A fuzzy soft set based novel method to destabilize the terrorist network

Abstract

This paper aims to select the appropriate node(s) to effectively destabilize the terrorist network in order to reduce the terrorist group’s effectiveness. Considerations are introduced in this literature as fuzzy soft sets. Using the weighted average combination rule and the D–S theory of evidence, we created an algorithm to determine which node(s) should be isolated from the network in order to destabilize the terrorist network. The paper may also prove that if its power and foot soldiers simultaneously decrease, terrorist groups will collapse. This paper also proposes using entropy-based centrality, vote rank centrality, and resilience centrality to neutralize the network effectively. The terrorist network considered for this study is a network of the 26/11 Mumbai attack created by Sarita Azad.

Keywords

Terrorist network mining (TNM)destabilization centralities fuzzy soft set social network analysis (SNA)global network efficiency average clustering coefficient Dempster–Shafer theory of evidence

1 Introduction

The problem facing old history is the threat to human civilization from extremists. Technological development has enabled the common people to rise and adversely affect society with the advanced techniques of these inhuman individuals. Law-enforcement agencies seek to deter potential attacks in this context. It is a reason to study and identify terrorist networks. Another effective solution is that law enforcement authorities utilize data mining. Social Network Analysis (SNA), which studies terrorist networks to recognize connections and relationships that may reside among terrorist nodes, is one such data mining method.

Social Network Analysis (SNA) uses the framework hypothesis to disrupt social networks between interpersonal organizations. The project includes hubs (performers, individuals, relationships, etc.) within the system and interactions between hubs (community, family relationships, conversation, money exchange, etc.). Social interactions can be attributed to informal, disconnected organizations (such as companionship, family relationships, correspondence, trade, and many more) or online interpersonal organizations (such as Facebook, Twitter, etc.). Different SNA steps were taken to talk to onscreen characters, inspect strong or weak relations, recognize key/focal actors and subgroups in the system, and detect topology and system consistency. Late SNA has become a standard in variousfields.

Soon after the 9/11 terrorist attack, SNA’s intensity in combating terrorism became prevalent. The SNA is broadly used across information and law enforcement agencies to comprehend and cover pioneers in terrorist and behavioral oppressor processes for the structure and processes of terrorism operations. In counter-terrorism, some common uses of SNA include key player detection, group exploration, interaction analysis, onscreen nodes (discovery personality) and dynamic network assessment, etc. [1].

Terrorist activity is the unlawful act of a certain number of citizens who take many innocent people’s lives. Families and children are deeply affected by systematic brutality, terrorism, and mental distress in the community, such as gunning down, killing civilians, or other attacks. An in-depth analysis of these terrorist attacks shows that people from various countries form a broad-based organization that works on different network layers, and these networks are also of different categories [2, 3].

As underground networks share certain features with civilians (explicit nodes), it is challenging to identify as they conceal their activities. Moreover, the secret/terrorist networks are now merged into many people’s lives, which is a potential problem that causes uncertainty in decision-making. Therefore, it is essential to develop tools to precisely identify people in underground networks to deal with uncertainty and use them more effectively to destabilize them [4].

Soft set theory was introduced by Molodtsov in 1999 [5] as a novel mathematical method for coping with uncertainties. Unlike conventional tools for coping with uncertainty, such as probability theory, fuzzy set theory [6], and rough set theory [7], the strength of soft set theory is that it is not constrained by the inadequacy of those theories’ parametrization tools.

In the past few years, in different engineering fields like forecasting [8], sound quality checking [9], supplier selection [10], simulation and modelling [11], decision making [12, 13], and data analysis [14], almost everywhere, soft set and fuzzy soft set theories have established their presence and applicability. However, with the same fuzzy soft-set decision-making dilemma, numerous results from different assessment bases and adequate level soft sets can be derived, making it hard for policymakers to determine which one is the optimal option. Thus, the most critical problem in solving such a sparse fuzzy soft-set related issue in decision-making [15] is to decrease the uncertainty created by individual personal reasoning to increase the degree of decision-making.

Dempster–proof Shafer’s principle (D-S principle) is defined as an optimal computational intelligence model to fuse information. Dempster first proposed the idea in 1967 [16], then perfected its current incarnation in 1976 [17] through his colleague Glenn Shafer. The principle of evidence could address and manage uncertainty differently than the theory of probability [18]. Dempster’s combination rule is the essential D-S theory instrument, with many essential numerical characteristics, such as associativity and commutativity [19]. The principle of evidence has been adapted all across today, like objective identification [20], decision-making [21], reliability study [22], etc. The ambiguity measure indicates proof of consistency or accuracy. Ambiguity measurement (AM) [23] is a simple way to represent uncertainty, which is sensitive to changing facts. AM was extensively used for decision-making challenges [24].

The methodology proposed here in this article is taken from Fuyuan Xiao’s work in the year 2020 [25], in which the author proposed integrating Dempster-Shafer’s theory with belief entropy Multi-Criteria Decision Making called EFMCDM. Each criterion is represented as evidence in this approach, and all possible alternatives are assembled as the frame of discernment within the context of Dempster-Shafer’s theory. The EFMCDM approach takes both subjective and objective weighting into account when solving MCDM problems to produce more suitable Basic Probability Assignments (BPAs) for the criteria.

Therefore, this article introduced a novel method, an integrated fuzzy soft set system of the Dempster–Shafer evidence principle and the ambiguity measures, to achieve a more precise decision of terrorist nodes to destabilize their network. The 26/11 Mumbai Attack dataset is considered for testing this approach and measuring the effectiveness of the neutralization of terrorist networks.

The paper is structured as follows: Passage 2 contains the Fuzzy Soft Set Preliminaries, the D-S Theory of Proof, and the Weighted-Averaging Combination of Evidence Law and Uncertainty Measure; Passage 3 contains the methodology. A method was applied to the 26/11 Mumbai attack dataset in Passage 4 to determine the necessary node (s) to delete and neutralize the network; The efficacy and efficiency of the suggested solution discussed in Passage 5; finally, the conclusion discussed in Passage 6 of this report.

2 Primaries

Most of the preliminaries of the fuzzy soft set, D-S Theory, evidence combination rule, and AM are briefly presented below in this segment.

2.1 Fuzzy soft set

Definition 2.1(a). [26] Suppose the objects are F, the parameters are E, and the universal set is U. A pair (F, E) is shown as a soft set over the universal set U, given that mapping exists between F and E, $F : E \to P (U)$ (1)

Where P(U) is U’s power set.

Let us take the following example to demonstrate this concept,

Example 1. Let, universe L ={ l₁, l₂, l₃, l₄ } is a set of Terrorist node(s) in a terrorist group, P ={ p₁, p₂, p₃, p₄ }. The equation is a set of criteria used to select the essential node(s) within the terrorist network. Consider that mapping A is a mapping of P in all sub-sets of set L. Consider a soft set (A, P). That is why a terrorist node in a terrorist network is so critical. The soft set (A, P) is indicated according to objective criteria, $\begin{matrix} {A, P} = \\ {\begin{matrix} (p_{1}, {l_{1}, l_{2}, l_{4}}), (p_{2}, {l_{2}, l_{3}}), (p_{3}, {l_{2}, l_{4}}), \\ (p_{4}, {l_{2}, l_{3}, l_{4}}) \end{matrix}} \end{matrix}$

Where, $\begin{matrix} A (p_{1}) = {l_{1}, l_{2}, l_{4}}, A (p_{2}) = {l_{2}, l_{3}}, \\ A (p_{3}) = {l_{2}, l_{4}} and A (p_{4}) = {l_{2}, l_{3}, l_{4}} . \end{matrix}$

In two-dimensional computer representation used to store the soft set (A, C), where for all entries t_ij, would be l_ij = 1, if l_i ∈ A (c_j), l_ij = 0 if not. Table 1 displays a table representing the soft set (A, P), which interprets that the terrorist l₂ has a higher probability of getting selected to eliminate from the network so that the network efficiency will decrease effectively.

Table 1

Soft set table of the application domain

T	p ₁	p ₂	p ₃	p ₄	Choice value
l ₁	1	0	0	0	1
l ₂	1	1	1	1	4
l ₃	0	1	0	1	2
l ₄	1	0	1	1	3

Definition 2.1(b). Let, Γ (U) is a set of all universe U fuzzy sets, C is all parameters. A set (A_i, C_i) has been described as a fuzzy soft set over U: $A_{i} : C_{i} \to Γ (U)$ (2)

Let, object o fits into criteria c with membership value A (c) (o). Then determination of A (c) given below: $A (c) = {o / A (c) (o) | o \in U}$ (3)

Example 2. Let, universe T ={ t₁, t₂, t₃, t₄ } is a set of Terrorist node(s) in a terrorist group, C ={ c₁, c₂, c₃, c₄ } Set C is a set of criteria used to select the essential node(s) within the terrorist network. Consider that mapping $\tilde{A}$ . This is a mapping of C in all sub-sets of set T. Consider a fuzzy soft set $(\tilde{A}, C)$ , It explains how the terrorist node in a terrorist network is relevant. The fuzzy soft set $(\tilde{A}, C)$ is given as per objective criteria, $\begin{matrix} \tilde{A} (c_{1}) = {t_{1} / 0.8, t_{2} / 1, t_{3} / 0.3, t_{4} / 0.9}, \\ \tilde{A} (c_{2}) = {t_{1} / 0.1, t_{2} / 0.9, t_{3} / 1, t_{4} / 0.4}, \\ \tilde{A} (c_{3}) = {t_{1} / 0.4, t_{2} / 1, t_{3} / 0.2, t_{4} / 0.8}, \\ \tilde{A} (c_{4}) = {t_{1} / 0.2, t_{2} / 0.8, t_{3} / 0.9, t_{4} / 1}, \end{matrix}$

Table 2 shows the fuzzy soft set (A, C).

Table 2

Fuzzy soft set table of the application domain

T	c ₁	c ₂	c ₃	c ₄	Choice value
t ₁	0.8	0.1	0.4	0.2	1.5
t ₂	1	0.9	1	0.8	3.7
t ₃	0.3	1	0.2	0.9	2.4
t ₄	0.9	0.4	0.8	1	3.1

Definition 2.1(c). If M is an approach to utilize a fuzzy soft set to solve decision-making problems. This approach (M) can then be determined in the following way:

$\begin{matrix} Ψ_{M} = \frac{1}{\sum_{i = 1}^{n} \sum_{(j = 1) (j \neq i)}^{n} | A (c_{i}) (o_{t}) - A (c_{j}) (o_{t}) |} \\ + \sum_{i = 1}^{n} A (c_{i}) (o_{t}) \end{matrix}$ (4)

where n number of criteria and A (c_i) (o_t) is a membership value of the terrorist node o_t for the criteria c_i. If we have two methods, then the high value of Ψ_M for the respective method will have better performance, compared to others.

2.2 Dempster–Shafer theory of evidence (D-S theory)

Let, $ℕ = {x_{1}, x_{2}, \dots, x_{n}}$ is a non-empty finite set consisting n elements and $2^{ℕ} = {Φ, {x_{1}},$ {x₂ } , …, { x_n } , { x₁, x₂ } , …, { x₁, x₂, …, x_n }} is a set of all possible subsets of $ℕ$ .

Basic Probability Assignment (BPA): $2^{ℕ} \to [0, 1]$ .

That satisfies, $\sum_{X \subseteq ℕ} m (X) = 1$ and m (Φ) = 0.

Body of Evidence (BOE): If m (X) > 0, X is known as the focal point. Set of all BOE focal elements.

Dempster’s combination rule for combined evidence: if multiple independent BOE’s are available, then Dempster’s combination rule can be determined as follows:

$m (X) = \frac{\sum m_{1} (X_{i}) m_{2} (X_{j})}{1 - K}, \forall X_{i} \subseteq ℕ, X_{j} \subseteq ℕ$ (5)

where X_i ∩ X_j = X and $K = \sum m_{1} (X_{i}) m_{2} (X_{j}), \forall X_{i} \cap X_{j} = Φ$

Here K is an evidence conflict measure.

Example 3. Let two experts E₁ and E_2, from a law enforcement agency discussing the leader of any terrorist group. The key person of the terrorist group can be either T₁, T_2, or T₃. The first expert, E_1, believes that T₁ is the leader with a probability of 0.7 or T₂ with a probability of 0.1 or T₃ with a probability of 0.2 (denoted by m₁ (T₁), m₁ (T₂) and m₁ (T₃) Respectively). The Second expert, E_2, believes that T₁ is the leader with a probability of 0.5 or T₂ with a probability of 0.2 or T₃ with a probability of 0.3 (denoted by m₂ (T₁), m₂ (T₂) and m₂ (T₃) Respectively). The result calculation of the combination rule is shown in Table 3.

Table 3

Combination rule table for example 3

			EXPERT-1
			T1	T2	T3
			0.7	0.1	0.2
EXPERT-2	T1	0.5	{T1}	{ø}	{ø}
			{0.7*0.5 = 0.35}	{0.1*0.5 = 0.05}	{0.2*0.5 = 0.1}
	T2	0.2	{ø}	{T2}	{ø}
			{0.7*0.2 = 0.14}	{0.1*0.2 = 0.02}	{0.2*0.2 = 0.04}
	T3	0.3	{ø}	{ø}	{T3}
			{0.7*0.3 = 0.21}	{0.1*0.3 = 0.03}	{0.2*0.3 = 0.06}

Expert-1 $\begin{matrix} m_{1} (T_{1}) = 0.7 \\ m_{1} (T_{2}) = 0.1 \\ m_{1} (T_{3}) = 0.2 \end{matrix}$

Expert-2 $\begin{matrix} m_{2} (T_{1}) = 0.5 \\ m_{2} (T_{2}) = 0.2 \\ m_{2} (T_{3}) = 0.3 \end{matrix}$

The combined probability is evaluated as follows:

T1 = 0.35/(1-(0.05 + 0.1 + 0.14 + 0.04 + 0.21 + 0.03))=0.814

T2 = 0.02/(1-(0.05 + 0.1 + 0.14 + 0.04 + 0.21 + 0.03))=0.0465

T3 = 0.06/(1-(0.05 + 0.1 + 0.14 + 0.04 + 0.21 + 0.03))=0.1395

Belief (lower bound): Given $B \in 2^{N}$ , The Belief of $B$ is determined as follows: $Bel (B) = \sum_{b \subseteq B} m (b)$ (6)

Plausibility (upper bound): Given $B \in 2^{N}$ , The Plausibily of $B$ is determined as follows: $Pl (B) = \sum_{b \cap B \neq Ø} m (b)$ (7)

$[Bel (B), Pl (B)]$ satisfy the following conditions for any proposition:

$Pl (B) = 1 - Bel (\bar{B})$ , where, $\bar{B}$ s the classical complement of $B$ . $Pl (B) ⩾ Bel (B)$

The uncertainty of $B$ is described below: $u (B) = Pl (B) - Bel (B)$ (8)

Example 4. Let, universe L ={ l₁, l₂, l₃ } is a set of Terrorist node(s) in a terrorist group, a BPA is given that m ({ l₁ }) = 0.2, m ({ l₁, l₂ }) = 0.5, m ({ l₂, l₃ }) = 0.2, m ({ l₁, l₂, l₃ }) = 0.1. The Plausibility, Belief, and uncertainty result are shown in Table 4.

Table 4

Plausibility, Belief, and uncertainty result

Fun.	{l₁}	{l₂}	{l₃}	{l₁, l₂}	{l₁, l₃}	{l₂, l₃}	{l₁, l₂, l₃}
m	0.2	0	0	0.5	0	0.2	0.1
Bel	0.2	0	0	0.7	0.2	0.2	1
Pl	0.8	0.8	0.3	1	1	0.8	1
u	0.6	0.8	0.3	0.3	0.8	0.6	0

2.3 Weighted-averaging evidence combination rule

Sometimes, when BOE’s have high conflict, the maximum chance is that Dempster’s combination rule will give illogical results [29, 30]. Deng et al. [31] and Fuyuan Xiao et al. [32] have implemented a new Weighted-Averaging combination rule to deal with this problem.

Evidence distance (dest_ij): Let, m_i and m_j Are two BPA’r t universe $ℕ$ . Then, evidence distance can be evaluated as follows: ${dest}_{ij} = \sqrt{\frac{{({\vec{m}}_{i} - {\vec{m}}_{j})}^{T} ℤ ({\vec{m}}_{i} - {\vec{m}}_{j})}{2}}$ (9) where $ℤ (X, Y) = \frac{X \cap Y}{X \cup Y}, \forall X, Y \in P (ℕ)$ , which is an 2ⁿ × 2ⁿ Matrix.

Similarity Measure (S_ij): Let, m_i and m_j are two BPA’s under the universe $ℕ$ similarity can be evaluated as follows: $S_{ij} = 1 - {dest}_{ij}$ (10)

Similarity Measure matrix (SimM): The agreement between BOE’s, m_i (i = 1, 2, …, k) represented in similarity measure matrix given below, $SimM = (\begin{matrix} \begin{matrix} 1 & \dots \end{matrix} \begin{matrix} S_{1 j} & \dots & S_{1 k} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} S_{i 1} & \dots \end{matrix} \begin{matrix} S_{ij} & \dots & S_{in} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} S_{in} & \dots \end{matrix} \begin{matrix} S_{nj} & \dots & 1 \end{matrix} \end{matrix})$

Support Degree (SD_{m
_i}): Support degree of evidence can be determined as follows: ${SD}_{m_{i}} = \sum_{j = 1, j \neq i}^{k} S_{ij}$ (11)

Information Volume (I_{m
_i}): The Belief entropy Π_{m
_i} of the BPA m_i (i = 1, 2, …, k) can be determined as follows: $I_{m_{i}} = e^{Π_{m_{i}}} = e^{- \sum_{B \subseteq N} m (B) log \frac{m (B)}{2^{| B |} - 1}}$ (12)

Normalized Information volume, ${\bar{I}}_{m_{i}} = \frac{I_{m_{i}}}{\sum_{l = 1}^{k} I_{m_{i}}}$ (13)

Credibility Degree (CD_{m
_i}): Credibility degree of evidence can be determined as follows: ${CD}_{m_{i}} = \frac{{SD}_{m_{i}}}{\sum_{l = 1}^{k} {SD}_{m_{l}}}$ (14)

Modified Credibility Degree (MCD_{m
_i}): Modified Credibility degree of evidence can be determined as follows: ${MCD}_{m_{i}} = {CD}_{m_{i}} \times {\bar{I}}_{m_{i}}^{(\frac{\sum_{l = 1}^{k} {CD}_{m_{l}}}{k} - {CD}_{m_{i}})}$ (15)

Normalized modified credibility degree, ${\bar{MCD}}_{m_{i}} = \frac{{MCD}_{m_{i}}}{\sum_{l = 1}^{k} {MCD}_{m_{l}}}$ (16)

Weighted Averaging evidence combination (W_m): It can be determined as follows: $W_{m} = \sum_{i = 1}^{k} {\bar{MCD}}_{m_{i}} \times m_{i}$ (17)

When there are k parts of the evidence, the classical Dempster’s rule Equation (5) can be used to aggregate the weighted mass average of k-1 times. $m_{W} = {({({(W_{m} \oplus W_{m})}_{1} \oplus \dots)}_{h} \oplus W_{m})}_{k - 1}$ (18)

2.4 Ambiguity measure

The disadvantages of measurement in aggregate uncertainty a overcome using the Ambiguity measure (AM). Let, $ℕ = {x_{1}, x_{2}, \dots, x_{n}}$ is a non-empty framework discernment consisting of k elements, and m is a BPA defined on $ℕ$ . The Ambiguity Measure [33] can be evaluated as follows: $ℚ_{m} = - \sum_{x \in ℕ} {BetP}_{m} (x) {log}_{2} ({BetP}_{m} (x))$ (19)

where pignistic distribution (BetP_m (a)) [34] can be fined as follows: ${BetP}_{m} (a) = \sum_{a \in B \subseteq ℕ} \frac{m (B)}{| B |}$ (20)

Example 5. Consider two BOE’s, E₁ and E₂, $\begin{matrix} E_{1} : m ({l_{1}}) = 0.2, m ({l_{1}, l_{2}}) = 0.5, \\ m ({l_{2}, l_{3}}) = 0.2, m ({l_{1}, l_{2}, l_{3}}) = 0.1 \\ E_{2} : m ({l_{1}}) = 0.5, m ({l_{1}, l_{2}}) = 0.2, \\ m ({l_{2}, l_{3}}) = 0.1, m ({l_{1}, l_{2}, l_{3}}) = 0.2 \end{matrix}$

Table 5 shows the result of the ambiguity measure for E₁ and E₂.

Table 5

Result of ambiguity measure for E1 and E2

Function	E ₁	E ₂
BetP _m	BetP₁ (l₁) = 0.4834	BetP₂ (l₁) = 0.6667
	BetP₁ (l₂) = 0.3834	BetP₂ (l₂) = 0.2167
	BetP₁ (l₃) = 0.1334	BetP₂ (l₃) = 0.1167
$ℚ_{m}$	1.4249	1.2297

3 Methodology

3.1 Social network analysis (SNA)

3.1.1 Entropy-based centrality [35]

Definition 3.1.1(a). $D_{i}^{in}$ denotes the in-degree node v_i, and $D_{i}^{out}$ out denotes out-degree of the same node v_i. Assume j (1, 2, …, k) is denoted as a direct neighbouring node to ithen the total In-degree Relation defined as $E_{i}^{in}$ and Out-degree Relation defined as $E_{i}^{out}$ of S_i are evaluated as follows: $E_{i}^{in} = \sum_{j = 1}^{k} D_{j}^{in}$ (21) $E_{i}^{out} = \sum_{j = 1}^{k} D_{j}^{out}$ (22)

Definition 3.1.1(b). The In-degree Relation and Out-degree Relation entropy centralities defined as ${RE}_{i}^{in}$ and ${RE}_{i}^{out}$ of node v_i are evaluated as follows: ${RE}_{i}^{in} = - \sum_{j = 1}^{k} \frac{D_{j}^{in}}{E_{i}^{in}} log \frac{D_{j}^{in}}{E_{i}^{in}}$ (23) ${RE}_{i}^{out} = - \sum_{j = 1}^{k} \frac{D_{j}^{out}}{E_{i}^{out}} log \frac{D_{j}^{out}}{E_{i}^{out}}$ (24)

The final relational entropy centrality RE_i is, ${RE}_{i} = {RE}_{i}^{in} + {RE}_{i}^{out}$ (25)

3.1.2 Resilience centrality [36]

Definition 3.1.2. Let, $A_{i j}$ is an adjacency matrix representation of a graph G = (N, E) representing a social network. $D_{i}^{in}$ denotes the in-degree node v_i, and $D_{i}^{out}$ out denotes out-degree of the same node v_i. Assume j (1, 2, …, k) is denoted as a direct neighbouring node to i. The resilience centrality quantifies the ability of a node i to affect system resilience after removing it from the network. It can be determined as follows:

$\begin{matrix} {RC}_{i} = \frac{\sum_{j = 1}^{k} (A_{i j} D_{i}^{in} + A_{j i} D_{i}^{out}) + D_{i}^{in} D_{i}^{out}}{\sum_{j = 1}^{k} D_{j}^{in} D_{j}^{out}} \\ - \frac{D_{i}^{in} + D_{i}^{out}}{\sum_{j = 1}^{k} D_{j}^{in}} \end{matrix}$ (26)

3.1.3 Vote rank centrality

The Vote Rank is an incremental technique to assess the best spreadable potential for a group of decentralised spreaders. With this method, each spreader voting node would reduce the voting power of neighbours of the selected spreader [37].

Definition 3.1.3(a). Let each node v_i ∈ N is related with a tuple (S_i, V_{a
_i}), where S_i is the voting score of node v_i which is evaluated by summing the voting ability of its adjacent neighbours and V_{a
_i} is represented to the voting ability of a node v_i. Initially, it set to S_i = 0 and V_{a
_i} = 1.

Definition 3.1.3(b). Vote Rank of a node v_i can calculate in following five phases,

Initialize the tuple (S_i, V_{a
_i}) as (0, 1)

Vote: Nodes vote for their neighbours, while their neighbours vote. Each node’s voting score is decided after the vote. Note that the voting module is set to 0 if it was previously appointed to avoid re-election.

Select: Pick the v_max is a node that receives the maximum votes compared to the vote scores determined in Phase 2. This node may not engage in following rotations of voting, i.e., va_{v
_max} may not be able to vote at all from now.

Update: The voting power of the nodes that voted for v_max Phase 2 is weakened.

Repeat Phase 2 through 4 before r spreaders are selected.

3.2 Method of destabilization of terrorist network

Let, Alternatives x_i (i = 1, 2, …, s) is a set of Terrorist node(s) in a terrorist group, c_j (j = 1, 2, …, n) are set of criteria such as Entropy-based centrality, Resilience centrality, and VoteRank centrality, which are parameters considered. $ℕ = {x_{1}, x_{2}, \dots, x_{n}}, C = {c_{1}, c_{2}, \dots, c_{n}}$ . Define $F : C \to 2^{ℕ}$ by F (c_i) (x_i) - 𝕦_ij (j = 1, 2, …, n : i = 1, 2, …, s). The implementation steps are as follows:

Step 1. Find the fuzzy soft set matrix $U = {(𝕦_{i j})}_{s \times n}$ of Terrorist node(s) × Criteria over $ℕ (F, C)$ where 𝕦_ij is membership value of i(1, 2, ... ,s) and j(1,2, ... ,n) evaluated in section 3.1. $U = (\begin{matrix} \begin{matrix} 𝕦_{11} & \dots \end{matrix} \begin{matrix} 𝕦_{1 j} & \dots & 𝕦_{1 n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} 𝕦_{i 1} & \dots \end{matrix} \begin{matrix} 𝕦_{i j} & \dots & 𝕦_{i n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} 𝕦_{s 1} & \dots \end{matrix} \begin{matrix} 𝕦_{s j} & \dots & 𝕦_{s n} \end{matrix} \end{matrix})$

Step 2. Construct the normalized matrix $\tilde{U} = {({\tilde{𝕦}}_{ij})}_{s \times n}$ where ${\tilde{𝕦}}_{1 j} = \frac{𝕦_{1 j}}{\sum_{i = 1}^{s} 𝕦_{ij}}$ . $\tilde{U} = (\begin{matrix} \begin{matrix} {\tilde{𝕦}}_{11} & \dots \end{matrix} \begin{matrix} {\tilde{𝕦}}_{1 j} & \dots & {\tilde{𝕦}}_{1 n} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} {\tilde{𝕦}}_{i 1} & \dots \end{matrix} \begin{matrix} {\tilde{𝕦}}_{ij} & \dots & {\tilde{𝕦}}_{in} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} {\tilde{𝕦}}_{s 1} & \dots \end{matrix} \begin{matrix} {\tilde{𝕦}}_{sj} & \dots & {\tilde{𝕦}}_{sn} \end{matrix} \end{matrix})$

Step 3. Calculate the uncertainty of each criterion c_j (j = 1, 2, …, n) using Equation (19) denoted as $ℚ (c_{j}) (j = 1, 2, \dots, n)$ . $ℚ (c_{j}) = - \sum_{i = 1}^{s} {\tilde{U}}_{ij} {log}_{2} {\tilde{U}}_{ij}$ (27) stop

Step 4. Normalize t Uncertainty $ℚ (c_{j})$ and determine the final degree of uncertainty for all criteria c_j, represent aa $\bar{ℚ} (c_{j}) (j = 1, 2, \dots, n)$ . $\bar{ℚ} (c_{j}) = \frac{ℚ_{c_{j}}}{\sum_{k = 1}^{n} ℚ_{c_{k}}}$ (28)

Step 5. Determine BPAs for each terrorist node(s) x_i (i = 1, 2, …, s) and $ℕ$ for each criterion c_j (j = 1, 2, …, n). $m_{c_{j}} (Ø) = 0$ (29) $m_{c_{j}} (x_{i}) = {\tilde{U}}_{ij} \times (1 - \bar{ℚ} (c_{j}))$ (30) $m_{c_{j}} (ℕ) = 1 - \sum_{i = 1}^{s} m_{c_{j}} (x_{i})$ (31)

For each j, $\sum_{C \subseteq ℕ} m_{c_{j}} (C) = 1$ , then m_{c
_j}, is BOE’s on $ℕ$ .

Step 6. Calculate Pearson correlation [38] for each terrorist node(s) x_i (i = 1, 2, …, s) and $ℕ$ for each criterion c_j (j = 1, 2, …, n). The similarity matrix is determined through Pearson correlation as follows: $SimM = (\begin{matrix} \begin{matrix} 1 & \dots \end{matrix} \begin{matrix} S_{1 j}^{P} & \dots & S_{1 k}^{P} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} S_{i 1}^{P} & \dots \end{matrix} \begin{matrix} S_{ij}^{P} & \dots & S_{in}^{P} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \end{matrix} \begin{matrix} ⋮ & ⋮ & ⋮ \end{matrix} \\ \begin{matrix} S_{in}^{P} & \dots \end{matrix} \begin{matrix} S_{nj}^{P} & \dots & 1 \end{matrix} \end{matrix})$

Step 7. Determine the support degree for each Terrorist node(s) using Equation (11). ${SD}_{m_{i}} = \sum_{j = 1, j \neq i}^{k} S_{ij}^{P}$ (32)

Step 8. Determine credibility degree for each Terrorist node(s) using Equation (14).

Step 9. Determine Normalized Information volume (believe entropy) for each Terrorist node(s) using Equation (12-13) $I_{m_{i}} {= e}^{Π_{m_{i}}} {= e}^{- Σ_{j} m_{c_{j}} {(x}_{i} {) \log}_{2} m_{c_{j}} {(x}_{i})}$ (33)

Step 10. Determine the Normalized Modified credibility degree for each Terrorist node(s) using Equation (15-16).

Step 11. The final BPA from each terrorist nodes x_i (i = 1, 2, …, s) could be obtained, which is explicitly regarded as a measure of confidence of each terrorist node(s) x_i (i = 1, 2, …, s). Rank the competitor terrorist node’s belief indicators and get the best terrorist node(s). Optimal choices can have more than one terrorist node when more nodes equate to the optimum.

4 Result and simulation

4.1 26/11 Mumbai dataset description 1

A team of terrorists recognizes the incident as one of India’s most significant terrorist attacks on 26 November 2008. The dataset consists of 13 terrorists (nodes), where ten terrorist operatives came to India for terrorist activities. The remaining three were in Pakistan and managed the terrorist activity. Zaki-Ur-Rehman Lakhvi and Kaahfa, including some senior LeT members, watched the terrorist group closely. The terrorist nodes are linked through a Guided Diagram with 13 nodes and 31 communication links. Some terrorist nodes are closely linked, while some nodes are highly external. Figure 1 depicts the terrorist communications matrix link network [39] involved in Mumbai Terrorist Attack 26/11, framed with the UCINET tool’s help [40].

Fig. 1

A network representation of the Mumbai terrorist attack on 26 November 2008.

4.2 Proposed approach implementation

Step 1. Construct the three columns matrix to demonstrate the criteria for decision-making. By the following the definitions of Sections 3.1.1, 3.1.2, and 3.1.3, respectively, values for three columns were evaluated, as shown in Table 6.

Table 6
Centralities calculated using equations in Sections 3.1.1, 3.1.2, and 3.1.3, respectively

NAME Entropy-based Resilience Vote rank

centrality centrality centrality

Abu Kaahfa 2.57 0.392 0.46

Wassi 5.16 0.524 1

Zarar 2.57 0.4 0.46

Hafiz Arshad 4 0.421 0.88

Javed 2.58 0.309 0.54

Abu Shoaib 2.58 0.313 0.34

Abu Umer 4 0.421 0.88

Abdul Rehman 0.99 0.135 0.12

Fahadullah 0.99 0.127 0.12

Ba Imran 0 0.311 0.26

Nasir 0 0.311 0.26

Ismail Khan 0 0.008 0

Ajmal Kasab 0 0.008 0

NAME	Entropy-based	Resilience	Vote rank
Abu Kaahfa	2.57	0.392	0.46
Wassi	5.16	0.524	1
Zarar	2.57	0.4	0.46
Hafiz Arshad	4	0.421	0.88
Javed	2.58	0.309	0.54
Abu Shoaib	2.58	0.313	0.34
Abu Umer	4	0.421	0.88
Abdul Rehman	0.99	0.135	0.12
Fahadullah	0.99	0.127	0.12
Ba Imran	0	0.311	0.26
Nasir	0	0.311	0.26
Ismail Khan	0	0.008	0
Ajmal Kasab	0	0.008	0

Step 2. Normalize the values matrix values column-wise and resultant values shown in Table 7.

Table 7

Normalize parameters values

NAME	Entropy-based	Resilience	Vote rank
	centrality	centrality	centrality
Abu Kaahfa	0.101022	0.1065	0.086466
Wassi	0.202830	0.1423	0.187969
Zarar	0.101022	0.1086	0.086466
Hafiz Arshad	0.157232	0.1144	0.165413
Javed	0.101415	0.0839	0.101503
Abu Shoaib	0.101415	0.0850	0.063909
Abu Umer	0.157232	0.1144	0.165413
Abdul Rehman	0.038915	0.0366	0.022556
Fahadullah	0.038915	0.0345	0.022556
Baba Imran	0	0.0845	0.04887
Nasir	0	0.0845	0.04887
Ismail Khan	0	0.0021	0
Ajmal Kasab	0	0.0021	0

Step 3. Calculate the uncertainty for each parameter; the result is shown in Table 8.

Table 8

Representation of $ℚ (c_{j})$

	Entropy-based	Resilience	Vote rank
	centrality	centrality	centrality
$ℚ (c_{j})$	3.00856884	3.3942573	3.18383

Step 4. Evaluate the degree of uncertainty for each parameter, the result shown in Table 9.

Table 9

Representation of $\bar{ℚ} (c_{j})$

	Entropy-based	Resilience	Vote rank
	centrality	centrality	centrality
$\bar{ℚ} (c_{j})$	0.31382881	0.35406063	0.332111

Step 5. Calculate the BPA’s m_{c
_j} (x_i), for each Terrorist node(s), The result represents in Table 10.

Table 10

Representation of m_{c
_j} (x_i)

NAME	Entropy-based	Resilience	Vote rank
	centrality	centrality	centrality
Abu Kaahfa	0.069318	0.068807	0.05775
Wassi	0.139176	0.091976	0.125543
Zarar	0.069318	0.070211	0.05775
Hafiz Arshad	0.107889	0.073897	0.110478
Javed	0.069588	0.054238	0.067793
Abu Shoaib	0.069588	0.05494	0.042685
Abu Umer	0.107889	0.073897	0.110478
Abdul Rehman	0.026702	0.023696	0.015065
Fahadullah	0.026702	0.022292	0.015065
Baba Imran	0	0.054589	0.032641
Nasir	0	0.054589	0.032641
Ismail Khan	0	0.001404	0
Ajmal Kasab	0	0.001404	0

Step 6. Generate the similarity matrix of size s × s using BPAs of each terrorist node(s). The result is shown in Table 11.

Table 11

Similarity measure matrix for all BPA’s 2

	A	B	C	D	E	F	G	H	I	J	K	L	M
A	1	0.4	1	0.1	–0.2	–0.3	0.1	–0.2	0.4	0.7	0.7	–0.2	–0.2
B	0.4	1	0.4	0.4	0.4	0.2	0.4	–0	–0	1	1	–0.4	–0.4
C	1	0.4	1	0.1	–0.2	–0.3	0.1	0.4	–0.2	0.7	0.7	–0.2	–0.2
D	0.1	0.4	0.1	1	0.5	0.8	1	–0.4	–0.4	0.4	0.4	–0.2	–0.2
E	–0.2	0.4	–0.2	0.5	1	1	0.5	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2
F	–0.3	0.2	–0.3	0.8	1	1	0.8	–0.3	–0.3	–0.2	–0.2	–0.2	–0.2
G	0.1	0.4	0.1	1	0.5	0.8	1	–0.4	–0.4	0.4	0.4	–0.2	–0.2
H	–0.2	–0	0.4	–0.4	–0.2	–0.3	–0.4	1	–0.1	–0.2	–0.2	–0.2	–0.2
I	0.4	–0	–0.2	–0.4	–0.2	–0.3	–0.4	–0.1	1	–0.2	–0.2	–0.2	–0.2
J	0.1	1	0.1	0.1	–0.2	–0.2	0.1	–0.2	–0.2	1	1	–0.1	–0.1
K	0.1	1	0.1	0.1	–0.2	–0.2	0.1	–0.2	–0.2	1	1	–0.1	–0.1
L	–0.2	–0.4	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2	–0.1	–0.1	1	1
M	–0.2	–0.4	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2	–0.1	–0.1	1	1

Step 7. Determine the support degree using Equation (32). The result is shown in Table 12 and Fig. 2.

Table 12

Support degree of each terrorist node

	SD _{m _i}
Abu Kaahfa	2.387
Wassi	3.185
Zarar	2.387
Hafiz Arshad	2.546
Javed	0.978
Abu Shoaib	0.757
Abu Umer	2.546
Abdul Rehman	–1.722
Fahadullah	–1.722
Baba Imran	1.695
Nasir	1.695
Ismail Khan	–0.963
Ajmal Kasab	–0.963

Fig. 2

Support Degree of each terrorist node.

Step 8. Determine the credibility degree CD_{m
_i}. The result is shown in Table 13 and Fig. 3.

Table 13

Credibility degree of each terrorist node

	CD _{m _i}
Abu Kaahfa	0.1863
Wassi	0.2487
Zarar	0.1863
Hafiz Arshad	0.1988
Javed	0.0763
Abu Shoaib	0.0591
Abu Umer	0.1988
Abdul Rehman	–0.1344
Fahadullah	–0.1344
Baba Imran	0.1323
Nasir	0.1323
Ismail Khan	–0.0751
Ajmal Kasab	–0.0751

Fig. 3

Credibility degree of each terrorist node.

Step 9. Determine the Normalised information volume ${\bar{I}}_{m_{i}}$ . The result is shown in Table 14 and Fig. 4.

Table 14

Normalized information volume

	${\bar{I}}_{m_{i}}$
Abu Kaahfa	0.087915
Wassi	0.1208
Zarar	0.0882
Hafiz Arshad	0.1079
Javed	0.0869
Abu Shoaib	0.0812
Abu Umer	0.1079
Abdul Rehman	0.0582
Fahadullah	0.0579
Baba Imran	0.0601
Nasir	0.0601
Ismail Khan	0.0412
Ajmal Kasab	0.0412

Fig. 4

Normalized information volume.

Step 10. Determine the Normalized Modified credibility degree ${\bar{MCD}}_{m_{i}}$ . The result is shown in Table 15 and Fig. 5.

