Simulations and Conclusions
5.3 Networks with Bernoulli Nodes
Next, we consider the network with unreliable nodes but with reliable links, i.e. 0 ≤ p1 < 1 and p2 = 1. This is the same model discussed in [6] and [14] in which nodes may break down with probability 1 − p1 independently after deployed. Figure 5.3 is two networks over the same 200 random nodes but with different value of r’s deployed in an unit-area disk. Similar to the r-disk graphs in Figure 5.1, black nodes represent active
b a
(a) riso= |ab| = 0.132620.
b a
(b) rcon= |ab| = 0.144876.
Figure 5.3: The r-disk graph over an unit-area disk with unreliable nodes.
nodes, and solid lines are links between two nodes. Moreover, the white nodes represent the broken nodes. Figure 5.3(a) is the network without isolated nodes, and ab is the CTR for without isolated nodes and with length 0.132620. Figure 5.3(b) is the network with connectivity, and ab is the CTR for connectivity and with length 0.144876. Figure 5.4 illustrates the c.d.f. corresponding to p1 = 0.9, p1 = 0.8, and p1 = 0.7, respectively. The average CTRs and inaccuracies of Riso, Rcon, and Rth are listed in Table 5.2.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) p1= 0.9 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) p1= 0.8 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(c) p1= 0.7 and p2= 1.
Figure 5.4: The c.d.f. of CTR’s over an unit-area disk with unreliable nodes.
Table 5.2: The average CTR’s corresponding to Figure 5.4.
Figure 5.4(a): p1 = 0.9, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1548 0.1706 0.1332 16.22% 28.12%
400 0.0853 0.0907 0.0753 13.23% 20.41%
1600 0.0466 0.0487 0.0416 12.13% 17.26%
Figure 5.4(b): p1 = 0.8, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1617 0.1795 0.1396 15.87% 28.60%
400 0.0895 0.0955 0.0792 13.04% 20.69%
1600 0.0492 0.0513 0.0437 12.35% 17.22%
Figure 5.4(c): p1 = 0.7, p2 = 1
n Riso Rcon Rth DRiso DRcon
100 0.1714 0.1905 0.1471 16.50% 29.47%
400 0.0957 0.1025 0.0837 14.33% 22.41%
1600 0.0519 0.0542 0.0464 12.04% 16.78%
Besides generating points over an unit-area disk, we also run similar simulations over an unit-area square. The c.d.f. corresponding to p1 = 0.9, p1 = 0.8, and p1 = 0.7 are illustrated by Figure 5.5. The average CTR’s and inaccuracies of Riso, Rcon, and Rth are listed in Table 5.3. Basically, the results are similar to results of random point sets over an unit-area disk, and our theory still keeps the accuracy.
Table 5.3: The average CTR’s corresponding to Figure 5.5.
Figure 5.5(a): p1 = 0.9, p2 = 1
n Riso Rcon Rth DRiso DRcon
100 0.1549 0.1720 0.1332 16.29% 29.15%
400 0.0853 0.0908 0.0753 13.20% 20.58%
1600 0.0461 0.0483 0.0416 10.82% 16.20%
Figure 5.5(b): p1 = 0.8, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1631 0.1805 0.1396 16.88% 29.30%
400 0.0897 0.0962 0.0792 13.27% 21.56%
1600 0.0483 0.0506 0.0437 10.34% 15.62%
Figure 5.5(c): p1 = 0.7, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1721 0.1928 0.1471 16.96% 31.06%
400 0.0957 0.1025 0.0837 14.30% 22.48%
1600 0.0509 0.0538 0.0464 09.80% 16.10%
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) p1= 0.9 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) p1= 0.8 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(c) p1= 0.7 and p2= 1.
Figure 5.5: The c.d.f. of CTR’s over an unit-area square with unreliable nodes.
In addition, we consider another scenario. Each node has probability p1 to stay in waking state, and may switch to sleeping state independently with probability 1 − p1. Nodes only do jobs in waking state, such as sending/receiving data, or being a member of the virtual backbone and relaying packet for other nodes. When in sleeping state, they do nothing but monitor a particular broadcasting channel, e.g. beacons in ZigBee networks [2]. Moreover, nodes in sleeping state need to have at least one waking neighbor to prevent from being isolated in the networks. For such networks, a node is isolated if it doesn’t have waking neighbors, and a network is connected if every nodes have at least one waking neighbors. Figure 5.6 shows two networks over the same 200 random nodes with defferent value of r’s deployed in an unit-area disk. Black nodes represent nodes in waking state, and white nodes represent nodes in sleeping state. Dotted lines are the listening links of white nodes to black nodes.
b a
(a) riso= |ab| = 0.129692.
a
b
(b) rcon= |ab| = 0.172918.
Figure 5.6: The r-disk graph with waking/sleeping nodes and listening links.
