The given parameters and decision variables of the problem are shown in Table 2-6 and Table 2-7.
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Table 2-6: Given Parameters Given parameter
Notation Description
V
Index set of nodes of both players’ network topologiesV
A Index set of nodes of player A’s network topologyV
B Index set of nodes of player B’s network topologyR
Index set of rounds in the attack and defense actionsV
ArIndex set of nodes of player A’s network topology in the range of player B’s combined local view in round r, where r ∈ R and VAr
⊆ V
AV
BrIndex set of nodes of player B’s network topology in the range of player A’s combined local view in round r, where r ∈ R and VBr ⊆ VB
E
ArIndex set of new explored nodes of player A’s network topology in round r, where r ∈ R and EAr
⊆ V
A-V
A-1E
BrIndex set of new explored nodes of player B’s network topology in round r, where r ∈ R and EBr
⊆ V
B-V
B-1K
A Index set of collaborative attackers of player AK
B Index set of collaborative attackers of player Bg
Ar Player A’s group of collaborative attackers in round r-1, where gAr ∈ KAand
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r ∈ R
g
BrPlayer B’s group of collaborative attackers in round r-1, where gBr ∈ KB
and r ∈ R
w
r The weight of the average DOD in round r, where r ∈ R Total budget of player ATotal budget of player B
T
kiAttacker k’s capacity of attack power on node i, where k ∈ KA
⋃ K
B and i∈ VAr
⋃ V
Brt
kiAttacker k’s attack power when attacking on node i in the last round, where
k ∈ g
Ar⋃ g
Br and i ∈ VAr⋃ V
BrC
kn The cooperative effect of attacker k with attacker n, where k, n ∈ KA⋃K
BL
rThe leadership of the leader in round r, which could be positive or negative, where r ∈ R
θ
i Existing defense resource allocated on node i, where i ∈ Ve
riRepair cost of players when node i is dysfunctional in round r-1, where i ∈
V
and r ∈ Rd
riThe discount rate of the resources that players reallocate on node i in round
r, where i ∈ V
and r ∈ R58
f
riThe reinforcement rate above 1 represents the players reallocate more resources on the repaired node i in round r, where i ∈ Vand r ∈ R
l
kriExploration cost of attacker k when information of node i is updated in round r, where k ∈ gAr
⋃ g
Br, i ∈ EAr⋃ E
Br, and r ∈ R. The cost is greatlyincreasing if the node is not adjacent with combined local view.
h
kriThe discount rate of attacker k’s attack power over node i in round r, which is gained from exploring, where k ∈ gAr
⋃ g
Br, i ∈ EAr⋃ E
Br, and r ∈ Ru
i The reward of compromising node i, where i ∈ Vδ
ri1 if node i is compromised in round r-1, 0 otherwise, where i ∈ VAr
⋃ V
Br and r ∈ Rσ
rThe discount rate of the attack power in round r (after-strike effect) after the PS of another player in round r-1, where r ∈ R
α
Player A’s weight of ADODβ
Player B’s weight of ADODλ
k The attenuation factor of the collaborative attacker k, where k ∈ KA⋃ K
Bο
k The recovery factor of the collaborative attacker k, where k ∈ KA⋃ K
B59
Table 2-7: Decision Variables Decision variable
Notation Description
y
riProactive defense budget allocation on node i in round r, where i ∈ V and r ∈ R
z
riPS defense budget allocation on node i in round r, where i ∈ VAr
⋃ V
Br and r ∈ Rρ
ArPlayer A’s group of collaborative attackers in round r, where ρAr ∈ KA
and r ∈ R
ρ
BrPlayer B’s group of collaborative attackers in round r, where ρBr ∈ KB
and r ∈ R
s
ri1 if node i is repaired by another player in round r, 0 otherwise, where i
∈ VAr
⋃ V
Br and r ∈ RA
r Total budget of player A in round r, where r ∈ RB
r Total budget of player B in round r, where r ∈ RA
rPlayer A’s budget allocation, which is a vector of cost A1, A2 to Ar in round r, where i ∈ V and r ∈ R
B
r Player B’s budget allocation, which is a vector of cost B1, B2, to Br in60
round r, where i ∈ V and r ∈ R
Player A’s Average DOD, which is considered under player A’s and player B’s budget allocation on player A’s network topology in round r, where r ∈ R
Player B’s Average DOD, which is considered under player A’s and player B’s budget allocation on player B’s network topology in round r, where r ∈ R
By using the above given parameters and decision variable, we then formulate the problems as the following min-max problems respectively for player A and player B:
Objective functions:
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Explanation of the objective functions：
(IP 1) This is the objective function of player A. The ratio of player A’s ADOD dividing by player B’s ADOD is considered in the objective function. For player A, the ratio would be the smaller the better. Furthermore, ɑ and β represent the importance of player A and player B respectively in each round. The objective function of player A is to minimize the maximum sum of the product of the ratio and weight in each round. The important degree of Average DOD value in each round is usually different, so the weight would be assigned to the Average DOD value in each round in this model.
