The concept of social stratification and hierarchy among human dates is back to the origin of human race. Presently, the growing reputation of social networks has given us with an opportunity to analyze these well-studied phenomena over different networks at different scales. Generally, a social network could be defined as a collection of actors and their interactions. In this work, we concern ourselves with a particular type of social networks, known as trust networks. In this type of networks, there is an explicit show of trust (positive interaction) or distrust (negative interaction) among the actors. In a social network, actors tend to connect with each other on the basis of their perceived social hierarchy. The emergence of such a hierarchy within a social community shows the manner in which authority manifests in the community. In the case of signed networks, the concept of social hierarchy can be interpreted as the emergence of a tree-like structure comprising of actors in a top-down fashion in the order of their ranks, describing a specific parent-child relationship, viz. child trusts parent. However, owing to the presence of positive as well as negative interactions in signed networks, deriving such “trust hierarchies” is a non-trivial challenge. We argue that traditional notions (of unsigned networks) are insufficient to derive hierarchies that are latent within signed networks.
Structural analysis of complex networks has been a dynamic and challenging area of interest among researchers for the past few decades [
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 [
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 [
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 explicit show of trust or distrust among actors. They can designate others as friends or foes [
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 depicting a specific relationship. Earlier studies on dominance relationship in animal societies, Bonabeau et al. suggest a process of self-organization of nodes depending on their roles and importance [
In 1984, Huseyn et al. [
Likewise, hierarchy is observed in certain types of collaboration networks too. Rowe et al. [
Liben-Nowell and E. Gilbert et al. [
Apart from these, attempts have been made lately to explore hierarchies as well. Helic D. and Strohmaier M. [
The proposed algorithm has two phase procedure to ensure that as much as the given tags are being connected to the main tree without the tree being fairly deep. In the first phase, it populates a forest of multiple trees with the most general node as the root node, iterates through the centrality list, identifies the most similar tag to the current tag in the tree computing the co-occurrence threshold and then appends the tag as a child to its most similar tags. It attaches a maximum of subcategories to a given category. Later the produced trees are sorted in descending order of their size (no: of categories they possess) and the largest tree is considered as the main tree. In the second phase, the algorithm appends the other trees to the main tree by connecting the root node of a particular tree to the most similar node in the main tree. In case the most similar mode consists of only one free sub-category spot, then a misc category is introduced into the free spot and then the given tree is appended to that misc category. However, the nesting of misc category is also necessary and cannot be avoided completely due to the very structure of tag-tag networks. Normally, in a typical power-law network, the nodes with high degree centrality are connected to a small number of high and mid degree nodes (high centrality) as well as to a large number of low degree nodes (low centrality). Such high centrality tags occupy the top positions of the hierarchy. Therefore, in the hierarchy building process, the algorithm first appends the adjacent high degree and the mid degree nodes as sub-categories to a given node using up all free sub-category spots followed by the addition of the low degree nodes through the misc category. It is to be noted that nested misc categories do not affect the semantics of the network but rather keep the tags away from the most related ones into its misc categories. The results and simulation studies illustrates that the proposed algorithm outperforms existing ones in constructing a tag hierarchy useful for better navigation.
Maiya and Berger-Wolf [
Gupte et al. [
In 2012, the problem has been examined and a universal hierarchy measure has recently been put forward by Mones et al. [
The concept of Global Reaching Centrality (GRC) [
GRC = ∑ i ∈ V [ C R max − C R ( i ) ] N − 1 (1)
V = set of nodes; C R ( i ) = local reaching centrality; C R max = highest local reaching centrality; N − 1 = maximum traversals possible.
For weighted undirected graphs, the generalization of GRC is quite straight- forward based on local reaching centrality as defined for unweighted direct graphs. In the case of weighted directed graphs, the sum of lengths of all out- going directed paths from node ‘i’ to node ‘j’ as well as the weight of edge along the path is taken into account. If there exist more than one directed shortest path from i to j, then the path with maximum weight (i.e. maximum connection strength) is considered. Similarly, for an undirected unweighted graph, GRC can be obtained by excluding computation of weights of shortest path between two nodes. Further, GRC is observed on an adjustable hierarchy (AH) model. In an AH model, all nodes in a directed tree is assigned to a level ‘l’ such that the level of the root node is equal to the total number of levels and those at the bottom level has l = 1. If a node has a level l, then the level of the child nodes would be l − 1. Thereafter an additional no: of random edges are included in the model in such a way that 1 − p proportion of edges is totally random. That is, two nodes chosen, say A and B, are connected if they were not already connected in the (AB) direction. The p proportion of the edges are connected as (AB) only if, to preserve hierarchy. Randomization of real networks is done by generating a random network with the same in and out degree with respect to the original model and followed by choosing two random edges AB and CD and then changing the endpoints to obtain AD and CB.
