Rajan Viva Voce PhD

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Investigation of Some Fundamental Issues of Connectivity and Topology Control Relevant

to Wireless Ad-Hoc Networks Including Network Partition and Cross Layer Interaction

and Design by Connectivity Index

Rajan.M.A,

Research Scholar,Department Of Computer Science,

Dravidian University, KuppamUnder the Guidance

ofDr. Lokanatha C. Reddy

andDr. Prakash S. Hiremath

2

Outline

Goal of the Research Brief Introduction to Ad-Hoc networks (MANETS) Key Fundamental Issues in MANET Network Modeling Concept Of Connectivity Index (CI) in MANET. Mobility and Transmission Power Effect on Topology K-Connected Networks Network Partition Detection CI based Cross Layer Design Methodology Conclusion Future Work Discussion

3

Goal of the Research Understand the Fundamental issues in

Wireless Adhoc Networks Connectivity, Scalability, Topology control

and Routing. Network modeling and simulation Network Partition Detection cross-layer design or cross layer

interaction

4

What is MANET?

5

MANET stands for Mobile Adhoc Network. Each node in an ad-hoc network is

equipped with a radio transmitter and receiver, which allow it to communicate with other nodes over wireless channels.

A special kind of ad-hoc network is the sensor network where the nodes forming the network do not or rarely moves. Further, the nodes of a sensor network are similar.

MANET

6

Characteristics

Self-organizing: without centralized control Scarce resources: bandwidth and batteries Dynamic network topology

7

Connected Adhoc Network

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

sink

X Axis in meters

Y A

xis

in m

eter

s

Connected Network with 200 nodes and 1324 links

8

Applications

Defense industry (battlefield) Academic institutions (conference and

meeting) Personal area networks and Bluetooth Home networking Embedding computing applications Health facilities Disaster recovery (search-and-rescue)

9

Fundamental Issues

Connectivity Scalability Routing Topology Control

10

Connectivity One of the fundamental and most important

issues of the MANETs. The important factor, which affects the

connectivity, is the transmission range of the nodes and the mobility of the nodes.

Some of the concepts of graph theory that are extensively used to study the connectivity issues are graph spanners, proximity graph sparsifications and spectral graph theory.

11

Scalability Scalability is the study of network stability,

whenever the number of nodes of the network undergoes changes.

Addition of nodes to the network may cause the network be disconnected to start with. This necessitates topology control, an important issue. Some of the fundamental questions that arise during topology change are how the performance of the network and routing will be affected

The graph theory concepts like graph clustering, graph partitioning, and graph evolution.

12

Routing The factors which can affect the routing

are Connectivity mobility of the nodes. Scalability of the network

Routing protocols in mobile ad-hoc networks are more complex than in static networks.

Graph theory concepts like planarity, graph colouring, graph spanners, clustering plays an imporrtant role in designing the routing algorithms.

13

Network Modeling

Graph Theory Concepts Graph Based Models Mobility Models

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Introduction to Graph theory

A graph G is a triplet consists of Vertex set , Edge set and a relation that associates with each edge, two vertices (end points). That is a

Graph G is denoted by G(V,E). The two variants of the graph are number of vertices and number of edges.

Two edges are said to be adjacent to each other if the one of the end vertex of the edges are same

With respect to the network, a vertex is a node and an edge is a link between two nodes

A graph G (N, n) with N vertices and n edges algebraically represented as adjacency matrix, whose entries are given by

15

Introduction to Graph theory contd..

A graph consists of number of vertices and edges, where an edge is an association between two vertices.

Mathematically, a graph G is a triplet consists of Vertex Set V(G) , Edge Set E(G) and a relation that associates with each edge, two vertices.

An edge between two nodes i and j is represented as (i,j) and using the usual notation can be written as

A Graph G is denoted as G(V,E)

Two vertices are said to be adjacent to each other, if there exist an edge between them.

If each edge of a graph is associated with some specific value (weight), graph is said to be weighted graph.

The number of edges associated with the vertex v is called degree of a vertex denote by d(v).

ij,ji,Vji,ji,GE and|

16

Introduction to Graph theory contd..

The minimum degree of a graph is the least degree of a vertex of a graph denoted by δ(G) and the maximum degree of a graph is the maximum degree of any vertex of a graph denoted by ∆(G).

A graph G is regular if and only if (iff) δ(G)=∆(G)

A graph G is said to be connected, if for every pair of vertices u, v belongs to G, there exist a path, otherwise Graph is disconnected.

