Social network analysis (SNA) basically is more interested on the arrangement of ties found in a group of actors who include; groups, individuals, nations, and organization of the stuffs resulting from the products of human activities (Liljeros et al, 2011). They also emanate from the cognition of webs such as the semantic concepts, websites, and others. Any social system or process which can be categorized as a group of units or any group of lines which connect pairs of units can be analyzed as a social network. Most of the social structures which have been emphasized in the studies as networks include the friendship found among the children in a school facility, hyperlinks between the websites, family relations which exist between members of a social elite, members sharing the same board of corporations and the traditional relation which exist between certain countries as substantiated by Liljeros et al 2001.
Problem statement
With the use of a modelling perspective, a network is depicted as a relatively simple object which mainly consists of nodes and links the real problem is mainly to have the links placed in a manner that they will reproduce the complexity and the apparent randomness of the systems that are real as substantiated by Newman, 2001. Networks are considered to be ever-present in our universe and most of the times it is difficult to have the network classified into a single grouping because they are interconnected. According to Newman, 2002, this happens to be the reason as to why the network theory has become an interdisciplinary research field and maybe the main reason as to why most of the networks do share similar generic characteristics. For instance there is the World Wide Web which would be viewed as a communication system or even as a social network. The availability of the electronic databases tends to establish many complex networks, studies on the serving experimental network, whereby all ought to be modelled via a theoretical study (Scott, 2017).
An important goal of network science is to come up with models which in an accurate manner reproduce characteristics of networks that are real as observed in systems that are real. The many networks that are encountered in nature tend not to have the regularity of comforts from the crystal lattice or the radial architecture of the spider web that is predictable. This research tends to critically compare and contrast, assess the random Poisson network model and the scale-free network model which were proposed by Albert and Barabasi.
Critical analysis of the theoretical background
The random Poisson network model
The random model is a theory that tends to embrace the feature of randomness via constructions which happens to be truly random. The model uses the random graph theory which has attested crucial while the internet is modelled and social networks are being constructed. In its formation, it has three major assumptions which include; the configuration of a given periphery tend to be self-regulating of the of other edges being formed, a chart exhibiting at most the edge existing between every two nodes, and lastly all the edges tend to form with a similar probability. A general agreement has been established showing that the model happens to be too rigid for it to capture the many networks that exist in the real world. There are surveys that elaborate more on the applications of two generation of scholars emphasizing more on phase transitions, power laws, and the scale-free networks. Whenever the appropriate alternatives assumptions are in place, any person is capable of deriving and illustrating the likelihood algorithm of a novel maximum to estimate the model parameters. Whereas these parameters are at hand, it is easier to discover statistically major linkage between any pair of nodes (Strogatz, 2001).
In real real life application, most of the graphs are basically derived from the multi-graphs. To have any analysis simplified, the multiple edges which are found between the two nodes in a multi-graph ought to be collapsed to a single edge. An example is the star graph movie where two players tend to be interlinked using an edge whenever they emerge in the exact movie some of the performer pairs are likely to emerge in a movie basically by probability. The other performer's pairs happen to be interlinked with many edges basically because they are essentially linked.
An observed potency of the replica is that it allows statistical significance appraisal whereby it assists in distinguishing functional and random connectivity. Stressing the investigative characteristics of Poisson multi-graph, its main reason is to basically investigate larger datasets for the structures which are hidden. Identification of the nodes pairs and the hub nodes with more edges happen to be the main goal (Wang, 2006).
Scale-free network model
A scale-free network is a network that has a degree distribution following the power law at least asymptotically. There are many networks which have classified to be scale-free though the statistical analysis always refutes most of them while questioning others. The scale-free models, on the other hand, are used to create the scale-free network (Ravasz " Barabási, 2003). Most of these models introduce mainly two ingredients which include preferential attachment and growth. Preferential attachment basically refers to the fact that new nodes connect to those nodes that have a large degree whereas growth depicts the notion that those nodes in the network system tend to increase in time.
Characteristics of the scale-free network
The main characteristics of a scale-free network happen to be the relative commonness of vertices which have a degree that exceeds the average greatly. The nodes which have the highest degree are known as hubs and they tend to serve a specific purpose in their networks through this depends mainly on the domain.
