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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 93-104

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.93

© The Korean Institute of Intelligent Systems

Competitions on Fuzzy Mixed Graph and its Application in Countries for Health and Disaster

Pabitra Kumar Gouri1,2, Bharti Saxena1,2, Rajesh Kedarnath Navandar3, Pranoti Prashant Mane4, Ramakant Bhardwaj5, Jambi Ratna Raja Kumar6, Surendra Kisanrao Waghmare7, and Antonios Kalampakas8

1Department of Mathematics, Chhotakhelna Surendra Smriti Vidyamandir, Maligram, India
2Department of Mathematics, Rabindranath Tagore University, Bhopal, India
3Department of Electronic & Telecommunication Engineering, JSPM Jayawantrao Sawant College of Engineering Hadaspar, Pune, India
4Department of Computer Engineering, MES’s Wadia College of Engineering, Pune, India
5Department of Mathematics, Amity University, Kolkata, India
6Computer Engineering Department, Genba Sopanrao Moze College of Engineering, Pune, India
7Department of Electronics and Telecommunication Engineering, G H Raisoni College of Engineering and Management, Pune, India
8College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

Correspondence to :
Bharti Saxena (bhartisaxena060@gmail.com)

Received: August 31, 2023; Revised: January 28, 2024; Accepted: June 25, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study introduces the concept of fuzzy mixed graphs (FMGs) to represent uncertain relationships in social networks such as Facebook, where connections can be friends, followers, or mutuals. These graphs are an extension of the mixed graph theory, accommodating ambiguity in user relationships. We propose FMGs in which each vertex and link is assigned a membership degree between 0 and 1, reflecting the uncertainty of the connections. A subtype, competition FMGs, is explored to model scenarios in which users vie for shared resources or objectives. Our investigation reveals insights into the dynamics of competition within these graphs, including the conditions for the existence and uniqueness of maximal competitors, interplay between competition and network connectivity, and influence of fuzziness on competition intensity. By applying our theoretical framework to real-world scenarios, we demonstrate its utility in health and disaster management systems. By identifying essential regions and stakeholders affected by disease or disaster proliferation, our approach offers a novel analytical tool that can be substantiated by numerical simulations.

Keywords: Fuzzy mixed graphs, Social network analysis, Uncertainty modeling, Resource competition, Health systems analysis, Disaster management

However, in reality, there may be situations where the data are not precise [1] and the concepts of crisp graphs are inadequate. To address such situations, Kaufman [2] introduced a fuzzy graph in which each node and edge has a fuzzy membership value (MV) that indicates the degree of belongingness. Based on this idea, the notion of a fuzzy competition graph was proposed [3] and applied to a food web under fuzzy conditions. Later, fuzzy graph extensions were developed [4, 5]. This inspired us to study the competitions [6] on mixed graphs under fuzzy settings.

Mixed graphs have both directed and undirected edges, and can capture the mixed nature of some real-world networks, such as the current Facebook network [7]. The concept of mixed graphs [8] was first introduced in 1970 and various aspects of these graphs have been studies such as coloring [9], matrices [10,11], and isomorphism [12]. Recently, Samanta et al. defined a semidirected graph [13], which is a special type of mixed graph in which two nodes can have directed and undirected edges between them. However, all these studies were based on crisp graphs. A few additional studies have been conducted [1417].

Das et al. [18] introduced the concept of picture fuzzy competition graphs, which model the competition among entities with uncertain and ambiguous information. They also presented some generalizations and applications of these graphs in education, ecology, business, and job markets. Deva and Felix [19] proposed a novel approach for designing a decision-making trial and evaluation laboratory method in a bipolar fuzzy environment, which can handle both positive and negative degrees of membership. Additionally, they illustrated the proposed method using a supplier selection case study. Akram and Sattar [20] investigated the notion of competition graphs under complex Pythagorean fuzzy information, which is a generalization of Pythagorean and complex intuitionistic fuzzy sets. They also explored some properties and characteristics of these graphs and reported potential applications in the field of ecology. Nithyanandham and Augustin [21] developed a novel technique for prioritizing coronavirus disease 2019 (COVID-19) vaccines based on a bipolar fuzzy p-competition graph and the additive ratio assessment method. They also compared the proposed technique with existing methods and demonstrated its validity and practicability. Narayanamoorthy et al. [22] introduced the concept of regular and totally regular bipolar fuzzy hypergraphs, which are generalizations of bipolar fuzzy graphs. They proved some mathematical properties and provided examples of these hypergraphs. Karthik et al. [23] proposed a material selection model based on spherical Dombi fuzzy graphs that could handle uncertain and ambiguous information. They applied this model to select the best material for a given application and compared it with other fuzzy models. Further details on the applications of fuzzy graphs can be found in [2438].

Therefore, we aim to study mixed graphs in fuzzy environments and propose different types of competitions for mixed graphs under this framework.

The motivation for this study was to extend the concept of the mixed graph theory to represent different types of relationships in social networks, as well as account for the uncertainty and fuzziness that may exist in these networks. We propose fuzzy mixed graphs (FMGs), which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, to capture the ambiguities in the following and connectedness. Additionally, we investigate the concept of competition on FMGs, which can model situations in which users compete for common resources or goals and explore their properties and characteristics. We applied this concept to analyze and improve health and disaster management systems by identifying the key regions and actors most affected by the spread of diseases or disasters. We aim to contribute to the field of fuzzy graph theory and its applications in real-world problems.

The novelties of this study are as follows:

  • • The introduction of FMGs, which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, captures the uncertainty in social networks.

  • • The investigation of competition on FMGs, which model situations where users compete for common resources or goals, and the exploration of its properties and characteristics.

  • • Competition is applied to FMGs to analyze and improve health and disaster management systems. More specifically, the application of competition enables FMGs to identify the key regions and actors that are most affected by the spread of diseases or disasters.

These novelties contribute to the field of fuzzy graph theory and its applications to real-world problems.

This paper is organized as follows. Section 1 introduces the study with a literature review. Section 2 describes some basic notions related to the study. Section 3 introduces competition fuzzy mixed graph. Section 4 introduces m-step competition fuzzy mixed graph. Section 5 proposes some related properties. Section 6 describes a numerical application. Section 7 concludes the study with future directions.

2.1 Graph

Let a graph G = (V, E) having n nodes and m links are represented by an adjacent matrix A = {aij} ∈ Rn×n, where aij = 1 if node i is linked with node j and aij = 0 otherwise. In social networks, two nodes are related by a link.

2.1.1 Competition graph

Definition 1. Let us consider a directed graph X = (N, L). The competition graph C(X) of X is an undirected graph that has an equal node set/vertex set and a link/edge between two distinct vertices if there exists a common out-neighbor.

2.2 Mixed Graph

Definition 2. Let N be a set of vertices or nodes of a graph and L = L1L2 where L1 = {(u, v) | u, vN} ⊂ N × N called a set of undirected edges of the graph and called a set of directed edges of the graph. The graph X = (N, L1, L2) is then said to form a mixed graph.

2.3 Fuzzy Graph

Let G = (V, E) be a graph with the pair of mappings σ : V → [0, 1] and μ : E → [0, 1] such that the condition μ(x, y) ≤ σ(x) ∧ σ(y) for all (x, y) ∈ EV × V, G = (V, E, μ, σ) is then called a fuzzy graph.

2.3.1 FMG

Definition 3. Graph X = (N, L1, L2, μ1, μ2, σ, δ) is called FMG if there exist functions σ : N → [0, 1], μ1 : L1 → [0, 1], μ2 : L2 → [0, 1] and μ : L2 → [0, 1] such that

μ1(x,y)σ(x)σ(y)for all (x,y)L1N×N,μ2(x,y)σ(x)σ(y)for all (x,y)L2N×N,σ(x,y)σ(x)-σ(y)for all (x,y)L2N×N,

where σ is MV of vertex, μ1 is MV of undirected edge, μ2 is MV of directed edge, and δ is measure of directionality.

Example 1. Let us consider an FMG with four nodes with two undirected and three directed edges.

2.4 UndirectedWalk and Path

Definition 4. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. An undirected walk is a sequence : v1e1v2e2v3 ... ekvk+1, where v1, v2, v3, ..., vk+1 are the vertices and e1, e2, ..., ek are the edges, such that all the edges are undirected and μ1(ei) > 0 for i = 1, 2, ..., k. An undirected walk from vertex u to v is said to be an undirected path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P(u,v)m.

Example 2. Let us consider an FMG (Figure 1). From Figure 1, v1-v4-v3 is an undirected path.

2.5 Directed Walk and Path

Definition 5. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. A directed walk is a sequence : v1 e1 v2 e2 v3 ... ek vk+1, where v1, v2, v3, ..., vk+1 are the vertices and e1, e2, ..., ek are the edges, such that all the edges are directed in the same direction, and μ2(ei) > 0 for i = 1, 2, ..., k. A directed walk from vertex u to v is said to be a directed path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P(u,v)m.

Example 3. Consider an FMG (Figure 1). From Figure 1, it follows that v3v2v4 is the directed path of the graph.

2.6 Neighborhood

Definition 6. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. The neighborhood of vertex u is denoted by NB(u) and is defined as the set

NB(u)={(v,μ1(u,v)):;vVand μ1(u,v)>0}.

Note.

  • • The out-neighborhood of vertex u is denoted by NB+(u) and is defined as the set NB+(u)={(v,μ2(u,v)+μ2(u,v)δ(u,v)):vNand μ2(u,v)>0}.

  • • The in-neighborhood of vertex u is denoted by NB(u) and is defined as the set NB-(u)={(v,μ2(u,v)+μ2(u,v)δ(u,v)):vNand μ2(u,v)>0}.

Example 4. Let us consider an FMG (Figure 1). Here, from Figure 1, it follows that

  • (a) The neighborhood of v1 is NB(v1) = {(v4, 0.4)}.

  • (b) The out-neighborhood of v1 is NB+(v1) = {(v2, 0.44)}.

  • (c) The in-neighborhood of v2 is NB (v2) = {(v1, 0.44), (v3, 0.66)}.

2.7 m-Step Neighborhood

Definition 7. Let X=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The m–step neighborhood of vertex u is denoted by NBm(u) and is defined as the set NBm(u)={(v,σm(v)):vNand σm(v)=min{μ1(x,y):(x,y)P(u,v)m}}.

Note.

  • • The m-step out-neighborhood of vertex u is denoted by NBm+(u), and is defined as the set

    NBm+(u)={(v,σm(u,v)):vNand σm(u,v)}=min{μ2(a,b)+μ2(a,b)δ(a,b):(a,b)P(u,v)m}}.

  • • The m-step in the neighborhood of vertex u is denoted by NBm-(u) and is defined as the set

    NBm-(u)={(v,σm(u,v)):vNand σm(u,v)}=min{μ2(a,b)+μ2(a,b)δ(a,b):(a,b)P(v,u)m}}.

Example 5. Let us consider an FMG (Figure 1). From Figure 1, it follows that

  • (a) 2-step neighborhood of v1 is NB2(v1) = {(v3, 0.3)}.

  • (b) 2-step out-neighborhood of v3 is NB2+(v3)={(v4,0.48)}.

  • (c) 2-step in-neighborhood of v4 is NB2-(v4)={(v1,0.44),(v1,0.48)}.

3.1 Definition of CFG

Definition 8. Let X=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The competition fuzzy graph (CFG) of X is denoted by C(X) and is defined as an undirected fuzzy graph in which N is also a vertex set. There exists an edge between vertices x, yN if NB+(x) ∩ NB+(y) ≠ ∅ with membership value α(x, y) is defined by

α(x,y)=minvNB+(x)NB+(y)μ2(x,z)+μ2(x,z)δ(x,z)-{μ2(y,z)μ2(y,z)δ(y,z)}.

3.2 Algorithm to Find the CFG of FMG and the Adjacency Matrix of CFG

Algorithm 1.

  • Step 1 : Consider an FMG with given memberships of vertices and edges.

  • Step 2: Find the neighbors for all xiN.

  • Step 3: Find the intersections NB+(xi) ∩ NB+(xj) for all xi, xjV and xixj .

  • Step 4: If NB+(xi) ∩ NB+(xj) = {xk : k = 1, 2, ..., }, then find μ2(xi,xm)δ(xi,xm)-μ2(xj,xm)δ(xj,xm) and same for all non-void intersections.

  • Step 5: Find α(xi,xj)=minxkN+(xi)N+(xj)|μ2(xi,xm)δ(xi,xm)-μ2(xj,xm)δ(xj,xm)|. The values obtained in this section are the membership values of the edges of the CFG.

  • Step 6: If aij denotes the ij–th element of adjacency matrix of the CFG, then

aij={α(xi,xj),if there is an edge between xi,xj,1,if there is no edge between xi,xj.

A flowchart of the algorithm is shown in Figure 2.

Example 6. Let us consider an FMG (Figure 1). Here, from Figure 1, the CFG is defined as Figure 3.

4.1 Definition of m-Step CFG

Definition 9. Let G=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The m-step CFG of G is denoted by Cm(G) and is defined as an undirected fuzzy graph in which N is also a vertex set, and there exists an edge between vertices x, yN if NBm+(x)NB+(y) with membership value βm(x, y) defined by

βm(x,y)=minvNBm+(x)NBm+(y)σm(x,v)-σm(y,v).

4.2 Algorithm to Find the m-Step CFG of FMG and the Adjacency Matrix of m-Step CFG

Algorithm 2.

  • Step 1: Consider an FMG with given memberships of vertices and edges.

  • Step 2: Find the neighbors for all xiN.

  • Step 3: Find the intersections NBm+(xi)NBm+(xj) for all xi, xjN and xixj.

  • Step 4: If NB+(xi) ∩ NB+(xj) = {xk : k = 1, 2, ..., }, then find σm(xi,xk)-σm(xj,xk) and same for all non-void intersections.

  • Step 5: Find βm(xi,xj)=minxkNB+(xi)NB+(xj)σm(xi,xk)-σm(xj,xk). The values obtained in this section are the membership values of the edges of the CFG.

  • Step 6: If aij denotes the ij-th element of the adjacency matrix of the CFG, then

aij={βm(xi,xj),if there is an edge between xi,xj,1,if there is no edge between xi,xj.

A flowchart of the algorithm is shown in Figure 4.

Example 7. Let us consider an FMG (Figure 1). From Figure 1, the 2-step CFG is defined as Figure 5.

Theorem 1. Let X = (N, L) be a fuzzy graph. Subsequently, FMG X′ with C(X′) = X. occurs.

Proof.

Given:

Here, X = (N, L) is a fuzzy graph.

To prove : C(X′) = X

Let (x, y) be an edge of a fuzzy graph. FMG X′ is created, where C(X′) = X.

Let x′, y′ ∈ X′ be the equivalent vertices of x, yX.

We can join two directed edges x′, y′ to a vertex z′ ∈ X′ such that z′ ∈ NB+(x′) ∩ NB+(y′).

Therefore, this can be done for all nodes and links of X.

Hence, C(X′) = X.

Theorem 2. The competition number (CN) of a circuit in a fuzzy graph is 1.

Proof.

Claim:

Let us consider a circuit of length n in a fuzzy graph.

To prove:CN of a circuit in a fuzzy graph is 1.

We drew all possible directions with the vertices of the circuit to fit a corresponding FMG whose competition graph is the circuit. However, it is impossible to draw without isolating at least one extra vertex from the circuit.

Hence, CN of a circuit in a fuzzy graph is 1.

Theorem 3. If Xm is the m–step graph of graph X, then C(Xm) = Cm(X).

Proof:

Given: Here, X is an FMG and Xm is an m-step graph of graph X.

To prove : C(Xm) = Cm(X)

Here, the vertex set of X and Xm are equal.

Let us take (u, v) ∈ C(Xm).

Here, (u, v) ∈ C(Xm),; therefore, there exist edges (u,w1),(v,w1);(u,w2),(u,w2);(u,wn),(v,wn) of some integer n.

Here, NB+(u) ∩ NB+(v) = {w1, w2, ..., wn}.

An edge (u,w1)Xm indicates that there occurs a u,w1m-length path from u to w1 and same as for (v,w1)Xm.

Hence,

(u,v)Xm(G).

Similarly, edge (x, y) ∈ Cm(X), implies that

(u,y)C(Xm).

Here, from (1) and (2), it follows that C(Xm) = Cm(X).

Note.

  • • In an FMG, if a vertex has only in-neighbors, then the vertex must be isolated from the corresponding competition graph.

  • • The competition graph on every fuzzy mixed circuit graph is a null graph.

Similar to competition in an ecosystem, economic competition among countries is considered here for representation. The competition between countries with respect to the health index (HI) and disaster risk index (DI) was evaluated. Therefore, we need to construct a network before evaluation and presentation.

  • • The advantages of this algorithm in health and disaster management are

    • ○ It captures the uncertainty and variability in the relationships among users, regions, and resources in a realistic manner using degrees of membership instead of binary values.

    • ○ It measures the intensity of competition among users, regions, and resources using the concept of maximal competitors and the competition index.

    • ○ It identifies the most vulnerable and influential regions and actors in the network using the concepts of competition centrality and degree of competition.

    • ○ It suggests optimal strategies for reducing the spread of diseases or disasters using the concepts of competition reduction and minimization.

6.1 Construction of a Network

To construct a mixed network, we consider 10 countries : Germany, India, the United Kingdom, France, Italy, Brazil, Canada, Russia, South Korea, and Spain.

All countries compete for health and disaster management (Figure 6). Therefore, there are direct links between health and disaster. Health and disaster index data were obtained from Wikipedia. All data with normalized values are listed in Table 1.

6.2 Algorithm

The results of competitions are calculated using the following steps.

  • Step 1: Consider a network of countries with health and disaster risks. All degrees of membership of the edge were considered as input.

  • Step 2: Find the out-neighbors of the countries and the intersections of all pairwise out-neighbors of the countries.

  • Step 3: Represent the network as FMG G = (V, E1, E2, μ1, μ2, σ), where V is the set of vertices (countries), E1 is the set of undirected edges (health relations), E2 is the set of directed edges (disaster relations), μ1 and μ2 are the membership functions for E1 and E2, respectively, and σ is the membership function for V. Adjacency matrices can be used to represent FMG1.

  • Step 4: Find the degree of membership of competing countries toward health and disaster using Table 1.

  • Step 5: The required result for the competition is obtained using Algorithm 2.

  • Step 6: Calculate the fuzzy mixed degree of each vertex, which is the sum of the degrees of membership of all edges incident to the vertex. The fuzzy mixed degree reflects the overall competitive performance of a country.

  • Step 7: Compare the fuzzy mixed degrees of the competing countries and rank them according to their values. The higher the fuzzy mixed degree, the better is the country’s performance.

6.3 Result

Table 2 presents the competition in health management, and Table 3 presents the competition in disaster management. The membership values for the corresponding competition graphs are listed in Table 4.

Thus, a real-life competition for health and disaster management among countries was presented using the concepts of the proposed algorithms. The following results were observed:

  • 1. The lower non-zero values in Table 4 indicate that competition among the corresponding countries is higher.

  • 2. The competition value should be used as the comparison between two items.

  • 3. Competition studies should be used as models for data analysis in marketing.

We presented the concept of FMGs as an extension of mixed graphs, and defined and analyzed the competition and competition numbers on FMGs. We also demonstrated the application of competition on FMGs to health and disaster scenarios. This study opens new avenues for future research on various topics in FMG theory, such as interval-valued FMGs, generalized FMGs, and fuzzy mixed planar graphs, and their potential applications in science and engineering problems.

Table. 1.

Table 1. Collections of data on health and disasters of countries from Wikipedia.

Sl. No.Country nameHINHIDINDI
1Germany73.320.8942.950.444
2India67.130.8196.641
3The United Kingdom74.460.9083.540.533
4France79.990.9762.620.395
5Italy66.590.8124.420.666
6Brazil56.290.6874.090.616
7Canada71.580.8733.010.453
8Russia57.590.7033.580.539
9South Korea81.9714.590.691
10Spain78.880.9623.050.459

Table. 2.

Table 2. Competition for health.

12345678910
100.0750.0140.0820.0820.2070.0210.1910.1060.039
20.07500.0890.1570.0070.1320.0540.1160.1810.036
30.0140.08900.0680.0960.2210.0350.2050.0920.053
40.0820.1570.06800.1640.2890.1030.2730.0240.121
50.0820.0070.0960.16400.1250.0610.1090.1880.043
60.2070.1320.2210.2890.12500.1860.0160.3130.168
70.0210.0540.0350.1030.0610.18600.170.1270.018
80.1910.1160.2050.2730.1090.0160.1700.2970.152
90.1060.1810.0920.0240.1880.3130.1270.29700.145
100.0680.1430.0540.0140.150.2750.0890.2590.0380.107

Table. 3.

Table 3. Competition for disaster.

12345678910
100.5560.0890.0490.2220.1720.0090.0950.2470.015
20.55600.4670.6050.3340.3840.5470.4610.3090.541
30.0890.46700.1380.1330.0830.080.0060.1580.074
40.0490.6050.13800.2710.2210.0580.1440.2960.064
50.2220.3340.1330.27100.050.2130.1270.0250.207
60.1720.3840.0830.2210.0500.1630.0770.0750.157
70.0090.5470.080.0580.2130.16300.0860.2380.006
80.0950.4610.0060.1440.1270.0770.08600.1520.08
90.2470.3090.1580.2960.0250.0750.2380.15200.232
100.0150.5410.0740.0640.2070.1570.0060.080.2320

Table. 4.

Table 4. Resultant competition.

12345678910
100.0750.0140.0490.0820.1720.0090.0950.1060.015
20.07500.0890.1570.0070.1320.0540.1160.1810.036
30.0140.08900.0680.0960.0830.0350.0060.0920.053
40.0490.1570.06800.1640.2210.0580.1440.0240.064
50.0820.0070.0960.16400.050.0610.1090.0250.043
60.1720.1320.0830.2210.0500.1630.0160.0750.157
70.0090.0540.0350.0580.0610.16300.0860.1270.006
80.0950.1160.0060.1440.1090.0160.08600.1520.08
90.1060.1810.0920.0240.0250.0750.1270.15200.145
100.0150.1430.0540.0140.150.1570.0060.080.0380

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Pabitra Kumar Gouri received an M.Sc. degree from Guru Ghasidas University, Bilaspur. He is the Headmaster & Secretary of Chhotakhelna Surendra Smriti Vidyamandir, India. His research interests are graph theory and fuzzy systems.

E-mail : pabitrakumargouri@gmail.com

Bharti Saxena received her Ph.D. from Rabindranath Tagore University, Bhopal in June 2019. She is an associate professor at Rabindranath Tagore University, Bhopal. Her areas of research are operation research, graph theory, and fuzzy sets.

E-mail : bhartisaxena060@gmail.com

Rajesh Navandar received his Ph.D. degree in electronics from NGBU University, India. He is an associate professor, Department of Electronic & Telecommunication Engineering, JSPM Jayawantrao Sawant College of Engineering Hadaspar, Pune. His research interests are VLSI design.

E-mail : navandarajesh@gmail.com

Pranoti Prashant Mane received her Ph.D. degree in electronics & telecommunications from Sant Gadage Baba Amravati University, Amravati, M.S., India. Dr. Pranoti Prashant Mane is an associate professor and the Head of Department, MES’s Wadia College of Engineering, Pune, India. Her research interests are image processing, machine learning, robotics, biomedical signal processing, and IOT.

E-mail : ppranotimane@gmail.com

Ramakant Bhardwaj received his Ph.D. from Barkatullah University Bhopal in Jan 2010. He received D.Sc. from APS University Rewa, MP in 2023. He is a professor of Department of Mathematics, Amity University, Kolkata, W.B. His research areas are non-linear analysis and computer-oriented mathematics (fuzzy set, soft set, and graph theory).

E-mail : rkbhardwaj100@gmail.com

Jambi Ratna Raja Kumar is Principal, Genba Sopanrao Moze College of Engineering, Balewodi, Pune, He has published more than 22 national and international papers (Scopus indexed and level of SCI 17) on artificial research and intelligence. research and and work. He has published seven machine patents. and has guided 75 UG projects, 25 PG projects, and three PhD students. He was a distinguished principal and was conferred the Dr. APJ Abdul Kailam Puroskar 2022 and the Innovative Leader of the Year—Maharashtra) awards at the Asia Education Summit & Awards 2019.

E-mail : ratnaraj.jambi@gmail.com

Surendra Kisanrao Waghmare is the Head of Department in E&TC Engineering, G H Raisoni College of Engineering and Management, Pune. He has 22 years of academic and research experience. He received B.E. and M.Tech. degrees in electronics engineering from S R T M University, Nanded, and a Ph.D. degree in Electronics Engineering from R T M Nagpur University. He has published 25 research articles in International Peer Journals and conducted a funded research project under BCUD, SP Pune University. His research interests include RF MEMS technology, embedded systems, VLSI design, neural networks, fuzzy logic, image processing, and automotive electronics.

E-mail : surendra.waghmare358@gmail.com, drssssamanta@gmail.com

Antonios Kalampakas received a Ph.D. in mathematics from Aristotle University of Thessaloniki, Greece. He is currently an associate professor of Mathematics at the American University of the Middle East, Kuwait. His research interests include graph theory, discrete mathematics, fuzzy graphs, graph neural networks, network optimization, hyperstructures, graph automata, and graph recognizability.

E-mail : antonios.kalampakas@aum.edu.kw

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 93-104

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.93

Copyright © The Korean Institute of Intelligent Systems.

Competitions on Fuzzy Mixed Graph and its Application in Countries for Health and Disaster

Pabitra Kumar Gouri1,2, Bharti Saxena1,2, Rajesh Kedarnath Navandar3, Pranoti Prashant Mane4, Ramakant Bhardwaj5, Jambi Ratna Raja Kumar6, Surendra Kisanrao Waghmare7, and Antonios Kalampakas8

1Department of Mathematics, Chhotakhelna Surendra Smriti Vidyamandir, Maligram, India
2Department of Mathematics, Rabindranath Tagore University, Bhopal, India
3Department of Electronic & Telecommunication Engineering, JSPM Jayawantrao Sawant College of Engineering Hadaspar, Pune, India
4Department of Computer Engineering, MES’s Wadia College of Engineering, Pune, India
5Department of Mathematics, Amity University, Kolkata, India
6Computer Engineering Department, Genba Sopanrao Moze College of Engineering, Pune, India
7Department of Electronics and Telecommunication Engineering, G H Raisoni College of Engineering and Management, Pune, India
8College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

Correspondence to:Bharti Saxena (bhartisaxena060@gmail.com)

Received: August 31, 2023; Revised: January 28, 2024; Accepted: June 25, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This study introduces the concept of fuzzy mixed graphs (FMGs) to represent uncertain relationships in social networks such as Facebook, where connections can be friends, followers, or mutuals. These graphs are an extension of the mixed graph theory, accommodating ambiguity in user relationships. We propose FMGs in which each vertex and link is assigned a membership degree between 0 and 1, reflecting the uncertainty of the connections. A subtype, competition FMGs, is explored to model scenarios in which users vie for shared resources or objectives. Our investigation reveals insights into the dynamics of competition within these graphs, including the conditions for the existence and uniqueness of maximal competitors, interplay between competition and network connectivity, and influence of fuzziness on competition intensity. By applying our theoretical framework to real-world scenarios, we demonstrate its utility in health and disaster management systems. By identifying essential regions and stakeholders affected by disease or disaster proliferation, our approach offers a novel analytical tool that can be substantiated by numerical simulations.

Keywords: Fuzzy mixed graphs, Social network analysis, Uncertainty modeling, Resource competition, Health systems analysis, Disaster management

1. Introduction

However, in reality, there may be situations where the data are not precise [1] and the concepts of crisp graphs are inadequate. To address such situations, Kaufman [2] introduced a fuzzy graph in which each node and edge has a fuzzy membership value (MV) that indicates the degree of belongingness. Based on this idea, the notion of a fuzzy competition graph was proposed [3] and applied to a food web under fuzzy conditions. Later, fuzzy graph extensions were developed [4, 5]. This inspired us to study the competitions [6] on mixed graphs under fuzzy settings.

Mixed graphs have both directed and undirected edges, and can capture the mixed nature of some real-world networks, such as the current Facebook network [7]. The concept of mixed graphs [8] was first introduced in 1970 and various aspects of these graphs have been studies such as coloring [9], matrices [10,11], and isomorphism [12]. Recently, Samanta et al. defined a semidirected graph [13], which is a special type of mixed graph in which two nodes can have directed and undirected edges between them. However, all these studies were based on crisp graphs. A few additional studies have been conducted [1417].

Das et al. [18] introduced the concept of picture fuzzy competition graphs, which model the competition among entities with uncertain and ambiguous information. They also presented some generalizations and applications of these graphs in education, ecology, business, and job markets. Deva and Felix [19] proposed a novel approach for designing a decision-making trial and evaluation laboratory method in a bipolar fuzzy environment, which can handle both positive and negative degrees of membership. Additionally, they illustrated the proposed method using a supplier selection case study. Akram and Sattar [20] investigated the notion of competition graphs under complex Pythagorean fuzzy information, which is a generalization of Pythagorean and complex intuitionistic fuzzy sets. They also explored some properties and characteristics of these graphs and reported potential applications in the field of ecology. Nithyanandham and Augustin [21] developed a novel technique for prioritizing coronavirus disease 2019 (COVID-19) vaccines based on a bipolar fuzzy p-competition graph and the additive ratio assessment method. They also compared the proposed technique with existing methods and demonstrated its validity and practicability. Narayanamoorthy et al. [22] introduced the concept of regular and totally regular bipolar fuzzy hypergraphs, which are generalizations of bipolar fuzzy graphs. They proved some mathematical properties and provided examples of these hypergraphs. Karthik et al. [23] proposed a material selection model based on spherical Dombi fuzzy graphs that could handle uncertain and ambiguous information. They applied this model to select the best material for a given application and compared it with other fuzzy models. Further details on the applications of fuzzy graphs can be found in [2438].

Therefore, we aim to study mixed graphs in fuzzy environments and propose different types of competitions for mixed graphs under this framework.

