1. Week 3 - Complex Networks and their Properties
Miniconference on the Mathematics of Computation
AM8002
Fall 2014
Week 3 - Complex Networks and their Properties
Dr. Anthony Bonato
Ryerson University

2. Complex Networks
web graph, social networks, biological networks, internet networks, …
Networks - Bonato

3. What is a complex network?
no precise definition
however, there is general consensus on the following observed properties
large scale
evolving over time
power law degree distributions
small world properties
other properties depend on the kind of network being discussed

4. Examples of complex networks
technological/informational: web graph, router graph, AS graph, call graph, graph
social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph
biological networks: protein interaction networks, gene regulatory networks, food networks

5. Example: the web graph nodes: web pages edges: links
one of the first complex networks to be analyzed
viewed as directed or undirected
Networks - Bonato

6. Example: On-line Social Networks (OSNs)
nodes: users on some OSN
edges: friendship (or following) links
maybe directed or undirected
Anthony Bonato - The web graph

7. Example: Co-author graph
nodes: mathematicians and scientists
edges: co-authorship
undirected

8. Example: Co-actor graph
nodes: actors
edges: co-stars
Hollywood graph
undirected

9. Heirarchical social networks
social networks which are oriented from top to bottom
information flows one way
examples: Twitter, executives in a company, terrorist networks

10. Example: protein interaction networks
nodes: proteins in a living cell
edges: biochemical interaction
undirected
Introducing the Web Graph - Anthony Bonato

11. Properties of complex networks
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

12. Properties of complex networks
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

13. Properties of Complex Networks
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
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)
Complex Networks

15. Interpreting a power law
Many low-degree nodes
Few high-degree nodes
Complex Networks

16. Binomial Power law Highway network Air traffic network
Complex Networks

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

18. Degree distribution (log-log plot) of a power law graph
Complex Networks

19. Power laws in OSNs
Complex Networks

20. Discussion Which of the following are power law graphs?
High school/secondary school graph. Nodes: students in a high school; edges: friendship links.
Power grids. Nodes: generators, power plants, large consumers of power; edges: electrical cable.
Banking networks. Nodes: banks; edges: financial transaction.

22. Graph parameters Wiener index, W(G) average distance:
clustering coefficient:
Wiener index, W(G)
Complex Networks

23. 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

24. Properties of Complex Networks
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

25. Nuit
Blanche
Ryerson
City of
Toronto
Four
Seasons
Hotel
Frommer’s
Greenland
Tourism

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:
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

28. Densification – Physics Citations
1.69
Complex Networks

29. Densification – Autonomous Systems
e(t)
1.18
n(t)
Complex Networks

30. Decreasing distances (Leskovec, Kleinberg, Faloutsos,05):
distances (diameter and/or average distances) decrease with time
(Kumar et al,06):
Diameter first, DPL second
Check diameter formulas
As the network grows the distances between nodes slowly grow
Complex Networks

31. Diameter – ArXiv citation graph
time [years]
Complex Networks

32. 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) …

33. Discussion
Compute the average distance of each of the following graphs.
A star with n nodes (i.e. a tree of order n with one vertex of order n-1, the rest degree 1)
A path with n nodes
A wheel with n+1 nodes, n>2.

35. Web Search
the web contains large amounts of information (≈ 4 zettabytes = 1021 bytes)
rely on web search engines, such as Google, Yahoo! Search, Bing, …

36. 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

37. Indexing
the search engine then indexes the information crawled from the web, storing and sorting it

38. User interface
users type in queries and get back a sorted list of web pages and links

39. Key questions How do search engines choose their rankings?
What makes modern search engines more accurate than the first search engines?
What does math have to do with it?

40. Challenges of web search
Massive size.
Multimedia.
Authorities.

41. 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!

42. Analogy: evil librarian
you are looking for a book on baseball in a library
evil librarian spends her time moving books to fool you

43. Then came

44. Google uses graph theory!
Google founders: Larry Page, Sergey Brin

45. Pagerank is the probability a random
surfer visits a page
PageRank models web surfing via a
random walk
surfer usually
moves via out-links
on occasion, the
surfer teleports to a
random page

46. 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

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

49. Example

50. Example, continued

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+1T = ztTP = …= z0TPt
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)

54. Example, continued
PageRank vector: