Comfortability of a Team in Social Networks
aa r X i v : . [ c s . S I] J un Comfortability of a Team in Social Networks
Lakshmi Prabha S a , T.N.Janakiraman a, ∗ a Department of Mathematics, National Institute of Technology, Trichy-620015, TamilNadu, India.
Abstract
There are many indexes (measures or metrics) in Social Network Analysis(SNA), like density, cohesion, etc. We have defined a new SNA index called“comfortability ”. In this paper, core comfortable team of a social networkis defined based on graph theoretic concepts and some of their structuralproperties are analyzed. Comfortability is one of the important attributes(characteristics) for a successful team work. So, it is necessary to find acomfortable and successful team in any given social network.It is proved that forming core comfortable team in any network is NP-Complete using the concepts of domination in graph theory. Next, we givetwo polynomial-time approximation algorithms for finding such a core com-fortable team in any given network with performance ratio O (ln ∆), where∆ is the maximum degree of a given network (graph). The time complexityof the algorithm is proved to be O ( n ), where n is the number of persons(vertices) in the network (graph). It is also proved that the algorithms givegood results in scale-free networks. Keywords:
Social networks, comfortability, less dispersive set, core ∗ Corresponding Author - T.N.Janakiraman.
Email addresses: [email protected] (Lakshmi Prabha S), [email protected] (T.N.Janakiraman)
Preprint submitted to ArXiv January 29, 2018 omfortable team, graph algorithms, performance ratio, k − domination.
1. Introduction
There are many factors, lack of which affect the group or team effec-tiveness. Team processes describe subtle aspects of interaction and patternsof organizing, that transform input into output. The team processes will bedescribed in terms of seven characteristics: coordination, communication, co-hesion, decision making, conflict management, social relationships and per-formance feedback. The readers are directed to refer Michan et al. [9] forfurther details of characteristics of team and Forsyth [3] for more details ongroup dynamics. In this paper, we discuss about an attribute or character-istic called “COMFORTABILITY ”, which is also essential for a successfulteam work. So, we defined it as a new SNA index in our paper [6].Since the beginning of Social Network Analysis, Graph Theory has beena very important tool both to represent social structure and to calculatesome indexes, which are useful to understand several aspects of the socialcontext under analysis. Some of the existing indexes (measures or metrics)are betweenness, bridge, centrality, flow betweenness centrality, centraliza-tion, closeness, clustering coefficient, cohesion, degree, density, eigenvectorcentrality, path length. Readers are directed to refer Martino et. al. [8] formore details on indexes in SNA. In our paper [6], we defined a new SNA indexcalled ‘comfortability ’. Based on this index, we have defined comfortableteam, better comfortable team and highly comfortable team in our paper [6]and totally comfortable team in our paper [7].2et the social network be represented in terms of a graph, with the vertexof the graph denotes a person (an actor) in the social network and an edgebetween two vertices in a graph represents relationship between two personsin the social network. All the networks are connected networks in this paper,unless otherwise specified. If the given network is disconnected, then eachconnected component of the network can be considered and hence it is enoughto consider only connected networks. Hereafter, the word ‘team’ representsinduced sub network (sub graph) of a given network (graph).Following are some introduction for basic graph theoretic concepts .Some basic definitions from Slater et al. [4] are given below.The graphs considered in this paper are finite, simple, connected andundirected, unless otherwise specified. For a graph G , let V ( G ) (or simply V ) and E ( G ) denote its vertex (node) set and edge set respectively and n and m denote the cardinality of those sets respectively. The degree of avertex v in a graph G is denoted by deg G ( v ). The maximum degree of thegraph G is denoted by ∆( G ). The length of any shortest path between anytwo vertices u and v of a connected graph G is called the distance between u and v and is denoted by d G ( u, v ). For a connected graph G , the eccen-tricity e G ( v ) = max { d G ( u, v ) : u ∈ V ( G ) } . If there is no confusion, wesimply use the notions deg ( v ), d ( u, v ) and e ( v ) to denote degree, distanceand eccentricity respectively for the concerned graph. The minimum andmaximum eccentricities are the radius and diameter of G , denoted by r ( G )and diam ( G ) respectively. A vertex with eccentricity r ( G ) is called a centralvertex and a vertex with eccentricity diam ( G ) is called a peripheral vertex .For v ∈ V ( G ), neighbors of v are the vertices adjacent to v in G . The3eighborhood N G ( v ) of v is the set of all neighbors of v in G . It is alsodenoted by N ( v ). N j ( v ) is the set of all vertices at distance j from v in G .A vertex u is said to be an eccentric vertex of v , when d ( u, v ) = e ( v ). If A and B are not necessarily disjoint sets of vertices, we define the distancefrom A to B as dist ( A, B ) = min { d ( a, b ) : a ∈ A, b ∈ B } . Cardinality ofa set D represents the number of vertices in the set D . Cardinality of D isdenoted by | D | .A vertex of degree one is called a pendant vertex . A walk of length j isan alternating sequence W : u , e , u , e , u , . . . , u j − , e j , u j of vertices andedges with e i = u i − u i . If all j edges are distinct, then W is called a trail .A walk with j + 1 distinct vertices u , u , . . . , u j is a path and if u = u j but u , u , . . . , u j are distinct, then the trail is a cycle . A path of length n isdenoted by P n and a cycle of length n is denoted by C n . A graph G is said tobe connected if there is a path joining each pair of nodes. A component of agraph is a maximal connected sub graph. If a graph has only one component,then it is connected, otherwise it is disconnected . A tree is a connected graphwith no cycles (acyclic).We say that H is a sub graph of a graph G , denoted by H < G , if V ( H ) ⊆ V ( G ) and uv ∈ E ( H ) implies uv ∈ E ( G ) . If a sub graph H satisfies the added property that for every pair u, v of vertices, uv ∈ E ( H )if and only if uv ∈ E ( G ), then H is called an induced sub graph of G . Theinduced sub graph H of G with S = V ( H ) is called the sub graph inducedby S and is denoted by h S | G i or simply h S i .Let k be a positive integer. The k th power G k of a graph G has V ( G k ) = V ( G ) with u, v adjacent in G k whenever d ( u, v ) ≤ k .4he concept of domination was introduced by Ore [10] . A set D ⊆ V ( G )is called a dominating set if every vertex v in V is either an element of D oris adjacent to an element of D . A dominating set D is a minimal dominatingset if D −{ v } is not a dominating set for any v ∈ D . The domination number γ ( G ) of a graph G equals the minimum cardinality of a dominating set in G .A set D of vertices in a connected graph G is called a k -dominating set ifevery vertex in V − D is within distance k from some vertex of D . The conceptof the k -dominating set was introduced by Chang and Nemhauser [1, 2] andcould find applications for many situations and structures which give rise tographs; see the books by Slater et al [4, 5]. So, dominating set is nothing but1-dominating set.Sampath Kumar and Walikar [11] defined a connected dominating set D to be a dominating set D , whose induced sub-graph h D i is connected .The minimum cardinality of a connected dominating set is the connecteddomination number γ c ( G ).The readers are also directed to refer Slater et al. [4] for further detailsof basic definitions, not given in this paper.Let us recall the terminologies as follows: The symbol ( → ) denotes “rep-resents ” • Graph → Social Network (connected) • Vertex of a graph → Person in a social network • Edge between two vertices of a graph → Relationship between twopersons in a social network • Induced subgraph of a graph → Team or Group of a social network.5iven a connected network of people. Our problem is to find a team (subgraph) which is less dispersive, highly flexible and performing better.