The motivation for this study was to extend the concept of the mixed graph theory to represent different types of relationships in social networks, as well as account for the uncertainty and fuzziness that may exist in these networks. We propose fuzzy mixed graphs (FMGs), which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, to capture the ambiguities in the following and connectedness. Additionally, we investigate the concept of competition on FMGs, which can model situations in which users compete for common resources or goals and explore their properties and characteristics. We applied this concept to analyze and improve health and disaster management systems by identifying the key regions and actors most affected by the spread of diseases or disasters. We aim to contribute to the field of fuzzy graph theory and its applications in real-world problems.

The novelties of this study are as follows:

  • • The introduction of FMGs, which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, captures the uncertainty in social networks.

  • • The investigation of competition on FMGs, which model situations where users compete for common resources or goals, and the exploration of its properties and characteristics.

  • • Competition is applied to FMGs to analyze and improve health and disaster management systems. More specifically, the application of competition enables FMGs to identify the key regions and actors that are most affected by the spread of diseases or disasters.

These novelties contribute to the field of fuzzy graph theory and its applications to real-world problems.

This paper is organized as follows. Section 1 introduces the study with a literature review. Section 2 describes some basic notions related to the study. Section 3 introduces competition fuzzy mixed graph. Section 4 introduces m-step competition fuzzy mixed graph. Section 5 proposes some related properties. Section 6 describes a numerical application. Section 7 concludes the study with future directions.

2. Basic Definitions

2.1 Graph

Let a graph G = (V, E) having n nodes and m links are represented by an adjacent matrix A = {aij} ∈ Rn×n, where aij = 1 if node i is linked with node j and aij = 0 otherwise. In social networks, two nodes are related by a link.

2.1.1 Competition graph

Definition 1. Let us consider a directed graph X = (N, L). The competition graph C(X) of X is an undirected graph that has an equal node set/vertex set and a link/edge between two distinct vertices if there exists a common out-neighbor.

2.2 Mixed Graph

Definition 2. Let N be a set of vertices or nodes of a graph and L = L1L2 where L1 = {(u, v) | u, vN} ⊂ N × N called a set of undirected edges of the graph and called a set of directed edges of the graph. The graph X = (N, L1, L2) is then said to form a mixed graph.

2.3 Fuzzy Graph

Let G = (V, E) be a graph with the pair of mappings σ : V → [0, 1] and μ : E → [0, 1] such that the condition μ(x, y) ≤ σ(x) ∧ σ(y) for all (x, y) ∈ EV × V, G = (V, E, μ, σ) is then called a fuzzy graph.

2.3.1 FMG

Definition 3. Graph X = (N, L1, L2, μ1, μ2, σ, δ) is called FMG if there exist functions σ : N → [0, 1], μ1 : L1 → [0, 1], μ2 : L2 → [0, 1] and μ : L2 → [0, 1] such that

μ1(x,y)σ(x)σ(y)for all (x,y)L1N×N,μ2(x,y)σ(x)σ(y)for all (x,y)L2N×N,σ(x,y)σ(x)-σ(y)for all (x,y)L2N×N,

where σ is MV of vertex, μ1 is MV of undirected edge, μ2 is MV of directed edge, and δ is measure of directionality.

Example 1. Let us consider an FMG with four nodes with two undirected and three directed edges.

2.4 UndirectedWalk and Path

Definition 4. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. An undirected walk is a sequence : v1e1v2e2v3 ... ekvk+1, where v1, v2, v3, ..., vk+1 are the vertices and e1, e2, ..., ek are the edges, such that all the edges are undirected and μ1(ei) > 0 for i = 1, 2, ..., k. An undirected walk from vertex u to v is said to be an undirected path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P(u,v)m.

Example 2. Let us consider an FMG (Figure 1). From Figure 1, v1-v4-v3 is an undirected path.

2.5 Directed Walk and Path

Definition 5. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. A directed walk is a sequence : v1 e1 v2 e2 v3 ... ek vk+1, where v1, v2, v3, ..., vk+1 are the vertices and e1, e2, ..., ek are the edges, such that all the edges are directed in the same direction, and μ2(ei) > 0 for i = 1, 2, ..., k. A directed walk from vertex u to v is said to be a directed path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P(u,v)m.

Example 3. Consider an FMG (Figure 1). From Figure 1, it follows that v3v2v4 is the directed path of the graph.

2.6 Neighborhood

Definition 6. Let X = (N, L1, L2, μ1, μ2, σ, δ) be an FMG. The neighborhood of vertex u is denoted by NB(u) and is defined as the set

NB(u)={(v,μ1(u,v)):;vVand μ1(u,v)>0}.

Note.

  • • The out-neighborhood of vertex u is denoted by NB+(u) and is defined as the set NB+(u)={(v,μ2(u,v)+μ2(u,v)δ(u,v)):vNand μ2(u,v)>0}.

  • • The in-neighborhood of vertex u is denoted by NB(u) and is defined as the set NB-(u)={(v,μ2(u,v)+μ2(u,v)δ(u,v)):vNand μ2(u,v)>0}.

Example 4. Let us consider an FMG (Figure 1). Here, from Figure 1, it follows that

  • (a) The neighborhood of v1 is NB(v1) = {(v4, 0.4)}.

  • (b) The out-neighborhood of v1 is NB+(v1) = {(v2, 0.44)}.

  • (c) The in-neighborhood of v2 is NB (v2) = {(v1, 0.44), (v3, 0.66)}.

2.7 m-Step Neighborhood

Definition 7. Let X=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The m–step neighborhood of vertex u is denoted by NBm(u) and is defined as the set NBm(u)={(v,σm(v)):vNand σm(v)=min{μ1(x,y):(x,y)P(u,v)m}}.

Note.

  • • The m-step out-neighborhood of vertex u is denoted by NBm+(u), and is defined as the set

    NBm+(u)={(v,σm(u,v)):vNand σm(u,v)}=min{μ2(a,b)+μ2(a,b)δ(a,b):(a,b)P(u,v)m}}.

  • • The m-step in the neighborhood of vertex u is denoted by NBm-(u) and is defined as the set

    NBm-(u)={(v,σm(u,v)):vNand σm(u,v)}=min{μ2(a,b)+μ2(a,b)δ(a,b):(a,b)P(v,u)m}}.