A dis-connected graph has number of components; each component being a connected graph.

17

Random Graph Models applicable to Ad-hoc Networks

A lot of research about ad-hoc networks is

carried out using the mathematical models and their simulation rather than experimenting on real mobile ad-hoc networks.

Several issues like node density, mobility of the nodes, link formation between nodes and packet routing between the nodes needs to be simulated.

To simulate MANETs concepts of graph theory (particularly random graph theory) is utilized.

Needless to say it is necessary to know the different models and decide upon the network model to be simulated.

18

Random Graph Models applicable to Ad-hoc Networks Contd..

Random graph is denoted by G(n,p).

G(n,p) = set of all random graphs with parameters n and p n nodes and each edge is included

with probability p. expected degree = p(n-1)(1-p)3

p3

p(1-p)2

p2(1-p)

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Random Graph Models Random static graph models.

the nodes are assumed to be almost stationary but located at arbitrary positions.

model is more suited for ad-hoc sensor networks, where the nodes in the network rarely moves.

Example: Erdo˝s and Re´nyi Model, Binomial Model, Random Geometric Graph Model and Poisson Cloning Model

Random mobility graph models. the movement of the node details is taken into considerations. model is more suitable for mobile ad-hoc networks, where the

nodes are moving. Individual mobility Models Group mobility models

Example: random walk mobility model, random waypoint mobility model, random direction model, boundless simulation area mobility model and Gauss-Markov mobility model.

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RANDOM STATIC GRAPH MODELS Erdo˝s and Re´nyi Model

Define a random graph as N labeled nodes connected by n edges, which are chosen randomly from the (n*(n-1))/2 possible edges

In total there are graphs with N nodes and n edges, forming a probability space in which every realization is equi-probable. The number of edges El in the random graph is then a random variable with the

expectation The random graph of Erdös and Rényi is one of the best

studied models of a network because of its simplicity. Model is not a realistic representation of a wireless ad-hoc

network. In ad-hoc networks, nodes at close range have a higher

probability of being connected than nodes at farther distances.

n

1NN

2

1NNpEE l

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Random Geometric Graph Model Graphs with distance-dependent links

between nodes and correlated links well suited for modeling ad-hoc networks

as in nodes at close range have a higher probability of being connected than nodes at farther distances.

Link between any two nodes is possible ,if their euclidean distance is atmost r

A variation of RGG is Unit Disc Graph(UDG) is a RGG with r=1

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Unit Disc Graph

A simple ad hoc wireless network of five wireless mobile hosts.

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RANDOM MOBILITY GRAPH MODELS Basically there are two types of mobility models

individual mobility models group mobility models

individual mobility models deals with the movement at the node level, where each

node is considered independently from others. Example:

Random Walk mobility model Random Waypoint mobility model Random Direction mobility model Boundless Simulation Area mobility model Gauss-Markov mobility model

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Group mobility models

the mobility of a node is computed relatively to the mobility of a reference point in the subset of nodes (group) it belongs to Example:

Reference Point group mobility model Exponential Correlated mobility model Nomadic Community mobility model Pursue mobility model

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Important Graph Theory Concepts

Graph Spanners Proximity

Unit Distance Graph (UDG) Nearest Neighbor Graphs (NNG) Minimum Spanning Trees (MST) Relative Neighborhood Graphs (RNG) Delaney Triangulation (DT) Gabriel Graphs

Spectral graph theory

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Concept Of Connectivity Index (CI)

Metric to study the connectivity of the network.

CI was proposed by Randić index in 1975 to study the physico- chemical properties properties of the chemical compounds..

CI of a network is defined by = CI for random graphs are redefined

for the first time novelly.

G )( )()(

1

GEuv GG vdud

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CI contd.. CI for Erdo˝s and Reńyi Model is

defined as Lemmas

connectivity index of k-regular Erdo˝s and Reńyi random graph is

The connectivity index of complete Erdo˝s and Reńyi random graph is

)( )()(

1,

GEuv GG vdudpG

pp

k1nk2 p1pk

1n

1n

2

n

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CI Simulation Analysis

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Mobility and Transmission Power Effect on Topology

Affects performance of the MANETs Connectivity (topology) Routing Power Consumption.