Note . Notation 1:
In all the figures of this paper, • { v , v , . . . , v n } represent the vertex set of the graph G , that is, V ( G ) = { v , v , . . . , v n } . • The numbers besides every vertex represents the eccentricity of thatvertex. For example, in Figure 1, in the graph G , e ( v ) = 5, e ( v ) = 4, e ( v ) = 3, e ( v ) = 3, e ( v ) = 4 and e ( v ) = 5. • The set notation D = { v , v , . . . , v n } represents only the individualpersons but does not represent the relationship between them. • The notation h D i represents the team. h D i is the induced sub graphof G , which represents the persons as well as the relationship betweenthem. So, the set D represents only the team members and the teamrepresents the persons with their relationship.The remaining part of the paper is organized as follows: • Section 2 discusses about prior work. • Section 3 defines core comfortable team and analyses the concept withsome examples. • In section 4, an approximation algorithm is given for finding core com-fortable team in any given network, with illustrations. Time complex-ity of the algorithm is analyzed. Also, correctness of the algorithm andperformance ratio of the algorithm are proved in this section.6
In Section 5, another approximation algorithm is given, based on con-nected dominating set, for finding core comfortable team in any givennetwork, with illustrations. Time complexity, correctness and perfor-mance ratio of this algorithm are discussed in this section. • In section 6, some advantages of the two algorithms are given. • Section 7 concludes the paper and discusses about some future work.
2. Prior Work
In our paper [6], we defined characteristics of a good performing team andmathematically formulated them and given approximation algorithm for find-ing such a good performing team.
In order to make this paper self-contained,we have given all the necessary definitions, examples and properties from ourpaper [6], which are needed for this paper , in this section.
Definition 1. [6] A team is said to be good performing or successful ifthe team is1. less dispersive2. having good communication among the team members3. easily accessible to the non- team members4. a good service provider to the non-team members (for the whole net-work).
Mathematical FormulationDomination → good service provider to the non-team members . Connectedness → good communication among team members .7 efinition 2. Less Dispersive Set: [6] A set D is said to be less dispersive,if e h D i ( v ) < e G ( v ), for every vertex v ∈ D . Definition 3. Less Dispersive Dominating Set: [6] A set D is said tobe a less dispersive dominating set if the set D is dominating, connected andless dispersive. The cardinality of minimum less dispersive dominating set of G is denoted by γ comf ( G ). A set of vertices is said to be a h γ comf − set i , if itis a less dispersive dominating set with cardinality γ comf ( G ). Definition 4. Comfortable Team: [6] A team h D i is said to be a comfort-able team if h D i is less dispersive and dominating. Minimum comfortableteam is a comfortable team with the condition: | D | is minimum. Example 1:
Consider the graph (network) G in Figure 1. Figure 1: A Network and its Comfortable Team
Here, G is a path of length six ( P ). D = { v , v , v , v } . The inducedsub graph h D i of G forms a path of length four ( P ) and so it dominates allthe vertices in V − D . Also, h D i forms the comfortable team of G , because e h D i ( v ) = 3 < e G ( v ) . ⇒ e h D i ( v ) < e G ( v ). Similarly, e h D i ( v i ) < e G ( v i )for every i = 3 , ,
5. Thus, D forms less dispersive set and hence h D i formsthe comfortable team of G . ⇒ γ comf ( P ) = 4.8o, the problem is coined as: Find a team which is dominating, connectedand less dispersive. It is to be noted that there are many graphs which donot have h γ comf − set i . So, we must try to avoid such kind of networks forsuccessful team work. Example 2:
Consider the graph G in Figure 2. Figure 2: A Network and its CC Team
Here, G is a cycle of length six ( C ). The vertices v and v dominate allthe vertices of G . So, with connectedness, we can take D = { v , v , v , v } .The set D dominates G , but D is not less dispersive, because, e h D i ( v ) = 3 = e G ( v ) and e h D i ( v ) = 3 = e G ( v ). The vertices v and v maintained the original eccentricity as in G . Thus, e h D i ( v ) < e G ( v ), for everyvertex V ∈ D is not satisfied . So, D is not less dispersive and hence h D i isnot a comfortable team.Also, D = { v , v , v } forms less dispersive set in G , (from Figure 2), but D is not dominating. The vertex v is left undominated. Note . From the above discussion in Example 2, we get, • the less dispersive set may not be dominating9 the dominating set may not be less dispersive.So, under one of these two cases, the graph G does not possess comfortableteam. It is to be noted that not only C , but all the cycles C n , do not possessa comfortable team. Also, there are infinite families of graphs which do notpossess comfortable team. Disadvantage of the comfortable team [6]:
As discussed in the Example 2, comfortable team does not exist in any givennetwork. Infinite families of networks do not possess comfortable team.The main aim of this paper is to find a team which is more comfortableand less dispersive , in any given social network .So, we define a core comfortable team with a modification in the comfortableteam in the next section.