Example 5. Let us consider an FMG (Figure 1). From Figure 1, it follows that

  • (a) 2-step neighborhood of v1 is NB2(v1) = {(v3, 0.3)}.

  • (b) 2-step out-neighborhood of v3 is NB2+(v3)={(v4,0.48)}.

  • (c) 2-step in-neighborhood of v4 is NB2-(v4)={(v1,0.44),(v1,0.48)}.

3. Competition Fuzzy Graph (CFG)

3.1 Definition of CFG

Definition 8. Let X=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The competition fuzzy graph (CFG) of X is denoted by C(X) and is defined as an undirected fuzzy graph in which N is also a vertex set. There exists an edge between vertices x, yN if NB+(x) ∩ NB+(y) ≠ ∅ with membership value α(x, y) is defined by

α(x,y)=minvNB+(x)NB+(y)μ2(x,z)+μ2(x,z)δ(x,z)-{μ2(y,z)μ2(y,z)δ(y,z)}.

3.2 Algorithm to Find the CFG of FMG and the Adjacency Matrix of CFG

Algorithm 1.

  • Step 1 : Consider an FMG with given memberships of vertices and edges.

  • Step 2: Find the neighbors for all xiN.

  • Step 3: Find the intersections NB+(xi) ∩ NB+(xj) for all xi, xjV and xixj .

  • Step 4: If NB+(xi) ∩ NB+(xj) = {xk : k = 1, 2, ..., }, then find μ2(xi,xm)δ(xi,xm)-μ2(xj,xm)δ(xj,xm) and same for all non-void intersections.

  • Step 5: Find α(xi,xj)=minxkN+(xi)N+(xj)|μ2(xi,xm)δ(xi,xm)-μ2(xj,xm)δ(xj,xm)|. The values obtained in this section are the membership values of the edges of the CFG.

  • Step 6: If aij denotes the ij–th element of adjacency matrix of the CFG, then

aij={α(xi,xj),if there is an edge between xi,xj,1,if there is no edge between xi,xj.

A flowchart of the algorithm is shown in Figure 2.

Example 6. Let us consider an FMG (Figure 1). Here, from Figure 1, the CFG is defined as Figure 3.

4. m-Step Competition Fuzzy Graph

4.1 Definition of m-Step CFG

Definition 9. Let G=(N,L1,L2,μ1,μ2,σ,δ) be an FMG. The m-step CFG of G is denoted by Cm(G) and is defined as an undirected fuzzy graph in which N is also a vertex set, and there exists an edge between vertices x, yN if NBm+(x)NB+(y) with membership value βm(x, y) defined by

βm(x,y)=minvNBm+(x)NBm+(y)σm(x,v)-σm(y,v).

4.2 Algorithm to Find the m-Step CFG of FMG and the Adjacency Matrix of m-Step CFG

Algorithm 2.

  • Step 1: Consider an FMG with given memberships of vertices and edges.

  • Step 2: Find the neighbors for all xiN.

  • Step 3: Find the intersections NBm+(xi)NBm+(xj) for all xi, xjN and xixj.

  • Step 4: If NB+(xi) ∩ NB+(xj) = {xk : k = 1, 2, ..., }, then find σm(xi,xk)-σm(xj,xk) and same for all non-void intersections.

  • Step 5: Find βm(xi,xj)=minxkNB+(xi)NB+(xj)σm(xi,xk)-σm(xj,xk). The values obtained in this section are the membership values of the edges of the CFG.

  • Step 6: If aij denotes the ij-th element of the adjacency matrix of the CFG, then

aij={βm(xi,xj),if there is an edge between xi,xj,1,if there is no edge between xi,xj.

A flowchart of the algorithm is shown in Figure 4.

Example 7. Let us consider an FMG (Figure 1). From Figure 1, the 2-step CFG is defined as Figure 5.

5. Basic Properties

Theorem 1. Let X = (N, L) be a fuzzy graph. Subsequently, FMG X′ with C(X′) = X. occurs.

Proof.

Given:

Here, X = (N, L) is a fuzzy graph.

To prove : C(X′) = X

Let (x, y) be an edge of a fuzzy graph. FMG X′ is created, where C(X′) = X.

Let x′, y′ ∈ X′ be the equivalent vertices of x, yX.

We can join two directed edges x′, y′ to a vertex z′ ∈ X′ such that z′ ∈ NB+(x′) ∩ NB+(y′).

Therefore, this can be done for all nodes and links of X.

Hence, C(X′) = X.

Theorem 2. The competition number (CN) of a circuit in a fuzzy graph is 1.

Proof.

Claim:

Let us consider a circuit of length n in a fuzzy graph.

To prove:CN of a circuit in a fuzzy graph is 1.

We drew all possible directions with the vertices of the circuit to fit a corresponding FMG whose competition graph is the circuit. However, it is impossible to draw without isolating at least one extra vertex from the circuit.

Hence, CN of a circuit in a fuzzy graph is 1.

Theorem 3. If Xm is the m–step graph of graph X, then C(Xm) = Cm(X).

Proof:

Given: Here, X is an FMG and Xm is an m-step graph of graph X.

To prove : C(Xm) = Cm(X)

Here, the vertex set of X and Xm are equal.

Let us take (u, v) ∈ C(Xm).

Here, (u, v) ∈ C(Xm),; therefore, there exist edges (u,w1),(v,w1);(u,w2),(u,w2);(u,wn),(v,wn) of some integer n.

Here, NB+(u) ∩ NB+(v) = {w1, w2, ..., wn}.

An edge (u,w1)Xm indicates that there occurs a u,w1m-length path from u to w1 and same as for (v,w1)Xm.

Hence,

(u,v)Xm(G).

Similarly, edge (x, y) ∈ Cm(X), implies that

(u,y)C(Xm).

Here, from (1) and (2), it follows that C(Xm) = Cm(X).

Note.

  • • In an FMG, if a vertex has only in-neighbors, then the vertex must be isolated from the corresponding competition graph.

  • • The competition graph on every fuzzy mixed circuit graph is a null graph.

6. Applications

Similar to competition in an ecosystem, economic competition among countries is considered here for representation. The competition between countries with respect to the health index (HI) and disaster risk index (DI) was evaluated. Therefore, we need to construct a network before evaluation and presentation.