Analysis on a few classical networks such as Erdős-Rényi (ER) graphs [
The proposed hierarchical visualization technique for large graphs assigns each node into different levels on the basis of a local quantity. For an unweighted digraph this local quantity is equal to the local reaching centrality. Therefore, an ER graph posses a two layered hierarchical structure and arborescence has many layers. The structure of an SF graph lies in between an ER graph and an arborescence with a few clearly separated layers. To avoid different hierarchical lay-outs for single graphs of same graph model, ensembles of ER, SF, directed AH and real networks are visualized. In short, the proposed hierarchy measure, GRC quantifies the heterogeneity of local reaching centrality in whole network by introducing bidirectional edges among equivalent nodes. It is free from the drawbacks of the hierarchy measures so far been suggested. Hence, it can be concluded that GRC is a more suitable measure for hierarchy in any network.
Networks could be of different types. Some of them include:
1) Physical networks comprising of physical entities and their interactions. Examples could be road network, world-maritime network etc., where cities/ ports are nodes and their routes are links,
2) Biological networks like protein-interaction networks, gene-regulatory network where proteins/genes form the nodes and their interactions form links,
3) Social networks where people or other entities become the nodes depending on the social context and their interactions being links.
Social network, where people represent nodes and the relationship between them represents links. A Link can be either directed (e.g. twitter where relationship is directional) or undirected (e.g. Facebook where relationship is mutual). Physical networks like Road network, World maritime network. Here nodes are represented by Cities and Ports respectively, and links are represented by routes. Biological networks: One of the examples is Protein-Protein interaction network where each protein is a node and interaction between them is represented by a link.
Among all the networks our focus is mainly on social networks. It shows some different properties in compare to other networks present like internet, World Wide Web. A network can be categorized into Infrastructure network and Interaction network. In infrastructure networks a link can be established even if the nodes do not interact with each other. Typical examples of infrastructure networks are Facebook, Orkut, Twitter, and etc. In Interaction network a link is setup among nodes when they interact with each other. Typical examples are Protein-Protein interaction network, Slashdot social network which is a result of interaction between nodes due to the threaded discussion. One major difference between the two networks is a link may lose its importance during the course of time in Infrastructure network i.e. a link might languish (or in other words stay static). For example, if one does not interact with a person on a regular basis then the link which connects both of them loses its importance with time. But in interaction network a link never loses its importance with time, as the nodes continue to interact with each other regularly.
A network can also be classified as signed and unsigned networks. In Unsigned networks, link between the nodes doesn’t say about the nature of the link. Online social networks like Facebook, twitter, friendster, and etc come under this category. As opposed to unsigned network, in signed network a link carries +1 sign which represents a positive relationship or −1 sign which represents negative relationship among nodes. Both signs can be interpreted differently in different networks. For example in Eopinions network, +1 represents Trust while −1 represents Distrust, while in Slashdot Zoo network +1 represents friendship and −1 represents Foe ship between people.
Studies so far reveal only certain typical statistical properties shared by most of the complex networks. Some of distinctive properties include small-world phenomena [
Connectedness is a property exhibited by all networks and it determines the arrangement of nodes within a network. Such an arrangement gives rise to different classes of nodes based on certain factors that serve as a measure. In online social networks, actors tend to connect with each other within and across different classes on the basis of their perceived social hierarchy. The concept of social hierarchy can be stated as the emergence of a tree-like structure comprising of actors in a top-down fashion in the order of their ranks, describing a specific parent-child relationship. The total prestige owned by an actor could be considered as a measure of status. Therefore, a social hierarchy conveys a structure of authority and could be latent in every social network and needs to be extracted. Different literatures present a variety of approaches and measures for mining hierarchy in complex networks. Attempts have also been made to mine hierarchy in social networks. These are further discussed in the related literature section. However, in signed networks the hierarchy is far less discernible. The presence of negative interactions in signed networks, pose an additional challenge in deriving trust hierarchies from signed networks. Hence, we argue that the traditional notions are insufficient to derive hierarchies underlying signed networks.
In order to extract hierarchies from signed networks, we have considered the Slashdot and Epinions networks [
In this work, we attempt to mine hierarchies that remain latent in a signed network that represents the trust of nodes from the bottom to the root. It also based on a node’s immediate neighborhood of trust relationships. Therefore, the trust hierarchy shows the nature of nodes trusting each other and at the same time preserves the locality of trust. These hierarchies are termed as locality-preserving trust hierarchies.
