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

An introduction to the concept of networks, discussing network elements such as nodes and edges, their directions and attributes. It also covers the representation of networks using adjacency matrices and lists, and calculating network metrics like graph density.

Typology: Slides

2012/2013

Uploaded on 04/23/2013

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saraswathi 🇮🇳

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Lecture 4:
Mathematics of 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