Complexity

Volume 2017, Article ID 3203615, 11 pages

https://doi.org/10.1155/2017/3203615

## Optimization of the Critical Diameter and Average Path Length of Social Networks

^{1}Center for Administration and Complexity Science, Xi’an Jiaotong University, Xi’an, Shanxi Province 710049, China^{2}Morrison Institute for Population and Resource Studies, Stanford University, Stanford, CA 94305, USA

Correspondence should be addressed to Marcus W. Feldman; ude.drofnats@namdlefm

Received 29 November 2016; Accepted 13 February 2017; Published 28 March 2017

Academic Editor: Katarzyna Musial

Copyright © 2017 Haifeng Du et al. 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

Optimizing average path length (APL) by adding shortcut edges has been widely discussed in connection with social networks, but the relationship between network diameter and APL is generally ignored in the dynamic optimization of APL. In this paper, we analyze this relationship and transform the problem of optimizing APL into the problem of decreasing diameter to 2. We propose a mathematic model based on a memetic algorithm. Experimental results show that our algorithm can efficiently solve this problem as well as optimize APL.

#### 1. Introduction

Following the introduction of models for small-world and scale-free networks, much research has been devoted to analyzing network characteristics [1–5]. In particular, there has been a focus on finding indices to quantify features of network structure such as structural entropy, robustness, or modularity [6–8]. These indices play an important role in measuring specific performance aspects of networks, and optimizing them can help to improve network performance.

Average path length (APL), the average shortest distance between all nodes in a network, is not only a measurement of static characteristics such as connectivity and robustness but also an important control variable in dynamic processes, such as the spread of diseases or target searching [9–11]. Optimizing APL has also attracted attention in the field of structural optimization. Decreasing APL by adjusting nodes or edges can effectively enhance the transfer efficiency and synchronization ability [12–17]. In addition, optimization of APL has also been widely used in urban planning and site selection [14, 18, 19]. Xuan et al. [20] proposed a simulated annealing model to optimize APL in order to speed up convergence. Keren [21] employed a spectral technique to reduce APL in binary decision diagrams.

In order to optimize APL, many scholars focus on adding a given number of edges to produce the largest decrease in APL. These added edges are called “shortcut” edges and the problem of finding the best set of shortcut edges is defined as the “shortcut-selection” problem [22]. A series of methods have been proposed to solve this problem. Meyerson and Tagiku proposed an approximation method, which involved finding a source node and then connecting other nodes to this node to decrease APL [22]. Parotsidis et al. analyzed the exact effect of a single edge insertion on APL and proposed the EdgeEffect Algorithm to maximize the effect of edge insertion [23]. A greedy algorithm, which adds edges one by one and which makes the maximum reduction of APL for each added edge, has proved to be efficient [24, 25]. These methods have solved the shortcut-selection problem to some extent. However, a common phenomenon has been ignored in the process of adding edges. In experiments to optimize APL by adding edges, we find that no matter which method is used to add edges, there always exists a turning point at which APL begins to decrease linearly as more edges are added. This phenomenon can be related to the network diameter.

In this paper, we define the network diameter at the turning point as the “critical diameter,” and analyze both this critical diameter and APL in the process of adding edges. We transform the problem of optimizing APL into the problem of optimizing the critical diameter. Specifically, we focus on adding the minimum number of shortcut edges to make the network diameter decrease to 2. Research on predicting missing links has attracted much attention in recent years, the algorithms of which can extract missing information or identify spurious interactions [26–29]. Gao et al. analyzed the feature of predicted network, and they found the network diameter and APL shows a negative linear relation to all of the tested prediction methods [29]. Therefore, our research can also provide some a priori knowledge in designing the method of link prediction. In the next section, we introduce the critical diameter and explore the special relationship between critical diameter and APL; the algorithm for optimizing the critical diameter is proposed in Section 3; Section 4 gives results of testing our method on generated networks; our conclusions and further work are presented in Section 5.

#### 2. Critical Diameter and APL

Network diameter, the maximum path length for all pairs of nodes, is closely related to APL; they both contain information about connectivity and transfer efficiency [30, 31]. Imase and Itoh gave the inequalities to describe the static relationship between network diameter, , and APL [32]. In the dynamic process of adding shortcut edges, there exists a turning point, as shown in Figure 1. APL declines nonlinearly with the number of added edges until a turning point and then decreases linearly as increases further. We compute the path length between every pair of nodes and find that the longest path length of the network is larger than 2 before reaches the turning point (i.e., the network diameter ); when reaches the turning point, the diameter equals 2 (). This is because if , a new added edge between a pair of nodes can only change the path lengths between these two nodes from 2 to 1 but cannot change the path lengths of other pairs of nodes, and the APL can be reduced by just for each added edge, which constitutes a linear decline.