Fig. 1.1 Simple architecture model of a cellular network and terminology employed (cell, terminal, base station, coverage area). 10

Fig. 1.2 Network formation tradeoff: cost versus benefit of collaboration. For the network tradeoff, the total cost and total gain, summed over all entities are considered. 17

Fig. 2.1 Examples of node infection models of interest. 38

Fig. 3.1 Simple epidemic model: SI infection paradigm for each member of the population. 42

Fig. 3.2 Simple epidemic model: Percentage of infected hosts as a time function. 43

Fig. 3.3 Kermack-McKendrick: Underlying infection model. 44

Fig. 3.4 State transitions in the two-factor spreading model. 47

Fig. 3.5 Two-factor model: numbers of infected and removed hosts. 49

Fig. 3.6 General epidemics infection model—state transition diagram. 56

Fig. 4.1 Mapping of malware diffusion problem to the behavior of a queuing system. The shaded nodes are susceptible legitimate neighbors of node i. The colored nodes are either malicious nodes or legitimate already infected neighbors of i. Node i is considered susceptible at the moment. 66

Fig. 4.2 Closed queuing systems modeling malware diffusion over a wireless SIS network. 68

Fig. 4.3 The Norton equivalent model for malware propagation in communications networks. The figure shows the instance where k nodes are currently infected. 70

Fig. 4.4 State diagram for the analysis of the two-queue closed network and for obtaining the expression of the steady-state distribution. 73

Fig. 4.5 Probability of no infected nodes πI(0). 77

Fig. 4.6 Probability of all nodes infected πI(N). 77

Fig. 4.7 Average number of infected nodes E[LI] versus λ/μλ/µ. 78

Fig. 4.8 Average number of infected nodes E[LI] versus legitimate N and malicious M nodes. 79

Fig. 4.9 Average throughput of noninfected queue E[γS]E[γs] versus legitimate N and malicious M nodes. 80

Fig. 4.10 Average throughput of noninfected queue E[γS]E[γs] versus infection λ and recovery μ rate. 81

Fig. 4.11 Norton equivalent of the closed queuing network model for a propagative system. Compared to Fig. 4.3, there is a difference in the infection rate due to the impact of attacker. 82

Fig. 4.12 Probability of zero nodes infected πI(0) (accurate-approximated). 85

Fig. 4.13 Probability of all nodes infected πI(N) versus λ/μλ/µ. 86

Fig. 4.14 Probability of all nodes infected πI(N) versus R. 87

Fig. 4.15 Average number of infected nodes E[LI] versus λ/μλ/µ. 88

Fig. 4.16 Average number of infected nodes E[LI] versus R. 89

Fig. 4.17 Average number of infected nodes E[LI] versus N. 90

Fig. 4.18 Average throughput of the noninfected queue E[γS]E[γs] versus λ. 91

Fig. 4.19 Average throughput of the noninfected queue E[γS]E[γs] versus R. 92

Fig. 4.20 Average throughput of the noninfected queue E[γS]E[γs] versus N. 93

Fig. 4.21 State-transition diagram for legitimate nodes in a network with churn. 95

Fig. 4.22 Queuing models for malware spreading in networks with churn. 97

Fig. 4.23 Percentage of susceptible and infected nodes versus network infection/ recovery strength and comparison with networks with no churn for complex networks. 102

Fig. 4.24 Expected percentage of susceptible, infected, and recovering nodes versus infection/recovery strength (simulation) for complex networks with 400 and 800 initial nodes. 103

Fig. 4.25 Percentages of susceptible and infected nodes as a function of infection to recovery strength (numerical) for wireless distributed (multihop) networks. 104

Fig. 4.26 Expected number of nodes in each state of a wireless distributed (multihop) network with respect to network density. 105

Fig. 4.27 Expected number of nodes in each state of a wireless distributed (multihop) network with respect to infection/recovery rates. 106

Fig. 4.28 Expected percentage variation of the total number of nodes with respect to node density and infection/recovery strength for wireless distributed (multihop) networks. 106

Fig. 5.1 Random Field (RF) terminology over a random network of n + 1 sites and three phases. 109

Fig. 5.2 Examples of complex network topologies of interest. 116

Fig. 5.3 Examples of neighborhood for the darkly blue shaded (black in print versions) node (site) s in topologies of interest. 118

Fig. 5.4 SIS malware-propagative chain network and MRF notation. 119

Fig. 5.5 Steady-state system distributions for T/JT/J = –0.2. 123

Fig. 5.6 Expected number of infected nodes. 124

Fig. 5.7 Lattice network and MRF malware diffusion model notation. 126

Fig. 5.8 ER random networks and malware modeling MRFs. 130

Fig. 5.9 MRF malware diffusion modeling for WS SW networks. 131

Fig. 5.10 MRF malware diffusion modeling for SF networks. 132

Fig. 5.11 MRF malware diffusion modeling for random geometric (multihop) networks. 133

