Closeness Centralization Measure for Two-mode Data of Prescribed Sizes^†^†thanks: This work was supported by PHC Proteus 26818PC, Slovenian ARRS bilateral projects BI-FR/12-13-PROTEUS-011 and BI-FR/14-15-PROTEUS-001.

Matjaž Krnc¹¹1Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Slovenia. matjaz.krnc@gmail.com. Jean-Sébastien Sereni²²2CNRS (LORIA), Vandœuvre-lès-Nancy, France. sereni@kam.mff.cuni.cz. This author’s work was partially supported by the French Agence Nationale de la Recherche under reference anr 10 jcjc 0204 01. Corresponding author,

+ 33354958640

. Riste Škrekovski³³3Department of Mathematics, University of Ljubljana, and Faculty of information studies, Novo Mesto, and FAMNIT, University of Primorska, Koper, Slovenia. Partially supported by ARRS Program P1-0383. Email: skrekovski@gmail.com Zelealem B. Yilma⁴⁴4Carnegie Mellon University Qatar, Doha, Qatar. E-mail: zyilma@qatar.cmu.edu. This author’s work was partially supported by the French Agence Nationale de la Recherche under reference anr 10 jcjc 0204 01.

July 29, 2019

Abstract

We confirm a conjecture by Everett, Sinclair, and Dankelmann [Some Centrality results new and old, J. Math. Sociology 28 (2004), 215–227] regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite graphs with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth $2$ , where neighbors of the root have an equal or almost equal number of children.

Keywords: centrality, closeness centrality, network, graph, complex network.

Closeness Centralization Measure for Two-mode Data of Prescribed Sizes

1 Introduction

A social network is often conveniently modeled by a graph: nodes represent individual persons and edges represent the relationships between pairs of individuals. Our work focuses on simple unweighted graphs: our graph only tells us, for a given (binary) relation $R$ , which pairs of individual are in relation according to $R$ .

Centrality is a crucial concept in studying social networks [8, 12]. It can be seen as a measure of how central is the position of an individual in a social network. Various node-based measures of the centrality have been proposed to determine the relative importance of a node within a graph (the reader is referred to the work of Koschützki et al. [9] for an overview). Some widely used centrality measures are the degree centrality, the betweenness centrality, the closeness centrality and the eigenvector centrality (definitions and extended discussions are found in the book edited by Brandes and Erlebach [5]).

We focus on closeness centrality, which measures how close a node is to all other nodes in the graph: the smaller the total distance from a node $v$ to all other nodes, the more important the node $v$ is. Various closeness-based measures have been developed [1, 2, 4, 13, 11, 14, 16, 13].

Let us see an example: suppose we want to place a service facility, e.g., a school, such that the total distance to all inhabitants in the region is minimal. This would make the chosen location as convenient as possible for most inhabitants. In social network analysis the centrality index based on this concept is called closeness centrality.

Formally, for a node $v$ of a graph $G$ , the closeness of $v$ is defined to be

CG(v)\coloneqq1∑u∈V(G)distG(v,u),

(1)

where ${dist}_{G} (u, v)$ is the distance between $u$ and $v$ in $G$ , that is, the length of a shortest path in $G$ between nodes $u$ and $v$ . We shall use the shorthand $WG(v)\coloneqq∑u∈V(G)d(v,u)$ . In both notations, we may drop the subscript when there is no risk of confusion.

While centrality measures compare the importance of a node within a graph, the associated notion of centralization, as introduced by Freeman [8], allows us to compare the relative importance of nodes within their respective graphs. The closeness centralization of a node $v$ in a graph $G$ is given by

C1(v;G)\coloneqq∑u∈V(G)[C(v)−C(u)].

(2)

Further, we set $C1(G)\coloneqqmax{C1(v;G):v∈V(G)}$ .

It is important to note that the parameter $C_{1}$ is really tailored to compare the centralization of nodes in different graphs. If only one graph is involved, then one readily sees that maximizing $C_{1} (v; G)$ over the nodes of a graph $G$ amounts to minimizing $W_{G}$ . Indeed, suppose that $G$ is a graph and $v$ a node of $G$ such that $W_{G} (v) ⩽ W_{G} (u)$ for every $u \in V (G)$ . Then for every node $x$ of $G$ ,

	$C_{1} (v; G) - C_{1} (x; G)$	$= (n - 1) (\frac{1}{W_{G} (v)} - \frac{1}{W_{G} (x)}) - (\frac{1}{W_{G} (x)} - \frac{1}{W_{G} (v)})$
		$= n (\frac{1}{W_{G} (v)} - \frac{1}{W_{G} (x)})$
		$⩾ 0.$

In what follows, we use the the following notation. The star graph of order $n$ , sometimes simply known as an $n$ -star, is the tree on $n + 1$ nodes with one node having degree $n$ . The star graph is thus a complete bipartite graph with one part of size $1$ . Everett, Sinclair, and Dankelmann [7] established that over all graphs with a fixed number of nodes, the closeness is maximized by the star graph.