(IP 1’) This is the objective function of player B. The ratio of player B’s ADOD dividing by player A’s ADOD is considered in the objective function. For player B, the ratio would be the smaller the better. Furthermore, ɑ and β represent the importance of player A and player B respectively in each round. The objective function of player B is to minimize the maximum sum of the product of the ratio and weight in each round. The important degree of Average DOD value in each round is usually different, so the weight would be assigned to the Average DOD value in each round in this model.
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Explanation of the constraint function：
(IP1.1) The constraint is for player A. In the left side of the inequality, it
represents the total budget of defense budget and attack budget in that round.
∈VA
i
y describes the budget of proactive defense,
ri
∈VA
the budget to repair compromised nodes combined with reactive defense
by reinforcing more defense budget, and the PS defense budget is expressed as
∈VBr
i
z ; in addition, the total attack budget in that round
rishould consider many factors. First, the cycle of attenuation and recovery of each collaborative attacker is expressed as
t
kiλ
k andt
kio
krespectively, which is decided on whether to attend the attack in the last round or not. Then, in order to consider the two discounts separately from learning new information in that round and form player B’s PS defense in the last round, we therefore multiply hkri for each collaborative attacker, and in the end multiply total collaborative attackers’ attack power by
σ
r. This complex item represents the total attack power used in that round; finally, the total exploration costs of each collaborative attacker in that round are also added to the sum. The sum in the left side should not exceed the sum of the total budget in that64
round, reallocated budget in that round, and the net rewards gained in that round from the calculation of total rewards obtained from compromising nodes in the last round minus the rewards retrieved back when repairing compromised nodes by player B in that round.
(IP1.2) The constraint is for player B. In the left side of the inequality, it
represents the total budget of defense budget and attack budget in that round.
∈VB
i
y describes the budget of proactive defense,
ri
∈VB
the budget to repair compromised nodes combined with reactive defense
by reinforcing more defense budget, and the PS defense budget is expressed as
∈VAr
i
z ; in addition, the total attack budget in that round
rishould consider many factors. First, the cycle of attenuation and recovery of each collaborative attacker is expressed as
t
kiλ
k andt
kio
krespectively, which is decided on whether to attend the attack in the last round or not. Then, in order to consider the two discounts separately from learning new information in that round and form player A’s PS defense in the last round, we therefore multiply hkri for each collaborative attacker, and in the end multiply total collaborative attackers’ attack power by
σ
r. This complex item represents the total65
attack power used in that round; finally, the total exploration costs of each collaborative attacker in that round are also added to the sum. The sum in the left side should not exceed the sum of the total budget in that round, reallocated budget in that round, and the net rewards gained in this round from the calculation of total rewards obtained from compromising nodes in the last round minus the rewards retrieved back when repairing compromised nodes by player A in that round.
(IP1.3) The constraint is for player A. Describe the sum of the total budgets in each round should not exceed the total budgets of player A.
(IP1.4) The constraint is for player B. Describe the sum of the total budgets in each round should not exceed the total budgets of player B.
(IP1.5) Describe the collaborative attacker’s current attack power over node i after attending the collaborative attack in the last round should not exceed his capacity of attack power on node i.
(IP1.6) Describe the collaborative attacker’s current attack power over node i after taking a rest in the last round should not exceed his capacity of attack power on node i.
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Chapter3 Solution Approach
In this paper, there exist two players. It is notable that the problem of player A and the problem of player B are exactly opposite; furthermore, we would have two Average DOD values to calculate respectively for player A’s and player B’s networks. For both players, they have their own Average DOD value according to the damage degree of their networks. Therefore, how to optimize resource allocation of each node on the network of their counterpart and on the network of their own and use the Average DOD value to evaluate the survivability of their networks are needed to be solved. Hence, the gradient method is adopted to calculate the Average DOD values and to find the optimal resource allocation strategies on each node for both networks. Besides, we would also combine the concept of game theory to find the optimal percentage resource allocation in each round for both players under their individual networks.
The detailed solution procedure would be illustrated in the first section. The concept of gradient method and the detail to calculate the Average DOD value would be introduced in the second section. Moreover, the combination with the notion of
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game theory would be further discussed in the third section. In the end of this chapter, the time complexity of the solution approach would be analyzed.