Toward a theoretical and substantial understanding of complex social networks - Essay Example

? TOWARD A THEORETICAL AND SUBSTANTIAL UNDERSTANDING OF COMPLEX SOCIAL NETWORKS By Location Social networks offer a wide range of behaviors that can be assessed and classified according to random and small-world network characteristics. In many cases, understanding these behaviors within the context of network models holds considerable promise for understanding, predicting and mitigating threats to both natural and man-made network systems. Both random and scale-free networks can be used to explain systemic processes according to rules of connectivity, which can in turn be used to construct dynamic models based on the manner in which nodes seek out and link with other nodes. Concepts of preferential connectivity, fitness and competitiveness impact the ways in which nodes interact and establish links under specific circumstances. Keywords: Social networks, random networks, scale-free networks, nodes, preferential connectivity CONTENTS 1. Introduction…………………………………………………………………………………..1 2. Social networks – characteristics……………………………………………………………..2 3. Friend networks……………………………………………………………………………….4 4. Health behaviors………………………………………………………………………………5 5. Mixed model…………………………………………………………………………………..6 6. Food web supplies…………………………………………………………………………….7 7. Competition and fitness in real world networks………………………………………………9 8. Criticism……………………………………………………………………………………...12 9. Conclusion……………………………………………………………………………………14 1 Toward a Theoretical and Substantial Understanding of Complex Social Networks 1. Introduction One of the best ways to determine the viability of a theory is to compare it to the parameters and processes of real world systems. Aristotle’s examination of the physical manifestations of natural phenomena helped lead him to his theory of universals, which holds that an object has its own immutable and innate form: a pear is a pear because it embodies that form. The observation of complex social networks, both great and small, yields invaluable information about how their processes affect form, systemic characteristics and interact with other systems. The study of real-world networks reveals a wealth of information about the relevance of the random network model and the theory of scale-free networks, as developed by Albert-Laszlo Barabasi and Reka Albert. Thus, by observing systems such as food web networks, human physiological systems and various social interactions we may determine to what extent the laws of connectivity predict how they behave under certain circumstances. By extension, we may also utilize systems that approximate real-world network tendencies, such as the worldwide web and power network grids. Both random and scale free networks exhibit characteristics that are identifiable in natural systems. In Linked: The New Science of Networks, Barabasi and Albert describe random networks in terms of human physiology, explaining that the more links that are added within a system, the more difficult it becomes to find an isolated node. Thus the networks around and within us are very dense, which explains why “all molecules in our body are integrated into a single complex cellular map” (Barabasi and Albert 2002, p. 19). In this way, Nature creates redundancy to ensure survival by “repeatedly and extravagantly (exceeding) the one-link minimum” (2002, p. 18). In developing the theory of scale-free network, Barabasi and Albert found that most real-world networks display what they 2 describe as “preferential connectivity” (Barabasi and Albert 2002), meaning that new vertices are more likely to connect to a vertex with a large number of connections. This contradicts the assumption made in random network models that the connection between any two vertices will be uniform (Barabasi and Albert year, p. 22). 2. Social networks - characteristics Social networks can be used to illustrate the lack of uniformity inherent in most real systems and to show that “networks continuously by the addition of new vertices…attach preferentially to already connected sites” (Barabasi and Albert 1999). In their groundbreaking 1999 paper, “Emergence of Scaling in Random Networks,” Barabasi and Albert contend that the concept of preferential connectivity is at work in actor networks. According to this construct, a new actor is much more likely to be paired with a more popular, well-known actor than with another new actor. This corresponds to the observable tendency among new web pages, which are more likely to establish links to frequently visited pages that have already achieved a high degree of connectivity (Barabasi and Albert 1999). In this way, the scale-free concept registers: The likelihood that a new vertex connecting to other vertices will not do so in a uniform manner A new vertex is significantly more likely to link to another highly connected vertex (p. 5) In 2002, two researchers from the University of Washington published a paper (“Statistical Tells Tails of Human Sexual Contacts”) in which they determined that the power law model that identifies a scale-free environment could not be applied to an outbreak of sexually transmitted disease in Uganda. According to the power law model, there should only be an epidemic transition in a small sampling of the parameter space labeled ?, however, statistical estimates placed ? well outside these limits in a sparsely populated region where a scale-free network is impossible (Jones and Handcock 2002, p. 5). The