Theorem 1.

If $G$ is a graph with $n$ nodes, then

C_{1} (u; S_{n - 1}) ⩾ C_{1} (G),

where $u$ is the node of $S_{n - 1}$ of maximum degree.

$v \in V (N)$ $C_{N} (v)$ $C_{1} (v, N)$ $S_{0}$ $1 / 10$ $0.1222$ $S_{1}$ $1 / 12$ $0.0055$ $S_{2}$ $1 / 12$ $0.0055$ $L_{0}$ $1 / 15$ $- 0.1111$ $L_{1}$ $1 / 15$ $- 0.1111$ $L_{2}$ $1 / 9$ $0.2000$ $L_{3}$ $1 / 15$ $- 0.1111$

Figure 1: A two-mode network

N

with

7

nodes (

3

in one part,

4

in the other) and

7

edges, with the corresponding values for

C_{N}

and

C_{1}

They also considered the problem of maximizing centralization measures for two-mode data [*]ESD04. In this context, the relation studied links two different types of data (e.g., persons and events) and we are interested in the centralization of one type of data only (e.g., the most central person). Thus the graph obtained is bipartite: its nodes can be partitioned into two parts so that all the edges join nodes belonging to different parts. A toy example is depicted in Figure 1, where one type of data consists of students and the other of classes: edges link the students to the classes they attended. (The sole purpose of this example is to make sure the reader is at ease with the definitions of $C$ and $C_{1}$ .) Closeness centrality is maximized at the student “ $S_{0}$ ” for one part and at the class “ $L_{2}$ ” for the other. An example of a real-world two-mode network $N$ on $89$ edges with partition sizes $| P_{1} | = 18$ and $| P_{2} | = 14$ , borrowed from [6] is depicted on Figure 2. On the figure, one can observe a frequency of interparticipation of a group of women in social events in Old City, 1936. On Tables 1 and 2, one can observe closeness centralization for partitions $P_{1}$ and $P_{2}$ and notice that closeness centrality (and hence centralization) is maximized at “Mrs. Evelyn Jefferson” and the event from “September 16th”, respectively.

Figure 2: A two-mode network

N

89

edges with partition sizes

n_{0} = 18

and

n_{1} = 14

. The network represents the participation of a given set of people in the social events from 1936 reported in Old City Herald, where circles represent social events while rectangles represent women (see Tables 1 and 2).

$v \in P_{1}$	$C_{N} (v)$	$C_{1} (v, N)$
Mrs. Evelyn Jefferson	$0.01667$	$0.07779$
Miss Theresa Anderson	$0.01667$	$0.07779$
Mrs. Nora Fayette	$0.01667$	$0.07779$
Mrs. Sylvia Avondale	$0.01613$	$0.06058$
Miss Laura Mandeville	$0.01515$	$0.02930$
Miss Brenda Rogers	$0.01515$	$0.02930$
Miss Katherine Rogers	$0.01515$	$0.02930$
Mrs. Helen Lloyd	$0.01515$	$0.02930$
Miss Ruth DeSand	$0.01471$	$0.01504$
Miss Verne Sanderson	$0.01471$	$0.01504$
Miss Myra Liddell	$0.01429$	$0.00160$
Miss Frances Anderson	$0.01389$	$- 0.01110$
Miss Eleanor Nye	$0.01389$	$- 0.01110$
Miss Pearl Oglethorpe	$0.01389$	$- 0.01110$
Mrs. Dorothy Murchison	$0.01351$	$- 0.02311$
Miss Charlotte McDowd	$0.01250$	$- 0.05555$
Mrs. Olivia Carleton	$0.01220$	$- 0.06530$
Mrs. Flora Price	$0.01220$	$- 0.06530$

Table 1: Nodes from the group of women and their closeness values.

$v \in P_{2}$	label on Fig. 2	$C_{N} (v)$	$C_{1} (v, N)$
September 16th	P8	$0.01923$	$0.15984$
April 8th	P9	$0.01786$	$0.11588$
March 15th	P7	$0.01667$	$0.07779$
May 19th	P6	$0.01562$	$0.04445$
February 25th	P5	$0.01351$	$- 0.02311$
April 12th	P3	$0.01282$	$- 0.04529$
April 7th	P12	$0.01282$	$- 0.04529$
June 10th	P10	$0.01250$	$- 0.05555$
September 26th	P4	$0.01220$	$- 0.06530$
February 23rd	P11	$0.01220$	$- 0.06530$
June 27th	P1	$0.01190$	$- 0.07459$
March 2nd	P2	$0.01190$	$- 0.07459$
November 21st	P13	$0.01190$	$- 0.07459$
August 3rd	P14	$0.01190$	$- 0.07459$

Table 2: Nodes from the partition of social events from 1936, reported in Old City Herald, and their closeness values.

Everett et al. formulated an interesting conjecture, which was later proved by Sinclair [15]. To state it, we first need a definition.

Definition 2.

Let $H (u; n_{0}, n_{1})$ be the tree with node bipartition $(A_{0}, A_{1})$ such that

$| A_{i} | = n_{i}$ for $i \in {0, 1}$ ;
there exists a node $u \in A_{0}$ such that $N_{G} (u) = A_{1}$ ; and
$deg (w) \in {1 + ⌈ \frac{n_{0} - 1}{n_{1}} ⌉, 1 + ⌊ \frac{n_{0} - 1}{n_{1}} ⌋}$ for all nodes $w \in A_{1}$ .

The node $u$ is called the root of $H (u; n_{0}, n_{1})$ .

The aforementioned conjecture was that the pair $(H (u; n_{0}, n_{1}), u)$ is an extremal pair for the problem of maximizing betweenness centralization in bipartite graphs with a fixed sized bipartition into parts of sizes $n_{0}$ and $n_{1}$ . Recall that for two-mode data, we are only interested in one type of data: in graph-theoretic terms, we look only at nodes that belong to the part of size $n_{0}$ , and we want to know which of these nodes has the largest closeness in the graph. In other words, letting $A_{0}$ be the part of size $n_{0}$ of $V (G)$ , we want to determine $max {C_{1} (v; G) : v \in A_{0}}$ .

Everett et al. also suggested that the same pair is extremal for closeness and eigenvector centralization measures. In this paper, we confirm the conjecture for the closeness centralization measure. That is, we prove that the pair $H (v; n_{0}, n_{1})$ is extremal for the problem of maximizing closeness centralization in bipartite graphs with parts of size $n_{0}$ and $n_{1}$ ,where $v$ is the root.

We point out that a similar study for the centrality measure of eccentricity was led recently [10]. In addition, Bell [3] worked on closely related notions, namely subgroup centrality measures. Similarly as for two-mode data, a susbet $S$ of the nodes is fixed (called a group) and the aim is to find a node in $S$ with largest centrality. However, unlike in the standard centrality notion, the centrality itself is computed using distances only to the nodes in $S$ (local centrality) or to the nodes outside $S$ (global centrality). Note that the standard notion, which is used in this work, takes into account the distances to all other nodes in the graph.

2 Bipartite Networks With Fixed Number of Nodes

Theorem 3.

Let $G$ be a bipartite graph with node parts $A_{0}$ and $A_{1}$ sizes $n_{0}$ and $n_{1}$ , respectively. Then for each $v \in A_{0}$ ,

C_{1} (u; H (u; n_{0}, n_{1})) ⩾ C_{1} (v; G) .

To prove Theorem 3, suppose that $G$ is a bipartite graph with bipartition $(A_{0}, A_{1})$ where $| A_{i} | = n_{i}$ for $i \in {0, 1}$ , and $u$ is a node in $A_{0}$ such that $C_{1} (u; G) ⩾ C_{1} (v; H (v; n_{0}, n_{1}))$ . We prove that this inequality must actually be an equality by showing that any such extremal pair $C_{1} (u; G)$ must satisfy the following three properties:

$G$ is a tree;
${deg}_{G} (u) = n_{1}$ ; and
$| {deg}_{G} (w_{1}) - {deg}_{G} (w_{2}) | ⩽ 1$ whenever $w_{1}, w_{2} \in A_{1}$ .

Property 1 is relatively straightforward to check and so is 3 if we assume that 2 holds. Thus the majority of the discussion below will be devoted to proving that 2 holds, which we do last. For convenience, we define $V$ to be $V (G)$ .

We start by establishing 1; namely, that the graph $G$ is a tree. Assume, for the sake of contradiction, that $G$ is not a tree and let $T$ be a breadth-first-search tree of $G$ rooted at $u$ . Note that $W_{G} (u) = W_{T} (u)$ and $W_{T} (x) ⩾ W_{G} (x)$ for any node $x \in V (G)$ . In addition, there exist at least two nodes for which the above inequality is strict. It follows that $C_{1} (u; T) > C_{1} (u; G)$ , a contradiction.

We now establish that 3 holds if 2 does. Thus we know that $G$ is a tree and we assume that $N_{G} (u) = A_{1}$ , therefore also all nodes from $A_{0} ∖ {u}$ are leaves. Suppose, for the sake of contradiction, that there exist nodes $w_{1}, w_{2} \in A_{1}$ such that $deg (w_{1}) ⩾ deg (w_{2}) + 2$ . Let $z$ be a neighbor of $w_{1}$ different from $u$ and consider the graph $G^{'}$ obtained by deleting the edge $w_{1} z$ and replacing it with $w_{2} z$ . Note that $W_{G^{'}} (u) = W_{G} (u)$ and that $W_{G^{'}} (x) = W_{G} (x)$ unless $x \in N_{G} [w_{1}] \cup N_{G} [w_{2}]$ , that is unless $x$ belongs to the closed neighborhood of either $w_{1}$ or $w_{2}$ . So

C_{1} (u; G^{'}) - C_{1} (u; G) = \sum x \in N_{G} [w_{1}] \cup N_{G} [w_{2}] \frac{1}{W_{G} (x)} - \sum x \in N_{G} [w_{1}] \cup N_{G} [w_{2}] \frac{1}{W_{G^{'}} (x)} .

(3)

Now, let $N_{G} (w_{1}) = {u, z, x_{1}, \dots, x_{t}}$ and $N_{G} (w_{2}) = {u, y_{1}, \dots, y_{s}}$ where, by assumption, $t > s$ .

Recalling that $G$ is a tree, observe that the following hold for every $i \in {1, \dots, t}$ and every $j \in {1, \dots, s}$ (for better illustration, see Figure 3).

$W_{G^{'}} (x_{i}) = W_{G} (x_{i}) + 2$ ;
$W_{G^{'}} (y_{j}) = W_{G} (y_{j}) - 2$ ;
$W_{G} (y_{j}) = W_{G} (x_{i}) + 2 (t - s + 1) > W_{G} (x_{i}) + 2$ ;
$W_{G^{'}} (z) = W_{G} (z) + 2 (t - s) > W_{G} (z)$ ;
$W_{G^{'}} (w_{1}) = W_{G} (w_{1}) + 2;$ and
$W_{G^{'}} (w_{2}) = W_{G} (w_{2}) - 2$ .

From (i)–(iii), we infer that for any $j \in {1, \dots, s}$ ,

\frac{1}{W_{G^{'}} (x_{j})} + \frac{1}{W_{G^{'}} (y_{j})} < \frac{1}{W_{G} (x_{j})} + \frac{1}{W_{G} (y_{j})},

and similarly by (v) and (vi),

\frac{1}{W_{G^{'}} (w_{1})} + \frac{1}{W_{G^{'}} (w_{2})} < \frac{1}{W_{G} (w_{1})} + \frac{1}{W_{G} (w_{2})} .

Thus the right side of (3) is greater than

\frac{1}{W_{G} (z)} - \frac{1}{W_{G^{'}} (z)} + t \sum j = s + 1 \frac{1}{W_{G} (x_{j})} - \frac{1}{W_{G^{'}} (x_{j})},

which is positive by (i) and (iv). This contradiction shows that 3 holds provided 2 does.

Figure 3: The subtree of

G

induced by

N_{G} [w_{1}] \cup N_{G} [w_{2}]

It remains to prove that 2 holds to complete the proof. First, if $n_{1} = 1$ , then the tree $G$ must be an $n_{0}$ -star, hence the second property is satisfied. Now consider the case where $n_{1} = 2$ . Then there is precisely one node $x$ that is adjacent to both nodes in $A_{1}$ . Moreover, $W_{G} (x) ⩽ W_{G} (w)$ if $w \in A_{0}$ since, if $w \in A_{0} ∖ {x}$ then $W_{G} (w) ⩾ 2 (n_{0} - 1) + 4 = 2 n_{0} + 2$ while $W_{G} (x) = 2 + 2 (n_{0} - 1) = 2 n_{0} + 1$ . Thus $u = x$ and hence ${deg}_{G} (u) = n_{1} = 2$ , as wanted.

From now on, we assume that $n_{1} ⩾ 3$ . As in the proof of 3, we argue that if 2 does not hold then $C_{1} (u; G)$ can be increased by altering the graph $G$ . In this case, however, we find it necessary to use our assumption that $C_{1} (u; G)$ itself is at least as large as $C_{1} (v; H (v; n_{0}, n_{1}))$ . This shall allow us to have a lower bound on $C_{1} (u; G)$ , by the next lemma.

Lemma 4.

$C_{1} (u; H (u; n_{0}, n_{1})) ⩾ \frac{n_{1} - 1}{2 (2 n_{1} - 1)}$ .

Proof.

We establish the inequality via a direct computation. Unfortunately, the expressions involved force a lengthy computation.

We set $m\coloneqqn0−1$ and we write $m = p n_{1} + r$ where $0 ⩽ r < n_{1}$ . Let us now calculate $W (x)$ for each node $x$ of $H (u; n_{0}, n_{1})$ .

$W (u) = n_{1} + 2 m$ .
Consider the neighbors of $u$ : there are
1. $r$ neighbors $x$ for which $W (x) = ⌈ m / n_{1} ⌉ + 1 + 2 (n_{1} - 1) + 3 (m - ⌈ m / n_{1} ⌉)$ ; and
2. $n_{1} - r$ neighbors $x$ for which $W (x) = ⌊ m / n_{1} ⌋ + 1 + 2 (n_{1} - 1) + 3 (m - ⌊ m / n_{1} ⌋)$ .
Consider the nodes at distance two from $u$ : there are
1. $r ⌈ m / n_{1} ⌉$ nodes $x$ for which $W (x) = 1 + 2 ⌈ m / n_{1} ⌉ + 3 (n_{1} - 1) + 4 (m - ⌈ m / n_{1} ⌉)$ ; and
2. $(n_{1} - r) ⌊ m / n_{1} ⌋$ nodes $x$ for which $W (x) = 1 + 2 ⌊ m / n_{1} ⌋ + 3 (n_{1} - 1) + 4 (m - ⌊ m / n_{1} ⌋)$ .

Since $⌊ m / n_{1} ⌋ = (m - r) / n_{1}$ and, for $r > 0$ , we have $⌈ m / n_{1} ⌉ = (m + n_{1} - r) / n_{1}$ , it follows that if $r > 0$ then

	$\begin{matrix} C_{1} (u) & = \frac{n_{1} + m}{n_{1} + 2 m} - \frac{r n_{1}}{3 m n_{1} - 2 m + 2 n_{1}^{2} - 3 n_{1} + 2 r} - \frac{n_{1} (n_{1} - r)}{} 3 m n_{1} - 2 m + 2 n_{1}^{2} - n_{1} + 2 r - \frac{r (m + n_{1} - r)}{4 m n_{1} - 2 m + 3 n_{1}^{2} - 4 n_{1} + 2 r} - \frac{(n_{1} - r) (m - r)}{4 m n_{1} - 2 m + 3 n_{1}^{2} - 2 n_{1} + 2 r} \end{matrix}$		(4)
		$⩾ \frac{n_{1} + m}{n_{1} + 2 m} - \frac{n_{1}^{2}}{3 m n_{1} - 2 m + 2 n_{1}^{2} - 3 n_{1} + 2 r} - \frac{n_{1} m}{4 m n_{1} - 2 m + 3 n_{1}^{2} - 4 n_{1} + 2 r},$		(5)

where we used that $n_{1} > 0$ to derive (5).

One notes that (5) is still true if $r = 0$ . Indeed, in this case $⌈ \frac{m}{n_{1}} ⌉ = ⌊ \frac{m}{n_{1}} ⌋ = \frac{m}{n_{1}}$ , so

C_{1} (u) = \frac{n_{1} + m}{n_{1} + 2 m} - \frac{n_{1}^{2}}{3 m n_{1} - 2 m + 2 n_{1}^{2} - n 1} - \frac{n_{1} m}{4 m n_{1} - 2 m + 3 n_{1}^{2} - 2 n_{1}},

so that (5) stays true.

As is seen from (4), if $n_{1}$ is fixed and $n_{0}$ tends to infinity (hence, so does $m$ ), then $C_{1} (u)$ approaches $1 / 2 - n_{1} / (4 n_{1} - 2) = \frac{n_{1} - 1}{4 n_{1} - 2}$ .