Figure 5.6(a) is the network without isolated nodes, and ab is the longest edge and with length 0.129692 that is corresponding to the CTR for without isolated nodes, and the r-disk graph is plotted with r = ka − bk. Figure 5.6(b) is the network with connectivity, and ab is the longest edge and with length 0.172918 that is corresponding to the CTR for connectivity, and the r-disk graph is plotted with r = ka − bk. Note that the CTR can be contributed by a dotted line. The c.d.f. corresponding to p1 = 0.9, p1 = 0.8 and p1 = 0.7 are illustrated by Figure 5.7.
The average CTR’s and inaccuracies of Riso, Rcon, and Rthare listed in Table 5.4. The results are similar to those of the former scenario.
5.4 Networks with Bernoulli Nodes and Links
In the real world, wireless signals may be blocked or reflected by geographic barriers and buildings, and interfered by other singals. Thus, communication links is not avail-able everytime. So, besides unreliavail-able nodes, we consider networks with unreliavail-able links.
Assume nodes may break down independently with probability 1 − p1, and links may be down independently with probability 1−p2. Figure 5.8 shows the instance of two networks over the same 200 nodes but with different value of r’s deployed in an unit-area disk with p1 = 0.8 and p2 = 0.8. Black nodes represent well-functioned devices, and white nodes represent failed ones. Edges denoted by solid lines between black nodes are up links, and edges denoted by dash lines are down links. Figure 5.8(a) is the network without isolated nodes, and ab is the longest edge and with length 0.097235 that is corresponding to the CTR such that every black nodes have at least one solid edge, and the graph is plotted
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) p1= 0.9 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) p1= 0.8 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(c) p1= 0.7 and p2= 1.
Figure 5.7: The c.d.f. of CTR’s with waking/sleeping nodes and listening links.
Table 5.4: The average CTR’s corresponding to Figure 5.7.
Figure 5.7(a): p1 = 0.9, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1557 0.1704 0.1346 15.71% 26.62%
400 0.0860 0.0910 0.0759 13.28% 19.83%
1600 0.0468 0.0486 0.0418 11.86% 16.24%
Figure 5.7(b): p1 = 0.8, p2 = 1
n Riso Rcon Rth DRiso DRcon 100 0.1662 0.1815 0.1427 16.42% 27.15%
400 0.0918 0.0972 0.0806 13.93% 20.70%
1600 0.0501 0.0519 0.0444 12.95% 16.98%
Figure 5.7(c): p1 = 0.7, p2 = 1
n Riso Rcon Rth DRiso DRcon
100 0.1786 0.1936 0.1526 17.06% 26.88%
400 0.0980 0.1032 0.0861 13.81% 19.81%
1600 0.0537 0.0557 0.0474 13.20% 17.34%
b a
(a) riso= |ab| = 0.097235.
b a
(b) rcon= |ab| = 0.135864.
Figure 5.8: The r-disk graph with unreliable nodes and links.
with r = ka − bk. Figure 5.8(b) is the network with connectivity, and ab is the longest edge and with length 0.135864 that is corresponding to the CTR such that black nodes and solid edges form a connected graph, and the graph is plotted with r = ka − bk. Figure 5.9 illustrates the c.d.f. of CTR’s corresponding to p1 = 0.9, and respectively p2 = 0.9, p2 = 0.8, and p2 = 0.7.
The average CTR’s and inaccuracies of Riso, Rcon, and Rth are listed in Table 5.5.
In addition, we consider another scenario in which every nodes independently stay in waking state with probability p1 and in sleeping state with probability 1 − p1, instead of breaking down. For such networks, a node is isolated if it doesn’t have an up link connecting to black node, and a network is connectied if awake nodes are connected by solid edge and every sleeping nodes have at least one solid edge. Figure 5.10 illustrates the c.d.f. of CTR’s corresponding to p1 = 0.9, and respectively p2 = 0.9, p2 = 0.8, and
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) p1= 0.9 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) p1= 0.8 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(c) p1= 0.7 and p2= 1.
Figure 5.9: The c.d.f. of CTR’s with unreliable nodes and links.
Table 5.5: The average CTR’s corresponding to Figure 5.8.
Figure 5.9(a): p1 = 0.9, p2 = 0.9
n Riso Rcon Rth DRiso DRcon 100 0.1619 0.1753 0.1404 15.33% 24.88%
400 0.0902 0.0943 0.0794 13.58% 18.78%
1600 0.0494 0.0509 0.0438 12.81% 16.10%
Figure 5.9(b): p1 = 0.9, p2 = 0.8
n Riso Rcon Rth DRiso DRcon 100 0.1704 0.1823 0.1489 14.43% 22.42%
400 0.0963 0.0991 0.0842 14.33% 17.69%
1600 0.0522 0.0532 0.0465 12.30% 14.50%
Figure 5.9(c): p1 = 0.9, p2 = 0.7
n Riso Rcon Rth DRiso DRcon
100 0.1859 0.1945 0.1592 16.79% 22.22%
400 0.1024 0.1048 0.0900 13.71% 16.44%
1600 0.0563 0.0570 0.0497 13.43% 14.68%
p2 = 0.7.