Being highly dynamic in nature, social networks have always reflected interesting patterns of connections among the nodes. These connections mostly lead to a parent-child relationship forming hierarchies among themselves. The hierarchical structure of a population in a social network often shapes the nature of the social interactions of individuals and, thus, provides insights into the underlying structure of the network. Understanding the mechanism by which hierarchies evolve is a fundamental question that still remains vague. Our approach could be relevant to a number of interesting current applications of social networks including information dissemination, community structure detection and a framework for local self-governance among the population. The crux of our work lies in the fact that we seek to mine hierarchies based on the trust locality of a node in a signed network. That is, the hierarchies should be an abstract portrayal of local community structure.
As discussed earlier, owing to the presence of positive (trust) as well as negative (distrust) interactions in signed networks (trust networks), the traditional notions of hierarchy were found to be inadequate to derive trust hierarchies. With the purpose of modeling both these interactions effectively into a hierarchy, we introduce two interpretations of trust or goodness into the trust networks. Trust is represented in terms of two different aspects namely, presence of trust and absence of distrust. In fact, these two interpretations could be considered as duals of trust signifying the degree of goodness of an actor. Presence of trust would imply how good an actor is where as an absence of distrust would imply how less bad the actor actually is. Consequently, the trust-based hierarchies thus obtained would consist of several actors arranged in the order of their degree of trust. This could be illustrated in
in turn indicating low badness when compared to those at the bottom of the hierarchy.
Alternatively, it could be viewed that ‘goodness’ decreases as we move down a trust hierarchy and ‘badness’ increases as we move down the distrust hierarchy. This view puts forward a question of the manner in which an actor is considered to be genuinely good or bad. The philosophy behind this view could be explained in terms of the collective opinion of the population. Social networks comprise of autonomous agents capable of expressing opinions on their own. These opinions solely are based on their independent cognitive processes or inferences. In other words, the opinion of an actor is not hampered by any party or an interest group in particular. Therefore, a collective opinion regarding the trustworthiness of an actor cannot be ruled out as a co-incidence. With time, an architecture entirely based on trust emerges. This emergent trust-based architecture eventually becomes acceptable to the whole population. Thus an actor who has earned the trust (distrust) through the unanimous opinion of the majority is considered to be genuinely trustworthy (untrustworthy). By means of this emergent architecture it is possible to gain new insights into patterns underlying a network. An interesting example in this regard could be the collaborative editing of content in Wikipedia pages. Readers are allowed to edit information related to a particular topic and over time, an information architecture evolves eventually reaching consensus among the editors.
However, so as to convey both trust and distrust effectively in a single hierarchy, the trust as well as the distrust earned by an actor are taken together in terms of their aggregate deserve values. That is, deserve of node u is the aggregate of the trust and distrust it earns from its neighbors. The trust/distrust from node v to node u is dampened based on its bias towards trusting or distrusting the population at large. Therefore, the higher the bias, the lower is the effect of v’s vote to u. Thus a consolidated hierarchy of actors is formed by way of a parent-child relationship, viz. child trusts parent. Here, the actors are arranged into different levels according to the deserve values within their neighborhood of trust. That is, the consolidated hierarchy not only portrays the aggregate trust earned but also preserves the locality of trust of an actor. Therefore, the hierarchy thus obtained is said to represent a locality-based social structure in the descending order of aggregate deserves.
The primary objective of our work was to mine locality-preserving trust hierarchies from signed networks. We discussed the different approaches adopted to mine hierarchies in complex networks in various fields ranging from sociology, biology to computer science. In addition to this, we explained why, unlike other networks, mining hierarchies in signed networks is novel. In trust-based networks, there is an explicit show of trust and distrust, very similar to real-world interactions. As a result, a trust-architecture evolves giving rise to an underlying social hierarchy. Then, we proposed the two interpretations associated with trust in signed networks and observed the nature of hierarchies derived from the above assumptions.
A future line of work could be a method for merging of hierarchies in order to arrive at the desired(smaller) number of hierarchies, without compromising on the locality-preservation to examine the patterns underlying it. Implementing other hypotheses relevant to locality-preserving hierarchy construction based on the application context. On a similar note, we plan to extend our approach and also analyze other signed networks such as Essembly etc.
Maktoubian, J., Noori, M., Amini, M. and Ghasempour- Mouziraji, M. (2017) The Hierarchy Structure in Directed and Undirected Signed Networks. Int. J. Communications, Network and System Sciences, 10, 209-222. https://doi.org/10.4236/ijcns.2017.1010012