Table 15

Normalized modified credibility degree

	${\bar{MCD}}_{m_{i}} .$
Abu Kaahfa	0.19
Wassi	0.253
Zarar	0.19
Hafiz Arshad	0.203
Javed	0.079
Abu Shoaib	0.061
Abu Umer	0.203
Abdul Rehman	–0.146
Fahadullah	–0.1467
Baba Imran	0.136
Nasir	0.136
Ismail Khan	–0.081
Ajmal Kasab	–0.081

Fig. 5

Normalized modified credibility degree.

Step 11. Finally, select the node with a high degree of modified credibility and direct neighbouring nodes with the most nearer credibility degree value to destabilise the network and eliminate them from the terrorist network. Table 16 and Fig. 6 shows the selected node to destabilise the network.

Table 16

Final selection of potential nodes for destabilization

Terrorist	Modified credibility degree
Abu Kaahfa	0.19
Wassi	0.253
Zarar	0.19
Hafiz Arshad	0.203
Abu Umer	0.203

Fig. 6

Final selection of potential nodes for destabilization.

5 Evaluation of destabilization through knowledge diffusion

In our proposed steps, we will assess the influence of individuals in two ways. First, without the isolated individual, we can contrast the relative resource congruence of the terrorist network. Second, we can compare the relative performance changes with and without these individuals regarding cluster coefficient, distribution, and capacity to respond to this shift for the terrorist network [41].

5.1 Clustering coefficient

The clustering coefficient is beneficial to knowledge dissemination [42]. The more significant clustering coefficient allows the structure to be transparent and allows the development of reputations and cooperation standards [43]. In a high cluster, the network’s coefficient, cooperative or non–cooperative activity, can rapidly spread to other network members, influence their future willingness to cooperate, exchange information more positively, and encourage knowledge-sharing [44].

Definition 5.1(a). Let n is some network nodes, the node, i cluster coefficient can be assessed as follows: $c_{i} = \frac{# of triangles connected to node n_{i}}{# of triples centered on node n_{i}}$ (34)

Where C_i = 0 for a node with degree wither 0 or 1. Figure 7 shows the clustering coefficient of individuals.

Fig. 7

The clustering coefficient of individuals.

The Average clustering coefficient [45] for the whole network can be determined as follows: $C = \frac{\sum_{i = 1}^{n} C_{i}}{n}$ (35)

The Average Clustering Coefficient of the network before and after the isolation of selected node for destabilization and compared with the existing method shown in Table 17, which prove the correctness of our hypothesis that after eliminating the selected terrorist node(s) from the network, it will decrease the network efficiency by the mean of decrement in behaviour transfer or information spreading.

Table 17

Average clustering coefficient of the network before and after the isolation of selected node for destabilization

	Clustering	Clustering
	coefficient before	coefficient after
	isolation of	isolation of
	selected nodes	selected nodes
Existing method [46]	0.330769231	0.00
proposed Method	0.330769231	0.00

5.2 Knowledge diffusion efficiency

The efficiency of information diffusion represents the pace and consistency of the exchange and distribution of knowledge through the entire network [47].

Definition 5.2(a). Let, a network $G$ with N number of nodes and h_ij Hop count with the shortest distance between node i and j within the network. The global average efficiency [48] of the network can evaluate as follows: $E (G) = \frac{\sum_{i \neq j \in G} \frac{1}{h_{ij}}}{N (N - 1)}$ (36)

Definition 5.2(b). The power ψ = [0, 1] of a member n_i ∈ N of the Terrorist Network can be evaluated as follows: $ψ = \frac{E (G) - E (G - n_{i})}{E (G)}$ (37) where $G - n_{i}$ is network after the elimination of n_i (i = 1, 2, … k) node(s) selected for the destabilization of the terrorist network.

Table 18 and Fig. 8 show a drop in knowledge diffusion. Here the value nearer to 1 interprets that higher drop-in network efficiency.

Table 18

Drop–in knowledge diffusion

	$E_{global} (G)$	$E_{global} (G)$	Drop-in
	before	after	Knowledge
	isolation	isolation	diffusion
Existing method [59]	0.3685	0.0892	0.75776
Proposed method	0.3685	0.0892	0.75776

Fig. 8

Impact of destabilization.

6 Conclusion

This paper analyzes the effect of fuzzy soft-fitting destabilization strategies on the 26/11 Mumbai Attack terrorist network. The fuzzy soft set concept played a prominent role in ranking the network nodes based on some parameters considered here with more accuracy and dealt with uncertainty positively. The results indicate that the terrorist network as distributed subnetworks is challenging to destabilize. The outcome shows that the only way to destabilize the network is to isolate the emerging leaders in the short term. The D-S Theory of Evidence plays a novel role in identifying the top five players in the terrorist network using the concept of weighted-averaging combination rule. Here we utilized a few novel centralities like entropy-information-based centrality, resilience centrality, and vote rank centrality for objective criteria selection to increase the accuracy of the proposed approach. The analysis of the results demonstrates the efficacy of our approach through knowledge diffusion within the network. Furthermore, it was discovered that after eliminating the selected node (s) for terrorist network destabilization, both the clustering coefficient and network efficiency decreased.

Most central nodes’ cost of isolation might be high, so future work should concentrate on destabilizing the network by eliminating the foot soldiers from the network, not the key player. So, the cost of destabilization can get reduced, and the approach’s impact would be more effective.

Conflict of interest

The authors declare no conflict of interest.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Footnotes

doi:

A: Abu Kaahfa, B: Wassi, C: Zarar, D: Hafiz Arshad, E: Javed, F: Abu Shoaib, G: Abu Umer, H: Abdul Rehman, I: Fahadullah, J: Baba Imran, K: Nasir, L: Ismail Khan, M: Ajmal Kasab.

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