1. Modelling and Searching Networks Lecture 2 – Complex Networks
Miniconference on the Mathematics of Computation
MTH 707
Modelling and Searching Networks Lecture 2 – Complex Networks
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, bitcoin 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. Bitcoin graph
nodes: users
edges: transactions
or protocols

12. 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

13. 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

14. 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

15. 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

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

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

18. 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

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

20. Power laws in OSNs
Complex Networks

21. Exercise 3.1 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.

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

24. 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

25. Properties of Complex Networks
Small world property
small world networks introduced by Watts and 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. Nuit
Blanche
Ryerson
City of
Toronto
Four
Seasons
Hotel
Frommer’s
Greenland
Tourism

27. Sample data: Flickr, YouTube, LiveJournal, Orkut
(Mislove et al,07): short average distances and high clustering coefficients
Complex Networks

28. 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

29. Densification – Physics Citations
1.69
Complex Networks

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

31. 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

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

33. 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) …

34. Exercise
3.2 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.