# Social Network Analysis and Churn Prediction in Telecommunications Using Graph Theory

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## Abstract

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## 1. Introduction

- We study a broad set of metrics to reveal various important telco network call graph structural properties;
- We define three sets of metrics that can precisely describe each element of the network call graph—directed and undirected metrics, as well as articulation point metrics; we will demonstrate that, even though telco networks are directed by nature, the undirected metrics as well can provide us with a meaningful and important additional insight regarding churn prediction;
- We introduce a new method based on graph theory metrics for detection of not valid (non-human) nodes of the network call graph;
- We present a model based on real-life data that can provide important business insights and help design better strategies for struggle against churn for mobile telco operators;
- We define a method for combining two separate clustering models results into a unified scale that presents a final model;
- Due to the limited scope of base variables analyzed in our paper (only number and duration of calls, as well as information regarding influencer churn), it is possible to combine our method with other state-of-art models that focus on other metrics (e.g., some customer satisfaction based churn model), so that the churn prediction success rate could be increased even more.

## 2. Related Work

## 3. Methodology

#### 3.1. Method Description

#### 3.2. Node Weights and Initial Graph Analysis

- Node degree is defined as the total number of edges that are incident to a specific node. Node degree can be calculated both for directed and undirected graphs. For undirected graphs, node degree would be equal to the number of nodes that are connected to a specific node On the other hand, for directed graphs node degree would be equal to the sum of node in-degree and node out-degree.
- Node in-degree in directed graphs is defined as the total number of in-edges incident to specific node.
- Node out-degree in directed graphs is defined as the total number of out-edges incident to specific node.
- Node first-order influence is a generalization of node degree metric that considers link weights of adjacent nodes. The general formula for first-order influence is:$${I}_{1}\left(u\right)=\frac{{\sum}_{v\in {N}_{u}}{w}_{uv}}{N}$$Node first-order influence can be calculated both for directed and undirected graphs. The main difference between these calculations is in definition of adjacent nodes, for directed graphs the neighbors are out-links.
- Node second-order influence is a generalization of node degree metric that considers link weights of nodes that are adjacent to adjacent nodes. The general formula for second-order influence is:$${I}_{2}\left(u\right)=\sum _{v\in {N}_{u}}{I}_{1}\left(v\right)$$Node second-order influence can be calculated both for directed and undirected graphs, similarly like in first-order influence case.
- Node eigenvector value is an extension of degree values in which centrality values are awarded for each node. Since not all nodes are equally important, a connection to a more important node should contribute more to centrality score than a connection to a less important node. The general formula for eigenvector value is:$$E\left(u\right)=\frac{1}{\lambda}\sum _{v\in N}{w}_{uv}E\left(v\right)$$
- Node authority value and node hub value represent measures of node importance. These values were introduced by Jon Kleinberg in his analysis of web pages importance and ranking [24]. This method can be used in telecommunications as well. We can say that more influential telco users are those who have a lot of incoming (or outgoing) calls (thus having larger node authority (or hub) value). The general formulas for authority and hub values are:$$A\left(u\right)=\alpha \sum _{v\in N}{w}_{uv}H\left(v\right)$$$$H\left(u\right)=\beta \sum _{v\in N}{w}_{vu}A\left(v\right)$$
- Node in a graph is an articulation point if its removal would cause an increase in the number of connected components. Each node in the large call graph will be awarded a value 1 if it is an articulation point, otherwise it will be awarded value 0. Node articulation point value can be calculated only for undirected graphs.

- Measures for directed graphs: Node in-degree, node out-degree, node first-order influence, node second-order influence, node authority value and node hub value. These measures are most important in telco graphs since it is important to distinguish originating and terminating number in a call.
- Measures for undirected graphs: Node degree and node eigenvector value. These measures are not as important as measures for directed graphs in telco graphs, but still carry some valuable information regarding call graph structure.
- Articulation point measure: Articulation point measure will be observed as separate measure because it represents a measure of graph structure and connectivity, but it does not mark a telco user as important by default. This will be explained in more detail in following example.

## 4. Data Clustering Process

#### 4.1. Datasets and Data Cleansing

- Network call graph contained 948 undirected connected components;
- Network call graph contained 2.8 million directed connected components;
- Network call graph has maximum clique size of 22.

- Network call graph contained 1003 undirected connected components;
- Network call graph contained 2.8 million directed connected components;
- Network call graph has maximum clique size of 22.

#### 4.2. MSISDN Clustering

#### 4.2.1. Data Sampling and Standardization

#### 4.2.2. Clustering

- Nodes which by their other characteristics (node degree value) belong to cluster 1 will be downgraded to cluster 1.
- All other nodes will remain at cluster 3 and will represent the real network core and most important nodes of the network call graph by undirected metrics criteria.

