# Cyber–Physical Correlation Effects in Defense Games for Large Discrete Infrastructures

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## Abstract

**:**

## 1. Introduction

- (a)
- knowledge about the infrastructure is available to the attacker which is sufficient to launch component attacks;
- (b)
- costs of attacks and reinforcements of components, denoted by ${L}_{A}({y}_{C},{y}_{P})$ and ${L}_{D}({x}_{C},{x}_{P})$, respectively, are not available to the other player;
- (c)
- components chosen by the provider to reinforce, and by the attacker to attack, are not revealed; and
- (d)
- incidents and results of attacks on components are known to the provider and attacker.

## 2. Related Work

## 3. Discrete System Models

#### 3.1. Cyber–Physical Structural Interactions

**Condition**

**1.**

**Cyber–Physical Correlation Function:**The survival probability a CPI is given by

- (a)
- OR Systems: A special class called the OR systems are defined in [4,5] to illustrate cases where cyber and physical parts can be independently analyzed. For these systems, the probability of failure of cyber or physical sub-infrastructure is ${P}_{\overline{C}\cup \overline{P}}={P}_{\overline{C}}+{P}_{\overline{P}}$ or equivalently ${P}_{\overline{C}\cap \overline{P}}=0$. That is, the failure of the physical sub-infrastructure is guaranteed not to cause the failure of the cyber sub-infrastructure. Thus, we have ${P}_{CP}={P}_{C}+{P}_{P}-1$ and $f\left(\right)open="("\; close=")">{P}_{C},{P}_{P}$. These systems are of mostly academic interest.
- (b)
- Linear Forms: The linear form$$f({P}_{C},{P}_{P})={a}_{C}(1-{P}_{C})+{b}_{C}$$
- (i)
- Statistical Independence: We have $f\left(\right)open="("\; close=")">{P}_{C},{P}_{P}$. That is, ${a}_{C}=1$ and ${b}_{C}=0$, so that ${P}_{\overline{C}\cap \overline{P}}={P}_{\overline{C}}{P}_{\overline{P}}$ or equivalently ${P}_{CP}={P}_{C}{P}_{P}$, and
- (ii)
- Failure Certainty: When physical failures lead to cyber failures with certainty, we have $f\left(\right)open="("\; close=")">{P}_{C},{P}_{P}$. That is, ${a}_{C}=0$ and ${b}_{C}=1$, such that ${P}_{CP}={P}_{C}$ (i.e., infrastructure survival probability solely depends on cyber sub-infrastructure).

More generally, if ${a}_{C}>1$ and ${b}_{C}\ge 0$, or ${a}_{C}\ge 1$ and ${b}_{C}>0$, the cyber failures are positively correlated to physical failures. That is, they occur with higher probability following physical failures (i.e., ${P}_{\overline{C}|\overline{P}}>{P}_{\overline{C}}$). If ${a}_{C}<1$ and ${b}_{C}\le 0$, or ${a}_{C}\le 1$ and ${b}_{C}<0$, i.e., $f\left(\right)open="("\; close=")">{P}_{C},{P}_{P}$, cyber failures are negatively correlated to physical failures (i.e., ${P}_{\overline{C}|\overline{P}}<{P}_{\overline{C}}$).

**Condition**

**2.**

**De-Coupled Reinforcement Effects:**The partial derivatives of ${P}_{CP}$ in Condition 1 satisfy the following conditions

#### 3.2. Sub-Infrastructure Survival Probabilities

**Condition**

**3.**

**Cyber and Physical Multiplier Functions:**The derivatives of survival probabilities of cyber and physical sub-infrastructures can be expressed as