The average CTR’s and inaccuracies of Riso, Rcon, and Rth are listed in Table 5.6.
Table 5.6: The average CTR’s corresponding to Figure 5.10.
Figure 5.10(a): p1 = 0.9, p2 = 0.9
n Riso Rcon Rth DRiso DRcon
100 0.1651 0.1778 0.1419 16.41% 25.35%
400 0.0911 0.0950 0.0801 13.74% 18.62%
1600 0.0497 0.0507 0.0441 12.63% 14.98%
Figure 5.10(b): p1 = 0.9, p2 = 0.8
n Riso Rcon Rth DRiso DRcon 100 0.1765 0.1856 0.1505 17.32% 23.35%
400 0.0964 0.0991 0.0849 13.55% 16.73%
1600 0.0525 0.0534 0.0468 12.30% 14.07%
Figure 5.10(c): p1 = 0.9, p2 = 0.7
n Riso Rcon Rth DRiso DRcon 100 0.1889 0.1966 0.1609 17.43% 22.21%
400 0.1039 0.1057 0.0908 14.44% 16.49%
1600 0.0568 0.0573 0.0500 13.60% 14.59%
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) p1= 0.9 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) p1= 0.8 and p2= 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(c) p1= 0.7 and p2= 1.
Figure 5.10: The c.d.f. of CTR’s with waking/sleeping nodes and unreliable links.
5.5 Secure Wireless Networks
The last simulations are for the m-composite key predistribution scheme [19] [20] [21].
In secure networks with key pool size K and key ring size k, at least m common keys are required for each pair of nodes to establish secured links. For example, Figure 5.11 is two secure networks over the same 200 random nodes but with defferent value of r’s.
a b
(a) riso= |ab| = 0.106341.
b a
(b) rcon= |ab| = 0.118285.
Figure 5.11: The secure networks with K = 40, k = 10, and m = 2.
Two nodes at the same position in Figure 5.11(a) and Figure 5.11(b) respectively own the same key ring which are randomly drawn from the key pool. Solid lines are secured links, and dashed lines are unsecured links. The edge ab marked by red line is corresponding to the CTR of each network. In the simulations, we assume K = 40 and k = 10. To focus our attention on the effect of the key predistribution scheme, we assume all nodes are active, i.e. p1 = 1. Figure 5.12 illustrates the c.d.f. of CTR’s corresponding to m = 1 and m = 2, respectively.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(a) m = 1.
p (%)
0 0.1 0.2 0.3
100
50 75
25
10 90
radius
(b) m = 2.
Figure 5.12: The c.d.f. of CTR’s of secure networks.
Table 5.7 shows the average CTRs and inaccuracies of Riso, Rcon, Rth. Note that Table 5.7: The average CTR’s corresponding to Figure 5.12.
Figure 5.12(a): K = 40, k = 10, m = 1 n Riso Rcon Rth DRiso DRcon 100 0.1482 0.1627 0.1300 13.97% 25.15%
400 0.0824 0.0874 0.0734 12.34% 19.18%
1600 0.0449 0.0467 0.0404 11.03% 15.57%
Figure 5.12(b): K = 40, k = 10, m = 2 n Riso Rcon Rth DRiso DRcon 100 0.1580 0.1698 0.1431 10.42% 18.62%
400 0.0879 0.0915 0.0808 08.83% 13.28%
1600 0.0482 0.0493 0.0445 08.31% 10.88%
K = 40, k = 10, and m = 1 is equivalent to p = 0.964555; and K = 40, k = 10, and m = 2 is equivalent to p = 0.795771.
5.6 Conclusions
The connectivity of wireless networks in which nodes and links are not reliable is investigated in this study by the distribution of the number of isolated nodes in the networks. We assume a wireless network is composed of a collection of wireless devices represented by an uniform point process or Poisson point process over the unit-area disk
or square. In the Bernoulli model, nodes are active independently with probability 0 <
p1 ≤ 1, and links are up independently with probability 0 < p2 ≤ 1.
We show result that if all nodes have the same transmission radius rn =
qln n+ξ πp1p2n for some constant ξ, then the total number of isolated nodes is asymptotically Poisson with mean e−ξ and the total number of isolated active nodes is also asymptotically Poisson with mean p1e−ξ. In the m-composite key predistribution schemes, let p denote the probability of the event that two neighbor nodes have a secure link. We show that if all nodes have the same transmission radius rn =
qln n+ξ
πpn for some constant ξ, then the total number of isolated nodes is asymptotically Poisson with mean e−ξ.
The convergence of the asymptotic CTR was verified by extensive simulations. Dif-ferent network models were considered, and the average and c.d.f. of CTR’s were in-vestigated. The problem whether vanishment of isolated nodes almost surely implies connectivity of networks or not is still open.
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