Understanding Network Elements & Metrics in Mathematics of Networks, Slides of Data Communication Systems and Computer Networks

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Lecture 4:

Mathematics of Networks

What are networks?

• Networks are collections of points joined by lines.

points lines Domain

vertices edges, arcs math

nodes links computer science

sites bonds physics

actors ties, relations sociology

2

“Network” ≡ “Graph” node

edge

Directed networks

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

4

Ada

Cora

Louise

Jean

Helen

Martha

Alice

Robin

Marion

Maxine

Lena

Hazel (^) Hilda

Frances Eva

Edna^ Ruth

Adele

Jane

Anna

Mary

Betty

Ella

Ellen

Laura

Irene

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?

5

Transcription regulatory

network in baker’s yeast

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

7

1

2

3

4 5

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

8

outdegree=

indegree=

degree=

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

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

B =

Collapsing to a one-mode network

• i and k are linked if they both link to j

• Pij = ∑k Bki Bkj

• P’ = B B

T

– the transpose of a matrix swaps Bxy and Byx

– if B is an n x m matrix, BT^ is an m x n matrix

i

j=

k

j=

B = B T^ =

Matrix multiplication

• general formula for matrix multiplication Z ij =

∑k X ik Ykj

• let Z = P’, X = B, Y = B

T

P’ =

1 1

1 2

1

= 11+1

• 10 + 1 = 2

Trees

• Trees are undirected graphs that contain no

cycles

examples of trees

• In nature

– trees

– river networks

– arteries (or veins, but not both)

• Man made

– sewer system

• Computer science

– binary search trees

– decision trees (AI)

• Network analysis

Kuratowski’s theorem

• Every non-planar network contains at least

one subgraph that is an expansion of K 5 or

K 3,3.

K 5 K 3,3 19

#s of planar graphs of different

sizes

1:

2:

3:

4:

Every planar graph

has a straight line

embedding