- (a)
- Statistically Independent Components: Let ${p}_{C|R}$ and ${p}_{C|N}$ denote the conditional survival probability of a cyber component with and without reinforcement, respectively. Under the assumption of statistical independence of component failures, the probabilities that the cyber and physical parts survive the attacks are given by [4]$${P}_{C}={p}_{C|R}^{{x}_{C}}{p}_{C|N}^{{N}_{C}-{x}_{C}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{P}={p}_{P|R}^{{x}_{P}}{p}_{P|N}^{{N}_{P}-{x}_{P}},$$
- (b)
- Contest Survival Functions: The contest survival functions are used to characterize ${P}_{C}$ and ${P}_{P}$ in [42] such that ${P}_{C}=\frac{\xi +{x}_{C}}{\xi +{x}_{C}+{y}_{C}}$, for which we have$$\frac{\partial {P}_{C}}{\partial {x}_{C}}={P}_{C}\left(\right)open="["\; close="]">\frac{{y}_{C}}{(\xi +{x}_{C}+{y}_{C})(\xi +{x}_{C})}$$

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Game-Theoretic Formulation

#### 4.1. Nash Equilibrium Conditions

#### 4.2. OR Systems

#### 4.3. Statistical Independence of Cyber and Physical Sub-Infrastructures

#### 4.4. NE Sensitivity Functions

**Theorem**

**1.**

**Proof:**

**Theorem**

**2.**

**Proof:**

#### 4.5. Sum-Form and Product-Form Utility Functions

#### 4.6. Survival Probabilities of Sub-Infrastructures

## 5. Application Examples

**Condition**

**4.**

#### 5.1. Cloud Computing Infrastructure

#### 5.2. Metro System

#### 5.3. Smart Power Grid Infrastructure

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Symbol | Explanation |
---|---|

${x}_{C},{x}_{P}$ | number of cyber and physical components reinforced, respectively |

${y}_{C},{y}_{P}$ | number of cyber and physical components attacked, respectively |

${P}_{CP}\left(\right)open="("\; close=")">{x}_{C},{x}_{P},{y}_{C},{y}_{P}$ | survival probability of the infrastructure |

${P}_{C}$, ${P}_{P}$ | marginal survival probabilities of cyber and physical sub-infrastructures, respectively |

$f({P}_{C},{P}_{P})$ | failure correlation function (i.e., the failure probability of cyber sub-infrastructure given the other’s failure) |

${\mathrm{\Lambda}}_{C}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$, ${\mathrm{\Lambda}}_{P}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | multiplier functions of cyber and physical sub-infrastructures |

${U}_{D}\left(\right)open="("\; close=")">{x}_{C},{x}_{P},{y}_{C},{y}_{P}$, ${U}_{A}\left(\right)open="("\; close=")">{x}_{C},{x}_{P},{y}_{C},{y}_{P}$ | provider’s and attacker’s composite utility function, respectively |

${F}_{D,G}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$, ${F}_{D,L}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | provider’s reward and cost multiplier functions, respectively |

${F}_{A,G}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$, ${F}_{A,L}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | attacker’s reward and cost multiplier functions, respectively |

${g}_{D}\left(\right)open="("\; close=")">{x}_{C},{x}_{P},{y}_{C},{y}_{P}$ | reward for rendering the infrastructure operational in the provider’s sum-form utility function |

${L}_{D}({x}_{C},{x}_{P})$, ${L}_{A}({y}_{C},{y}_{P})$ | provider’s and attacker’s total cost of cyber and physical attacks, respectively |

${G}_{D}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$, ${G}_{A}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | provider’s and attacker’s reward, respectively |

${a}_{C}$, ${b}_{C}$ | coefficients in the linear correlation function |

${p}_{C|R}$, ${p}_{C|N}$ | conditional survival probability of a cyber component with and without reinforcement, respectively |

${p}_{P|R}$, ${p}_{P|N}$ | conditional survival probability of a physical component with and without reinforcement, respectively |

${p}_{C|R}^{i}$, ${p}_{P|R}^{j}$ | survival probabilities of reinforced cyber component of type i and reinforced physical component of type j, respectively |

${p}_{C|N}^{i}$, ${p}_{P|N}^{j}$ | survival probabilities of cyber component of type i and physical component of type j without reinforcement, respectively |

${N}_{C}^{i}$, ${N}_{P}^{j}$ | number of type i cyber components and type j physical components, respectively |

$\xi $ | coefficient of inherent robustness of cyber component |

$\alpha $ | coefficient representing a partial effect of cyber–physical correlation |