Feng Xue and P.R. Kumar[1]:The study demonstrates that each node in a network with n randomly placed nodes should be connected to at least nearest neighbors

Random Geometric Graph (RGG) Model is adopted to analyze wireless networks.

nlog

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Mobility and Transmission Power Effect on Topology

Contd..

RGG model Random Walk Mobility graph

05

1015

20

5

10

154

6

8

10

12

Time(in Seconds)

Mobility of two nodes in Random walk mobility model

X-LOCATION

Y-L

OC

AT

ION

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Mobility Effect on Links of the Network

Mobility Vs CI Mobility Vs Link

0 10 20 30 40 50 60 70 80 90 100900

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1300

Time(in Seconds)

Lin

ks o

f th

e N

etw

ork

Effect of Mobility on Number of Links of the network at Tx=30

0 10 20 30 40 50 60 70 80 90 10048.8

48.9

49

49.1

49.2

49.3

49.4

49.5

49.6

Time(in Seconds)

CI

Effect of Mobility of Nodes on CI of Network with 100 Nodes at Tx=30

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Transmission Power Effect on Links of the Network

Tx Power Vs CI Tx Power Vs Links

0 20 40 60 80 100 120 140 0

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4500

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Tx Power

Links

Effect of Transmission Power on Links of the 100 nodes Network

0 20 40 60 80 100 12010

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Transmission Range of the Nodes

Conn

ectiv

ity In

dex

Effect of Transmission Range on Connectivity Index of the Network with 100 Nodes

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Link Power Vs CI Tx Power Vs Links

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Tx Power

Con

nect

ivity

of

the

Net

wor

k

Effect of TX Power onConnectivity of the 100 Nodes Network

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

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110NumberOf Links Vs Connectivity Index of the Network with 100 Nodes

Links

Conn

ectiv

ity In

dex

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Link Power Vs CI Tx Power Vs Links

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

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110NumberOf Links Vs Connectivity Index of the Network with 100 Nodes

Links

Conn

ectiv

ity In

dex

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Link Power Vs CI Nodes Vs Degree @Tx=30 units

10 20 30 40 50 60 70 80 90 1000

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20Degree Distribution of the nodes of the network atTransmiision Range 20

Nodes of the Network

Degr

ee of

the N

ode

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500

1000

1500

2000

2500

3000

3500

4000

4500

5000Connectivity Vs Links of a 100 node wireless Network under Varying Tx Power

Connectivity

Link

s

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Topology @ Tx=20 units Topology @ Tx=10 units

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100Network Topology at Tx=10, Edges=148 connectivity=0

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Degree Distribution0 10 20 30 40 50 60 70 80 90 100

0

10

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60

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100Tx=80Network Topology at Tx=80, Edges=4079 connectivity=51

0 10 20 30 40 50 60 70 80 90 1000

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100Fully Connected Network Topology at Tx=126, Edges=4950 connectivity=99

10 20 30 40 50 60 70 80 90 1000

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2Effect Of Variation of Transmission Range of the Nodes on the Degree distribution

10 20 30 40 50 60 70 80 90 1000

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Nodes of the Network

Degree of the Nodes

Tx=1

Tx=25

Tx=50

Tx=75

Tx=100

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K-Connected Networks Between any two nodes of the

network K disjoint paths exist. K-1 Routes fault tolerant networks.

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K-Connected Networks Contd..

Lemma

1. Minimum number of edges required for K-connected graph is or N-1

2. Minimum number of edges that needs to be added to construct a K+1 connected graph G from K-connected graph G is

3. The number of possible K-connected graphs with minimum number of is at-most

or with K=1 and K>1 4. The probability of constructing random K-

connected graph ( K > 1) is less than or equal to

2

KN

2

N

2

KN

1N

1NN

2

1KNNN

2

12

1

2

2

1

1NN

n n

NN

KN

NN

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K-Connected Networks Contd..

5. The maximum number of packets transmitted from the nodes in order to send a packet from one node to another node in an N-1 connected network through flooding is

6. Let G be a graph with vertices and . Suppose for some

K,Such that

then G is K-connected.