study concluded that persistent variables in sexually transmitted disease epidemics, such as the Ugandan 3 scenario, negated the scale-free model. “Given the fact that Uganda has one of the highest HIV incidence rates in the world, the scale-free power law model clearly fails to describe Ugandan sexual networks” (2002, p. 5). Thus, since “the epidemic threshold parameter increases linearly with the variance under behavioral heterogeneity, the epidemic threshold should always be exceeded in…a free-scale system” (2002, p. 2). The epidemiological example presents other problems for the application of the scale-free model. In a scale-free network, epidemics may “reproduce with a considerably lower number of infected persons at each point in time, than in a random network” (Liljeros, Edling & Amaral 2003, p.194). However, one notable characteristic of scale-free networks holds interesting implications for preventing the spread of sexually transmitted disease. It has been noted that the maintenance of highly connected nodes is crucial to the functioning of a scale-free network. In an affected population, “if only a few very active persons are removed (or change their behavior) the network very soon falls apart in separated components, thus preventing the emergence of epidemics” (p. 194). Just as the entire scale-free network disintegrates when highly connected nodes fail, so too an STD epidemic may come undone when its “highly connected” transmitters are negated. Interconnectedness may hold the key for successfully applying social network analysis to the very building blocks upon which human life depends. This interconnectedness transcends many levels of activity within human body chemistry. Barabasi and Albert use the effect of cancer upon human cell structure, noting that “one can remove many nodes from this key cellular network without risk of killing the organism. If…a drug or an illness shuts down the genes encoding the most connected proteins, the cell will not survive” (Barabasi and Albert 2002, p. 118). And yet by understanding scale-free networks within the context of the human physiological system, medical science may one day obtain a minute 4 knowledge of the “precise wiring diagram of a cell” and develop medical therapies capable of determining the strength of a cell and how it may react to various drugs (p. 194). In their study of complex networks, Paul Erdos and Alfred Renyi postulated a model by which a random network behaves in such a way that it exhibits behavior reminiscent of a phase transition, or the transformation of one physical state into another, such as water changing into ice (Malkevitch, 2012). Applying their theory to a random graph, Erdos and Renyi found that if the probability p used in applying an edge to a random graph is a function of the number of vertices n, a substantive transition takes place when p achieves a critical value (2012). According to the Erdos-Renyi model, all nodes have the same probability of being linked to other nodes. However, many researchers have determined that the natural world can be used to refute this contention. 3. Friend networks In their analysis of complex real-world networks, Matthias Dehmer and Frank Emmert-Streib used a typical social network of friends as an example. “The number of friends will depend on characteristics of the individual, such as being out-going, rich, introverted, gregarious, or other factors that result in a linking probability” (Dehmer & Emmert-Streib 2009, p. 39). Thus, the sheer complexity of real-world networks, with their wide variation of factors and variables, problematize the notion of uniformity in random networks. Social networks, in particular, present an especially challenging variety of generation mechanisms, which necessitate the presence of multiple indicators (2009, p. 401). The “friends” network proposed in Dehmer and Emmert-Streib requires one to take into account the concept of preferential connectivity, which in such a network/community is occasioned by variables such as personality and wealth. A particularly interesting presentation of random graphing in friend networks was published in 2010 by Stanford University researchers Benjamin Golub and Yair Livne. Their paper, “Strategic Random Networks: Why Social Networking Technology Matters,” showed how significant changes in 5 behavior can be influenced by social networking; specifically, how individuals establish a network and how their choices, and the consequences of those choices, are affected by what Golub and Livne call “the costs and benefits of socializing” (2010, p. 2). The benefit of a direct connection made through networking might include finding out about a job from a friend, while an indirect connection may result in finding out about a potential job from a friend’s co-worker (2010, p. 2). Social networking technology is identified as a networking site such as Facebook or LinkedIn, which add value to the benefit of establishing and maintaining “friends of friends.” The social networks the study describes embody the complexities of real-world systems while mirroring random network models (2010, p. 24). This theory suggests that links in such a network would exhibit a degree of uniformity, with highly sociable individuals, in the capacity of nodes, establishing many friendship links. Conversely, less socially inclined actors in this model would be expected to uniformly establish fewer direct links, thus resulting in fewer indirect (or “friends of friends”) links. Golub and Livne show that random networks comprising a symmetrical equilibrium indicate a radical transition from a low-intensity to a high-intensity network when the value of the “friends of friends” connection exceeds a “cost-dependent threshold” (2010, p. 2). Here again we see evidence of that condition, described by Erdos and Renyi, in which a random network parallels transitional change in a physical, real-world network. 