h-Type indices, partial sums and the majorization order

Egghe, Leo; Rousseau, Ronald

doi:10.1162/qss_a_00005

Abstract

We study the array of partial sums, P_X, of a given array X in terms of its h-type indices. Concretely, we show that h(P_X) can be described in terms of the Lorenz curve of the array X and obtain a relation between the sum of the components of P_X and the Gini index of X. Moreover, we obtain sharp lower and upper bounds for h-type indices of P_X.

partial sums of an array, h-index, g-index, R-index, Gini index, Lorenz curve

1. INTRODUCTION

h-type indices such as the h-index itself and the g-index have interesting mathematical properties as shown, for example, in Egghe and Rousseau (2019a), although they are only probably approximately correct (PAC) in research evaluation exercises (Bouyssou & Marchant, 2011; Waltman & van Eck, 2012; Rousseau, 2016). In this investigation we continue our theoretical investigation of the mechanism leading to h-type indices. Concretely, we study properties related to h-type indices of the array of partial sums of a given array X. We recall that these partial sums form the basis of the Lorenz curve and the related Gini index. As a consequence, we also obtain relations with the Gini index and the Lorenz curve of the original array X. We will further derive sharp lower and upper bounds for h-type indices of P_X.

In the following section we recall the definitions we will use in this investigation.

2. DEFINITIONS

Let (R⁺)^N be the set of all arrays of length N with nonnegative real values. An array X = (x_r)_r=1,2,…,N in (R⁺)^N is said to be decreasing if, for all r = 1, 2, …, N, x_r ≥ x_r+1. The set (R⁺)^N has a natural partial order defined by X ≤ Y if, for all r = 1, 2, …, N, x_r ≤ y_r. Equality between X and Y only occurs if x_r = y_r for all r. We denote the set of all decreasing arrays in (R⁺)^N with at least one component larger than or equal to 1 by Φ^N.

Next we recall the definition of some h-type indices for arrays in (R⁺)^N.

2.1. The h-index (Hirsch, 2005)

Let X = (x_r)_r=1,2,…,N ∈ Φ^N. The h-index of X, denoted h(X), is the largest natural number such that the first h coordinates have each at least a value h. If all components of a decreasing array X are strictly smaller than 1, then h(X) = 0. Such arrays are not considered further on because we will work with arrays in Φ^N. If x_N, the last element in X, is larger than or equal to N, then h(X) = N.

Recall that an h-index (and similarly for the other h-type indices defined further on) can only be defined for decreasing arrays. Moreover, for r ≤ h(X), x_r ≥ r; conversely, if for all r ≤ n, x_r ≥ n, then n ≤ h(X). Further, if r > h(X), then x_r < h(X) + 1.

2.2. The g-index (Egghe, 2006a, b)

Let X = (x_r)_r=1,2,…,N ∈ Φ^N. The g-index X, denoted g(X), is defined as the highest natural number g such that the sum of the first g coordinates is at least equal to g². If the sum of all coordinates of X is strictly larger than N², then we extend the array X with coordinates equal to zero, making it into an array in Φ^M, M > N, until it is possible to apply the definition.

2.3. The R-index (Jin et al., 2007)

Let X = (x_r)_r=1,2,…,N ∈ Φ^N. The R-index of X is defined as the square root of the sum of all coordinates up to and including the one with index h(X). Omitting the square root yields the R²-index. As it is easier to work with R² than with R if their properties are for our purposes the same, all concrete examples will be given for R².

2.4. Kosmulski’s h⁽²⁾ Index (Kosmulski, 2006)

Let X = (x_r)_r=1,2,…,N ∈ Φ^N. The h⁽²⁾ or Kosmulski index of X, denoted h⁽²⁾(X), is the largest natural number h⁽²⁾ such that the first h⁽²⁾ coordinates have each at least a value (h⁽²⁾)².

2.5. The Majorization Order (Hardy et al., 1934)

Let X, Y ∈ Φ^N, where N is any finite number in N = {1, 2, 3, …}. The array X is majorized by Y, or X is smaller than or equal to Y in the majorization order, denoted as X ≤ Y if for all i = 1, …, N

\{\begin{array}{c} \sum_{i = 1}^{N} x_{i} = \sum_{i = 1}^{N} y_{i} and \\ \sum_{j = 1}^{i} x_{j} \leq \sum_{j = 1}^{i} y_{j}; \forall i = 1, \dots, N . \end{array}

We note that this definition is also valid for arrays in which all values are between zero (included) and 1 (not included).

3. AN INEQUALITY RELATED TO THE g-INDEX AND THE MAJORIZATION ORDER

Theorem 1. ∀ N ∈ N: X, Y ∈ Φ^N: X − < Y ⇒ g(X) ≤ g(Y) (*)

Proof. Although this theorem is implied in Egghe (2009, p. 487) we present here two short proofs.

First proof: For each i ≤ g(X), $\sum_{j = 1}^{i} y_{j}$ ≥ $\sum_{j = 1}^{i} x_{j}$ ≥ i². This implies that g(Y) ≥ g(X).

Second proof: For each i > g(Y), $\sum_{j = 1}^{i} x_{j}$ ≤ $\sum_{j = 1}^{i} y_{j}$ < i². Also this inequality implies that g(X) ≤ g(Y).