${N}_{L}$ | number of trains running on a line, or the number of sensors connected using a communication node |

${N}_{S}$ | number of servers connected through a fiber |

${f}_{P}$ | normalization factor in the survival probability of metro system and smart power grid infrastructure |

${f}_{C}$ | normalization factor in the survival probability of cloud computing infrastructure |

${L}_{G,L}^{D}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | composite gain–cost term |

${F}_{G,L}^{D,B}({x}_{C},{x}_{P},{y}_{C},{y}_{P})$ | provider’s gain–cost gradient with respect to ${x}_{B}$, where $B=C,P$, for cyber and physical components, respectively |

${\mathsf{\Theta}}_{C}\left(\xb7\right)$, ${\mathsf{\Theta}}_{P}\left(\xb7\right)$ | cyber and physical scaled gain–cost gradients, respectively |

${x}_{C}^{T}$, ${x}_{C}^{S}$ | number of reinforced control centers and signals in metro system, respectively |

${x}_{C}^{S}$, ${x}_{C}^{R}$ | number of reinforced servers and routers in cloud computing infrastructure, respectively |

${x}_{C}^{S}$, ${x}_{C}^{M}$ | number of reinforced communication nodes and smart meters in smart power grid infrastructure, respectively |

${P}_{A}^{S}$, ${P}_{A}^{M}$ | probabilities of an attack on a communication node and smart meter in smart power grid infrastructure, respectively |

${\mathit{F}}_{\mathit{D},\mathit{G}}$ | ${\mathit{G}}_{\mathit{D}}$ | ${\mathit{F}}_{\mathit{D},\mathit{L}}$ | |
---|---|---|---|

sum-form: ${U}_{D+}$ | $\left(\right)$ | ${g}_{D}$ | 1 |

product-form: ${U}_{D\times}$ | 0 | 0 | $\left(\right)$ |

**Table 3.**Gain and cost terms and their multipliers for sum-form and product-form utilities of the provider.

${\mathit{F}}_{\mathit{D},\mathit{G}}$ | ${\mathit{G}}_{\mathit{D}}$ | ${\mathit{F}}_{\mathit{D},\mathit{L}}$ | ${\mathit{L}}_{\mathit{D}}$ | $\frac{\mathit{\partial}{\mathit{F}}_{\mathit{D},\mathit{G}}}{\mathit{\partial}{\mathit{P}}_{\mathbf{CP}}}$ | $\frac{\mathit{\partial}{\mathit{G}}_{\mathit{D}}}{\mathit{\partial}{\mathit{x}}_{\mathit{B}}}$ | $\frac{\mathit{\partial}{\mathit{F}}_{\mathit{D},\mathit{L}}}{\mathit{\partial}{\mathit{P}}_{\mathbf{CP}}}$ | |
---|---|---|---|---|---|---|---|

sum-form: ${U}_{D+}$ | $\left(\right)$ | ${g}_{D}$ | 1 | ${L}_{D}$ | −1 | 0 | 0 |

product-form: ${U}_{D\times}$ | 0 | 0 | $\left(\right)$ | ${L}_{D}$ | 0 | 0 | −1 |

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## Share and Cite

**MDPI and ACS Style**

Rao, N.S.V.; Ma, C.Y.T.; He, F.; Yau, D.K.Y.; Zhuang, J.
Cyber–Physical Correlation Effects in Defense Games for Large Discrete Infrastructures. *Games* **2018**, *9*, 52.
https://doi.org/10.3390/g9030052

**AMA Style**

Rao NSV, Ma CYT, He F, Yau DKY, Zhuang J.
Cyber–Physical Correlation Effects in Defense Games for Large Discrete Infrastructures. *Games*. 2018; 9(3):52.
https://doi.org/10.3390/g9030052

**Chicago/Turabian Style**

Rao, Nageswara S. V., Chris Y. T. Ma, Fei He, David K. Y. Yau, and Jun Zhuang.
2018. "Cyber–Physical Correlation Effects in Defense Games for Large Discrete Infrastructures" *Games* 9, no. 3: 52.
https://doi.org/10.3390/g9030052