This document provides an introduction to networks. It discusses how networks are used to model relationships between entities in various domains, including social networks, protein interactions, and infrastructure networks. It also describes some key concepts in network analysis, such as degree distribution, shortest paths, centrality measures, and topological properties like the small-world and scale-free networks commonly seen in real-world systems.

1. An Introduction to Networks
Francesco Gadaleta, PhD.

2. Networks are around us

3. Spreading consensus
The model
• few peers spreading a message
(advertising)
• others sharing to their friends
(if they don’t already know)

8. The network of relationships

9. Economy network

10. Protein-protein interaction network

11. Political network

12. Not-so-recent graph of the Internet

13. Solving the problem of DNA sequencing
(c) 2014 www.worldofpiggy.com
Definition:
Each read is an edge
Nodes are prefix and suffix of the
string that connects them
Solution:
Find a cycle in such a graph: reading the
superstring that contains all reads with
maximum overlap.
Hey! That’s an
Eulerian cycle

14. Being a freeloader with networks
Homeless
Visit a place
Doesn’t repeat a node

15. Networks today
• Social relationships
• Professional networks (boss, employees)
• Power grids
• Internet
• Biology (cells, genes, proteins, diseases…)

16. Power grids

18. Facebook at 10am

19. Graph Theory
(Mathematics)
Social Network Analysis
1920
economic transactions
trades among nations
communications between
groups

20. Complexity of networks
• irregular structure
• evolution in time
• dimension

21. Complexity of networks
• irregular structure
• evolution in time
• dimension
time = t0 time = t

22. Nature(1998) Small-world networks
Watts D., Strogatz S.

23. Science(1999) Scale-free networks
Barabasi, Albert

24. Topology
Related to the structure of the network
eg. how nodes are connected

25. Topology: Modules
Subnetworks with specific properties

26. Some definitions
graph:
N nodes
E edges
directed
undirected

27. Neighbours of node i
(of order k) neigh(i,k)
neigh(3,1) = ?
neigh(2,2) = ?
{2,4}
{1,3,4,5}
Example

28. Reachability
of two nodes i and j
walk: alternating sequence of nodes
and edges from i to j
eg. (1-2-3-4-3)
trail: a walk with no repeated edges
eg. (1-2-3-4-5-2)
path: a walk with no repeated nodes
eg. (1-2-3-4-6)

30. Connectivity matrix
(also known as adjacency matrix)
A =
Size
binary or weighted

33. Node degree
d(4) = ?
d(6) = ?
3
1

34. Degree distribution
Determines the statistical properties of uncorrelated networks

35. Degree distribution
Determines the statistical properties of uncorrelated networks

36. Degree distribution
Determines the statistical properties of uncorrelated networks

37. Shortest path
• Indicates the distance between i and j in terms of geodesics
(unweighted)
• Can define the structure of a network
Transport and communication
p(1,3) = {1-5-4-3}
{1-5-2-3}
{1-2-5-4-3}
{1-2-3}
Warning: the “longest” path can be
the shortest (weighted graph)

38. Diameter
• Indicates the maximum number of hops between i and j
(unweighted)
• global property of a network

39. Average Shortest Path - ASP
•
• global property of a network
Problem?
i and j are disconnected
Solution (efficiency)

40. Betweenness centrality
# SPs from j to k via i
# SPs from j to k

41. Which node is the most important?

42. Communities/Clusters
• Local properties are shared only by a subset of the nodes

43. Facebook

44. Facebook
(again)

45. Network components

46. Network components
• define the topology
• locally
• globally
(how many triads/pendants/dyads…)

47. example: count the number of triads in
a network for comparison

49. Topologies: small-world
Random shortcuts
ASP
each node is connected to
any other node in only
log(N) steps

50. Topologies: scale-free
Degree distribution follows power-law
Fact!
most real networks follow a power-law

51. Topologies: scale-free
Degree distribution follows power-law
• the sizes of earthquakes
• craters on the moon
• solar flares
• the foraging pattern of various species
• the sizes of activity patterns of
neuronal populations
• the frequencies of words in
most languages
• frequencies of family names
• sizes of power outages
• wars
• criminal charges per convict
• and many more…

52. Topologies: random
Nodes are statistically independent

53. Networks
static (1)
given a degree distrib. -> connect
dynamic (2)
structural changes are governed by
evolution of the system
(gene-gene, web, social net.)
(1) given , assign uniform prob. to all random graphs
with a number of nodes with degree k (Aiello et. al)
N and k are fully determined
(2) Prob. of link j connected to existing node i is proportional to

54. Weighted networks
A =
3 1
2 5
8
2
0 5 0 0 1 0
5 0 8 0 6 0
6
0 8 0 2 0 0
0 0 2 0 3 2
1 6 0 3 0 0
0 0 0 2 0 0

55. Weighted networks
node strength
strength of nodes of
degree k
(independence between weight and topology)
average weight
with correlation

56. “The richest people in the world look for and
build networks. Everyone else looks for work.”
–Robert Kiyosaki