Network properties

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Lecture 2 Network properties
CS 790g Complex Networks
Slides are modified from Networks Theory and
Application by Lada Adamic
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Outline

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

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What are networks?

• Networks are collections of points joined by
  lines.

Network Graph
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Network elements edges

• Directed (also called arcs)
• A -gt B
• A likes B, A gave a gift to B, A is Bs child
• Undirected
• A lt-gt 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

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Directed networks

• girls school dormitory dining-table partners
  (Moreno, The sociometry reader, 1960)
• first and second choices shown

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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 enemys enemy my friend?

Transcription regulatory network in bakers yeast
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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
A
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Adjacency lists

• Edge 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
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Nodes

• Node network properties
• from immediate connections
• indegreehow many directed edges (arcs) are
  incident on a node
• outdegreehow many directed edges (arcs)
  originate at a node
• degree (in or out)number of edges incident on a
  node

indegree3
outdegree2
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Node degree from matrix values
2
3
1
4
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

• 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
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Characterizing networksIs everything connected?
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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

• Weakly connected components
• A B C D E
• G H F

• In undirected networks one talks simply about
  connected components

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network metrics size of giant component

• if the largest component encompasses a
  significant fraction of the graph, it is called
  the giant component

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Outline

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

15
Structural Metrics

• Degree distribution
• Average path length
• Centrality
• Betweenness
• Closeness
• Graph density
• Clustering coefficient
• Several other graph metrics exist
• Assortativity
• Modularity

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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)

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Structural MetricsDegree distribution
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Characterizing networksHow far apart are things?
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Structural metrics Average path length
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Characterizing networksWho is most central?
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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.

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Centrality closeness

• How close the vertex is to others
• depends on inverse distance to other vertices

• Normalization

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network metrics graph density

• Of the connections that may exist between n nodes
• directed graph emax n(n-1)
• undirected graphemax 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)?

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Structural MetricsClustering coefficient
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Outline

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

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Four structural models

• Regular networks
• Random networks
• Small-world networks
• Scale-free networks

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Regular networks fully connected
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Regular networks Lattice
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Regular networks Lattice ring world
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modeling networks random networks

• Nodes connected at random
• Number of edges incident on each node is Poisson
  distributed

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Random networks
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Erdos-Renyi random graphs

• What happens to the size of the giant component
  as the density of the network increases?

http//ccl.northwestern.edu/netlogo/models/run.cgi
?GiantComponent.884.534
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Random Networks
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modeling networks small worlds

• 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
http//creativecommons.org/licenses/by-nc/2.0/dee
d.en
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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 friends 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 393440-442.
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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?

http//projects.si.umich.edu/netlearn/NetLogo4/Sma
llWorldWS.html
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Small-world networks
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Scale-free networks
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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 numberof
edges
number of edges
no typical number of edges
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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)
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Scale-free networks