# Tie persistence analysis

## Related Work

• Structure and tie strengths in mobile communication network - Onnela, Barabasi - PNAS 2007
```   In this paper they studied coupling between tie strengths and local network structure.
They also compared information diffusion through strong ties with the same through weak ties.
```
• The dynamics of a mobile phone network - Hidalgo et. al. ScienceDirect Jan 2008
```   They studied the relation between structure of mobile network and link persistence
They proposed rule based prediction technique to predict which ties will remain in future.
```

## Problem Formulation

• We divided the 6 months of phone call data into time panels of period 15 days each.
• Given the links and network structure in panel 1, we predict which links will persist in panels 2,3,4 etc.

## Term definitions

• Tie persistence: It is the stability of ties across time as number of panels in which a link is observed, over the total number of panels.

Pij = Sum_t (Aij(t)/m),

where Pij is the persistence of tie eij , Aij is 1 if users i and j communicated in time panel t and 0 otherwise, and m is the total number of panels.

• User perseverance: Perseverance of a user is defined as the average of the persistences of all his/her ties.

Pi = 1/Ki Sum_j (Pij) , where Pi is the perseverance of user i, Ki is i’s degree (number of neighbors), and Pij is the persistence of tie eij as defined above.

• Tie Attributes
``` * Reciprocity (R): R is a Boolean attribute which denotes whether the tie between i and j is reciprocated during a given time period. That is, R is 1 if both edges eij and eji exist, and 0 otherwise.
* Topological Overlap (TO):
TO(i, j)= sqrt(Oi,j^2 / (Ki * Kj)), where Oi,j is the number of common neighbors of
node i and node j, and Ki denotes the degree of node i.
```
• Node Attributes
```  * Degree (K): Ki denotes the number of neighbors of a given node i.
* Cluster Coefficient (C):
Ci = 2 (num triangles node i is part of) / Ki * (Ki−1)
* User Reciprocity (r): ri the fraction of ties of a
```

given user i that are reciprocated.

## Methodology

• We first find the correlation between different tie attributes and Link persistence. We use Pearson correlation coefficient to measure this.
• We also find correlation between node attributes and user perseverence.
• For predicting whether a link in panel 1 will persist in future panel, we divide the data into train and test.
• We then learn Logistic regression function for the tie persistence using train data and we measure the F1 accuracy of the learnt model on test data.
• We also compare results with rule based system proposed in earlier work.

## Conclusions

• Local network attributes such as clustering coefficient and tie attributes such as reciprocity help to predict whether ties will persist in the future.
• Our prediction results using logistic regression show that tie attributes give better accuracy than node attributes and using both types of attributes together yields the best prediction accuracy.
• Regression techniques give better accuracy than rule based techniques.