3. Core Comfortable Team
As discussed in Note 2, the less dispersive set may not be dominating.So, we define less dispersive set with k ∗ − domination. Definition 5. Less Dispersive k ∗ − Dominating Set:
A set D is said to be a ‘less dispersive k ∗ -dominating’ set if D is a lessdispersive set and a k ∗ − dominating set. That is,1. e h D i ( v ) < e G ( v ), for every vertex v ∈ D (less dispersive)2. dist ( D, V − D ) ≤ k ∗ ( k ∗ − domination). This implies that k ∗ ≤ diam ( G ), (because any vertex in V − D can reach D at a distanceof at most diam ( G )). 10inimum cardinality of a ‘less dispersive k ∗ -dominating’ set of G is denotedby γ k ∗ comf ( G ) and maximum cardinality of a ‘less dispersive k ∗ -dominating’set of G is denoted by Γ k ∗ comf ( G ). Definition 6. Core Comfortable Team:
A team h D i is said to be a CoreComfortable (CC) team if h D i is less dispersive k ∗ -dominating. • Min CC team is a CC team with the condition: | D | and k ∗ are minimum • Max CC team is a CC team with the condition: | D | is maximum and k ∗ is minimum. Example 3:
Consider the graph G ( C ) in Figure 2. In C , D = { v , v , v } forms a less dispersive 2-dominating set, because1. e h D i ( v i ) < e G ( v i ), for i = 1 , , k ∗ = 2, because v is reachable from D by distance two, v and v arereachable from D by distance one. ⇒ dist ( D , V − D ) ≤ C and hence γ comf ( G ) = Γ comf ( G ) = 3. Theorem 1.
Forming core comfortable team in a given network is NP-complete.Proof.
Let D be a minimum less dispersive k ∗ − dominating set of G . ⇒ D is a connected k ∗ − dominating set of G (since any less dispersive setis a connected set). ⇒ D is a connected dominating set of G k ∗ (by definition of the graph G k ).Finding γ c ( G ), for any graph G is NP-complete (by Slater et al. [4]). ⇒ Finding γ c ( G k ∗ ) is NP-complete. 11 Finding minimum less dispersive k ∗ − dominating set of G is NP-complete(by above points).Thus forming core comfortable team in a given network is NP-complete.Next, we give two polynomial time approximation algorithms for findingcore comfortable team in any given network.
4. Approximation Algorithm 1
In this section, we give a polynomial-time approximation algorithm forfinding CC team from a given network. • D → minimum less dispersive, k ∗ -dominating set. • D → minimal less dispersive, k -dominating set, (output of our algo-rithm). ⇒ | D | ≥ | D | . • k ∗ → the distance between two sets D and V − D , that is, dist ( D, V − D ) ≤ k ∗ . • k → the distance between two sets D and V − D , that is, dist ( D , V − D ) ≤ k . • Instant D → The set D at a particular iteration. • Performance ratio = | minimal set || minimum set | .12 .2. Algorithm GOCOM A polynomial time approximation algorithm for finding CC team is givenbelow.