4. Health behaviors The idea that social networking can affect behaviors within a group has gained considerable coinage in recent years, particularly in the area of individual health practices. However, the size and composition of such a network may be quite different from the model envisioned by researchers. Whereas scientists have traditionally concluded that networks with many distant relationships (in other words, where an individual knows many other people but not well) yield rapid transformative change, new research from MIT indicates that when it comes to learning and practicing beneficial health behavior a smaller scale network is actually preferable. When an individual is surrounded by a group of people he 6 or she knows well, that individual is more likely to internalize and practice new information and ideas (Dizikes 2010). According to Professor Damon Centola, the MIT study revealed that a disease can spread as a simple contagion, but behaviors capable of preventing its spread may be disseminated only in complex terms. Therefore, widespread diffusion throughout a population is apt to be more successful in a smaller, more clustered network than in a random network model (2010). David Lazer from the Harvard School of Government added that “You need wide bridges to transmit complex information like health data, and that is different from the traditional picture of how things spread in a network.” Centola cites colonoscopy as an example. Simply hearing about it and how it works would probably not be enough to convince someone to have one, whereas knowing people who have had the procedure and benefited from it is much more likely to impact the individual’s behavior choices (Dizikes 2010). For one thing, the individual is much more likely to trust the word of a close contact. Professor Centola came to that conclusion after conducting an experiment using subjects who were part of networks with long ties and those who were part of a large, centrally clustered group. Fifty-four percent of those who elected to participate in a health forum belonged to clustered networks, while less than 40 percent of those with long ties in a social network chose to take part in the forum (2010). 5. Mixed model Centola’s study appears to find corroboration in a paper that preceded the MIT research by several years. In this exposition, entitled “Meeting Strangers and Friends: How Random are Social Networks,” it was determined that clustering coefficients are denser and larger in a centrally concentrated group than in random networks (Jackson and Rogers 2004, p. 2). As such, some distributions exhibited scale-free characteristics. The study showed that highly linked nodes are more likely to be linked to other high-degree nodes. Here we see preferred connectivity at work in the formation of new networks, but also the occurrence of models “that mix random meetings and preferential attachment” (2004, p. 6). 7 The mixed model that emerges from the study indicates that the fraction of transitive triples and the coefficient that emerges from clustering are positive; in practical terms, a node is apt to link to two nodes that are already linked for the simple reason that they are linked (Jackson and Rogers 2004, p. 14). This describes the mixed, or “network-based meeting model” which, according to the study, accounts for why it differs from random and scale-free models, where the random and preferentially attached aspects and not connected (2004, p. 14). Jackson and Rogers were able to show that the characteristics implied by their model matched the behaviors elicited by several real-world networks, including a network of friends among prison inmates, a network of romantic relationships among high school students and a “web” of ham radio contacts made during a period of one month (2004, p. 18). 6. Food web supplies As seen in the Jackson-Rogers study, network modeling is useful for assessing, even predicting the behaviors of many real-world systems. All are instructive, but some are more useful than others in sheer terms of utility and of benefit to human beings. Such is the case with food web networks, which approximate the predilections of scale-free networks in terms of vulnerability. As mentioned previously, scale-free networks are typified by a relatively few highly connected nodes, which are unlikely to fail but will result in the destruction of the entire network if they do so. Thus it is in food webs: those with scale-free degree distributions are, in general, the most vulnerable to attack (Norbert & Cumming 2008, p. 105). The study of network modeling offers a means for understanding the topological properties of food webs. Such knowledge can lead to a greater understanding of the predations that take place within a food web network, predations that may well cause the disastrous collapse of an entire food supply. In the study of food webs, where vertices equate to species, network graphing studies have revealed that there is an abundance of predator-prey motifs, which expand the field of study beyond the classical construction in which omnivores