google • Facebook: > 1 billion nodes; Twitter: > 500 million
nodes – much denser (ie higher average degree) than the
web graph • protein interaction networks: order in thousands
11
Properties of complex networks
2. Evolving: networks change over time
• web graph: billions of nodes and links appear and disappear each day
• Facebook: grew to 1 billion users – denser than the web graph
• protein interaction networks: order in the thousands
– evolves much more slowly
12
Complex Networks 13
Properties of Complex Networks 3. Power law degree distribution
• for a graph G of order n and i a positive integer, let Ni,n
denote the number of nodes of degree i in G
• we say that G follows a power law degree distribution if for some range of i and some b > 2, • b is called the exponent of the power law
niN bni
−≈,
Complex Networks 14
Properties of Complex Networks • power law degree distribution in the web
graph:
(Broder et al, 01) reported an exponent b = 2.1 for the in-degree distribution (in a 200 million vertex crawl)
17
Notes on power laws
• b is the exponent of the power law • note that the law is
– approximate: constants do not affect it – asymptotic: holds only for large n – may not hold for all degrees, but most
degrees (for example, sufficiently large or sufficiently small degrees)
Complex Networks
Discussion
Which of the following are power law graphs?
1. High school/secondary school graph. Nodes: students
in a high school; edges: friendship links. 2. Power grids. Nodes: generators, power plants, large
consumers of power; edges: electrical cable. 3. Banking networks. Nodes: banks; edges: financial
transaction.
20
Complex Networks 22
Graph parameters
• average distance:
• clustering coefficient:
∑∈
−
=
)(,
1
2),()(
GVvu
nvudGL
∑=
=
∈
−
)(
1-1
)()( ,2
)deg(|))((| )(
GVxxcnGC
xxNExc
Wiener index, W(G)
Examples • Cliques have average distance 1, and clustering
coefficient 1 • Triangle-free graphs have clustering coefficient 0 • Clustering coefficient of following graph is 0.75.
• Note: average distance bounded above by diameter
23
Complex Networks 24
Properties of Complex Networks 4. Small world property
• small world networks introduced by social
scientists Watts & Strogatz in 1998 – low distances
• diam(G) = O(log n) • L(G) = O(loglog n)
– higher clustering coefficient than random graph with same expected degree
Complex Networks 26
Sample data: Flickr, YouTube, LiveJournal, Orkut
• (Mislove et al,07): short average distances and high clustering coefficients
Complex Networks 27
Other properties of complex networks – many complex networks (including on-line
social networks) obey two additional laws: 1. Densification Power Law (Leskovec,
Kleinberg, Faloutsos,05): – networks are becoming more dense over
time; i.e. average degree is increasing |(E(Gt)| ≈ |V(Gt)|a
where 1 < a ≤ 2: densification exponent
Complex Networks 30
2. Decreasing distances (Leskovec, Kleinberg,
Faloutsos,05): • distances (diameter and/or average distances)
decrease with time
(Kumar et al,06):
Other properties • Connected component structure: emergence of
components; giant components
• Spectral properties: adjacency matrix and Laplacian matrices, spectral gap, eigenvalue distribution
• Small community phenomenon: most nodes belong to small communities (ie subgraphs with more internal than external links)
…
32
Discussion
Compute the average distance of each of the following graphs.
1. A star with n nodes (i.e. a tree of order n
with one vertex of order n-1, the rest degree 1)
2. A path with n nodes 3. A wheel with n nodes, n>2.
33
Web Search
• the web contains large amounts of information (≈ 0.5 zettabyte = 1021 bytes) – rely on web search engines, such as Google,
Yahoo! Search, Bing, …
35
Search Engines
• search engines are tools designed to hunt for information on the web
• they do this by first crawling the web by making copies of pages and their links
36
Indexing
• the search engine then indexes the information crawled from the web, storing and sorting it
37
Key questions
1. How do search engines choose their rankings?
2. What makes modern search engines more accurate than the first search engines?
3. What does math have to do with it?
39
Text based search • first search engines ranked
pages using word frequency – eg: if “baseball’’ appears
many times on page X, then X is ranked higher on a search for “baseball’’
• easily spammed: insert “baseball” 100s of times on page!
41
Analogy: evil librarian
42
• you are looking for a book on baseball in a library
• evil librarian spends her time moving books to fool you
45
• PageRank models web surfing via a random walk
• surfer usually moves via out-links
• on occasion, the surfer teleports to a random page
• Pagerank is the probability a random surfer visits a page
How PageRank addresses the challenges of web search
• PageRank can be computed quickly, even for large matrices
• PageRank relies only on the link structure – popular pages are those with many in-links, or
linked to other popular pages • “authorities” have higher PageRank
46
47
Google random walk
• this modification of the usual random walk is called the Google random walk
• note that it takes place on a directed graph
48
The Google Matrix • given a digraph G with nodes {1,…,n}, define the matrix P1
• form P2 by replacing any zero rows of P1 by 1/nJ1,n
• define the Google matrix P as
-c in (0,1) is the teleportation constant
51
Motivation
• P1 corresponds to the random walk using out-links
• P2 takes care of spider traps: nodes with zero out-degree
• P(G) adds in the teleportation: – 85% of the time follow out-links, 15% of the
time use jump to a new node chosen at random from all nodes
52
PageRank defined Theorem (Brin, Page, 2000) The Google random
walk converges to a stationary distribution s, which is the dominant eigenvector of P(G).
That is, the PageRank vector s solves the linear
system:
P(G)s = s.
53
Power method • for a fixed integer n > 0, let z0 be the stochastic vector
whose every entry is 1/n
• define zt+1
T = ztTP = …= z0
TPt
Lemma 6 (Power Method): The limit of the sequence of (zt : t ≥ 0) is the dominant eigenvector. • gives a simple method of computing Pagerank: multiply
by powers of P(G)