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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
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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
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What is MANET?
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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
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Characteristics
Self-organizing: without centralized control Scarce resources: bandwidth and batteries Dynamic network topology
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Connected Adhoc Network
0 10 20 30 40 50 60 70 80 90 1000
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100
sink
X Axis in meters
Y A
xis
in m
eter
s
Connected Network with 200 nodes and 1324 links
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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)
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Fundamental Issues
Connectivity Scalability Routing Topology Control
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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.
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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.
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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.
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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
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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|
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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.
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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.
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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 Re´nyi Model is
defined as Lemmas
connectivity index of k-regular Erdo˝s and Re´nyi random graph is
The connectivity index of complete Erdo˝s and Re´nyi 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
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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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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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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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Transmission Power Effect on Links of the Network
Link Power Vs CI Tx Power Vs Links
20 40 60 80 100 1200
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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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Links
Conn
ectiv
ity In
dex
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Transmission Power Effect on Links of the Network
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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Transmission Power Effect on Links of the Network
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
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2000
2500
3000
3500
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5000Connectivity Vs Links of a 100 node wireless Network under Varying Tx Power
Connectivity
Link
s
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Transmission Power Effect on Links of the Network
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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100Network Topology at Tx=20, Edges=501 connectivity=1
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100Network Topology at Tx=30, Edges=975 connectivity=1
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100Network Topology at Tx=40, Edges=1612 connectivity=1
Topology @ Tx=30 units Topology @ Tx=40 units
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Transmission Power Effect on Links of the Network
Topology @ Tx=20 units Topology @ Tx=80 units
Degree Distribution0 10 20 30 40 50 60 70 80 90 100
0
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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
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2Effect Of Variation of Transmission Range of the Nodes on the Degree distribution
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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
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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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NS2 Simulation of K-Connected Network
Time vs Throughput (1-Connected)
0.94
0.95
0.96
0.97
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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)
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Time (Seconds )
Dro
pp
ed
Packets
Drop for AODV
Drop for DSDV
Drop for DSR
Time vs Throughput (3-Connected)
0.93
0.94
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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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NS2 Simulation of K-Connected Network
Time vs Throughput (5-Connected)
0.88
0.9
0.92
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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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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NS2 Simulation of K-Connected Network
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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Clustered Network Partition Detection (post)
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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Clustered Network Partition Detection (post)
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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CI based Cross Layer Design (CLD) Methodology
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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CI based Cross Layer Design (CLD) Methodology
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 Cross Layer Design (CLD) Methodology
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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CI based Cross Layer Design (CLD) Methodology
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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CI based Cross Layer Design (CLD) Methodology
.
Algorithm to compute the CIP of a path
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CI based Cross Layer Design (CLD) Methodology
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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CI based Cross Layer Design (CLD) Methodology
.
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.