1)2(1 NN

,.......,, 321 Nxxxx

).(.....)()( 21 Nxdxdxd ,0 NK

,1)( Kjxd j )(11 1 KNxdNjfor

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Simulation of K-Connected Network

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Analytical Simulation of K-Connected Network

K-Connectivity Vs number of possible K-Connected networks

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5x 10

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Connecitvity

No o

f P

ossib

le N

etw

ork

s

Connectivity Vs Number of Possible Networks(nodes=10)

Connectivity Vs number of K-Connected networks

1 2 3 4 5 6 7 8 90

0.01

0.02

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0.1

Connectivity

Pro

bability

Probability Distribution of Possible K-Connected Networks(with 10 Nodes)

Connectivity Vs CI, energy,links Connectivity Vs CI, energy ,links

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NS2 Simulation of K-Connected Network

QoS Parameters Packet delivery ratio (PDR) End to end delay Routing overhead Drop packets Dequeue packets Average number of

packets generated.

Number of Packets generated Request packets Reply packets Acknowledgement

packets Send packets Receive packets Forward packets Routing Protocols

AODVDSRDSDV

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Time vs Throughput (1-Connected)

0.94

0.95

0.96

0.97

0.98

0.99

1

10 20 30 40 50

Time(Seconds)

Th

rou

gh

pu

t Throughput forAODV

Throughput forDSDV

Throughput forDSR

Time vs Dropped Packets (1-Connected)

0

50

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250

10 20 30 40 50

Time (Seconds )

Dro

pp

ed

Packets

Drop for AODV

Drop for DSDV

Drop for DSR


0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

10 20 30 40 50

Time (Seconds)

Th

rou

gh

pu

t Throughput forAODV

Throughput forDSDV

Throughput forDSR

Time vs Dropped Packets(3-Connected)

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Time (Seconds)

Dro

pp

ed

Packets

Drop for AODV

Drop for DSDV

Drop for DSR

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0.88

0.9

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0.94

0.96

0.98

1

1.02

10 20 30 40 50

Time (Seconds)

Th

ro

ug

hp

ut

Throughput forAODV

Throughput forDSDV

Throughput forDSR

Time vs Dropped Packets(5-Connected)

0100200300400500600700800

10 20 30 40 50

Time (Seconds )

Dro

pp

ed

Packets

Drop for AODV

Drop for DSDV

Drop for DSR

Average Packet Graph (DSDV)

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10000

15000

20000

25000

Connectivity

Pac

kets

REQUEST

REPLY

ACK

Send

Receive

Drop

Dequeue

Forward

Average Packet Graph (AODV)

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15000

20000

25000

Connectivity

Pac

kets

REQUEST

REPLY

ACK

Send

Receive

Drop

Dequeue

Forward

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Average Packet Graph (DSR)

05000

100001500020000250003000035000

Connectivity

Packets REQUEST

REPLY

ACK

Send

Receive

Drop

Dequeue

Forward

QoS Throughput

0.980.9820.9840.9860.9880.99

0.9920.9940.996

AODVThroughput

DSDVthroughput

DSR Throughput

Protocol

Th

rou

gh

pu

t

Series1

Packet Delivery Ratio

96.9

97

97.1

97.2

97.3

97.4

97.5

97.6

AODV DSDV DSR

Protocols

PD

R Packet DeliveryRatio

End to end Delay

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

AODV DSDV DSR

Protocols

Tim

e

End to endDelay

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Network Partition Detection

Partition Detection pre-detection

Prediction Algorithms Watch Dogs Adoptive routing

algorithms Overhead processing

post-detection Unstable till partition

detected Congestion Less overhead processing

Network Partition The failure of set of links or

nodes Network breaks away into

two or more components or clusters.

Cause Mobility Of Nodes

Dynamic topology changes

Effect Performance of the

network Throughput PDR Routing performance

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Clustered Network Partition Detection (post)

using CI Complete Product of

Two Graphs

Direct Sum of Graphs

0 10 20 30 40 50 60 70 80 9010

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Cluster1 Cluster2

Common Link

0 10 20 30 40 50 60 70 8010

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30

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70Complete Prodcut of Two Clusters

Cluster2

Cluster1

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using CI Lemmas

Detection Methods Spectral graph theory

Demands high processing time.

Needs distributed algorithm. Processing time

Exponentially increases with size of the network.

Connectivity Index Approach Less processing time Simple distributed

algorithm. Complexity increases with

the links rather than the nodes.

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using CI

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Network Partition Detection

Algorithm Input: two minimal 1-connected clustered networks with N1 and

N2 as number of nodes in clusters G1 and G2 respectively . Let G is a network obtained by the union of two clusters G1 and

G2 Steps

Compute the parameters: CI (G), eigenvalue2 and graph energy. Choose two nodes u and v randomly, where, and then add the

link between them. Compute energy of a graph, eigenvalue2 and connectivity index. Progressively add the links in each cluster. Partition the network by dropping the link (u, v) between two

clusters and compute the parameters mentioned in step 1 and restore back the link.