Input: G . Output: D , which is a less dispersive, k -dominating set, so that h D i is acore comfortable team.GOCOM(G)1. Choose a central vertex v (ties can be broken arbitrarily) and add it to D .2. If diam ( G ) is even, then choose all the vertices in N j ( v ), for j ≤ diam ( G )2 and add them to D .else choose all the vertices in N j ( v ), for j ≤ ( diam ( G ) − D .3. Put i = (cid:22) diam ( G )2 (cid:23) .4. If e h D i ( v ) < e G ( v ), for every vertex v ∈ D , then Goto next step (step5), else Goto Step 7.5. Put i = i + 1.6. Choose all vertices from N i ( v ) and add it to D . Then GOTO step 4.7. Remove suitably some vertices from D (say from D ∩ N i − ( v ), from D ∩ N i − ( v ), and so on) such that the condition in step 4 is satisfied.8. Print D .9. Stop. Note . At each iteration after forming D , we check up the condition: e h D i ( v ) < e G ( v ) , f or every vertex v ∈ D . (1)13f the condition 1 is not satisfied in Step 4, then the Step 7 is executed inthe algorithm. We remove some vertices from D until the condition 1 issatisfied.If the condition 1 is satisfied, then we add some vertices to D . Theremay be a question: why should the process be continued? We add somevertices to D , in order to minimize k . Our aim is to minimize k as well asto satisfy the condition 1.So, in Step 6, we add some vertices to D and check up the condition 1.If the condition 1 is satisifed, then we proceed to add vertices to D . But,we can not go on adding vertices to D , because at one stage, the condition1 will not be satisfied. Then the step 7 will be executed.After Step 7 is executed, the condition 1 is satisfied. So, no furtheraddition and deletion of vertices are done. The algorithm prints D andends. Note . The algorithm GOCOM finds a CC team. If Min CC team is needed,some vertices can be removed from the output set such that k ∗ is minimum.If Max CC team is needed, then some vertices could be added to the outputsuch that condition 1 is satisified and k ∗ is minimum. Consider the network (graph) G as in the Figure 3. In this network, diam ( G ) = 5.Inititally, let us choose the central vertex v and add them to D . As diam ( G ) is odd, let us choose up to diam ( G ) − = 2 neighborhoods of v andadd them to D . Now, instant D = { v , v , v , v , v , v , v , v } . But, we cansee in Figure 4 that vertices v , v and v of D at an intermediate interation14 igure 3: Illustration Figure violate the condition 1. In order to make D maintain the condition 1, weremove some vertices from D .We can either remove those vertices v , v and v or the vertices v and v , so that the condition 1 is satisfied. So, we get two different outputs. Letus make the first one as D = { v , v , v , v , v } and the other one as D = { v , v , v , v , v , v } . Refer Figure 5. Both outputs satisfy the condition 1and hence both h D i and h D i are core comfortable teams. But, for D , weget k = 2 and for D , we get k = 1. D is a maximal CC team for k = 2 and hence Γ comf ( G ) = 5. Also, for k = 2, we see that D = { v , v } forms a minimal CC team. The vertices v and v are suuficient and every vertex in V − D is reachable from D by adistance of at most two. Also, D satisfies the condition 1. Refer Figure 5.Thus, D is a minimal CC team and hence γ comf ( G ) = 2.15 igure 4: Intermediate Iteration showing vertices violating the condition (1) For k = 1, D is a minimal as well as a maximal CC team and hence γ comf ( G ) = Γ comf ( G ) = 6. Figure 5: Three Different Outputs
Note . From the Illustration 4.3, we can observe that the core comfortableteam is not unique for a given network. A social network may have many corecomfortable teams. We can choose one team among all the teams whichever16s suitable for a particular situation. We can choose any CC team (minimumor maximum) according to our need for a particular situation.