Comments

A.
This theorem proves that g is an order-preserving mapping from (R⁺)^N with the majorization order to the positive real numbers with their natural order.
B.
The converse of inequality (*) does not hold. Consider for instance X = (5, 2, 2) and Y = (4, 4, 1). Then g(X) = g(Y) = 3 but neither X ≤ Y nor Y ≤ X holds.
C.
The inequality (*) can be strict. Indeed, take X = (2, 1, 1) and Y = (2, 2, 0). Then X ≤ Y, but g(X) = 1 and g(Y) = 2.
D.
Yet, inequality (*) cannot be strict for N = 2. Indeed, consider X = (x₁, x₂) and Y = (y₁, y₂), with X ≤ Y. Then 1 ≤ x₁ ≤ y₁ (hence g(X), g(Y) ≥ 1) and x₁ + x₂ = y₁ + y₂. As N = 2, this sum completely determines the value of the g-index. Hence this value must be equal for X and Y. We note that even here there is no upper bound to the value of g(X) = g(Y).
E.
If it were allowed that x₁ < 1 then the previous Comment D is not valid. Indeed, take X = (½, ½) and Y = (1, 0) then X ≤ Y, and g(X) = 0 and g(Y) = 1.
F.
Inequality (*) does not hold for the h- or the R²-index. Consider X = (3, 3, 3) and Y = (5, 2, 2). Then X ≤ Y but h(X) = 3 > h(Y) = 2. Moreover, R²(X) = 9 > R²(Y) = 7.

For small N we even have the opposite relation for the h-index. This is shown in the next proposition.

Proposition 1.

(a)
For X, Y ∈ Φ² and X ≤ Y, h(X) ≥ h(Y)
(b)
For X, Y ∈ Φ³, X ≤ Y and if the components of X and Y are strictly positive natural numbers, then h(X) ≥ h(Y).

Proof.

(a).
N = 2, X ≤ Y then x₁ ≤ y₁ and x₁ + x₂ = y₁ + y₂. Hence x₂ ≥ y₂. If now h(X) = 1, then 1 ≤ y₁ and 2 > x₂ ≥ y₂. This implies that h(Y) = 1 = h(X). The case h(X) = 2 is trivial: As N = 2, it follows that h(Y) ≤ 2 = h(X).
(b).
N = 3 and X ≤ Y, then x₁ ≤ y₁; x₁ + x₂ ≤ y₁ + y₂ and x₁ + x₂ + x₃ = y₁ + y₂ + y₃. This already implies that x₃ ≥ y₃. We now consider three cases: h(Y) = 3, h(Y) = 2 and h(Y) = 1.

Assume first that h(Y) = 3. Then y₃ ≥ 3. Hence we see that x₁ ≥ x₂ ≥ x₃ ≥ y₃ ≥ 3, from which we derive that h(X) = 3 = h(Y).

Assume next that h(Y) = 2. Then y₃ < 3, hence y₃ = 2 or y₃ = 1, and y₁ ≥ y₂ ≥ 2. We first consider the case y₃ = 1. We know already that x₃ is at least equal to 1. So, if x₃ is equal to 1, then x₁ + x₂ = y₁ + y₂. As x₁ ≤ y₁ the previous equality implies that y₂ ≤ x₂. Now h(Y) = 2 leads to 2 ≤ y₂ ≤ x₂, or h(X) ≥ 2 = h(Y). Still with y₃ = 1 we now consider the case that x₃ > 1. Then x₁ ≥ x₂ ≥ x₃ ≥ 2, leading to h(X) ≥ 2 = h(Y).

Next we consider the case y₃ = 2. Then x₁ ≥ x₂ ≥ x₃ ≥ y₃ = 2, which implies that h(X) ≥ 2 = h(Y).

Finally, as components are assumed to be strictly positive natural numbers, h(X) and h(Y) are at least equal to 1. Hence h(Y) = 1 implies h(X) ≥ h(Y).

Comments

A.
Proposition 1(b) is not valid if some of the components are zero. This is illustrated as follows. Let X = (2, 1, 1) and Y = (2, 2, 0). Then X ≤ Y, but h(X) = 1 and h(Y) = 2.
B.
Proposition 1(b) is also not valid if some of the components are not natural numbers. Indeed, let X = (2, 1.5, 1.5) and Y = (2, 2, 1). Then X ≤ Y, but h(X) = 1 and h(Y) = 2.
C.
Propositions 1(a) and 1(b) are not valid for R².
- (a)
  N = 2. Consider X = (1, 1) and Y = (2, 0). Then X ≤ Y, h(X) = h(Y) = 1; R²(X) = 1 and R²(Y) = 2.
- (b)
  N = 3. Consider X = (2, 2, 2) and Y = (3, 2, 1). Then X ≤ Y, h(X) = 2 = h(Y) and R²(X) = 4 < R²(Y) = 5.
D.
D. Proposition 1 with N ≥ 4 is not valid for the h-index h.

Consider X = (6, 5, 2, 2) and Y = (6, 5, 3, 1). Then X ≤ Y, h(X) = 2 < h(Y) = 3. We further remark that R²(X) = 11 < R²(Y) = 14.

4. INTRODUCING THE ARRAY OF PARTIAL SUMS

Now we come to the main part of this article. First we introduce some notation. Let X = (x_r)_r=1,2,…,N ∈ Φ^N and consider the partial sums:

\sum_{j = 1}^{i} x_{j}, i = 1, \dots, N .

Ranking these partial sums again in decreasing order leads to the array P_X. The ith component of P_X, denoted as y_i, is equal to

\sum_{j = 1}^{N - i + 1} x_{j}

⁠. An example: Let X = (4, 3, 2, 1). Then P_X = (10, 9, 7, 4) and h(P_X) = 4.

Remarks

1.
(P_X)₁ = A = $\sum_{j = 1}^{N} x_{j}$ ⁠; (P_X)_N = x₁.
2.
If X ends with p zeros, then P_X starts with p + 1 As.
3.
Clearly X ≤ P_X, as (P_X)_N = x₁. Hence h(X) ≤ h(P_X), g(X) ≤ g(P_X) and R(X) ≤ R(P_X). (Egghe & Rousseau, 2019a; Proposition 2).

If X denotes the number of received citations of an author’s publications, then the indicator value h(P_X) shows how many of the less cited publications can be removed so that the total number of the remaining items of X is higher than the rank of this total in the array P_X. This is another way of describing the impact of the most cited publications. Contrary to the case of h(X), h(P_X) may increase if a publication in X’s h-core, not necessarily the most cited one, increases its number of citations. We provide an example: let X = (4, 2, 0, 0, 0, 0, 0, 0). Then h(X) = 2; P_X = (6, 6, 6, 6, 6, 6, 6, 4) and h(P_X) = 6. Consider now X′ = (4, 3, 0, 0, 0, 0, 0, 0). Then h(X′) = 2; P_X′ = (7, 7, 7, 7, 7, 7, 7, 4) and h(P_X′) = 7.

5. A RELATION WITH THE GINI INDEX

We recall (Rousseau et al., 2018, formula (4.19)) that the Gini concentration index of a decreasing array X of nonnegative real numbers, (x_j)_j=1,…,N is obtained as

G (X) = \frac{1}{N} (N + 1 - \frac{2}{A} \sum_{j = 1}^{N} j x_{j}),

(1)

where A =

\sum_{j = 1}^{N} x_{j}

⁠. Consider now P_X. The sum of all components of P_X, denoted as S(X), is

\sum_{j = 1}^{N} x_{j} + \sum_{j = 1}^{N - 1} x_{j} + \dots + \sum_{j = 1}^{1} x_{j} = \sum_{j = 1}^{N} (N - j + 1) x_{j} = (N + 1) A - \sum_{j = 1}^{N} {j x}_{j} .

From this result we obtain a relation between G(X) and S(X):

G (X) = \frac{1}{N} (N + 1 - \frac{2}{A} ((N + 1) A - S (X))) = \frac{1}{N} (N + 1 - 2 (N + 1) + \frac{2 S (X)}{A}) = \frac{1}{N A} (2 S (X) - A (N + 1)) .

(2)

Conversely

S (X) = \frac{A}{2} (N (1 + G (X)) + 1) .

(3)

An example: If X = (a, a, a, a), a > 0, then G(X) = 0 (by definition), N = 4, A = 4a and S(X) = 10a. Now we check and find that, indeed, 10a =

\frac{4 a}{2}

(4 + 1).

6. A GEOMETRIC INTERPRETATION OF h(P_X) IN TERMS OF THE LORENZ CURVE L_X

For the decreasing array X of nonnegative real numbers, (x_j)_j=1,…,N and for a_j = $\frac{x_{j}}{\sum_{k = 1}^{N} x_{k}}$ = $\frac{x_{j}}{N * \bar{x}}$ ⁠, the Lorenz curve of X, denoted as L_X, connects points with coordinates $(s = \frac{i}{N}, \sum_{j = 1}^{i} a_{j})$ ⁠. The average of array X is denoted as $\bar{x} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}$ ⁠.

Now h(P_X) is equal to the largest natural number i such that y_i =

\sum_{j = 1}^{N - i + 1} x_{j}

≥ i, which is also equal to (N + 1) minus the smallest natural number i such that y_N−i+1 =

\sum_{j = 1}^{i} x_{j}

≥ N − i + 1. Dividing by the sum of all elements in X this yields

\sum_{j = 1}^{i} a_{j} \geq \frac{N - i + 1}{N * \bar{x}} = \frac{1}{\bar{x}} (1 - s + \frac{1}{N}) .

Consequently, h(P_X) is equal to N (1 − the smallest s such that L_X(s) ≥

\frac{1}{\bar{x}} (1 - s + \frac{1}{N})

⁠) + 1.

An illustration: If X = (3, 2, 1, 0), N = 4, $\bar{x}$ = $\frac{3}{2}$ ⁠, P_X = (6, 6, 5, 3) and h(P_X) = 3. Now L_X $(\frac{1}{4})$ = $\frac{3}{6}$ < $\frac{2}{3} (1 - \frac{1}{4} + \frac{1}{4})$ = $\frac{2}{3}$ ⁠; but L_X $(\frac{2}{4})$ = $\frac{5}{6}$ ≥ $\frac{2}{3} (1 - \frac{2}{4} + \frac{1}{4})$ = $\frac{2}{4}$ ⁠. Hence, the smallest s is equal to 2/4 and h(P_X) = 4(1 − 2/4) + 1 = 2 + 1 = 3.

In Egghe and Rousseau (2019b) we studied h(P_X) and its relation with the Lorenz curve in a continuous context. This led to a new geometric interpretation of the h-index.

7. BOUNDS ON h-TYPE INDICES

In the next sections we derive bounds for h-type indices of P_X. This is of importance for the following reason. A function relates an input to a unique output. In this way the standard h-index is a function which maps an array to a natural number. Yet it is not an explicit function, such as the function that maps the real number x to x² + 4x + 7 or the function which maps a finite array to its sum. Finding an h-index needs a procedure and hence it is not possible to study properties in an analytical way (e.g., using integrals). The bounds obtained in this article are explicit functions which can be studied using analytical methods.

We denote by ⌊a $⌋$ ⁠, the floor function of a (i.e., the largest integer smaller than or equal to a). We note that a ≥ ⌊a⌋ > a − 1. Using the notation just introduced we come to the following interesting theorem.