- Nodes which by their other characteristics (in-degree, out-degree, first-order influence and second-order influence value) belong to cluster 1 or cluster 2 will be downgraded to cluster 1 or cluster 2, respectively.
- All other nodes will remain at cluster 4 and will represent the real network core and the most important nodes of the network call graph by directed metrics criteria.

#### 4.2.3. Execution Times

## 5. Empirical Findings

#### 5.1. Data Analysis and Prediction Models

- In connected components size 2, both nodes are marked as Important;
- In other connected components, a node is marked as Important if it is an articulation point;
- All other nodes from smaller connected components are marked as Unimportant.

#### 5.2. Verification of the Model

- Disconnection reasons for all monitored telco operator postpaid users were examined and some reasons were ignored and not treated as external churn. For example, if a customer switches from monitored telco operator Postpaid to monitored telco operator Prepaid, then from the company’s point of view it did not perform any external churn.
- Each disconnected node was analyzed and we calculated the number of its adjacent nodes that were deactivated in the forthcoming period.

- It can be noted that churn statistics (percentage of deactivated users) of nodes from the total population as well as Standard and Leader nodes are very similar. For the reason of data confidentiality, we can show only relations between different clusters.
- The percentage of deactivated users is smaller in Core nodes. Core nodes are tightly connected to their adjacent nodes, and as such are more difficult to be “taken” from our operator. On the other hand, Followers are loosely connected to the rest of the network and are more prone to leave our operator.
- The percentage of deactivated users that have deactivated adjacent nodes is highest for Leader nodes. This means that if any Leader leaves our network, there is a 75% probability that some adjacent node will leave our network in the forthcoming period. This percentage is much smaller for Standard and Follower nodes.
- Next important statistics is the average percentage of deactivated adjacent nodes per one deactivated node. This is the measure of how much one deactivated node actually degrades its surroundings. Again we can see that Leader nodes are most prone to degrade the graph structure. It should be noted that Core network nodes do not tend to degrade the graph structure. This is the case because Core nodes and their neighbors are in most cases connected to other Core nodes and as such are less probable to churn.
- The last verification statistics displayed was the number of deactivated adjacent nodes per one deactivated node. For the reason of data confidentiality, we can show only relations between different clusters. Again, we can see that Leaders cause great degradation, and that Followers cause the smallest disruption of the graph.
- Important nodes should be observed separately as they relate to a very small part of the population. We can see that they account for a large percentage of deactivated users, and that every deactivated user will destroy its small connected component thus confirming our theory.

#### 5.3. Clustering Results

## 6. Discussion

- Linear Perceptron (LP);
- Multilayer Perceptron (MLP);
- Radial Basis Function (RBF) Networks.

- Unequal Width and Height (UN);
- Unequal Widths (UW);
- Equal Width and Height (EQ);
- Equal Widths (EW);
- Equal Height (EH);
- Equal Volumes (EV).

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CART | Classification and Regression Trees |

CCC | Cubic Clustering Criterion |

CDR | Call Data Record |

CHAID | Chi-square Automatic Interaction Detector |

EH | Equal Height |

EQ | Equal Width and Height |

EV | Equal Volumes |

EW | Equal Widths |

LP | Linear Perceptron |

MLP | Multilayer Perceptron |

MSISDN | Mobile Station International Subscriber Directory Number |

PSF | Pseudo F statistics |

PST2 | Pseudo-T2 Statistics |

RBF | Radial Basis Function |

RSQ | R-Squared |

UN | Unequal Width and Height |

UW | Unequal Widths |

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Node | Undirected | Directed | Articulation | ||||||
---|---|---|---|---|---|---|---|---|---|