Let us now subtract $\frac{n_{1} - 1}{4 n_{1} - 2}$ from the right side of (5) and show that the difference is non-negative. After cross-multiplying and simplifying, we obtain a fraction with positive denominator (since each denominator in the right side of (5) is positive), and with numerator equal to

m^{2} (10 n_{1}^{4} - 44 n_{1}^{3} + 12 n_{1}^{2} r + 30 n_{1}^{2} - 8 n_{1} r - 4 n_{1}) + m (15 n_{1}^{5} - 77 n_{1}^{4} + 38 n_{1}^{3} r + 74 n_{1}^{3} - 54 n_{1}^{2} r - 14 n_{1}^{2} + 8 n_{1} r^{2} + 8 n_{1} r) + (6 n_{1}^{6} - 35 n_{1}^{5} + 22 n_{1}^{4} r + 45 n_{1}^{4} - 48 n_{1}^{3} r - 12 n_{1}^{3} + 12 n_{1}^{2} r^{2} + 14 n_{1}^{2} r - 4 n_{1} r^{2}) .

(6)

This expression increases with $n_{1}$ and is clearly positive when $n_{1} = 6$ (to see it quickly just compare, in each parenthesis, every (maximal) sequence of consecutive negative terms with the (maximal) sequence of positive terms preceding it). Further, a direct calculation ensures that (6) is actually positive even when $n_{1} = 5$ .

However, if $n_{1} \in {3, 4}$ , then (6) could take on negative values for certain values of $m$ . To deal with these two cases we revert back to the initial equation (4).

Assume that $n_{1} = 3$ . Then subtracting $\frac{n_{1} - 1}{4 n_{1} - 2}$ from both sides of (4) yields that $C_{1} (u) - \frac{n_{1} - 1}{4 n_{1} - 2}$ is at least

\frac{m + 3}{2 m + 3} - \frac{3 r}{7 m + 9 + 2 r} - \frac{9 - 3 r}{7 m + 15 + 2 r} - \frac{r (m + 3 - r)}{10 m + 15 + 2 r} - \frac{(3 - r) (m - r)}{10 m + 21 + 2 r} - \frac{1}{5} .

(7)

Placing (7) under one (positive) denominator, the numerator becomes

1540 m^{4} + 2 m^{3} (9075 - 1016 r + 588 r^{2}) + 6 m^{2} (10605 - 1047 r + 937 r^{2} + 112 r^{3}) + m (88155 - 3816 r + 9828 r^{2} + 2408 r^{3} + 96 r^{4}) + (42525 + 1350 r + 6174 r^{2} + 2280 r^{3} + 184 r^{4}),

(8)

which is clearly positive as $r ⩽ n_{1} - 1 = 2$ .

A similar calculation yields the conclusion when $n_{1} = 4$ . In this case, the difference of (4) and $\frac{n_{1} - 1}{4 n_{1} - 2}$ yields that $C_{1} (u) - \frac{n_{1} - 1}{4 n_{1} - 2}$ is at least

\frac{m + 4}{2 m + 4} - \frac{2 r}{5 m + 10 + r} - \frac{8 - 2 r}{5 m + 14 + r} - \frac{r (m + 4 - r)}{14 m + 32 + 2 r} - \frac{(4 - r) (m - r)}{14 m + 40 + 2 r} - \frac{3}{14},

whose numerator, when placed under a common (positive) denominator, is

1855 m^{4} + 4 m^{3} (5855 - 82 r + 100 r^{2}) + 2 m^{2} (52090 + 206 r + 1405 r^{2} + 80 r^{3}) + 4 m (49180 + 2022 r + 1793 r^{2} + 194 r^{3} + 4 r^{4}) + 3 (44800 + 4080 r + 2204 r^{2} + 332 r^{3} + 13 r^{4}) .

This is non-negative as $r ⩽ n_{1} - 1 = 3$ . This concludes the proof. ∎

It remains to demonstrate that 2 holds. To this end, we consider the tree $G$ to be rooted at $u$ and, for a node $x$ , we let $T_{x}$ be the subtree of $G$ rooted at $x$ . To avoid unnecessary notation later, let us observe immediately that if ${deg}_{G} (u) = 1$ then 2 holds. For otherwise, $n_{1} ⩾ 2$ and there exists a node $u^{'}$ at distance two from $u$ such that ${deg}_{G} (u^{'}) ⩾ 2$ . As a result, $W_{G} (u) ⩾ W_{G} (u^{'}) + | V (T_{u^{'}}) | - 1 > W_{G} (u^{'})$ , which implies that $C_{1} (u^{'}; G) > C_{1} (u; G)$ , a contradiction.

