Int. J. Communications, Network and System Sciences, 2017, 10, 209-222 http://www.scirp.org/journal/ijcns ISSN Online: 1913-3723 ISSN Print: 1913-3715 The Hierarchy Structure in Directed and Undirected Signed Networks Jamal Maktoubian1*#, Mohebollah Noori2, Mahta Amini3, Mehran Ghasempour-Mouziraji4 1International School of Information Management (ISIM), University of Mysore, Mysore, India 2Zarghan Branch, Islamic Azad University, Zarghan, Iran 3Department of Computer Science, Shahid Beheshti University, Tehran, Iran 4Department of Engineering, Islamic Azad University of Sari, Sari, Iran How to cite this paper: Maktoubian, J., Abstract Noori, M., Amini, M. and Ghasempour- Mouziraji, M. (2017) The Hierarchy Struc- The concept of social stratification and hierarchy among human dates is back ture in Directed and Undirected Signed to the origin of human race. Presently, the growing reputation of social net- Networks. Int. J. Communications, Network works has given us with an opportunity to analyze these well-studied pheno- and System Sciences, 10, 209-222. https://doi.org/10.4236/ijcns.2017.1010012 mena over different networks at different scales. Generally, a social network could be defined as a collection of actors and their interactions. In this work, Received: August 30, 2017 we concern ourselves with a particular type of social networks, known as trust Accepted: October 8, 2017 networks. In this type of networks, there is an explicit show of trust (positive Published: October 11, 2017 interaction) or distrust (negative interaction) among the actors. In a social Copyright © 2017 by authors and network, actors tend to connect with each other on the basis of their perceived Scientific Research Publishing Inc. social hierarchy. The emergence of such a hierarchy within a social commu- This work is licensed under the Creative Commons Attribution International nity shows the manner in which authority manifests in the community. In the License (CC BY 4.0). case of signed networks, the concept of social hierarchy can be interpreted as http://creativecommons.org/licenses/by/4.0/ the emergence of a tree-like structure comprising of actors in a top-down fa- Open Access shion in the order of their ranks, describing a specific parent-child relation- ship, viz. child trusts parent. However, owing to the presence of positive as well as negative interactions in signed networks, deriving such “trust hierar- chies” is a non-trivial challenge. We argue that traditional notions (of un- signed networks) are insufficient to derive hierarchies that are latent within signed networks. Keywords Unsigned Network, Signed Network, Global Reaching Centrality (GRC), Hierarchy, Social Network #First author *Corresponding author DOI: 10.4236/ijcns.2017.1010012 Oct. 11, 2017 209 Int. J. Communications, Network and System Sciences J. Maktoubian et al. 1. Introduction Structural analysis of complex networks has been a dynamic and challenging area of interest among researchers for the past few decades [1]. In a generic sense, a network is a collection of nodes associated to the other through links [2]. Several graph theoretic approaches over such networks have revealed certain fundamental facts. Evidently, network analysis could provide us with better in- sights in understanding the hidden aspects of individuals or groups involved within a network, the pattern of relationships, how they evolve etc [3]. Any net- work could be represented as a graph consisting of a collection of nodes (units) and edges (interactions) [4]. In a network, the manner in which one node inte- racts with the other displays an important feature, the connectedness among nodes. The nature of connectedness underlying a network also determines its complex topology. In other words, network complexity is an intrinsic property of any physical, chemical, biological or social system characterized by various nodes and their interactions [5]. Examples include organizational networks, neural networks, protein interaction networks, Internet, the World Wide Web and social networks to name but a few. The past decade witnessed a tremendous rise in the popularity of online social networks such as Twitter, Digg, Youtube, Delicious, Livejournal, Facebook etc. Our study mainly focuses on the analyses of similar online social networks in order to understand the underlying mechanism of the connections involved as well as to verify the existence of certain social phenomena within the networks. Broadly speaking, a social network could be directed or undirected depending on the type of edges present in them. Directed social networks are distinguished from undirected ones by the presence of directed edges between actors [6]. An example (Figure 1) for directed network could be followership in Twitter where an actor simply ‘follows’ another. Alternatively, undirected social networks comprise of undirected edges between actors. Facebook is an example for undi- rected networks with edges depicting only mutual