Fig. 5.12 Scaling of percentage of infected nodes with respect to network density: the sparse network regime. 135

Fig. 5.13 Scaling of percentage of infected nodes with respect to network density: the moderate-density regime. 135

Fig. 5.14 Scaling of percentage of infected nodes with respect to network density: the dense network regime. 136

Fig. 6.1 Transitions: S, I, R, D, respectively, represent fraction of the susceptible, infective, recovered, and dead. v(t) is the dynamic control parameter of the malware. 144

Fig. 6.2 Evaluation of the optimal controller and the corresponding states as functions of time. The parameters are time horizon: T=10T=10, initial infection fraction: I₀ = 0.1, contact rate: β=0.9β = 0.9, instantaneous reward rate of infection for the malware: f(I)=0.1If(I) = 0.1I, reward per each killed node: κ=1. Also, we have taken Q(S,I)≡0.2, and B(S,I)≡0 in the left and B(S,I)≡0.2 in the right figures. That is, in the left figure, patches can only immunize the susceptible nodes but in the right figure, the same patch can successfully remove the infection, if any, and immunize the node against future infection. We can see that when patching can recover the infective nodes too (right figure), then the malware starts the killing phase earlier. This makes sense as deferring the killing in the hope of finding a new susceptible is now much riskier. 149

Fig. 6.3 The jump (up) point of optimal v, i.e. the starting time of the slaughter period, for different values of the patching and rates. For both curves, we have taken the recovery rate of the susceptible nodes, i.e. Q(S, I) as γ, and the recovery rate of the infective nodes, i.e. B(S, I), once as zero and once as the same as Q(S, I) where γ is varied from 0.02 to 0.7 with steps of 0.02. The rest of the parameters are f(I)=0.1If(I) = 0.1I, κ=1, T=10, β=0.9β = 0.9, and I0=0.1. Note that when B(S,I)≡γ, then for γ ≥ 0.6, the malware starts killing the infective nodes from time zero. 150

Fig. 7.1 State transitions. uNi(t) and uNr(t) are the control parameters of the network while uM(t) is the control parameter of the malware. 159

Fig. 7.2 State evolution and saddle-point strategies. The parameters of the game are as follows: κI=10, κD=13, κu=10, κr=5, KI=KD=0, β2=β1=β0=4.47, π=1, and initial fractions I0=0.15,R0=0.1,D0=0, and T=4. 167

Fig. 9.1 Methodology for studying optimal attacks. 183

Fig. 9.2 Zero-level contours of g(x1,x2). 187

Fig. 9.3 Optimal E[LI] versus λ/μ. 188

Fig. 9.4 Optimal E[LI] versus N. 189

Fig. 9.5 Optimal E[γS]E[γs] versus (λ,μ). 190

Fig. 9.6 Optimal E[γS]E[γs] versus N. 191

Fig. 9.7 Contemporary wireless complex communication network architecture depicting all the considered and converged types of networks, including interconnections to wired backhauls. 194

Fig. 9.8 IDD in regular lattice (HoMPC;pw=0), ER (HoMEC;pw=1), SF (HeMUC), and SW (HoMUC) networks for different pw with mean degree equal to 10, λ=0.01, and N = 2500. 203

Fig. 9.9 IDD in dynamic MANET with R=2m, λ=1, and k¯=2.51,1.26,0.63,and0.13, respectively. 206

Fig. 9.10 The IDD in wireless complex networks (cyber-physical systems) consisting of both long-range and broadcast dissemination patterns. 208

Fig. 9.11 IDD in hybrid (HoMEC and HoMPC) complex networks of propagating information in both delocalized and broadcast fashions, where ke¯=6,kb¯=3, and λ=0.05λ = 0.05. 209

Fig. 9.12 Average number of infected users of the legitimate network E[LI] as a function of λ/μ. 212

Fig. 9.13 Average number of infected nodes E[LI] as a function of N (numerical result). 213

Fig. B.1 A generic independent queuing system. 236

Fig. B.2 Graphical presentation of the relation between the arrival-departure counting processes and visual explanation of Littleʼs law. 239

Fig. B.3 State diagram for the birth-death process. 243

Fig. B.4 Two queues in tandem. 249

Fig. B.5 A simple two-queue closed network. 251

Fig. B.6 State diagram of a two-queue closed queuing network with state-dependent service rates. 253

Fig. C.1 Analogy between functions, functionals, and extreme values. 260