We also note that if ${dist}_{G} (u, x) ⩽ 2$ for all $x \in V (G)$ , then 2 is satisfied. So assume that there exists some child of $u$ whose subtree has depth at least $2$ . Among all such children of $u$ , let $z$ be such that $| V (T_{z}) |$ is maximum, that is,

| V (T_{z}) | = max {| V (T_{v}) | : v child of~{} u and T_{v} has depth at least~{} 2} .

We now give some notations, which are illustrated in Figure 4. Let $y_{1}, \dots, y_{t}$ be the nodes of $T_{z}$ with depth $2$ and set $Y\coloneqq∪ti=1V(Tyi)$ . Note that, by definition, $t ⩾ 1$ and ${dist}_{G} (u, y_{i}) = 3$ whenever $1 ⩽ i ⩽ t$ . Let $p_{1}, \dots, p_{ℓ}$ be the children of $z$ (in $T_{z}$ ) with degree more than $1$ and set $P\coloneqq{p1,…,pℓ}$ . Let $P^{'}$ be the set of children of $z$ with degree $1$ and set $k\coloneqq|P′|$ .

Note that for any $w \in N (u)$ , the definition of $z$ ensures that $T_{w}$ is a star whenever $| V (T_{w}) | > | V (T_{z}) |$ . The graph $G^{'}$ is obtained from $G$ as follows. (An illustration is given in Figure 5.) For convenience, we set $n\coloneqqn0+n1=|V(G)|$ .

Figure 4: Figurative view of the subsets of nodes of

G

. Recall that

S′\coloneqqV(Tw)∖{w}

S = \emptyset

For each $i \in {1, \dots, t}$ , the edge $u y_{i}$ is added.
For each $i \in {1, \dots, ℓ}$ , the edge $z p_{i}$ is removed and all other edges incident to $p_{i}$ but one are removed. Thus the vertices $p_{1}, \dots, p_{ℓ}$ become leaves of $G^{'}$ , each being attached to one of the vertices $y_{1}, \dots, y_{t}$ .
If there exists a child $w$ of $u$ different from $z$ with $| V (T_{w}) | ⩾ n / 2$ , then we select an arbitrary set $S \subset V (T_{w}) ∖ {w}$ of size $| V (T_{w}) | - ⌊ n / 2 ⌋$ and we set $S′\coloneqqV(Tw)∖(S∪{w})$ . Then for each $s \in S$ , we replace the edge $s w$ by the edge $s z$ .
If there is no node $w$ as in 3, then we let $w$ be a child of $u$ different from $z$ such that $| V (T_{w}) |$ is as large as possible, and we define $S^{'}$ to be $V (T_{w}) ∖ {w}$ . (Recall that ${deg}_{G} (u) ⩾ 2$ , hence such a child always exists.) Moreover, we set $S\coloneqq∅$ for convenience.

As noted earlier, if 3 applies then $T_{w}$ is a star. Moreover, if $S \neq \emptyset$ , then one can see that $W_{G} (w) < W_{G} (u)$ and hence $C_{1} (w; G) > C_{1} (u; G)$ . However, this is not a contradiction since $C_{1} (u; G) = max {C_{1} (v; G) : v \in A_{0}}$ and $w \in A_{1}$ .

Regardless of whether 3 or 4 applies, . Actually, it is important to notice that, in $G^{'}$ , no child of $u$ different from $z$ has more than $⌊ n / 2 ⌋ - 1$ children itself. Even more, for any such child $x$ we know that $| V (T_{x}) | ⩽ ⌊ n / 2 ⌋$ . This follows from our previous remark if $T_{x}$ has depth at most $2$ , and from the fact that $| V (T_{x}) | ⩽ | V (T_{z}) |$ otherwise. Also, setting $R\coloneqqV∖V(Tz)∪V(Tw)$ , we observe that for every node $p_{i} \in P$

{dist}_{G} (p_{i}, x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} {dist}_{G} (u, x) - 2 & if x \in V (T_{p_{i}}) {dist}_{G} (u, x) + 2 & if x \in R \cup V (T_{w}) {dist}_{G} (u, x) & otherwise. \end{matrix}

Therefore, $W (p_{i}) ⩽ W (u) - 2 (∣ ∣ V (T_{p_{i}}) ∣ ∣ - (| R | + | V (T_{w}) |))$ . Since the definition of $u$ implies that $W (p_{i}) ⩾ W (u)$ , it follows that the size of $V (T_{p_{i}})$ is at most $⌊ n / 2 ⌋$ .