Theorem 2. Let X = (x_r)_r=1,2,…,N ∈ Φ^N, then

min (N, A) = min (N, N \bar{x}) \geq h (P_{X}) \geq ⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋,

(4)

Before proving Theorem 2 we make three remarks:

1.
The first inequality, namely min(N, A) ≥ h(P_X) is easy to see because, on the one hand, an h-index can never be larger than the length of the array and on the other 1 ≤ h(P_X) ≤ $\sum_{j = 1}^{N - h (P_{X}) + 1} x_{j}$ ≤ A.
2.
h(P_X) = N if and only if x₁ ≥ N.
3.
h(P_X) = 0 can never occur in our context. Indeed, this may only occur if all components are strictly smaller than 1, which is excluded. Yet, in Egghe and Rousseau (2019c) we showed that formula 4 is also correct in cases for which h(P_X) = 0.

Proof of Theorem 2. We only have to show the second inequality. By definition we know that h(P_X) is equal to the largest index i such that y_i = $\sum_{j = 1}^{N - i + 1} x_{j}$ ≥ i. We know that y_i = $\sum_{j = 1}^{N - i + 1} x_{j}$ = (N − i + 1).(the average of (x₁, x₂, …, x_N−i+1)) ≥ (N − i + 1). $\bar{x}$ (as the array X is ranked in decreasing order).

Now, if (N − i + 1).

\bar{x}

≥ i then certainly y_i ≥ i. Solving this inequality for i leads to

i \leq (N + 1) \frac{\bar{x}}{\bar{x} + 1} .

As the index i is a natural number it follows that h(P_X) ≥

⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋

⁠. This proves this theorem.

In order to make these bounds more concrete we provide a table (Table 1) for some values of N and $\bar{x}$ (or A), showing how sharp these bounds often are. Largest differences occur when the average number of items is one.

Table 1.

Some specific bounds for h(P_X) according to

N	$\bar{x}$
N	0.1	0.5	1	2	5	10
10	1 ≥ h(P_X) ≥ 1	5 ≥ h(P_X) ≥ 3	10 ≥ h(P_X) ≥ 5	10 ≥ h(P_X) ≥ 7	10 ≥ h(P_X) ≥ 9	10 ≥ h(P_X) ≥ 10
30	3 ≥ h(P_X) ≥ 2	15 ≥ h(P_X) ≥ 10	30 ≥ h(P_X) ≥ 15	30 ≥ h(P_X) ≥ 20	30 ≥ h(P_X) ≥ 25	30 ≥ h(P_X) ≥ 28
100	10 ≥ h(P_X) ≥9	50 ≥ h(P_X) ≥ 33	100 ≥ h(P_X) ≥ 50	100 ≥ h(P_X) ≥ 67	100 ≥ h(P_X) ≥ 84	100 ≥ h(P_X) ≥ 91
200	20 ≥ h(P_X) ≥ 18	100 ≥ h(P_X) ≥ 67	200 ≥ h(P_X) ≥ 100	200 ≥ h(P_X) ≥ 134	200 ≥ h(P_X) ≥ 167	200 ≥ h(P_X) ≥ 182

N	$\bar{x}$
N	0.1	0.5	1	2	5	10
10	1 ≥ h(P_X) ≥ 1	5 ≥ h(P_X) ≥ 3	10 ≥ h(P_X) ≥ 5	10 ≥ h(P_X) ≥ 7	10 ≥ h(P_X) ≥ 9	10 ≥ h(P_X) ≥ 10
30	3 ≥ h(P_X) ≥ 2	15 ≥ h(P_X) ≥ 10	30 ≥ h(P_X) ≥ 15	30 ≥ h(P_X) ≥ 20	30 ≥ h(P_X) ≥ 25	30 ≥ h(P_X) ≥ 28
100	10 ≥ h(P_X) ≥9	50 ≥ h(P_X) ≥ 33	100 ≥ h(P_X) ≥ 50	100 ≥ h(P_X) ≥ 67	100 ≥ h(P_X) ≥ 84	100 ≥ h(P_X) ≥ 91
200	20 ≥ h(P_X) ≥ 18	100 ≥ h(P_X) ≥ 67	200 ≥ h(P_X) ≥ 100	200 ≥ h(P_X) ≥ 134	200 ≥ h(P_X) ≥ 167	200 ≥ h(P_X) ≥ 182

The next theorem shows that the second inequality in Theorem 2 becomes an equality for the array $\bar{X}$ = (⁠ $\bar{x}$ ⁠, $\bar{x}$ ⁠, …, $\bar{x}$ ⁠) ∈ (R⁺)^N.

Theorem 3. Let X = (x_r)_r=1,2,…,N ∈ Φ^N, then

min (N, A) = min (N, N \bar{x}) \geq h (P_{\bar{X}}) = ⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋,

(5)

Proof. We see that $P_{\bar{X}}$ = (N $\bar{x}$ ⁠, (N − 1) $\bar{x}$ ⁠, …, 2 $\bar{x}$ ⁠, $\bar{x}$ ⁠). Then h(⁠ $P_{\bar{X}}$ ⁠) is the largest natural number i such that (N − i + 1) $\bar{x}$ ≥ i. We observe that then h(⁠ $P_{\bar{X}}$ ⁠) is equal to the largest natural number i such that i ≤ (N + 1) $\frac{\bar{x}}{\bar{x} + 1}$ and hence h(⁠ $P_{\bar{X}}$ ⁠) = $⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋$ ⁠. This proves Theorem 3.

We next present some examples, illustrating different aspects of the previous results.

Example 1. Returning to the example introduced before, we have X = (4, 3, 2, 1), with $\bar{x}$ = 2.5 and P_X = (10, 9, 7, 4). Now N = h(P_X) = 4 > $⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋$ = $⌊5 \times \frac{2.5}{2.5 + 1}⌋$ = ⌊3.571⌋ = 3. This illustrates that the second inequality in Theorem 2 can be strict. Continuing now with $\bar{X}$ we see that N = 4 > h(⁠ $P_{\bar{X}}$ ⁠) = h(10, 7.5, 5, 2.5) = 3 = $⌊5 \times \frac{2.5}{3.5}⌋$ ⁠.