Deg. | EV | Deg. In | Deg. Out | Inf. 1 | Inf. 2 | Hub | Auth. | ||

A | 6 | 0.96 | 4 | 3 | 0.27 | 0.27 | 1.00 | 0.63 | 1 |

B | 1 | 0.30 | 1 | 0 | 0.00 | 0.00 | 0.00 | 0.41 | 0 |

C | 1 | 0.30 | 0 | 1 | 0.09 | 0.27 | 0.30 | 0.00 | 0 |

D | 1 | 0.30 | 0 | 1 | 0.09 | 0.27 | 0.30 | 0.00 | 0 |

E | 2 | 0.33 | 1 | 1 | 0.09 | 0.27 | 0.30 | 0.00 | 1 |

F | 3 | 0.87 | 2 | 1 | 0.09 | 0.27 | 0.16 | 0.67 | 0 |

G | 4 | 1.00 | 3 | 2 | 0.18 | 0.36 | 0.62 | 1.00 | 0 |

H | 2 | 0.57 | 1 | 2 | 0.18 | 0.45 | 0.64 | 0.33 | 0 |

I | 4 | 0.84 | 2 | 3 | 0.27 | 0.36 | 0.80 | 0.33 | 1 |

J | 1 | 0.26 | 1 | 0 | 0.00 | 0.00 | 0.00 | 0.33 | 0 |

K | 1 | 0.10 | 0 | 1 | 0.09 | 0.09 | 0.00 | 0.00 | 0 |

Cluster | Undirected | |

Percentage of Members in Sample | Percentage of Members in Population | |

1 | 53.48 | 56.89 |

2 | 46.13 | 42.72 |

3 | 0.39 | 0.39 |

Cluster | Directed | |

Percentage of Members in Sample | Percentage of Members in Population | |

1 | 42.53 | 39.90 |

2 | 41.04 | 45.25 |

3 | 16.07 | 14.47 |

4 | 0.36 | 0.38 |

Directed | Undirected | Articulation | Percentage of Members | Value | Unique Cluster Name |
---|---|---|---|---|---|

1 | 1 | 0 | 28.0125 | 3 | FOLLOWER |

1 | 1 | 1 | 12.0505 | 3.5 | |

1 | 2 | 0 | 0.0094 | 4 | |

1 | 2 | 1 | 0.0111 | 4.5 | |

2 | 1 | 0 | 7.9166 | 5 | STANDARD |

2 | 1 | 1 | 9.1662 | 5.5 | |

2 | 2 | 0 | 10.4609 | 6 | |

2 | 2 | 1 | 17.7645 | 6.5 | |

2 | 3 | 0 | 0.0540 | 7 | LEADER |

2 | 3 | 1 | 0.0230 | 7.5 | |

3 | 2 | 0 | 3.4634 | 8 | |

3 | 2 | 1 | 11.0023 | 8.5 | |

3 | 3 | 0 | 0.0023 | 9 | CORE |

3 | 3 | 1 | 0.0040 | 9.5 | |

4 | 2 | 0 | 0.0003 | 10 | |

4 | 2 | 1 | 0.0012 | 10.5 | |

4 | 3 | 0 | 0.0314 | 11 | |

4 | 3 | 1 | 0.0265 | 11.5 |

Unique Cluster Name | Total % of Users | % of Deac. Users | % of Deac. Users that Have Deac. Adj. Nodes | Average % of Deac. Adj. Nodes per Deac. Node | Number of Deac. Adj. Nodes per Deac. Node |
---|---|---|---|---|---|

FOLLOWER | 40.05 | 1.27 B | 31.3 | 25.7 | A |

STANDARD | 45.31 | 0.8 B | 48.9 | 30.4 | 4.09 A |

LEADER | 14.54 | 0.88 B | 74.7 | 38.2 | 16.35 A |

CORE | 0.07 | 0.32 B | 57.1 | 2.6 | 1.01 A |

IMPORTANT | 0.03 | 3.38 B | 88.2 | 100 | - |

Total population | 100 | B | 43.2 | 33.0 | 4.08 A |

Dataset | Total Percentage of Users | Lift |
---|---|---|

Interact with Important | 0.002 | 45.80 |

Interact with Leader | 4.543 | 3.88 |

“Churn” | 4.545 | 3.89 |

“Churn” + Interact with Follower | 6.815 | 3.67 |

“Churn” + Interact with Follower and Standard | 10 | 2.80 |

Decision Tree Type | Tree Assesment Method | Max. Depth | Max. Branch | Number of Nodes | Lift |
---|---|---|---|---|---|

SAS Default Tree | SAS Default | 6 | 2 | 65 | 2.38 |

SAS Default Tree | Average Square Error | 6 | 2 | 65 | 2.42 |

CHART | Average Square Error | 10 | 2 | 317 | 2.46 |

CHART | Average Square Error | 6 | 2 | 101 | 2.48 |

CHAID | SAS Default | 6 | 6 | 83 | 2.48 |

CHAID | Average Square Error | 6 | 4 | 73 | 2.49 |

Neural Network Type | Combinational Functions | Lift |
---|---|---|

LP | N/A | 2.47 |

MLP | N/A | 2.56 |

ORBF | EQ | 2.41 |

ORBF | UN | 2.48 |

NRBF | EQ | 2.57 |

NRBF | EV | 2.51 |

NRBF | EH | 2.65 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kostić, S.M.; Simić, M.I.; Kostić, M.V.
Social Network Analysis and Churn Prediction in Telecommunications Using Graph Theory. *Entropy* **2020**, *22*, 753.
https://doi.org/10.3390/e22070753

**AMA Style**

Kostić SM, Simić MI, Kostić MV.
Social Network Analysis and Churn Prediction in Telecommunications Using Graph Theory. *Entropy*. 2020; 22(7):753.
https://doi.org/10.3390/e22070753

**Chicago/Turabian Style**

Kostić, Stefan M., Mirjana I. Simić, and Miroljub V. Kostić.
2020. "Social Network Analysis and Churn Prediction in Telecommunications Using Graph Theory" *Entropy* 22, no. 7: 753.
https://doi.org/10.3390/e22070753