The property for the scale-free strongly correlates with the robustness of the networks to fail. The protocol is that the major hubs mostly are followed by those that are small. Thereafter, the smaller hubs tend to be followed by the other nodes which have even a smaller degree. The hierarchy created makes it possible for the fault tolerant behavior. Whenever a failure is experienced at random with the vast majority of the nodes being the ones with smaller degrees, the chances of the hubs being affected is considered to be nil or negligible. If there are chances that a hub failure occurs, the network generally does not lose its connectedness and this is because of the remaining hubs. Additionally, if a few hubs are taken out of the network, then the network is changed into a set instead of an isolated graph as substantiated by Liu et al, 2011.
According to Jeong et al, 2003 hubs basically are the strengths and the weaknesses of scale-free networks. Their properties have mainly been studied analytically by the use of percolation theory by Cohen and Callaway. Cohen proved that for every broad range of scale-free networks, the critical percolation threshold is p_c=0. The meaning to this is that removal of any fraction of nodes from the network will lead to the destruction of the network.
Another characteristic of the scale-free network that seems to be important is the clustering coefficient distribution which tends to decrease when the degree of the nodes is decreasing. The above distribution also happens to follow the power law. It is an implication that the nodes which have a low degree are comprised in the sub-graphs which are very dense and the sub-graphs are then connected to each other via the hubs. When a social network is considered whereby links are acquaintance relationship which exists between people and nodes are people, it becomes easier to note that people mostly from communities. These are smaller groups whereby each person knows everyone as substantiated by Dezső " Barabási, 2002. Additionally, the members of the community tend to have few acquaintance relationships to those people who are outside of that community. However, there are people who are connected to the larger number of the population such the politicians and musicians. These celebrities can be considered as the hubs necessary for the small world phenomenon.
According to Davidsen et al, 2002, the current world, the scale-free network’s characteristics which are more specific tend to differ with the generative apparatus which is to create them. For example, those networks that are generated by superior attachment tend to lay their vertices which are of high degree in between the network, which connects them formulate a core using the progressively low degree of nodes, thus, forming up the regions among the periphery and the core.
A final characteristic is depicted on the standard distance which exists among the two vertices which exist in any network. When the messy networks including the small world network model are put into consideration, the distance tends to be very small when compared to the networks which are highly ordered such as the lattice graph (Carrington et al, 2005).
Usage of the models
The models have frequently been used in the real world and they have been bringing accurate results. An example of this is the modeling of lattice cellular networks with the use of the Poisson process. This was initiated due to the extensive deployment and the upgrading of the system for it cope with rising user traffic. During the development of the model Poisson network gave provision to log-normal shadowing which was sufficiently high and good approximation was obtained for the standard deviation of the shadowing as substantiated by Bornholdt " Ebel, (2001).
Analysis and evaluation of results
As observed above, the properties illustrated by the scale-free networks tend to have a slight difference from the strictly random. According to Barabasi " Oltvai, 2004, random networks basically resemble the highway system of the US where it consists of nodes which have randomly placed connections. In those connections, a plot of the distribution of the linkages by the nodes happens to follow a bell-shaped curve while most of the odes taking approximately a similar number of links. In contrast, the scale-free networks have a resemblance to the US airline system whereby they contain hubs nodes which tend to have a large amount of links. In this form of a network, the distribution of nodes linkages happens to follow the power law whereby most of the nodes will entail a few tremendous numbers of the existing links.
The information above shows that the scale-free model and the Poisson random The illustration above shows that the scale-free networks tend to be forceful against failure which is not the case in the random networks. The network in the scale-free is capable of staying connected when compared to the random network after the randomly chosen nodes have been removed. Also, the scale-free networks tend to be more vulnerable when compared to the other networks when it comes to the non-random attacks. Unlike the random network, the scale-free network is more likely to disintegrate whenever the nodes have removed on the basis of their degree. When the scale-free and the pure random models are compared, the scale-free tends to have shorter average path lengths whereas that of the Poisson random models is longer as substantiated by Albert " Barabási, (2002).
Conclusion
In collusion, the models illustrated shows complete differences in the way they are used and their characteristics. Despite their differences, they are both useful in the real world situation due to their different strengths. Where one of the models cannot be used, typically the other model can be relied. They might have some disadvantages but they tend to offer accurate information when used in the real world situation.
List of references
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