Network Questions: Structural

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Network Questions Structural

1. How many connections does the average node have?
2. Are some nodes more connected than others?
3. Is the entire network connected?
4. On average, how many links are there between
   nodes?
5. Are there clusters or groupings within which the
   connections are particularly strong?
6. What is the best way to characterize a complex
   network?
7. How can we tell if two networks are different?
8. Are there useful ways of classifying or
   categorizing networks?

slides from David P. Feldman
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Network Questions Communities

1. Are there clusters or groupings within which the
   connections are particularly strong?
2. What is the best way to discover communities,
   especially in large networks?
3. How can we tell if these communities are
   statistically significant?
4. What do these clusters tell us in specific
   applications?

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Network Questions Dynamics of

1. How can we model the growth of networks?
2. What are the important features of networks that
   our models should capture?
3. Are there universal models of network growth?
   What details matter and what details dont?
4. To what extent are these models appropriate null
   models for statistical inference?
5. Whats the deal with power laws, anyway?

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Network Questions Dynamics on

1. How do diseases/computer viruses/innovations/
   rumors/revolutions propagate on networks?
2. What properties of networks are relevant to the
   answer of the above question?
3. If you wanted to prevent (or encourage) spread of
   something on a network, what should you do?
4. What types of networks are robust to random
   attack or failure?
5. What types of networks are robust to directed
   attack?
6. How are dynamics of and dynamics on coupled?

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Network Questions Algorithms

1. What types of networks are searchable or
   navigable?
2. What are good ways to visualize complex networks?
3. How does google page rank work?
4. If the internet were to double in size, would it
   still work?

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Network Questions Algorithms

• There are also many domain-specific questions
• Are networks a sensible way to think about gene
  regulation or protein interactions or food webs?
• What can social networks tell us about how people
  interact and form communities and make friends
  and enemies?
• Lots and lots of other theoretical and
  methodological questions...
• What else can be viewed as a network? Many
  applications await.

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Network Questions Outlook

• Advances in available data, computing speed, and
  algorithms have made it possible to apply network
  analysis to a vast and growing number of
  phenomena.
• This means that there is lots of exciting, novel
  work being done.
• This work is a mixture of awesome, exploratory,
  misleading, irrelevant, relevant, fascinating,
  ground-breaking, important, and just plain wrong.
• It is relatively easy to fool oneself into seeing
  thing that arent there when analyzing networks.
• This is the case with almost anything, not just
  networks.
• For networks, how can we be more careful and
  scientific, and not just descriptive and
  empirical?

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Lecture 3 Mathematics of Networks
CS 765 Complex Networks
Slides are modified from Networks Theory and
Application by Lada Adamic
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What are networks?

• Networks are collections of points joined by
  lines.

Network Graph
node
edge
points lines Domain
vertices edges, arcs math
nodes links computer science
sites bonds physics
actors ties, relations sociology
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Network elements edges

• Directed (also called arcs)
• A -gt B (EBA)
• 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
• Multiedge multiple edges between two pair of
  nodes
• Self-edge from a node to itself

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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
• If self-loop, Aii?
• Aij Aji if the network is undirected,or if i
  and j share a reciprocated edge

j
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 0 0 0 1
1 1 0 0 0
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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HyperGraphs

• Edges join more than two nodes at a time
  (hyperEdge)
• Affliation networks
• Examples
• Families
• Subnetworks
• Can be transformed to a bipartite network

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Bipartite (two-mode) networks

• edges occur only between two groups of nodes, not
  within those groups
• for example, we may have individuals and events
• directors and boards of directors
• customers and the items they purchase
• metabolites and the reactions they participate in

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in matrix notation

• Bij
• 1 if node i from the first group links
  to node j from the second group
• 0 otherwise
• B is usually not a square matrix!
• for example we have n customers and m products

i
j
1 0 0 0
1 0 0 0
1 1 0 0
1 1 1 1
0 0 0 1
B
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going from a bipartite to a one-mode graph
group 1

• Two-mode network

• One mode projection
• two nodes from the first group are connected if
  they link to the same node in the second group
• naturally high occurrence of cliques
• some loss of information
• Can use weighted edges to preserve group
  occurrences

group 2
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Collapsing to a one-mode network
i
k

• i and k are linked if they both link to j
• Pij ?k Bki Bkj
• P B BT
• the transpose of a matrix swaps Bxy and Byx
• if B is an nxm matrix, BT is an mxn matrix

j1
j2
1 0 0 0
1 0 0 0
1 1 0 0
1 1 1 1
0 0 0 1
1 1 1 1 0
0 0 1 1 0
0 0 0 1 0
0 0 0 1 1

B
BT
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Matrix multiplication

• general formula for matrix multiplication Zij ?k
  Xik Ykj
• let Z P, X B, Y BT

1 1 1 1 0
1 1 1 1 0
1 1 2 2 0
1 1 2 4 1
0 0 0 1 1
1 0 0 0
1 0 0 0
1 1 0 0
1 1 1 1
0 0 0 1
1 1 1 1 0
0 0 1 1 0
0 0 0 1 0
0 0 0 1 1


P
1
1 1 1 1
1
1
0
0
1
1111 10 10 2
1
1
2
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Collapsing a two-mode network to a one
mode-network

• Assume the nodes in group 1 are people and the
  nodes in group 2 are movies
• The diagonal entries of P give the number of
  movies each person has seen
• The off-diagonal elements of P give the number
  of movies that both people have seen
• P is symmetric

1 1 1 1 0
1 1 1 1 0
1 1 2 2 0
1 1 2 4 1
0 0 0 1 1
1
1
P
1
1
2