Let us discuss the time complexity of the algorithm as follows:The definition of CC team and the algorithm GOCOM is dependent oneccentricity of every vertex. So, we have to find eccentricity of every vertexof G . By Performing Breadth-First Search (BFS) method from each vertex,one can determine the distance from each vertex to every other vertex. Theworst case time complexity of BFS method for one vertex is O ( n ). As theBFS is method is done for each vertex of G , the resulting algorithm hasworst case time complexity O ( n ). As eccentricity of a vertex v is defined as e ( v ) = max { d ( u, v ) : u ∈ V ( G ) } , finding eccentricity of vertices of G takesat most O ( n ).Thus, the total worst case time complexity of the algorithm is at most O ( n ). In order to prove that the algorithm GOCOM yields a CC team, it issufficient to prove that the condition in Step 4 of the algorithm is satisfied.As proof follows from Note 3, we state the following theorem without proof.
Theorem 2. e h D i ( v ) < e G ( v ) , for every vertex v ∈ D , where D is theoutput of our Algorithm GOCOM.4.6. Performance Ratio of the Algorithm GOCOM It is to be noted that the algorithm has two parameters, namely, | D | and k ∗ . The set D represents the team members of CC team, and k ∗ represents17he distance between the team members and non-team members. We findperformance ratio of the algorithm for finding Min CC team . So, it isnecessary that both D and k ∗ should be minimized simultaneously.The following theorem gives the performance ratio for finding the mini-mum less dispersive k ∗ − dominating set. Performance ratio of the algorithmfor finding the minimum CC team is equal to | D | / | D | . Theorem 3.
The performance ratio of the algorithm for finding Min CCteam is at most O (ln ∆( G )) .Proof. Let D be a minimal less dispersive k − dominating set of G andlet D be a minimum less dispersive k ∗ − dominating set of G . ⇒ D is a connected k ∗ − dominating set of G (since any less dispersive setis a connected set). ⇒ D is a connected dominating set of G k ∗ (by definition of the graph G k ∗ ).Performance ratio for finding γ c ( G ) is at most O (ln ∆( G )). ⇒ Performance ratio for finding γ c ( G k ∗ ) is at most O (ln ∆( G k ∗ )).But, ∆( G k ∗ ) ≤ (∆( G )) k ∗ . ⇒ Performance ratio for finding γ c ( G k ∗ ) is at most O (ln(∆( G )) k ∗ ) = O ( k ∗ ln ∆( G )) = O (ln ∆( G )).Thus, the performance ratio of the algorithm for finding Min CC team is atmost O (ln ∆( G )).Next, let us give the performance ratio for finding k ∗ . Performance ratioof the algorithm for finding k ∗ is equal to k/k ∗ . Theorem 4.
The performance ratio of the Algorithm GOCOM for finding k ∗ is at most r ( G ) . roof. Let us recall from Notation 2: dist ( D, V − D ) ≤ k ∗ and dist ( D , V − D ) ≤ k , that is, k and k ∗ denote the minimal parameter and the minimumparameter respectively.In the Algorithm GOCOM, we start from a central vertex. So, the outputset D always contains at least one central vertex. This implies that anyvertex in V − D is reachable from D by distance at most r ( G ). Thus, k ≤ r ( G ).Also, as 1-dominating set is possible in many cases, k ∗ ≥ kk ∗ ≤ r ( G ).
