A secure and robust multilayer network with optimum inter layer links under budget constraints

Abstract

Most of the complex systems consist of multiple subsystems and can be modeled as multilayer networks. These networks are prone to random or strategical attacks and end up disintegrating the entire multilayer network. These attacks can’t be avoided, but a model can be proposed to restore the network to make it secure and robust. In multilayer networks, each inter-layer link has its own cost (in terms of money) associated with it. The total cost of the inter-layer links is considered as the available budget. In the present work, a method is proposed to introduce the optimal number of inter-layer links (under budget constraints) to maintain the robustness of the multilayer network. For the simulation purpose, three variants of the artificial multilayer networks and data set EU-Air Transport Networks are considered. Simulation results reveal that for the multilayer network constructed with random network layers, approx. 65% of the available budget is utilized. Nearly 70% of the total nodes are connected to a mutually connected giant component (MCGC) via inter-layer links. However, for the Configuration model, using the almost total available budget, nearly all the nodes from the considered network layers are present in MCGC. Finally, in the case of the empirical dataset, by using approx. 75% of the available budget, almost 80% of the total nodes from the considered network layers are connected to MCGC via inter-layer links.

Appendix

The optimization problem of determining the number of optimal inter-layer links is mapped to the Knapsack-Problem. The structure of the considered problem is described as follows: Weights of the items are the cost of inter-layer links. The number of items is inter-layer links. The total cost of the items is the available budget for the considered inter-layer links. Thus, the optimization problem is to determine the optimal number of inter-layer links connecting the nodes from the network layers that maximize the total degrees of nodes at respective network layers. Higher degree nodes at the different network-layer enable to increase the size of MCGC (ϕ) and enhance the robustness of the considered multilayer networks.

The method to obtain optimal set $\mathcal {E}^{\star }_{\alpha ,{\beta }}$ of ILLs is presented as follows:

Let $V_{\alpha }^{\star } \subseteq {V}_{\alpha }$ and $V_{\beta }^{\star } \subseteq {V}_{\beta }$ be the optimal sets of nodes such that degree $k_{i}^{\alpha }$ of $i_{\alpha } \in V_{\alpha }^{\star }$ and $k_{i}^{\beta }$ of $i_{\beta } \in V_{\beta }^{\star }$ maximize the values of Q_α and Q_β respectively. Then, optimal set of ILLs $\mathcal {E}^{\star }_{\alpha ,{\beta }}$ is obtained from $\mathcal {E}_{\alpha ,{\beta }}$ by choosing the ILLs connecting the node $i_{\alpha } \in V_{\alpha }^{\star }$ to the node $j_{\beta } \in V_{\beta }^{\star }$ which maintains the robustness ϕ of the MLN. Therefore, the considered problem is formulated into an optimization problem with objective function given by,

$$ \begin{array}{@{}rcl@{}} &&{Maximize} {\sum}_{i=1}^{l}k_{i}^{\alpha} x_{i}^{\alpha}{}\\ &&Subject to {\sum}_{i=i}^{l} c_{i}^{\alpha,{\beta}}x_{i}^{\alpha}\leq b\\ &&c_{i}^{\alpha,{\beta}}\geq 0 , c_{i}^{\alpha,{\beta}}\leq C* \eta_{max} \end{array} $$

(8)

where, $x_{i}^{\alpha } \in \{0,1\}$ is a decision variable indicating the decision of choosing a node i_α with degree $k_{i}^{\alpha }$ from the set V_α and l is the number of nodes in the set ${V}^{\star }_{\alpha }$. Optimal set $V_{\beta }^{\star }$ is obtained in similar way. Sets of nodes $V_{\alpha }^{\star }$ and $V_{\beta }^{\star }$ are obtained by using the Algorithm 1. A brief description of the Algorithm 1 is provided below (Fig. 13).

• The Algorithm 1 takes four input parameters, namely set of costs of ILLs ($\mathcal {C}$), set of the degree of nodes ($\mathcal {D}$), available budget (b) and number of nodes N and returns the optimal set of nodes V^⋆.

• In step 1, two tables T[N,$\mathcal {C}$], K[N,$\mathcal {C}$] are initialized with all the entries 0. Table T is updated with the values of the cost of ILLs depending upon the conditions in steps 6-10 and table K is updated with entries of 1.

• Variable c_i holds the cost of i th ILL, while d_i keeps the degree of i th node as shown in steps 4 and 5 respectively.

• In step 6, if the condition holds true then tables T and K are updated in step 7 and 8 respectively.

• In the steps 13-17, optimal set of nodes V^⋆ is obtained according to (6).

Table 11 Meaning of the acronyms used in the manuscript

Now, the working of Algorithm 1 is provided with an example. Let D = {2, 3, 4}, $\mathcal {C}=\{1,2,3\}$, N = 3 and b = 3. For i = 1, c₁ = 1, c = 0 and d₁ = 2 and condition in step 6 is not satisfied. Therefore according to step 7, T[1,0]= 0. Now, for i = 1 (in the outer loop in step 2), c₁ = 1, c = 1 and d₁ = 2 (inner loop in step 3) the condition in step 6 is satisfied and T[1,1]= 2 and K[1,1]= 1. In this way, in each iteration, the values of the tables T and K are updated. The optimal set V^⋆ is recovered from the table K in the following way. According to step 12, the value of $b^{\prime }=3$ and K[3,3]= 0. Hence, the condition in the step 14 is not satisfied and V^⋆ is not appended. Now, the value of i = 2 and K[2,3]= 1. Therefore, the condition in the step 14 is satisfied and V^⋆ is updated with 2 and so on. Thus, for the considered example, V^⋆ = {1, 2}.

Fig. 13

Flow chart of Algorithm 1. Input parameters are, set of costs of ILLs ($\mathcal {C}$), set of the degree of nodes ($\mathcal {D}$), available budget (b) and number of nodes N. Output is the optimal set nodes V^⋆

The meaning of acronyms and variables used in the paper are mentioned in Tables 11 and 12 respectively.

Table 12 Meaning of the variables used the manuscript