friendships. Another type of classification termed as the trust networks deals with nature of interactions (positive or negative) involved in social networks. In this type of classification, a social network could be categorized as either signed or unsigned. Unsigned networks are described by the presence of a single type of interaction, usually being positive in nature. That is, in unsigned networks all actors are same, either friends or strangers. Generally, social networks are largely found to be unsigned in nature [7]. Followership in Twitter and friendship in Facebook Figure 1. Examples of directed and undirected network connectivity. DOI: 10.4236/ijcns.2017.1010012 210 Int. J. Communications, Network and System Sciences J. Maktoubian et al. are typical examples. But in the real world, the relationships need not always be positive in nature. Signed networks, capture this aspect of society allowing expli- cit show of trust or distrust among actors. They can designate others as friends or foes [8]. In this scenario, an actor is said to trust the other if an actor approves of one’s opinion among themselves. At the same time, an actor is said to distrust the other if an actor disapproves of one’s opinion. E-opinions, Slashdot Zoo network are some of the examples of signed networks that indicate trust/friends or distrust/foes explicitly among themselves using an edge-weight of +1 and −1 respectively. Mathematically, a signed network can be defined as a directed graph, G = (V, E) where i) V is the set of actors in a network, E ⊆ V × V is the set of edges such that (u, v) indicates a link between u ∈ V and v ∈ V s: E → {+1, −1} assigns the edge weight [9]. Consider the following illustration in Figure 2. If node A is connected to node B as a friend, there should be a directed edge from node A to node B with a trust score of +1. Meanwhile, if A is connected to B as a foe, there should be an edge directed from A to B with a score of −1. 2. Background and Prior Work Various aspects of hierarchy have been studied in many literatures till date. The general idea behind the concept of hierarchy can be stated as the emergence of a tree-like structure in a top-down fashion in the order of their ranks further de- picting a specific relationship. Earlier studies on dominance relationship in ani- mal societies, Bonabeau et al. suggest a process of self-organization of nodes de- pending on their roles and importance [10]. This lead to the identification of important or ‘leader’ nodes within a community. Such nodes occupy the higher positions in the hierarchy. Therefore, it can be argued that in a hierarchy the higher node indicates a greater influence than the lower ones. Using the direc- tional correlation function analysis, M. Nagy et al. found that similar dominance hierarchies exist in the case of pigeon flocks [11]. In 1984, Huseyn et al. [12] suggested that hierarchy is found in numerous complex systems. Hierarchical organization is also studied in different real net- works such as actor network, language network, the Internet and World Wide Web by Ravasz and Barabási in 2003 [13]. They proposed that many real net- works are scale free and transitive in nature which can be seen as a consequence of the hierarchy underlying the network. Small groups of nodes rearrange them- selves to form a hierarchy of larger groups. In order to examine the presence of hierarchical structure in real networks, they argued that the scaling law for the clustering coefficient Ck, is sufficient to quantify the existence of hierarchy of nodes [14]. Figure 2. Examples for signed network connectivity. DOI: 10.4236/ijcns.2017.1010012 211 Int. J. Communications, Network and System Sciences J. Maktoubian et al. Likewise, hierarchy is observed in certain types of collaboration networks too. Rowe et al. [15], proposed a novel algorithm to find social hierarchy in e-mail networks by introducing a social score S. This score is computed for each user as a weighted combination of several other measures including the number of e-mails exchanged. Several studies came up with different hierarchy measures that lacked universal applicability on all network types. Instead of employing different measures, the need for a single efficient measure for quantifying hie- rarchy in complex networks was inevitable. Liben-Nowell and E. Gilbert et al. [16] [17] Studied on social networks dealt with link-prediction and tie-strength prediction. They addressed the link-pre-diction issue and discuss certain achievements based on proximity measures of nodes in a network. Rather than considering the network evolution, a static snapshot of the network along with some specific node attributes are taken into study. Link-prediction can be applied to social network analysis to find out interesting or promising interactions within its members. In 2009, E. Gilbert et al. provided a predictive model for tie strength. The model effectively distinguishes between strong and weak ties with over 85% accuracy.