5. Approximation Algorithm 2
In this section, we give another approximation algorithm for finding corecomfortable team and analyze its time complexity, correctness and perfor-mance ratio.Algorithm CONCOMF:Input: G .Output: D , which is a less dispersive k − dominating set, so that h D i is aCC team.CONCOMF( G )1. Find a connected k − dominating set of G (using an approximationalgorithm).2. Store all the vertices in D .3. If e h D i ( v ) < e G ( v ), for every vertex v ∈ D , then GOTO Step 5 elseGoto Step 4 (next step). 19. Remove suitably some vertices from D such that the condition in step3 is satisfied .5. Print D .6. Stop. Note . The algorithm CONCOMF finds a CC team. If Min CC team isneeded, in Step 1, find a minimal k − dominating set of G . If Max CC teamis needed, in Step 1, find a maximal connected k − dominating set of G .Executing the Algorithm CONCOMF in graph G of Figure 3, we get D ismaximal CC team and D is minimal CC team for k = 2 and D is a minimaland maximal CC team for k = 1. The worst case time complexity of the approximation algorithm for find-ing connected dominating set is at most O ( n ). Also, as discussed in theSection 4.4, the definition of CC team is dependent on eccentricity of everyvertex and finding eccentricity of vertices of G takes at most O ( n ) in worstcase.Thus, the total worst case time complexity of the algorithm is at most O ( n ). At the end of Step 4 in Algorithm CONCOMF, the output D satisfiesthe condition 1. So, we state the following theorem without proof. Theorem 5. e h D i ( v ) < e G ( v ) , for every vertex v ∈ D , where D is theoutput of our Algorithm GOCOM. .3. Performance Ratio Of the Algorithm CONCOMF As discussed in the Section 4.6, for finding a Min CC team, it is necessarythat the two parameters | D | and k ∗ should be minimized simultaneously.The following theorem gives the performance ratio for finding the mini-mum less dispersive k ∗ − dominating set. As proof follows form Theorem 3,we state the following theorem without proof. Theorem 6.
The performance ratio of the Algorithm CONCOMF for findingMin CC team is at most O (ln ∆( G )) . Next, we give performance ratio of the algorithm CONCOMF for finding k ∗ . Theorem 7.
The performance ratio of the Algorithm CONCOMF for finding k ∗ is at most diam ( G ) .Proof. As 1-dominating set is possible in many cases, k ∗ ≥ D is reachable form V − D by a distance of at most diam ( G ). This implies that k ≤ diam ( G ).Thus, kk ∗ ≤ diam ( G ).
6. AdvantagesAdvantage 1:
The algorithms give good results in scale free net-works for finding core comfortable team.
Explanation:
The performance ratio of the Algorithms GOCOM and CON-COMF for finding k ∗ , are dependent on r ( G ) and diam ( G ) respectively (byTheorems 7 and 4). It is known that the growing scale-free networks have21lmost constant diameter in practice. So, the algorithms give constant per-formance ratio in scale free networks. Advantage 2:
The algorithms can be applied in any random networks for finding CC team.
Explanation:
From the theorems 3, 4, 6 and 7, it is clear that the perfor-mance ratio of the algorithm for finding | D | and k ∗ is dependent on diam ( G )and ∆( G ). As both these terms can be expressed in terms of the probability p , the performance ratio of the algorithm can be easily obtained for randomnetworks in terms of p . Advantage 3:
CC team can be obtained in disconnected networks alsousing the algorithms.
Explanation:
If the network (graph) is disconnected, then as mentioned inthe Section 1, algorithm can can be applied to each connected component ofthe network. Thus, algorithm can be applied to find CC team in any givennetwork.
7. Conclusion
In this paper, core comfortable team of a social network is defined. Itis proved that forming core comfortable team in any given network is NP-complete. Two polynomial time approximation algorithms are given for find-ing a CC team in any given network and the time complexity of those algo-rithms are given to be O ( n ), where n is the number of vertices of G . Thecorrectness of the algorithms are analyzed. The performance ratio of thealgorithms for finding Min CC team is proved to be O (ln ∆), where ∆ isthe maximum degree of G . The performance ratio of the two algorithms for22nding k ∗ is proved to be at most r ( G ) and diam ( G ) respectively. It is alsoproved that the algorithms give good results in scale free networks. The algorithms can be applied in a particular social network, for example,Poisson network and can be tried to reduce the performance ratio in thatnetwork. Algorithms can be implemented to get exact values also in someparticular networks, for example scale free networks and so on. Also, asdiscussed in Note 5, a social network can have many CC teams. So, we cananalyse the different situations for which the CC teams are suitable.
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