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Direct Cluster Network Partition Detection

Algorithm Input: two minimal 1-connected clustered networks with N1 and

N2 as number of nodes in clusters G1 and G2 respectively . Let G is a network obtained by the union of two clusters G1 and

G2 Steps1. Compute the parameters: CI (G), eigenvalue2 and graph energy.2. Choose two nodes u and v randomly, where, and then add the link

between them.3. Compute energy of a graph, eigenvalue2 and connectivity index.4. Progressively add the links in each cluster.5. Partition the network by dropping the link (u, v) between two

clusters and compute the parameters mentioned in step 1 and restore back the link.

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Direct Cluster Network Partition Detection

6. At some jth iteration, drop a link in each cluster and compute the parameters mentioned in step 1. Restore back the links and go to step 3.

7. If the resultant cluster topologies are not fully connected then go to step 3.

8. Plot the graphs.9. Step 4 and 5 are executed in order to demonstrate

the post detection of network partition by using CI. Step 6 is executed in order to study the effect of dropping of two links from each cluster which do not cause network partition.

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Direct Cluster Network Partition Detection Result

Partition Detection using spectral value

Higher peak-partition Lower peak-connected

Partition Detection using CI Higher peak-partition Lower peak-connected

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Direct Product Cluster Network Partition Detection Result

Partition Detection using spectral value

Higher peak-partition Lower peak-connected

Partition Detection using CI Higher peak-Connected Lower peak-partitioned

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CI based Cross Layer Design (CLD) Methodology

CLD Applicability1. Resource optimization2. Topology control3. Self healing based on

node 4. Energy management5. Load balancing6. Congestion control7. Mobility management8. Scalability9. Optimized Routing.10. Fault tolerant11. QoS efficient networks12. Video streaming

CLD Methodology. Information is

exchanged between different protocol layers dynamically.

Exploits interaction between layers Promotes adaptability QoS efficient.

Collaboration across layers. networking signal processing information theory

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PHYSICAL

LINK

NETWORK

TRANSPORT

Synchronization Adjacent nodes Discovery

TransmissionScheduling

remoteradios

simulatedPHY

Node interconnection

End-to-end transport protocols…

Routing-structuremaintenance

Packet forwarding

APPLICATION Collaborative applications…

Typical Protocol Stack of an Adhoc Network

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Functional Protocol Stack of an Adhoc Network

Application Layer Topology Control AlgorithmServer LocationNetwork Map

Transport Layer Congestion windowTimeout clockPacket Losses Rate

Network Layer Routing AffinityRouting LifetimeMultiple Routing

MAC/Link Layer Link BandwidthLink QualityMac Packet Delay

Physical Layer Node’s LocationMobility patternRadio Transmission RangeSNR Information

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CI based CLD Methodology Based on location and neighbor information of the

nodes, compute the CI of the paths (CIP). Higher the CI

Lesser will be the signal interference across the path. Lesser the probability of collisions due to transmission from

hidden nodes across the path. Network layer can adopt the routing based on CIP only

when application layer demands the high quality Qos Packet.

A Connectivity Index of a Path is denoted as CIP and is defined as

nn iiiin vvvvP121

.......

nP

1

11

1nj

j ii

n

jjvdvd

PCIP

.

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v1

v3

v4 v5

v6

v2

Path1

Path2

Path3

21 vd

42 vd

33 vd

34 vd

45 vd 26 vd

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.

Algorithm to compute the CIP of a path

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PHYSICAL

LINK

NETWORK

TRANSPORT

APPLICATION

Synchronization Neighborhood Discovery

TransmissionScheduling

remoteradios

simulatedPHY

Node interconnection

Collaborative applications…

End-to-end transport protocols…

Routing-structuremaintenance

Packet forwarding based on Q bit

CIP routing

Protocol Stack with proposed Cross Layer Design Architecture

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.

Algorithm to compute the CIP of a path

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Future Work Implementation and Evaluation of

CI based CLD. New Protocol stack design based

on CI-CLD. Extending the work for Cognitive

Radio Networks. Further study on applications of

graph theory in CRN

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Open For Discussions.

Rajan Viva Voce PhD

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