Modeling wireless sensor networks using random graph
theoryDepartment of Control Science and Engineering, Huazhong
University of Science and Technology, Wuhan 430074,Received 2 March
2007; received in revised form 9 September 2007Available online 10
January 2008
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AbstractA critical issue in wireless sensor networks (WSNs) is
represented by limited availability of energy within network nodes.
Therefore, making good use of energy is necessary in modeling
sensor networks. In this paper we proposed a new model of WSNs on a
two-dimensional plane using site percolation model, a kind of
random graph in which edges are formed only between neighbouring
nodes. Then we investigated WSNs connectivity and energy
consumption at percolation threshold when a so-called phase
transition phenomena happen. Furthermore, we proposed an algorithm
to improve the model; as a result the lifetime of networks is
prolonged. We analyzed the energy consumption with Markov process
and applied these results to simulation. WSN---WSN..
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Sensor networks are composed of a large number of tiny sensor
nodes that include sensing, data processing and communicating
components. These sensors autonomously establish connections via
multi-hop wireless communication. Sensor networks gather
information in a certain area by sensors and transmit back to the
observer through a sink node [13]. There is no pre-configured
network infrastructure (such as base stations) or centralized
control in the area covered with sensors. The quantity of sensors
is huge or the monitored area sometimes is unreachable, so
arranging sensors by manpower is unrealistic. . [ 13 ]sink ) [ 4 ]
MAC [ 5 ]
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Percolation theory was introduced in 1957 by Broadbent and
Hammersley. This kind of random graph has been widely applied in
various fields, including economics [9], biology [10], sociology
[11] and communication [12]. Our original motivation of this paper
is to apply phase transition in percolation theory to sensor
networks and to obtain a new way of energy conservation. Indeed,
percolation theory has been used in the past to study connectivity
of wireless. Broadbent Hammersley 1957 [ 11 ] [ 12 ] [ 15 ] ref.[
16 ] omnidirectional percolation -
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G(V, E) V E G V E Gi Vi Ei Fig.1 Gi
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2. GAF algorithm selects certain nodes as a cluster based on
geographical location of nodes. First, the monitored area is
divided into several virtual square lattices and nodes in the same
square lattice form a cluster. The size of a virtual lattice should
meet the needs of communication between two possible farthest nodes
in any two adjacent lattices, see Fig. 2. GAF2.2.1Geographical
adaptive fidelity (GAF) algorithm
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2.1 Geographical adaptive fidelity (GAF) algorithm r 2R
-
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2.2. Random graphs A random graph consists of vertices and
edges. Any two vertices share an edge with the same probability p.
In Ref. [18], Erdos and Renyi studied that the probability of a
random graph being connected tends to 1 if E is greater than pc(E)
= N/ log N 2 (N is the number of vertices and E is the number of
edges). This is what we call a phasetransition in random graphs
that implies a sudden large change of network performance at pc. In
other words, the value of pc is a threshold beyond which the random
graph is very connected. P18Erdos and Renyi E NE 1 Pc pc
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2.3 , p1 p Fig.3 p PC p(p < pcs) (p > pcs)
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3. Gi10*10Gi100GAFGiGiGisinksink3.1
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3.2 , ID Fig.43.2.1
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3.2.2 After network initialization, each nodes in a cluster
creates a random number whose value is between 0 and 1. Choose a
node as cluster head randomly and let its = 1. All other nodes
increase their under the inactive state. does not increase under
the active state (include sensing state, receiving state and
transmitting state). When of a node reaches 1, a new cluster head
selection begins and the one with the higher energy wins. of the
losing onechanges to 0 and then increases as described above again.
, 0 1 = 1 1 0
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3.2.3 N(i j)(N(i 1, j ), N(i + 1, j ), N(i, j 1), N(i, j + 1))5
P
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3.3 If a cluster head is active, it wakes its cluster members to
gather information. We consider it as an open site. If a cluster
head is inactive, all sensors in this lattice will do nothing.We
consider it as a closed site. That cluster head is anopen site or a
closed site is equivalent to a lattice is ON or OFF because only
the cluster head can represent its square lattice to communicate
with nodes (also cluster head nodes) from other lattices. Then we
obtain a site percolation model in which the site is a
lattice..
ON OFF
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4. It is easy to find that under the same situation, if all
lattices in Gi are ON with probability p = 1, the Gi is connected
and data delivering ratio is 100% but lifetime of WSNs is the
shortest. Conversely, if we randomly shut down part of the sensors
in each step, lifetime of Gi is prolonged but Gi may be
disconnected and delivering ratio is reduced. From phase transition
we know that if p is less than 1 but more than pcs , the network
still has good connectivity because all ON lattices are connected
with each other. In other words, any gathered data could transmit
to the sink node by multi-hop. Energy consumption is reduced under
this situation because part of the lattice is OFF for saving
energy. Gi ONp = 1Gi 100%WSNs , Gi Gi p 1 PcsON Gi pPcs
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4. It is easy to find that under the same situation, if all
lattices in Gi are ON with probability p = 1, the Gi is connected
and data delivering ratio is 100% but lifetime of WSNs is the
shortest. Conversely, if we randomly shut down part of the sensors
in each step, lifetime of Gi is prolonged but Gi may be
disconnected and delivering ratio is reduced. From phase transition
we know that if p is less than 1 but more than pcs , the network
still has good connectivity because all ON lattices are connected
with each other. In other words, any gathered data could transmit
to the sink node by multi-hop. Energy consumption is reduced under
this situation because part of the lattice is OFF for saving
energy. Gi ONp = 1Gi 100%WSNs , Gi Gi p 1 PcsON Gi pPcs
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4.1 S1S2S36
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4.2 . S1S2S3S4sink7.
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4.3NiKE0
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4.4 E(i, j ), E(i 1, j ), E(i + 1, j ), E(i, j 1), E(i, j +
1)A(i, j ) = (E(i 1, j ) + E(i + 1, j ) + E(i, j 1) + E(i, j +
1))/4IF E(i, j ) is more than a _ A(i, j )THEN Lattice with energy
E(i, j ) is ONELSEIF E(i, j ) is less than b _ A(i, j )THEN Lattice
with energy E(i, j ) is OFFELSE Lattice with energy E(i, j ) is ON
with probability pcs .END(a, b are controllable parameters that
affect connectivity of WSNs).a,bWSN
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5. 10*10GiGisink180mAh600mAh22K
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5.
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5.
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6. Unlike other wireless networks, the use of sensor network is
limited by sensor energy. So modeling and analysis of it is quite
different from other ad hoc networks. Generally speaking, we want
to get the most abundant information and the longest lifetime of
WSNs, which seems to be a dilemma. In order to find a balance
between these two demands we brought the concept of random graphs
to model sensor networks and then proposed a model with site
percolation a kind of random graph in which the edges are formed
only between adjacent sites. This model has less active
WSNs WSNs WSNs
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