Example 2. Consider X = (4, 2, 1, 1) with $\bar{x}$ = 2 and P_X = (8, 7, 6, 4). Now N = h(P_X) = 4 > $⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋$ = $⌊5 \times \frac{2}{2 + 1}⌋$ = $⌊\frac{10}{3}⌋$ = 3. Continuing with $\bar{X}$ we see that N = 4 > h(⁠ $P_{\bar{X}}$ ⁠) = h(8, 6, 4, 2) = 3 = $⌊5 \times \frac{2}{3}⌋$ = $⌊\frac{10}{3}⌋$ ⁠. This example illustrates that the floor function is really needed, because 3 < 10/3.

Example 3. Let X = (4, 0, 0, 0) with $\bar{x}$ = 1 and P_X = (4, 4, 4, 4). Now N = h(P_X) = 4 > $⌊5 \times \frac{1}{1 + 1}⌋$ = $⌊5 \times \frac{1}{2}⌋$ = ⌊2.5⌋ = 2. This is another example that the second inequality in Theorem 2 can be strict. Continuing with $\bar{X}$ we see that N = A = 4 > h(⁠ $P_{\bar{X}}$ ⁠) = h(4, 3, 2, 1) = 2 = $⌊5 \times \frac{1}{2}⌋$ = ⌊2.5⌋. This is not only another example that the floor function is really needed, but it also illustrates that the first inequality in Theorem 3, and hence also in Theorem 2, can be strict.

Example 4. In the previous examples h(P_X) = N. Next we present an example where h(P_X) < N. Let X = (3, 2, 1, 0). Then $\bar{x}$ = 3/2 and P_X = (6, 6, 5, 3). Now N = 4 > h(P_X) = 3 ≥ $⌊5 \times \frac{3 / 2}{(3 / 2) + 1}⌋$ = $⌊5 \times \frac{3}{5}⌋$ = ⌊3⌋ = 3. Continuing with $\bar{X}$ we see that N = 4 > h(⁠ $P_{\bar{X}}$ ⁠) = h(6, 4.5, 3, 1.5) = 3 = $⌊5 \times \frac{3}{5}⌋$ = 3.

Example 5. Finally, we present an example where min(N, A) = A < N. Let X = (2, 0, 0, 0). Then $\bar{x}$ = ½ and P_X = (2, 2, 2, 2). Now A = 2 = min(N, A) = h(P_X) = 2 ≥ $⌊5 \times \frac{1 / 2}{(1 / 2) + 1}⌋$ = $⌊5 \times \frac{1}{3}⌋$ = 1. Continuing with $\bar{X}$ we see that A = 2 = min(N, A) > h(⁠ $P_{\bar{X}}$ ⁠) = h(2, 3/2, 1, 1/2) = 1 = $⌊5 \times \frac{1}{3}⌋$ = 1.

Corollaries

A.
If $\bar{x}$ ≥ N, then h(P_X) = N.

Proof. As lim_t⇒∞

\frac{t}{t + 1}

= 1, there exist a number t₀ such that for all t > t₀.

N \leq (N + 1) \frac{t}{t + 1} < N + 1 .

This double inequality clearly holds if we take t₀ = N. With

\bar{x}

in the role of t we see that in these circumstances

⌊(N + 1) \frac{\bar{x}}{\bar{x} + 1}⌋

= N and thus by Theorem 2, Corollary A is proved.

B.
${lim}_{\bar{x} \Rightarrow \infty}$ h(P_X) = N

This follows immediately from Corollary A.

Remark

When applied to publications, corollaries A and B show that for large $\bar{x}$ we only need those publications in X with the highest citations to determine h(P_X). This is in accordance with the principle and meaning of an h-index.

8. PARTIAL SUMS AND THE G-INDEX

Using the same notations as before, we next prove the analogue of Theorem 2 for the g-index. We recall that the g-index has no upper limit.

Theorem 4. Let X = (x_r)_r=1,2,…,N ∈ Φ^N. Then

g (P_{X}) \geq \{\begin{array}{c} ⌊(2 N + 1) \frac{\bar{x}}{\bar{x} + 2}⌋ if \sum_{j = 1}^{N} y_{j} < N^{2} (6 a) \\ ⌊\sqrt{N (N + 1) \frac{\bar{x}}{2}}⌋ if \sum_{j = 1}^{N} y_{j} \geq N^{2} (6 b) \end{array}

Proof.

g (P_{X}) = \{\begin{array}{c} max \{i \in N; \sum_{j = 1}^{i} y_{j} \geq i^{2}\} if \sum_{j = 1}^{N} y_{j} < N^{2} \\ max \{i \in N; \sum_{j = 1}^{N} y_{j} \geq i^{2}\} if \sum_{j = 1}^{N} y_{j} \geq N^{2} \end{array}

Now,

\sum_{j = 1}^{i} y_{j} = \sum_{j = 1}^{i} (\sum_{k = 1}^{N - j + 1} x_{k}) = \sum_{s = 1}^{N} x_{s} + \sum_{s = 1}^{N - 1} x_{s} + \dots + \sum_{s = 1}^{N - i + 1} x_{s}

≥ N

\bar{x}

+ (N − 1)

\bar{x}

+ … + (N − i + 1)

\bar{x}

(because X is ordered decreasingly) =

\bar{x}

×

(\frac{N (N + 1)}{2} - \frac{(N - i) (N - i + 1)}{2})

=

\frac{\bar{x}}{2}

i(2N − i + 1).

