Comparison of the Random Network and the Scale-Free Network Proposed by Barabasi and Albert - Essay Example

? COMPARISON OF THE RANDOM NETWORK AND THE SCALE-FREE NETWORK PROPOSED BY BARABASI AND ALBERT This paper critically assesses the Random Network and the Scale-Free Network, and compares the concept of the Random Network and the Scale-Free Network, as proposed by Reka Albert and Albert-Laszlo Barabasi. Examples of Random Network and Scale-Free Networks are provided here, and the paper goes ahead to discus the areas and the extent to which these models offer an accurate representation of real-world networks, with the aid of some examples. This paper also compares random networks to scale-free networks according to the type of connectivity and the degree distribution inherent in these network models. An examination of the vulnerability of scale-free networks is also discussed in this paper, and the effect of power law distribution on the network topology is analyzed. The removal of nodes in these network models and the effects of such removal are discussed. The contrast between Scale-free networks and random networks in the area of resisting failures is analyzed, as it has been suggested that the strongly connected nodes are responsible for the failure of scale-free networks. This paper also includes some theoretical syntheses, the proposal of new and exploratory conceptual models, theoretically grounded discussions of methodology, the analysis of historical developments with clear implications for current and future theory, theoretically relevant discussions of timely and important network issues, and comprehensive literature reviews with strong theoretical implications. INTRODUCTION In recent history, evolving networks have been seen as a relevant and very popular area of research among physicists. Reka Albert and Albert-Laszlo Barabasi introduced a concept of evolving networks that is based on preferential attachment, in order to understand the areas from which the ubiquity of scale-free distributions in real networks originates. Reka Albert and Albert-Laszlo Barabasi studied a highly connected network model which was later called the scale-free network. “Networks have become a general tool for describing the structure of interaction or dependencies in such disparate systems as cell metabolism, the internet, and society.” (Barabasi A-L, Albert R 2002) With scale free networks, even in very large networks, nodes can be selected arbitrarily and connected through other nodes which serve as the intermediary nodes. “There are features that the scale-free network contains that are lacking in the random network. In a scale free network, a small number of nodes contribute heavily to connectivity. These nodes are called hubs. In a random network, each node contributes approximately the same to the overall connectivity of the network.”(Barabasi, Albert-Laszlo 2002) In a scale-free network, the network is self-similar, in that different parts of the network are statistically similar throughout the entire network. This self similarity is a major feature of fractals. “The term "scale-free" was first coined by physicist Albert-Laszlo Barabasi and his colleagues at the University of Notre Dame in Indiana. In 1998, they mapped the connectedness of the World Wide Web and found, to their surprise, that the web did not have an even distribution of connectivity (so-called "random connectivity"). Instead, a very few network nodes (also referred to as hubs) were far more connected than other nodes. In general, they found that the probability P (k) that a node in the network connects with k other nodes was, in a given network, proportional to k??. They named this kind of network connectivity "scale-free". They also argued that there is a simple explanation for this behavior. Many networks expand through the addition of nodes to an existing network, and those nodes attach preferentially to nodes already well-connected. When this is the case, a scale-free network naturally arises.” (Watts, D.W 2003) Although a scale-free network may be robust, it can still function with a random displacement of some nodes in the network. It should be noted though, that the failure of any hub in the network can lead to the whole system becoming fragmented into a number of smaller units, which can effectively cause a failure in connectivity. In scale-free networks there is high-speed transfer of energy or information, and the hubs possess a combination of high-level global connectivity as well as highly developed local clustering. This leads to rapid synchronization of distant nodes in such a network. “Clustered small world network architectures like the Watts-Strogatz model can also be described as scale-free with the characteristic power law distribution of links. According to Strogatz: "disparate networks show the same three tendencies: short chains, high clustering, and scale-free link distributions. The challenge now is to decode the underlying meaning of small-world and scale-free architecture.” (Albert-Laszlo Barabasi and Reka Albert 1999) A scale-free network can be seen as a specific type of network for which connectivity distribution is very uneven. In a scale-free network, there are various nodes which function as extremely connected hubs by the use of a power-law distribution. In this type of connection, the situation affects the manner in which the network operates, as well as influencing the response to catastrophic events. Many large-scale networks like the Internet are categorized as scale-free networks. “A power law has a characteristic (constant) exponent which is sometimes called a dimension. No matter what size (magnification) the network, the dimension stays the same. Thus the term scale-free.” (Barabasi, A.L., Albert, R (1999) The term "scale-free" was initially used by Albert-Laszlo Barabasi and Albert when they tried to map the connectedness of the internet in 1998 and surprisingly found that the internet network possessed an uneven distribution of connectivity (also referred to as random connectivity). Usually, a network expands by the addition of nodes to the already existing