1. Network properties
Slides are modified from Networks: Theory and Application by Lada Adamic

2. How do you characterize it?
Outline
What is a network?
a bunch of nodes and edges
How do you characterize it?
with some basic network metrics
Network models

3. What are networks? Networks are collections of points joined by lines.
“Network” ≡ “Graph”
node
edge
points
lines
vertices
edges, arcs
math
nodes
links
computer science
sites
bonds
physics
actors
ties, relations
sociology

4. Network elements: edges
Directed (also called arcs)
A -> B
A likes B, A gave a gift to B, A is B’s child
Undirected
A <-> B or A – B
A and B like each other
A and B are siblings
A and B are co-authors
Edge attributes
weight (e.g. frequency of communication)
ranking (best friend, second best friend…)
type (friend, relative, co-worker)
properties depending on the structure of the rest of the graph: e.g. betweenness

5. Directed networks
girls’ school dormitory dining-table partners (Moreno, The sociometry reader, 1960)
first and second choices shown 2 1
Ada
Cora
Louise
Jean
Helen
Martha
Alice
Robin
Marion
Maxine
Lena
Hazel
Hilda
Frances
Eva
Ruth
Edna
Adele
Jane
Anna
Mary
Betty
Ella
Ellen
Laura
Irene

6. Edge weights can have positive or negative values
One gene activates/ inhibits another
One person trusting/ distrusting another
Research challenge:
How does one ‘propagate’ negative feelings in a social network?
Is my enemy’s enemy my friend?
Transcription regulatory network in baker’s yeast

7. Representing edges (who is adjacent to whom) as a matrix
Adjacency matrices
Representing edges (who is adjacent to whom) as a matrix
Aij = 1 if node i has an edge to node j = 0 if node i does not have an edge to j
Aii = 0 unless the network has self-loops
Aij = Aji if the network is undirected, or if i and j share a reciprocated edge j i i i j 1 2 3 4
Example: 5 1
A =

8. Adjacency lists Edge list Adjacency list
2 3
2 4
3 2
3 4
4 5
5 2
5 1
Adjacency list
is easier to work with if network is
large
sparse
quickly retrieve all neighbors for a node 1:
2: 3 4
3: 2 4
4: 5
5: 1 2 2 3 1 4 5

9. Node network properties
Nodes
Node network properties
from immediate connections
indegree how many directed edges (arcs) are incident on a node
outdegree how many directed edges (arcs) originate at a node
degree (in or out) number of edges incident on a node
indegree=3
outdegree=2
degree=5

10. Node degree from matrix values2
Node degree from matrix values 3 1 4 1 5
Outdegree =
A =
example: outdegree for node 3 is 2, which we obtain by summing the number of non-zero entries in the 3rd row 1
Indegree =
A =
example: the indegree for node 3 is 1, which we obtain by summing the number of non-zero entries in the 3rd column

11. Characterizing networks: Is everything connected?

12. Network metrics: connected components
Strongly connected components
Each node within the component can be reached from every other node in the component by following directed links B F G
Strongly connected components
B C D E A
G H F C A H D E
Weakly connected components: every node can be reached from every other node by following links in either direction A B C D E F G H
Weakly connected components
A B C D E
G H F
In undirected networks one talks simply about ‘connected components’

13. network metrics: size of giant component
if the largest component encompasses a significant fraction of the graph, it is called the giant component

14. How do you characterize it?
Outline
What is a network?
a bunch of nodes and edges
How do you characterize it?
with some basic network metrics
Network models

15. Several other graph metrics exist
Structural Metrics
Degree distribution
Average path length
Centrality
Betweenness
Closeness
Graph density
Clustering coefficient
Several other graph metrics exist
Assortativity
Modularity …

16. degree sequence and degree distribution
Degree sequence: An ordered list of the (in,out) degree of each node
In-degree sequence:
[2, 2, 2, 1, 1, 1, 1, 0]
Out-degree sequence:
[2, 2, 2, 2, 1, 1, 1, 0]
(undirected) degree sequence:
[3, 3, 3, 2, 2, 1, 1, 1]
Degree distribution: A frequency count of the occurrence of each degree
In-degree distribution:
[(2,3) (1,4) (0,1)]
Out-degree distribution:
[(2,4) (1,3) (0,1)]
(undirected) distribution:
[(3,3) (2,2) (1,3)]

17. Structural Metrics: Degree distribution

18. Characterizing networks: How far apart are things?

19. Structural metrics: Average path length

20. Characterizing networks: Who is most central?

21. Centrality: betweenness
The fraction of all directed paths between any two vertices that pass through a node
paths between j and k that pass through i
betweenness of vertex i
all paths between j and k
Normalization
undirected: (N-1)*(N-2)/2
directed graph: (N-1)*(N-2) e.g.

22. Centrality: closeness
How close the vertex is to others
depends on inverse distance to other vertices
Normalization

23. network metrics: graph density
Of the connections that may exist between n nodes
directed graph emax = n*(n-1)
undirected graph emax = n*(n-1)/2
What fraction are present?
density = e/ emax
For example, out of 12 possible connections,
this graph has 7, giving it a density of 7/12 = 0.583
Would this measure be useful for comparing networks of different sizes (different numbers of nodes)?

24. Structural Metrics: Clustering coefficient

25. How do you characterize it?
Outline
What is a network?
a bunch of nodes and edges
How do you characterize it?
with some basic network metrics
Network models

26. Four structural models
Regular networks
Random networks
Small-world networks
Scale-free networks

27. Regular networks – fully connected

28. Regular networks – Lattice

29. Regular networks – Lattice: ring world

30. modeling networks: random networks
Nodes connected at random
Number of edges incident on each node is Poisson distributed
Poisson distribution

31. Random networks

32. Erdos-Renyi random graphs
What happens to the size of the giant component as the density of the network increases?

33. Random Networks

34. modeling networks: small worlds
a friend of a friend is also frequently a friend
but only six hops separate any two people in the world
Arnold S. – thomashawk, Flickr;

35. Duncan Watts and Steven Strogatz
Small world models
Duncan Watts and Steven Strogatz
a few random links in an otherwise structured graph make the network a small world: the average shortest path is short
regular lattice:
my friend’s friend is
always my friend
small world:
mostly structured
with a few random
connections
random graph:
all connections
random
Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

36. Watts Strogatz Small World Model
As you rewire more and more of the links and random, what happens to the clustering coefficient and average shortest path relative to their values for the regular lattice?

37. Small-world networks

38. Scale-free networks

39. Many real world networks contain hubs: highly connected nodes
Scale-free networks
Many real world networks contain hubs: highly connected nodes
Usually the distribution of edges is extremely skewed
many nodes with few edges
number of nodes with so many edges
fat tail: a few nodes with a very large number of edges
number of edges
no “typical” number of edges

40. But is it really a power-law?
A power-law will appear as a straight line on a log-log plot:
A deviation from a straight line could indicate a different distribution:
exponential
lognormal
log(# nodes)
log(# edges)

41. Scale-free networks