Now we require that this expression is larger than or equal to i². This leads to:

i \leq \frac{(2 N - i + 1) \bar{x}}{2} .

Solving for i yields: i ≤ (2N + 1)

\frac{\bar{x}}{\bar{x} + 2}

⁠. Taking into account that g(P_X) is an integer, we obtain that if

\sum_{j = 1}^{N} y_{j}

< N² then g(P_X) ≥

⌊(2 N + 1) \frac{\bar{x}}{\bar{x} + 2}⌋

⁠.

If $\sum_{j = 1}^{N} y_{j}$ ≥ N² then we have to study $\sum_{j = 1}^{N} y_{j}$ ≥ i². In the same way as above we find that $\sum_{j = 1}^{N} y_{j}$ ≥ $\bar{x}$ ⁠. $(\frac{N (N + 1)}{2})$ ≥ i² (is all we need). Hence i ≤ $\sqrt{\frac{\bar{x}}{2} N (N + 1)}$ or i_max = $⌊\sqrt{\frac{\bar{x}}{2} N (N + 1)}⌋$ ⁠, where i_max denotes the maximal value the index i can take here.

Consequently, if $\sum_{j = 1}^{N} y_{j}$ ≥ N² then g(P_X) ≥ $⌊\sqrt{\frac{\bar{x}}{2} N (N + 1)}⌋$ ⁠.

Similar to the theory for the h-index, the next theorem shows that inequality in Theorem 4 becomes an equality for the array $\bar{X}$ ⁠.

Theorem 5. Let X = (x_r)_r=1,2,…,N ∈ Φ^N, then

g (P_{\bar{X}}) = \{\begin{array}{c} ⌊(2 N + 1) \frac{\bar{x}}{\bar{x} + 2}⌋ if \bar{x} < \frac{2 N}{N + 1} (7 a) \\ ⌊\sqrt{N (N + 1) \frac{\bar{x}}{2}}⌋ if \bar{x} \geq \frac{2 N}{N + 1} (7 b) \end{array}

Proof. Now: $\sum_{j = 1}^{N} y_{j} = \sum_{j = 1}^{N} (\sum_{k = 1}^{N - j + 1} \bar{x}) = \sum_{s = 1}^{N} \bar{x} + \sum_{s = 1}^{N - 1} \bar{x} + \dots + \sum_{s = 1}^{1} \bar{x}$ = N $\bar{x}$ + (N − 1) $\bar{x}$ + … + 1. $\bar{x}$ = $\bar{x}$ ⁠. $(\frac{N (N + 1)}{2})$ ⁠.

Hence, $\sum_{j = 1}^{N} (N - j + 1) \bar{x} < N^{2} \Leftrightarrow \bar{x} < \frac{2 N}{N + 1}$ ⁠.

Similarly, $\sum_{j = 1}^{N} (N - j + 1) \bar{x} \geq N^{2} \Leftrightarrow \bar{x} \geq \frac{2 N}{N + 1}$ ⁠.

Comment. Also here we can make the remark that lower bounds for g(P_X) and g(⁠ $P_{\bar{X}}$ ⁠) depend only on N and $\bar{x}$ ⁠.

Examples

Example 1. Take X = (4, 4, 4, 4), $\bar{x}$ = 4 and P_X = (16, 12, 8, 4). Then g(P_X) = 6 (as 40 > 6² and 40 < 7²). As this is a case where 40 > 4² we have to check formula 6b. This formula states that 6 = g(P_X) ≥ $⌊\sqrt{\frac{4 \times 4 \times 5}{2}}⌋$ = $⌊\sqrt{40}⌋$ = 6. Thanks to the use of the floor function we obtain an equality.

Example 2. Take X = (4, 3, 2, 1), $\bar{x}$ = 2.5 and P_X = (10, 9, 7, 4). Then g(P_X) = 5 (as 30 > 5² and 30 < 6²). Also here we have to check formula 6b. We see that 5 = g(P_X) ≥ $⌊\sqrt{\frac{2.5 \times 4 \times 5}{2}}⌋$ = $⌊\sqrt{25}⌋$ = 5. This is an example where the floor function is not necessary.

Example 3. For X = (4, 0, 0, 0), $\bar{x}$ = 1 and P_X = (4, 4, 4, 4). Here the sum, namely 16, is larger than or equal to N² = 4²; hence we have to check formula 6b. This leads to 4 = g(P_X) ≥ $⌊\sqrt{12 \times \frac{1}{2}}⌋$ = $⌊\sqrt{6}⌋$ = 2. This is another case where we have strict inequality.

Example 4. For X = (2, 0, 0, 0), $\bar{x}$ = 0.5 and P_X = (2, 2, 2, 2). Here the sum, namely 8 < 4², hence we have to check formula (6a). This leads to 2 = g(P_X) ≥ $⌊\frac{9 * 0.5}{2.5}⌋$ = 1. Also here we have strict inequality.

e) Finally we consider a case for which N ≠ 4. Let X = (5, 4, 3, 2, 1), $\bar{x}$ = 3 and P_X = (15, 14, 12, 9, 5). Here the sum namely 55 > 5²; hence we check formula 6b. We first note that g(P_X) = 7 (55 > 7² and 55 < 8²). Now 7 = g(P_X) ≥ $⌊\sqrt{\frac{3 \times 5 \times 6}{2}}⌋$ = $⌊\sqrt{45}⌋$ = 6. This is again a case with a strict inequality.

Corollary

{lim}_{\bar{x} \Rightarrow \infty} g (P_{X}) = \infty

Proof. If $\bar{x}$ is large, then we have to consider formula 6b. Then the right-hand side of formula 6b becomes unlimited large and hence this also holds for g(P_X). This result confirms the fact that the g-index has no upper limit.