network. Also these kinds of nodes tend to preferentially attach themselves to other already well-connected nodes. In this situation, there is natural occurrence of a scale-free network. “Although many real-world networks are thought to be scale-free, the evidence remains inconclusive, primarily because the generative mechanisms proposed have not been rigorously validated against the real-world data. As such, it is too early to rule out alternative hypotheses. A few examples of networks claimed to be scale-free include Social networks, including collaboration networks. An example that has been studied extensively is the collaboration of movie actors in films.” (Rozenfeld, A.F, Cohen, D. Havlin, S. 2002) Albert-Laszlo Barabasi and Albert called this type of network connectivity scale-free network connectivity, and projected their thoughts that there was a simple explanation for this kind of behavior. “A scale-free network is a noteworthy kind of complex network because many "real-world networks" fall into this category. Real-world here refers to any of various observable phenomena that exhibit network theoretic characteristics (see e.g., social network, computer network, neural network, epidemiology). In scale-free networks, some nodes act as "highly connected hubs" (high degree), although most nodes are of low degree. Scale-free networks' structure and dynamics are independent of the system's size N, the number of nodes the system has. In other words, a network that is scale-free will have the same properties no matter what the number of its nodes is.” (Watts, D.W (2003) It is notable that most networks, even networks that describe object interrelationships, are scale-free networks. Scale-free networks have thus been identified in connection with air travel connections, as well as various types of computer networks. There have been various studies on collaboration networks, with nodes representing people, and the collaboration between people represented by the links between the nodes. Most of these collaboration networks are also seen as to be scale-free networks. This kind of research has not been limited to collaboration between real people. For instance, a study was carried out involving Marvel comic book characters, in which the nodes were represented by comic characters, and links stood for appearing simultaneously in the same comic book. It was discovered that their inter-relationships were also scale-free. Scale-free computer networks are significantly different from networks of random connectivity in the presence of failure. “If nodes fail randomly, scale-free networks behave even better than random connectivity networks, because random failures are unlikely to harm an important hub. However, if the failure of nodes is not random, scale-free networks can fail catastrophically. For example, an intelligent attacker can essentially destroy an entire scale-free network by intelligently identifying and attacking its key hubs. Thus, the realization that certain networks are scale-free is important to security.” (Dorogovtsev, S.N., Mendes, J.F.F. 2002) With random networks, such as the small world network model, the average distance between two vertices in the network is very small relative to a highly ordered network such as a lattice. The clustering coefficient of scale-free networks can vary significantly depending on other topological details, and there are now generative mechanisms that allow one to create such networks that have a high density of triangles. An example of a random network is the network of land roads in the United States, which also has a bell shaped connectivity distribution. Contrastingly, airports in the United States form a scale-free network which has several hubs that connect a large number of airports nationwide. Examples of other scale-free networks are: Social webs, scientific citations, Communication networks like the internet, and Biological networks like interactions between proteins in human body, Ecological interrelations and food webs. With random networks, every node has approximately the same number of connections, and connectivity is scale-dependent. It has been observed that many large networks follow a scale-free power-law distribution. This happens as a result of generic mechanisms. Firstly, networks expand continuously as new vertices are added, and secondly, that these new vertices attach preferentially to areas that are already well connected. Scale-free networks tend to break down in a manner that is quite different from that of random networks. “A scale free network model presumes that social networks arise through a process that results in a robust and highly structured hierarchical system that is extremely resistant to disruptive events. Path lengths are short because a small number of highly connected nodes dominate the distribution, with many nodes having a small number of ties. These networks may also have local clusters, making them also small worlds. Empirical observation of social networks has found that few resemble pure scale free networks. Instead, the distribution of ties follows a power law distribution with a “fat tail,” and thus some analysts call them “truncated scale free” networks.” (Aldrich, Howard E. 2008) In random networks, the removal of nodes from the network causes a slow, steady decay in the connectedness of the network until this gets to a point where the network is broken into smaller, separate domains with limited communication. In contrast to random networks, Scale-free networks tend to resist random failures, as it is statistically not likely for strongly connected nodes to fail under random conditions, and these strongly connected nodes are responsible for the failure of scale-free networks. Connectivity in scale-free networks is maintained under random conditions. Scale-free networks may completely fail when there is a total wipeout of the hubs, which can only occur after several random failures. The degree of resistance to random failures is quite good, although catastrophic failure in a scale-free network due to targeted attacks can be quite negative. “The scale-free networks belong