  • • The advantages of this algorithm in health and disaster management are

    • ○ It captures the uncertainty and variability in the relationships among users, regions, and resources in a realistic manner using degrees of membership instead of binary values.

    • ○ It measures the intensity of competition among users, regions, and resources using the concept of maximal competitors and the competition index.

    • ○ It identifies the most vulnerable and influential regions and actors in the network using the concepts of competition centrality and degree of competition.

    • ○ It suggests optimal strategies for reducing the spread of diseases or disasters using the concepts of competition reduction and minimization.

6.1 Construction of a Network

To construct a mixed network, we consider 10 countries : Germany, India, the United Kingdom, France, Italy, Brazil, Canada, Russia, South Korea, and Spain.

All countries compete for health and disaster management (Figure 6). Therefore, there are direct links between health and disaster. Health and disaster index data were obtained from Wikipedia. All data with normalized values are listed in Table 1.

6.2 Algorithm

The results of competitions are calculated using the following steps.

  • Step 1: Consider a network of countries with health and disaster risks. All degrees of membership of the edge were considered as input.

  • Step 2: Find the out-neighbors of the countries and the intersections of all pairwise out-neighbors of the countries.

  • Step 3: Represent the network as FMG G = (V, E1, E2, μ1, μ2, σ), where V is the set of vertices (countries), E1 is the set of undirected edges (health relations), E2 is the set of directed edges (disaster relations), μ1 and μ2 are the membership functions for E1 and E2, respectively, and σ is the membership function for V. Adjacency matrices can be used to represent FMG1.

  • Step 4: Find the degree of membership of competing countries toward health and disaster using Table 1.

  • Step 5: The required result for the competition is obtained using Algorithm 2.

  • Step 6: Calculate the fuzzy mixed degree of each vertex, which is the sum of the degrees of membership of all edges incident to the vertex. The fuzzy mixed degree reflects the overall competitive performance of a country.

  • Step 7: Compare the fuzzy mixed degrees of the competing countries and rank them according to their values. The higher the fuzzy mixed degree, the better is the country’s performance.

6.3 Result

Table 2 presents the competition in health management, and Table 3 presents the competition in disaster management. The membership values for the corresponding competition graphs are listed in Table 4.

Thus, a real-life competition for health and disaster management among countries was presented using the concepts of the proposed algorithms. The following results were observed:

  • 1. The lower non-zero values in Table 4 indicate that competition among the corresponding countries is higher.

  • 2. The competition value should be used as the comparison between two items.

  • 3. Competition studies should be used as models for data analysis in marketing.

7. Conclusions

We presented the concept of FMGs as an extension of mixed graphs, and defined and analyzed the competition and competition numbers on FMGs. We also demonstrated the application of competition on FMGs to health and disaster scenarios. This study opens new avenues for future research on various topics in FMG theory, such as interval-valued FMGs, generalized FMGs, and fuzzy mixed planar graphs, and their potential applications in science and engineering problems.

Fig 1.

Figure 1.

Fuzzy mixed graph.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Fig 2.

Figure 2.

Flowchart for Algorithm 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Fig 3.

Figure 3.

Competition fuzzy graph.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Fig 4.

Figure 4.

Flowchart of Algorithm 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Fig 5.

Figure 5.

The 2-step competition fuzzy graph.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Fig 6.

Figure 6.

Competing countries.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 93-104https://doi.org/10.5391/IJFIS.2024.24.2.93

Table 1 . Collections of data on health and disasters of countries from Wikipedia.

Sl. No.Country nameHINHIDINDI
1Germany73.320.8942.950.444
2India67.130.8196.641
3The United Kingdom74.460.9083.540.533
4France79.990.9762.620.395
5Italy66.590.8124.420.666
6Brazil56.290.6874.090.616
7Canada71.580.8733.010.453
8Russia57.590.7033.580.539
9South Korea81.9714.590.691
10Spain78.880.9623.050.459

Table 2 . Competition for health.

12345678910
100.0750.0140.0820.0820.2070.0210.1910.1060.039
20.07500.0890.1570.0070.1320.0540.1160.1810.036
30.0140.08900.0680.0960.2210.0350.2050.0920.053
40.0820.1570.06800.1640.2890.1030.2730.0240.121
50.0820.0070.0960.16400.1250.0610.1090.1880.043
60.2070.1320.2210.2890.12500.1860.0160.3130.168
70.0210.0540.0350.1030.0610.18600.170.1270.018
80.1910.1160.2050.2730.1090.0160.1700.2970.152
90.1060.1810.0920.0240.1880.3130.1270.29700.145
100.0680.1430.0540.0140.150.2750.0890.2590.0380.107

Table 3 . Competition for disaster.

12345678910
100.5560.0890.0490.2220.1720.0090.0950.2470.015
20.55600.4670.6050.3340.3840.5470.4610.3090.541
30.0890.46700.1380.1330.0830.080.0060.1580.074
40.0490.6050.13800.2710.2210.0580.1440.2960.064
50.2220.3340.1330.27100.050.2130.1270.0250.207
60.1720.3840.0830.2210.0500.1630.0770.0750.157
70.0090.5470.080.0580.2130.16300.0860.2380.006
80.0950.4610.0060.1440.1270.0770.08600.1520.08
90.2470.3090.1580.2960.0250.0750.2380.15200.232
100.0150.5410.0740.0640.2070.1570.0060.080.2320

Table 4 . Resultant competition.

12345678910
100.0750.0140.0490.0820.1720.0090.0950.1060.015
20.07500.0890.1570.0070.1320.0540.1160.1810.036
30.0140.08900.0680.0960.0830.0350.0060.0920.053
40.0490.1570.06800.1640.2210.0580.1440.0240.064
50.0820.0070.0960.16400.050.0610.1090.0250.043
60.1720.1320.0830.2210.0500.1630.0160.0750.157
70.0090.0540.0350.0580.0610.16300.0860.1270.006
80.0950.1160.0060.1440.1090.0160.08600.1520.08
90.1060.1810.0920.0240.0250.0750.1270.15200.145
100.0150.1430.0540.0140.150.1570.0060.080.0380

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