9. PARTIAL SUMS, THE R(R²)-INDEX AND KOSMULSKI’S INDEX H⁽²⁾(X)

In the previous sections we studied the h-index and the g-index. As a final case we mention the R²-index and Kosmulski’s h⁽²⁾-index. For proofs of the results we refer the reader to Egghe and Rousseau (2019c).

Theorem 6. Let X = (x_r)_r=1,2,…,N ∈ Φ^N. Then

R^{2} (P_{X}) > \frac{\bar{x} \times (N + 1)}{2 {(\bar{x} + 1)}^{2}} \times (N \times {\bar{x}}^{2} + 2 N \times \bar{x} - \bar{x} - 2) .

(8)

Theorem 7. If (N + 1)

\frac{\bar{x}}{\bar{x} + 1}

is a natural number and X = (x_r)_r=1,2,…,N ∈ Φ^N, then

R^{2} (P_{\bar{X}}) = \frac{{(\bar{x})}^{2} (N + 1) (N \bar{x} + 2 N + 1)}{2 {(\bar{x} + 1)}^{2}} .

(9)

Finally, we extend our results to the case of Kosmulski’s index, denoted as h⁽²⁾, referring to Egghe and Rousseau (2019c) for proofs.

Theorem 8. Let X = (x_r)_r=1,2,…,N ∈ Φ^N. Then

min (N, \sqrt{A}) \geq h^{(2)} (P_{X}) \geq ⌊\frac{1}{2} (\sqrt{{(\bar{x})}^{2} + 4 (N + 1) \bar{x}} - \bar{x})⌋

(10)

Similarly to Theorem 3 we have

Theorem 9. Let X = (x_r)_r=1,2,…,N ∈ Φ^N. Then

min (N, \sqrt{A}) \geq h^{(2)} (P_{\bar{X}}) = ⌊\frac{1}{2} (\sqrt{{(\bar{x})}^{2} + 4 (N + 1) \bar{x}} - \bar{x})⌋

(11)

10. DISCUSSION AND CONCLUSION

In this article we studied arrays of partial sums, P_X, of a given array X in terms of their h-type indices. We showed that h(P_X) can be described in terms of the Lorenz curve of the array X. Moreover, we obtained sharp lower and upper bounds for these h-type indices. We found bounds that only depend on N, the length of the array, and the average of array X, or equivalently, on the length of the array and the total sum of all items in the array.

As h(P_X) is an h-index it is not surprising that it is not strictly independent in the sense of Bouyssou and Marchant (2011). This means that if h(P_X) < h(P_Y) and if one adds to X and Y the same items (X becomes X′, and Y becomes Y′) then it is possible that h(P_X′) > h(P_Y′). An example: Let X = (2, 0, 0, 0, 0) and Y = (1, 1, 1, 1, 1). Then P_X = (2, 2, 2, 2, 2) with h(P_X) = 2, and P_Y = (5, 4, 3, 2, 1) with h(P_Y) = 3, hence h(P_X) < h(P_Y). Adding 5 times 1 to each of them yields X′ = (2, 1, 1, 1, 1, 1, 0, 0, 0, 0), P_X′ = (7, 7, 7, 7, 7, 6, 5, 4, 3, 2) with h(P_X′) = 6, and Y′ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1), P_Y′ = (10, 9, 8, 7, 6, 5, 4, 3, 2, 1) with h(P_Y′) = 5, hence h(P_X′) > h(P_Y′).

A reviewer asked if h(P_X) can be described in terms of Vannucci’s (2010) dominance dimension. It can: Using Vannucci’s notation we see that h(P_X) = $dom [{({proj}_{n}^{n} (\sum_{j = 1}^{N + 1 - n} x_{j}))}_{n}| {(n^{1})}_{n}]$ ⁠, where X = (x_n)_n=1,…,N is an array of length N.

Our investigation illustrated the rich mathematical structure hidden in the mechanism leading to h-type indices (see also Egghe & Rousseau, 2019d). In this article we considered the discrete case, requiring the floor function in order to get the correct results. In further research we intend to study the continuous case, where by definition no floor function will be needed. Then, bounds will be differentiable and integrable functions.

AUTHOR CONTRIBUTIONS

Leo Egghe: conceptualization; formal analysis; investigation; methodology; writing—original draft; writing—review and editing. Ronald Rousseau: validation; writing—review and editing.

COMPETING INTERESTS

The authors have no competing interests.

FUNDING INFORMATION

No funding has been received.

DATA AVAILABILITY

Not applicable.

ACKNOWLEDGMENTS

The authors thank anonymous reviewers for useful suggestions to improve the presentation of this article.

REFERENCES

Bouyssou

,

D.

, &

Marchant

,

T.

(

2011

).

Ranking scientists and departments in a consistent manner

.

Journal of the American Society for Information Science and Technology

,

62

(

9

),

1761

–

1769

.

Google Scholar

Egghe

,

L.

(

2006a

).

An improvement of the h-index: The g-index

.

ISSI Newsletter

,

2

(

1

),

8

–

9

.

Google Scholar

Egghe

,

L.

(

2006b

).

Theory and practise of the g-index

.

Scientometrics

,

69

(

1

),

131

–

152

.

Google Scholar

Egghe

,

L.

(

2009

).

An econometric property of the g-index

.

Information Processing & Management

,

45

(

4

),

484

–

489

.

Google Scholar

Egghe

,

L.

, &

Rousseau

,

R.

(

2019a

).

Infinite sequences and their h-type indices

.