to the family of networks known as “small-world” networks that are based on the notion that there are only 6 degrees of separation between any two people in the world. Thus, it doesn’t take many hops to get from one node to another in such networks. Therefore, in a scale-free network, there is a high probability that many transactions would take place through one of the well-connected nodes known as the hubs of the network. Taking the World Wide Web as an example, more daily transactions take place through Web hubs like Yahoo Web portal and Google Web portal than those which take place through other less-connected Web portals.” (Rozenfeld, A.F, Cohen, D. Havlin, S. 2002) In comparison to a random network, a scale-free network is different in that it has a different type of connectivity due to the fact that the degree distribution is determined by a power law distribution, and not the Poisson distribution that is associated with the random network. “In a power law distribution, most nodes have relatively few links but a few nodes (called hubs) have a high number of links. “a fragmented network model presumes a highly clustered world in which people's searches for new ties are highly circumscribed by their environments, with people’s ties connecting them mainly to others in their same social context, while a small world network model posits a world that is fragmented into clusters, but in which the clusters are linked by bridging ties. Such ties serve as short cuts connecting many local clusters to other clusters, potentially reducing average path lengths to those found in random worlds.” (Aldrich, Howard E. 2008) The contribution of the hubs to the overall connectivity is very high. The connectivity contribution of the nodes with fewer links is much lower. This model corresponds to the Internet in which most web pages have only a few links connecting to them, but sites like Google and Yahoo have a very large number of hyperlinks pointing to them.” (A.F. Rozenfeld, R. Cohen, D. ben-Avraham, S. Havlin 2002) When considered in the context of network theory, an ideal scale-free network can be a random network that possesses a degree distribution following the distribution of scale-free ideal density. “Scale free networks have the special property of reproducing the city-size distribution and electoral results unraveling the size distribution of social groups with information theory on complex networks, when a competitive cluster growth process is applied to the network. Thanks to the modellization of the scale-free ideal network it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation.” (Dangalchev, C. 2004). The degree to which the scale-free network offers an accurate representation of real-world networks is evident in several networks, including the internet, social networks, protein networks and citation networks. A lower number of highly clustered hubs contain more links, while the high numbers of nodes have few links. Link clustering around some hubs due to preferential attachment is a feature of scale-free networks. These preferential attachments become visible due to legacy or merit. For instance, biological systems, including sub-atomic systems and ecosystems are known to represent scale-free networks in which energy efficiency is the basis of preferential attachment. “Now it has more than three billion. Most networks have expanded similarly. Hollywood had only a handful of actors in 1890, but as new people joined the trade, the network grew to include more than half a million, with the rookies connecting to veteran actors. The Internet had only a few routers about three decades ago, but it gradually grew to have millions, with the new routers always linking to those that were already part of the network.” (Hani Hazaa. 2008) The fact that emerging scale-free properties can be found in many real world networks suggests that random networks are actually general rules of evolving networks. This provides a clue as to the question whether the network follows the random model or Barabasi and Albert’s scal-free model. Barabasi and Albert came to the conclusion that scale-free networks are resilient to random failures, although they are fragile and can bcome fragmented into several disconnected smaller networks when faced with targeted attacks. In the scale-free network model, the larger hubs are highlighted. This is the norm with systems that are characterized by power law distribution and the foremost characteristic of scale-free networks is the networks relative commonness of vertices to an extent that surpasses the average. In this kind of network model, the highest degree nodes, also known as hubs, are believed to perform specific functions in the network, although this characteristic is very much dependent on the domain in question. In scale-free networks, the hubs are the greatest strength of the network, as well as being the greatest point of vulnerability. The power law distribution greatly affects the network topology. The major hubs in the network are followed closely by smaller hubs. These small hubs are in turn, followed by other nodes with a smaller degree, and so on. This hierarchical arrangement allows for a fault tolerant behavior. The vulnerability of scale-free networks raises an interesting issue of ascertaining the size of nodes that may be deemed to be essential in the network. Previous studies suggest that the simultaneous elimination of as low as 10 percent of the nodes can crash the network. Reliance on hubs has its advantages, as well as disadvantages, and depends on the system in question. “Resistance to random breakdown is good news for both the Internet and the cell. In addition, the cell's reliance on hubs provides pharmaceutical researchers with new strategies for selecting drug targets, potentially leading to cures that would kill only harmful cells or bacteria by selectively targeting their hubs, while leaving healthy tissue unaffected. But the ability of a small group of well-informed hackers to crash the entire communications infrastructure by targeting its hubs is a major reason for concern.” (Hani Hazaa. 