# Measuring Time-Dynamics and Time-Stability of Journal Rankings in Mathematics and Physics by Means of Fractional p-Variations

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## Abstract

**:**

## 1. Introduction

#### 1.1. Measuring Stability of Journal Rankings

#### 1.2. A Specific Model for Ensuring the Plausibility of Journal Rankings

- (1)
- The prestige of a given journal for a scientific community is supposed to be relatively stable. At least, it must change following a long-term pattern.
- (2)
- The position of a journal in a prestige-based list may increase or decrease over a long period, but a great amount of fluctuations in it must be understood as an anomalous behavior.
- (3)
- Consequently, a lot of significant changes in the position of a journal in a list is not a plausible behavior. Such a fact must not be interpreted as a fail in the policy of the journal, but in the measuring tool.
- (4)
- We obtain the following conclusion: an ordered impact list having an excessive rate of fluctuations in the positions of the journals must not be used as a proxy for the prestige of the journals in which the papers are published.

## 2. Materials and Methods

#### 2.1. p-Variations of Ordered Lists: Definition and Fundamental Properties

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. The Test Example: Eleven-Year Series in Mathematics and Applied Physics

## 3. Results

#### 3.1. The p-Fractional p-Variation and the p-Adequacy Degree of a Time Series

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

**Remark**

**1.**

**Example of application to curricula vitae (CV) evaluation.**To finish this section, let us explain a different application using the indices presented here to introduce a confidence interval in the evaluation of single curricula vitae. As we have explained in the previous sections, the main application of our results is to provide a criterium for rejecting the use of a particular impact list. However, it can also be used for rejecting the result in case that an automatic evaluation using such a list gives an unreliable result. Assume that such an evaluation procedure based on the use of the 2-year Impact Factor has been chosen by a research institute. In order to simplify the comparison, consider the problem of having just a pair of candidates for a given position in a research institute. Suppose that we have a simple system for evaluating the papers of both candidates consisting of giving $5-t$ points for each paper published in a journal that is in the quartile t of the list of the last year of a series S (recall that we call the first quartile to be the top one). Suppose that the adequacy degree of the list (for quartiles) is ${k}_{1,4}=66.6\%$. Let us consider two cases.

- (a)
- Suppose that Candidate 1 has published three papers in journals that are in the first quartile and one in the fourth, and Candidate 2 has published six in the second quartile. The marks that they get are 13 (Candidate 1) and 12 (Candidate 2), and so the institute will contract Candidate 1. However, using the probabilistic interpretation that we give to the index, we know that the lowest value that Candidate 1 should get is $2\times 4+1\times 3+1=12$, and the upper value for Candidate 2 should be $4\times 2+2\times 3=14$. Consequently, the institute cannot distinguish among both candidates using this system, and must find another procedure.
- (b)
- Consider now a different situation. Suppose that Candidate 1 has published six papers in journals that appear in the fourth quartile. The evaluation system gives $6\times 1=6$. The value of ${k}_{1,4}$ indicates that two of the six papers should be in the third quartile, and so the biggest mark obtained with the evaluation system should be $4\times 1+2\times 2=8.$

#### 3.2. Time Series of Impact Factor Lists of Mathematics and Applied Physics

## 4. Discussion

- Using the complete list, we obtain $P{I}_{1}\left(M\right)=0.1078.$
- If the 20 top journals are removed from the list, we get $P{I}_{1}\left(M\right)=0.1210.$
- If the 30 top journals are removed, then $P{I}_{1}\left(M\right)=0.1306.$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**p-variation of the points $(t,1-t)$ for: (

**left**) $p=2;$ (

**middle**) $p=1$; and (

**right**) $p=1/2$.

**Figure 4.**Behavior of the journals by quartiles (Q) in the list of mathematics. (

**top-left**) Q1; (

**top-right**) Q2; (

**bottom-left**) Q3; and (

**bottom-right**) Q4.

**Figure 5.**Behavior of the journals by quartiles (Q) in the list of applied physics. (

**top-left**) Q1; (

**top-right**) Q2; (

**bottom-left**) Q3; and (

**bottom-right**) Q4.

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**MDPI and ACS Style**

Ferrer-Sapena, A.; Díaz-Novillo, S.; Sánchez-Pérez, E.A.
Measuring Time-Dynamics and Time-Stability of Journal Rankings in Mathematics and Physics by Means of Fractional *p*-Variations. *Publications* **2017**, *5*, 21.
https://doi.org/10.3390/publications5030021

**AMA Style**

Ferrer-Sapena A, Díaz-Novillo S, Sánchez-Pérez EA.
Measuring Time-Dynamics and Time-Stability of Journal Rankings in Mathematics and Physics by Means of Fractional *p*-Variations. *Publications*. 2017; 5(3):21.
https://doi.org/10.3390/publications5030021

**Chicago/Turabian Style**

Ferrer-Sapena, Antonia, Susana Díaz-Novillo, and Enrique A. Sánchez-Pérez.
2017. "Measuring Time-Dynamics and Time-Stability of Journal Rankings in Mathematics and Physics by Means of Fractional *p*-Variations" *Publications* 5, no. 3: 21.
https://doi.org/10.3390/publications5030021