Journal of Informetrics

,

13

(

1

),

291

–

298

.

Google Scholar

Egghe

,

L.

, &

Rousseau

,

R.

(

2019b

).

A geometric relation between the h-index and the Lorenz curve

.

Scientometrics

,

119

(

2

),

1281

–

1284

.

Google Scholar

Egghe

,

L.

, &

Rousseau

,

R.

(

2019c

).

h-Type indices, partial sums and the majorization order: A first approach

.

E-LIS, http://hdl.handle.net/10760/34395

Google Scholar

Egghe

,

L.

, &

Rousseau

,

R.

(

2019d

).

Solution by step functions of a minimum problem in L²[0, T], using generalized h- and g-indices

.

Journal of Informetrics

,

13

(

3

),

785

–

792

.

Google Scholar

Hardy

,

G. H.

,

Littlewood

,

G. E.

, &

Pólya

,

G.

(

1934

).

Inequalities

.

Cambridge

:

Cambridge University Press

.

Hirsch

,

J. E.

(

2005

).

An index to quantify an individual’s scientific research output

.

Proceedings of the National Academy of Sciences of the United States of America

,

102

(

46

),

16569

–

16572

.

Google Scholar

Jin

,

B. H.

,

Liang

,

L. M.

,

Rousseau

,

R.

, &

Egghe

,

L.

(

2007

).

The R- and AR-indices: Complementing the h-index

.

Chinese Science Bulletin

,

52

(

6

),

855

–

863

.

Google Scholar

Kosmulski

,

M.

(

2006

).

A new Hirsch-type index saves time and works equally well as the original index

.

ISSI Newsletter

,

2

(

3

),

4

–

6

.

Google Scholar

Rousseau

,

R.

(

2016

).

Citation data as proxy for quality or scientific influence are at best PAC (Probably Approximately Correct)

.

Journal of the Association for Information Science and Technology

,

67

(

12

),

3092

–

3094

.

Google Scholar

Rousseau

,

R.

,

Egghe

,

L.

, &

Guns

,

R.

(

2018

).

Becoming Metric-Wise. A Bibliometric Guide for Researchers

.

Kidlington

:

Chandos-Elsevier

.

Vannucci

,

S.

(

2010

).

Dominance dimension: A common parametric formulation for integer-valued scientific impact indices

.

Scientometrics

,

84

(

1

),

43

–

48

.

Google Scholar

Waltman

,

L.

, &

van Eck

,

N.-J.

(

2012

).

The inconsistency of the h-index

.

Journal of the American Society for Information Science and Technology

,

63

(

2

),

406

–

415

.

Google Scholar

Author notes

Handling Editor: Vincent Larivière

2019

Leo Egghe and Ronald Rousseau. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.

This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a full description of the license, please visit https://creativecommons.org/licenses/by/4.0/legalcode.

h-Type indices, partial sums and the majorization order

Abstract

1. INTRODUCTION

2. DEFINITIONS

2.1. The h-index (Hirsch, 2005)

2.2. The g-index (Egghe, 2006a, b)

2.3. The R-index (Jin et al., 2007)

2.4. Kosmulski’s h⁽²⁾ Index (Kosmulski, 2006)

2.5. The Majorization Order (Hardy et al., 1934)

3. AN INEQUALITY RELATED TO THE g-INDEX AND THE MAJORIZATION ORDER

4. INTRODUCING THE ARRAY OF PARTIAL SUMS

5. A RELATION WITH THE GINI INDEX

6. A GEOMETRIC INTERPRETATION OF h(P_X) IN TERMS OF THE LORENZ CURVE L_X

7. BOUNDS ON h-TYPE INDICES

8. PARTIAL SUMS AND THE G-INDEX

9. PARTIAL SUMS, THE R(R²)-INDEX AND KOSMULSKI’S INDEX H⁽²⁾(X)

10. DISCUSSION AND CONCLUSION

AUTHOR CONTRIBUTIONS

COMPETING INTERESTS

FUNDING INFORMATION

DATA AVAILABILITY

ACKNOWLEDGMENTS

REFERENCES

Author notes

Email alerts

Cited By

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h-Type indices, partial sums and the majorization order

Abstract

1. INTRODUCTION

2. DEFINITIONS

2.1. The h-index (Hirsch, 2005)

2.2. The g-index (Egghe, 2006a, b)

2.3. The R-index (Jin et al., 2007)

2.4. Kosmulski’s h(2) Index (Kosmulski, 2006)

2.5. The Majorization Order (Hardy et al., 1934)

3. AN INEQUALITY RELATED TO THE g-INDEX AND THE MAJORIZATION ORDER

4. INTRODUCING THE ARRAY OF PARTIAL SUMS

5. A RELATION WITH THE GINI INDEX

6. A GEOMETRIC INTERPRETATION OF h(PX) IN TERMS OF THE LORENZ CURVE LX

7. BOUNDS ON h-TYPE INDICES

8. PARTIAL SUMS AND THE G-INDEX

9. PARTIAL SUMS, THE R(R2)-INDEX AND KOSMULSKI’S INDEX H(2)(X)

10. DISCUSSION AND CONCLUSION

AUTHOR CONTRIBUTIONS

COMPETING INTERESTS

FUNDING INFORMATION

DATA AVAILABILITY

ACKNOWLEDGMENTS

REFERENCES

Author notes

Email alerts

Cited By

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2.4. Kosmulski’s h⁽²⁾ Index (Kosmulski, 2006)

6. A GEOMETRIC INTERPRETATION OF h(P_X) IN TERMS OF THE LORENZ CURVE L_X

9. PARTIAL SUMS, THE R(R²)-INDEX AND KOSMULSKI’S INDEX H⁽²⁾(X)