2008) As failures in the network will occur at random, and most nodes have a small degree, the possibility that a hub will be affected is very low, and even when such an event occurs, the network usually maintains its connectivity because of the remaining hubs. “Generally, scale-free networks display an amazing robustness against accidental failures, a property that is rooted in their inhomogeneous topology. The random removal of nodes will take out mainly the small ones because they are much more plentiful than hubs and the elimination of small nodes will not disrupt the network topology significantly, because they contain few links compared with the hubs, which connect to nearly everything. Reliance on hubs has a serious drawback: vulnerability to attacks.” (Hani Hazaa. 2008) On the other hand, if the major hubs are targeted and removed from the network, then the network will simply fall apart and become a set of rather isolated graphs. Another important aspect of scale-free networks is the clustering co-efficient distribution, which decreases as the degree of the node increases. This distribution also follows a power law. This entails that the low-degree hubs belong to extremely dense sub-graphs that are inter-connected. For instance, in a social network in which the nodes are represented by people ad links are the acquaintance relationships between people. It can be seen that as people tend to form small groups and communities in which everybody knows everyone else in the community. Such a community of people can be visualized as a complete graph. Additionally, the people in such a community will also have some acquaintances outside that particular community. Some other people may be related or connected to popular people like politicians or celebrities, thus ensuring that they are connected to a large number of communities. In such a situation, these people can be considered to represent the hubs responsible for the ‘small world’ phenomenon. Presently, the more specific characteristics of scale-free networks can be viewed in terms of the generative mechanism through which the network was created, or in the context of any real-world network that is believed to be scale-free. For example, when a network is created by preferential attachment, high-degree vertices are typically placed in the middle of such a network, and this connects the vertices to form a core having progressively lower-degree nodes that make up the regions in between the periphery and the core. There have been some interesting results reached through this subclass of scale-free networks, such as the random removal of large fractions of vertices which has little effect on the overall connectivity of the network. This suggests that such topologies might be useful for security purposes. It should be noted that a lot of social networks are scale-free. “Collaboration between scientists from Boston University and Stockholm University, for instance, has shown that a network of sexual relationships among people in Sweden followed a power law although most individuals had only a few sexual partners during their lifetime, a few (the hubs) had hundreds. A recent study led by Stefan Bornholdt of the University of Kiel in Germany concluded that the network of people connected by e-mail is likewise scale free. Sidney Redner of Boston University demonstrated that the network of scientific papers, connected by citations, follows a power law as well.” (Hani Hazaa. 2008) Scale-free networks that place the high-degree vertices at the periphery may not exhibit such characteristics, and many results about scale-free networks are argued to apply to the physical structure of the world-wide-web, although some internet engineers and researchers have refuted this point of view. “A random network model presumes a highly individualized world in which everyone has nearly unlimited access to everyone else, constrained only by limits on the resources that can be devoted to the search for new social ties. Paths between distant people are short because they are not constrained regarding who they can interact with and so everyone is available as an intermediary or broker.” (Aldrich, Howard E. 2008) The discovery of the Scale-Free Network model is significant in that scale-free networks have been successfully applied in various complicated real-time networks, in which scale-free networks have been proven to be valid. This successful application of scale-free network model has rendered the random network model to be at best, questionable. Also, the fact that emerging scale-free properties are present in numerous real-world networks tends to suggest that random networks are not only dependent on the individual system characteristics, but are actually general rules of evolving networks. . FINDINGS AND CONCLUSION In comparison to a random network, a scale-free network is different in that it has a different type of connectivity due to the fact that the degree distribution is determined by a power law distribution, and not the Poisson distribution that is associated with the random network. In recent times, there have been recognized networks in many situations, including networks in nature, social networks and computer networks. In this spectrum also, can be found planned networks (that were designed by human beings) and unplanned networks (which that naturally evolve become organized into complex forms which are then discovered through science and research after being a part of our daily life for some time already). This is one reason why planned networks can be easily mapped, as the number and location of nodes in these networks is already well known, as well as their patterns of connectivity. Unplanned networks on the other hand, can be quite difficult to map due to the fact that they have to be felt or discovered after having being already established. A scale-free network can be seen as a specific type of network for which connectivity distribution is very uneven. In a scale-free network, there are various nodes which function as extremely connected hubs by the use of a power-law distribution. In this type of connection, the situation affects the