5.1 Finding Betweenness centrality utilizing Map Reduce

The whole node acts as a bridge as betweenness centrality develops over the shortest path connecting the two nodes. The following analysis examines betweenness vertex v at the graph $G= (V,E)$ including $V$ vertices:

1. Determine which node pair's shortest paths are $(s,t)$.

2. Calculate the percentage of the shortest path that leads from vertex $v$ to each of the node pairs $(s, t)$.

3. Total vertex pairs $(s,t)$ add up to this ratio.

The betweenness centrality indicated as follows:

where $\rho_{st}$ implies entire shortest paths through `$s$' node $t$ and $\rho_{st}(v)$ represents overall paths to pass from $v$

Algorithm 1: Map of betweeness centrality job

Algorithm 2: Betweeness centrality minimize job

5.2 Map reduce dependent nearness centrality finding

A node's closeness to all other nodes increases with its center position, this is labeled in equation (2).

The less path distance amid the $v$ and $u$ vertices is expressed as $dist(v,u)$, here $N$ refers total nodes in the network that is shown.

Algorithm 3: Closeness centrality map job

Algorithm 4: Closeness centrality decrease job

5.3 Map reduce dependent centrality degree finding

A simple method of calculating centrality that counts a node's neighbors is called degree centrality. This centrality can be mathematically stated in (3)

where deg($v$) implies the quantity of node real connection node $v$, and $n-1$ is the count of predictable link of every vertex.

Algorithm 5: The degree of centrality on a map job

Algorithm 6: Decrease the centrality degree for the job

5.4 Computing Level of Influence

Equations (4), (5), and (6) define the centrality ratios. The aggregation which refers to as the DI and is normalized among 0 and 1 that is exhibited in equation (7). Consider $\alpha$, $\beta=0.4$, and $\gamma =0.2$.

• DI(vi) - DI for node vi.

• $\alpha$, $\beta$, $\gamma$ - weight variables $\beta \ge \gamma > \lambda$ with $\alpha +\beta + \gamma =1$.

6.1 Relative Distance

Node vi's relative distance, or $\rho$I, is defined as the shorter distance between nodes v${}_{i}$ with high DI values is depicted in equation (8),

Typically with huge density for the node that take $\rho_{i} = \max_{j}(d_{ij})$.

Structural Centrality

The structural centers can be identified by their greater DI value for node vi is labelled in eqn (9).

The metric essentially ignores this situation when adjacent nodes along the better level of effect are recognized as structural centers.

Algorithm 7: Identify the leader nodes

7.1 Node Vector Set Generation

The leader nodes found in the previous step were used to create the node vectors for each node on the chosen network. Fig. 3 shows the node vectors for a certain nodes in the Karathe network.

Fig. 3. Node vector including seed nodes for karathe network.

7.2 Generation of Soft Community Clusters

Algorithm 8: Map_Reduce_FCM

Algorithm 9: Cluster_Centre_Mapper

Algorithm 10: Cluster_Center_Reducer

Algorithm 11: Membership_Mapper

Algorithm 12: Membership_Reducer

7.3 Allocating leader nodes towards the communities

The community cluster whose centroid has the greatest similarity to the leader node receives the allocation of the leader node. This is labeled in equation (10),

V${}_{leader}$ - Leader node,

C${}_{i}$ - Centroid for $i$th cluster.

8.1 Description of the Dataset

The datasets for our experiments include Dolphins, Football, Karate, Political Blogs, Power, Yeast, Political Book Network, Netscience, Jazz, and Email networks. Detailed information is provided in Table 1, indicating the number of vertices (|V|), links (|E|), average degree (Davg), clustering coefficient (C), and average path length (PLavg).

8.2 Outcomes based upon datasets

This section examines the proposed and existing CORPA, LBCD, LC, SLPA, LFM, and CPM methods under 10 real-world databases. The results produced by these techniques are tabulated in Table 2. Fig. 5 displays the membership matrix for the karate network. Fig. 6 reveals nodes with a membership value of 1 for one cluster, identifying them as leader nodes.

The relation among EQ and $\beta$ is displayed in Fig. 6. The value of EQ is raised with $\beta$ variable then it attains higher value0.4 to 0.45 in small datasets as shown in Fig. 6(a). and it reaches the greater value 0.35 to 0.4 in large datasets as shown in Fig. 6(b).

Table 3 tabulates the implementation time of proposed approach. This is represented in Fig. 7.

Fig. 5. Membership Matrix of karathe Network.}

Fig. 7. Implementation period of four real-world datasets.