Characterizing Networks: Metrics, Components, and Eigenvectors, Slides of Data Communication Systems and Computer Networks

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Lecture 5:

Mathematics of Networks (Cont)

Characterizing networks:

How far apart are things?

Network metrics: paths

Network metrics: shortest paths

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A

B C

E^ D

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2

2

3

3

Eulerian Path

• Euler’s Seven Bridges of Königsberg
  • one of the first problems in graph theory
• Is there a route that crosses each bridge only once and returns to the starting point?

Source: http://en.wikipedia.org/wiki/Seven_Bridges_of_KönigsbergImage 1 – GNU v1.2: Bogdan, Wikipedia; http://commons.wikimedia.org/wiki/Commons:GNU_Free_Documentation_License Image 2 – Image 3 – GNU v1.2: Booyabazooka, Wikipedia;GNU v1.2: Riojajar, Wikipedia; http://commons.wikimedia.org/wiki/Commons:GNU_Free_Documentation_License http://commons.wikimedia.org/wiki/Commons:GNU_Free_Documentation_License

Eulerian and Hamiltonian paths

• Hamiltonian path is self avoiding If starting point and end point are the same: only possible if no nodes have an odd degree as each path must visit and leave eachshore If don’t need to return to starting point can have 0 or 2 nodes with an odd degree

Eulerian path: traverse each edge exactly once

Hamiltonian path: visit each vertex exactly once

Network metrics: components

• If there is a path from every vertex in a network to every other, the network is connected - otherwise, it is disconnected
• Component : A subset of vertices such that there exist at least one path from each member of the subset to others and there does not exist another vertex in the network which is connected to any vertex in the subset - Maximal subset
• A singeleton vertex that is not connected to any other forms a size one component
• Every vertex belongs to exactly one component

components in directed networks

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A

B C

E^ D

F (^) G

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Weakly connected components A B C D E G H F

 Strongly connected components  Each node within the component can be reached from every other node in the component by following directed links

Strongly connected components B C D E A G H F

 Weakly connected components : every node can be reached from every other node by following links in either direction

A

B C

E^ D

F (^) G

H

network metrics: size of giant

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

bowtie model of the web

• The Web is a directed graph:
  • webpages link to other webpages
• The connected components tell us what set of pages can be reached from any other just by surfing - no ‘jumping’ around by typing in a URL or using a search engine
• Broder et al. 1999 – crawl of over 200 million pages and 1.5 billion links. (^) Docsity.com^14

Cut Sets and Maximum Flow

• A minimum cut set is the smallest cut set that will disconnect a specified pair of vertices - Need not to be unique
• Menger’s theorem: If there is no cut set of size less than n between a pair of vertices, then there are at least n independent paths between the same vertices. - Implies that the size of min cut set is equal to maximum number of independent paths (for both edge and vertex independence) (^16)

Graph Laplacian

Eigenvalues and eigenvectors

• In words, this deceptively simple equation says that for the square matrix A , there is a vector x such that the product of Ax such that the result is a SCALAR, λ, that, when multiplied by x , results in the same product. The multiplication of vector x by a scalar constant is the same as stretching or shrinking

Ax = λ x (E.01)

Ax = λ x

The vector x is called an eigenvector and the scalar λ, is called an eigenvalue.

Do all matrices have real eigenvalues?

No, they must be square and the determinant of A- λ I must equal zero. This is easy to show:

This can only be true if det( A- λ I )=| A- λ I |=

Are eigenvectors unique?

No, if x is an eigenvector, then β x is also an eigenvector and β λ is an eigenvalue.

Ax −λ x = 0 x A ( −λ I ) = 0

A (β x )= β Ax = βλ x = λ (β x )

(E.02) (E.03)

(E.04)