Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7420281, 9 pages

https://doi.org/10.1155/2017/7420281

## Quasi-Closeness: A Toolkit for Social Network Applications Involving Indirect Connections

School of Securities and Futures, Southwestern University of Finance and Economics, Chengdu, China

Correspondence should be addressed to Yahui Liu

Received 11 March 2017; Accepted 11 May 2017; Published 6 August 2017

Academic Editor: Yakov Strelniker

Copyright © 2017 Yugang Yin and Yahui Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We come up with a punishment in the form of exponential decay for the number of vertices that a path passes through, which is able to reconcile the contradictory effects of geodesic length and edge weights. This core thought is the key to handling three typical applications; that is, given an information demander, he may be faced with the following problems: choosing optimal route to contact the single supplier, picking out the best supplier between multiple candidates, and calculating his point centrality, which involves indirect connections. Accordingly, three concrete solutions in one logic thread are proposed. Firstly, by adding a constraint to Dijkstra algorithm, we limit our candidates for optimal route to the sample space of geodesics. Secondly, we come up with a unified standard for the comparison between adjacent and nonadjacent vertices. Through punishment in the form of exponential decay, the attenuation effect caused by the number of vertices that a path passes through has been offset. Then the adjacent vertices and punished nonadjacent vertices can be compared directly. At last, an unprecedented centrality index, quasi-closeness, is ready to come out, with direct and indirect connections being summed up.

#### 1. Introduction

Among many social network analysis (SNA) methods, point centrality has received particular focus from network researchers. Point centrality undertakes the task of identifying important and insignificant actors, which is the key application in graph theory. As early as 1934, Moreno tried to distinguish “stars” (people who drive more attention in a network) and “outsiders” (people who are neglected by others in a network) quantitatively [1]. Over the years, network researchers have developed many kinds of centrality variants [2, 3]. Measuring centrality in various aspects, these indices have been proved to be of great value in understanding the roles of vertices in networks [4]. Numerous efforts have been made to classify centrality indices. In an influential research, Freeman [5] gave more importance to degree [5–7], closeness [8–10], and betweenness [11, 12]. Along with eigenvector centrality [13], the four indices have become the most famous ones in measuring the centrality of a point.

In SNA terminology, geodesic refers to the shortest path between a given pair of vertices, and there may exist more than one geodesic. To some extent, geodesic is the most effective way for a vertex to communicate with other vertices. A number of centrality indices based on shortest path are widely used, such as closeness, betweenness, and Harmonic centrality [14] to deal with unconnected graphs (graphs with vertex having no path to any others). As a rapid expanding interdisciplinary field, SNA has encountered a variety of unprecedented application contexts, which makes the existing centrality indices powerless. Theoretically, there is no centrality index taking direct and indirect connection (if the geodesic distance between a given pair of vertices equals 1, they are directly connected; otherwise they are indirectly connected) into account in undirected valued graphs. Specifically, strength centrality is able to deal with valued graphs but loses indirect connection information. Closeness and betweenness involve direct and indirect connections; however they are not feasible in valued graphs.

We propose a quasi-closeness centrality index which is capable of being applied for valued graphs. Bavelas [15] and Leavitt [16] showed that distance is positively correlated with communication efficiency, which means that transition fades with distance. Following this thread, many centrality indexes based on shortest path are proposed by researchers. In line with other shortest-path centrality indices, our index is based on geodesic as well. We assume that, given a pair of unordered vertices in an undirected valued graph, the optimal path for information spreading or communication must be a geodesic. This premise is born from the idea that information attenuation caused by longer distance is serious enough to neutralize the information augment arising from more weight.

The centrality algorithm we propose can be divided into three parts and corresponds to three applications. Firstly, inspired by the spirit raised by Beauchamp [17] that geodesic distance can seriously affect the communication efficiency; we propose an algorithm which is different from the Dijkstra’s [18] and allows for the effect of geodesic distance. The algorithm is applicable in finding the optimal route between given vertices, which is a common situation in network analysis.

Secondly, as is frequently the case, an information searcher needs to choose the optimal supplier among many adjacent or nonadjacent candidates. In line with Hubbell [19] and Friedkin [20] who noted that evaluating the importance of a node in a network needs both direct connections and indirect connections, we advocate that indirect connections should also be considered. To achieve this, we come up with a unified standard to compare the relative importance of adjacent vertices and that of nonadjacent vertices and then get the priority order of all candidate information suppliers. Specifically, we set penalty in form of exponential decay for geodesics distance, and the decay indices are flexible values depending on concrete applications. This unified standard can also be applied to find the optimal partner in a network.

Finally, by summing up the exponential decayed indirect connections and direct connections, we can get an unprecedented point centrality index, namely, quasi-closeness centrality. This centrality takes both indirect and direct connections into account and is applicable for weighted networks, such as the citation network, biological network, or logistics network.

The rest of the paper is organized as follows. Section 2 gives some preliminary notions closely related to our theme. Section 3 describes the algorithm. Section 4 describes simulation results. Section 5 concludes.

#### 2. Preliminaries

Following Bollobás [21] and Diestel [22], we use standard graph theory terminology as follows.

An undirected graph consists of a vertex (also called node) set and a set of undirected edges (also called links or connections). An edge represents the tie between a pair of disorder vertices. If there is an edge between vertices and , we say that and are adjacent. And the graph is connected if every pair of vertices is linked by a path.

Given a pair of vertices in a connected graph, if they are adjacent, the distance between them equals one. Otherwise they must be linked by other vertices. In this paper we focus on geodesic between vertices. A geodesic is defined as follows.

In Figure 1, vertices 1 and 5 are not adjacent, but they are linked through other vertices. Geodesic is defined as the shortest path; here it refers to path (1, 2, 4, 5), rather than (1, 2, 3, 4, 5). Let denote the geodesic distance or simply the distance between vertices 1 and 5, which is the number of edges; then *.*