Figure 5: Obtaining

G^{'}

from

G

. Recall that

S′\coloneqqV(Tw)∖{w}

S = \emptyset

Note that $G^{'}$ is a tree, which we see rooted at $u$ , and $G$ and $G^{'}$ have the same node set, which we call $V$ . In addition, $G$ and $G^{'}$ have the same bipartition $(A_{0}, A_{1})$ . Our next task is to compare the total distance of nodes in $G$ and in $G^{'}$ , that is, we compare $W_{G} (x)$ and $W_{G^{'}} (x)$ . For readability purposes, let us set $W(x)\coloneqqWG(x)$ , $W′(x)\coloneqqWG′(x)$ , and let $T_{x}^{'}$ be the subtree of $G^{'}$ rooted at $x$ . We now make a few statements about $W (x)$ and $W^{'} (x)$ for various nodes. We shall often use that

Lemma 5.

The following hold.

If $x \in R$ , then $W (x) - W^{'} (x) = 2 | Y |$ .
If $x \in {z} \cup P^{'}$ , then $W^{'} (x) ⩾ W (x) - 2 | S |$ .
If $x \in {w} \cup S^{'}$ , then $W^{'} (x) = W (x) + 2 | S | - 2 | Y |$ .
If $x \in P \cup S$ , then $W^{'} (x) ⩾ W (x)$ .
If $S \neq \emptyset$ , then $W (x_{1}) > W (x_{2})$ and $W^{'} (x_{1}) > W^{'} (x_{2})$ whenever $x_{1} \in P^{'}$ and $x_{2} \in S^{'}$ .
If $x \in Y$ , then $W^{'} (x) ⩽ W (x)$ .
$W^{'} (x) ⩾ W^{'} (u)$ for every node $x \in Y \cup R \cup S^{'} \cup {w}$ .

Proof.

We prove all the statements in order.

1. If $x \in R$ , then the distance from $x$ to any node not in $Y$ is unchanged. In addition, ${dist}_{G^{'}} (x, y) = {dist}_{G} (x, y) - 2$ whenever $y \in Y$ , hence the conclusion.

2. If $x \in {z} \cup P^{'}$ , then ${dist}_{G^{'}} (x, v) ⩾ {dist}_{G} (x, v)$ for each $v \in V ∖ S$ . In addition, if $s \in S$ , then ${dist}_{G^{'}} (x, s) = {dist}_{G} (x, s) - 2$ , which yields the conclusion.

3. It suffices to observe that if $x \in {w} \cup S^{'}$ , then

{dist}_{G^{'}} (x, v) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} {dist}_{G} (x, v) & if v \in V ∖ (S \cup Y) {dist}_{G} (x, v) - 2 & if v \in Y {dist}_{G} (x, v) + 2 & if v \in S . \end{matrix}

4. First note that if $x \in P$ , then the definition of $G^{'}$ ensures that ${dist}_{G^{'}} (x, v) ⩾ {dist}_{G} (x, v)$ for each $v \in V$ , which implies that $W^{'} (x) ⩾ W (x)$ .

Now let $x \in S$ . Observe that if $v \in V$ , then ${dist}_{G^{'}} (x, v) ⩾ {dist}_{G} (x, v) - 2$ . In addition, if $v \in S^{'} \cup {w}$ , then ${dist}_{G^{'}} (x, v) = {dist}_{G} (x, v) + 2$ . Consequently,

W^{'} (x) - W (x) ⩾ 2 ∣ ∣ S^{'} \cup {w} ∣ ∣ - 2 ∣ ∣ V ∖ ({x, w} \cup S^{'}) ∣ ∣,

which is non-negative since when $S \neq \emptyset$ , and $x \notin S^{'} \cup {w}$ .

5. Let $x_{1} \in P^{'}$ and $x_{2} \in S^{'}$ . First note that every node in $V (T_{w}) ∖ {x_{1}}$ is two units closer to $x_{1}$ than to $x_{2}$ . Similarly, every node in

Closeness Centralization Measure for Two-mode Data of Prescribed Sizes††thanks: This work was supported by PHC Proteus 26818PC, Slovenian ARRS bilateral projects BI-FR/12-13-PROTEUS-011 and BI-FR/14-15-PROTEUS-001.

Abstract

1 Introduction

Theorem 1.

Definition 2.

2 Bipartite Networks With Fixed Number of Nodes

Theorem 3.

Lemma 4.

Proof.

Lemma 5.

Proof.

Closeness Centralization Measure for Two-mode Data of Prescribed Sizes^†^†thanks: This work was supported by PHC Proteus 26818PC, Slovenian ARRS bilateral projects BI-FR/12-13-PROTEUS-011 and BI-FR/14-15-PROTEUS-001.