manner in which the network operates, as well as influencing the response to catastrophic events. Many large-scale networks like the Internet are categorized as scale-free networks. In a scale-free network the nodes are not randomly connected, but are strongly connected, and are known as the connectivity hubs that determine the manner in which the network actually operates. Scale-free networks do not have a fixed size, and can increase steadily with time. As the size of the network increases, the ratio of the strongly connected nodes to the number of nodes inside the rest of the network remains constant. Random connectivity models on the other hand, postulated that there would be an absence of well-connected nodes or only a few statistically insignificant well-connected nodes. In a random network, most of the nodes would have a number of connections hovering around a small mean value. In a random network, as the number of nodes continues to increase, the relative number of strongly connected nodes will continue to fall. Scale-free networks also tend to break down in a manner that is quite different from that of random networks. In random networks, the removal of nodes from the network causes a slow, steady decay in the connectedness of the network until this gets to a point where the network is broken into smaller, separate domains with limited communication. In contrast, Scale-free networks tend to resist random failures, as it is statistically not likely for strongly connected nodes to fail under random conditions, and these strongly connected nodes are responsible for the failure of scale-free networks. In a scale-free network, the network is self-similar, in that different parts of the network are statistically similar throughout the entire network. This self similarity is a major feature of fractals a scale-free network may be robust, it can still function with a random displacement of some nodes in the network. It should be noted though, that the failure of any hub in the network can lead to the whole system becoming fragmented into a number of smaller units, which can effectively cause a failure in connectivity. In scale-free networks there is high-speed transfer of energy or information, and the hubs possess a combination of high-level global connectivity as well as highly developed local clustering. Scale-free networks like the world-wide-web are quite vulnerable to attacks. In a situation where a malicious attack is able to remove a portion of the well-connected nodes, the network would disintegrate. It is also notable that Scale-free networks remain very vulnerable to spreading viruses, as the nodes or hubs in the network can pass these viruses on a massive scale to other connected multiple nodes. Most networks, even networks that describe object interrelationships, are scale-free networks. Scale-free networks have thus been identified in connection with air travel connections, as well as various types of computer networks. There have been various studies on collaboration networks, with nodes representing people, and the collaboration between people represented by the links between the nodes. Scale-free computer networks are significantly different from networks of random connectivity in the presence of failure. With random networks, such as the small world network model, the average distance between two vertices in the network is very small relative to a highly ordered network such as a lattice. Scale-free networks may completely fail when there is a total wipeout of the hubs, which can only occur after several random failures. The degree of resistance to random failures is quite good, although catastrophic failure in a scale-free network due to targeted attacks can be quite negative. In the scale-free network model, the larger hubs are highlighted. This is the norm with systems that are characterized by power law distribution and the foremost characteristic of scale-free networks is the networks relative commonness of vertices to an extent that surpasses the average. The clustering co-efficient distribution in scale-free networks decreases as the degree of the node increases. This distribution also follows a power law. This entails that the low-degree hubs belong to extremely dense sub-graphs that are inter-connected. The degree to which the scale-free network offers an accurate representation of real-world networks is evident in several networks, including the internet, social networks, protein networks and citation networks. In contrast to random networks, Scale-free networks tend to resist random failures, as it is statistically not likely for strongly connected nodes to fail under random conditions, and these strongly connected nodes are responsible for the failure of scale-free networks. Connectivity in scale-free networks is maintained under random conditions. Scale-free networks may completely fail when there is a total wipeout of the hubs, which can only occur after several random failures. The degree of resistance to random failures is quite good, although catastrophic failure in a scale-free network due to targeted attacks can be quite negative. This happens when worse is directed at hubs. If all the very-connected nodes are taken out of the network, the network stops functioning immediately. It can thus be said that that scale-free networks are quite dangerous in the advent of targeted attacks. In any network following Barabasi and Albert’s model, the prevention of the spread of failure requires that close attention is paid to the protection of the hubs and not necessarily the numerous nodes that form the entire network. REFERENCES Aldrich, Howard E. (2008) Facilitating a Rational Process Model of Entrepreneurial Team Formation through Designing Effective Social Networks. University of North Carolina Barabasi, A.L.(2002) How everything is Connected to everything Else and What it means for business, Science and everyday life, Penguin, London. Ch2 R. Albert; A.-L. Barabasi (2002) Statistical mechanics of complex networks. Reviews of Modern Physics 74: 47–97. doi:10.1103/RevModPhys.74.47. 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