that devour other species (as well as the food eaten by their 8 prey) are traced (Caldarelli 2007, p. 37). Recent studies indicate that different food web networks embody a variety of network models, one exhibiting non-random clustering, while a larger food web studied revealed the opposite. Particular attention has been paid to scale-free networks in the study of food webs. In 2002, a study of three food supplies (the Ythan estuary, Silkwood and Little Rock lake food webs) revealed that the distribution of connections P(k) showed long tails, indicative of power-law scaling (Montoya & Sol, 2000). The implications for these findings are that food web communities may be organized in a non-random manner and, as such, could provide valuable information about the ability of such systems to resist potentially fatal shocks to their environments (2000). In general, the food web systems studied exhibited resiliency characteristic of small-world networks. As Barabasi and Albert have theorized, this may explain the ability of food web systems to survive naturally occurring threats to their existence (2002, p. 53). The Montoya/Sol study determined that though the three ecological food webs studied (see above) were relatively small, they have the same non-random network properties of larger food webs in common (such as Chesapeake Bay). Montoya’s results yielded a degree distribution that was in line with the power law expressed by a small-world exponent, ? ? 1.1. Barabasi and Albert note in “Statistical Mechanics of Complex Networks” that “the well-documented existence of key species that play an important role in food web topology points toward the existence of hubs (a common feature of scale-free networks)…” (2002, p. 53). Other studies focusing on larger samples have shown that while they vary significantly in terms of factors like number of species, each food web contains a variety of species that are no more than three edges separated from each other. From a real-world standpoint, the true value in food web studies is to be found in the degree to which they reveal the effect of change on the entire network. Specifically, this refers to the loss of species within an ecosystem. The resulting equation from the Montoya study measured the size S of the 9 dominant cluster, the average size {s} of other species clusters and the number of species that are marginalized due to the loss of species on which they rely (2000). This equation was used to show that scale-free networks are imbued with a significant degree of resistance when it comes to the occurrence of random error (loss of species). The study of food web networks and other complex systems show that scale-free random graphs are far less susceptible to random perturbations than random models, such as those proposed by Erdos and Renyi (Barabasi and Albert 2002, p. 91). However, they do display a vulnerability to attack, a familiar trait of scale-free networks, which are prone to thoroughgoing failure on the (comparatively) rare occasion when a highly connected node is destroyed. These models offer the potential for predicting the breakdown of food webs and other ecosystems. Food webs and other networks that occur in the natural world often exhibit behaviors that are reflected in man-made social networks. The worldwide web, Internet search engines and online social networking sites in many ways function according to principles that govern connectivity of natural systems, such as human cellular and ecological networks. Intrigued by the meteoric rise of Google, an ascension that defied observable rules concerning scale-free networks, Barabasi and Albert and their colleagues discovered a kind of Internet-style robustness among certain new nodes that has widespread applicability across a range of man-made applications. As a still-emerging technology, the Internet and its various permutations present researchers with a wealth of research possibilities against which to test and modify network models. 7. Competition and fitness in real-world networks In their description of Google’s remarkable success, Barabasi and Albert use the phrase “new kid on the block” to describe the effect of “dominant latecomers,” which they note are present in most networks (2002, p. 95). However, in other scale-free models the nodes were found to be identical, though there was variance in terms of the number of links. This is a function of timing, Barabasi and Albert explain, since newer nodes are subordinate to older, more established nodes that have accumulated more 10 links (2002, p. 95). The apparent anomaly represented by Google flies in the face of this theory because Google quickly surpassed other, more established search engines, such as Alta Vista and Yahoo. Consequently, Barabasi and Albert sought to develop a theory that would account for variation in the characteristics of nodes previously thought to be invariant. Further research revealed that the “new kid on the block” phenomenon was present among networks in competitive situations, such as might be found in any business environment. This led to a construct that introduced the concept of “fitness” among nodes (Barabasi and Albert 2002, p. 96). In this context, fitness refers not just to the number of links a node can claim but to a formula that assumes preferential attachment is “driven by the product of the node’s fitness and the number of links it has” (2002, p. 96). New nodes decide where to link based on this fitness connectivity product, meaning that nodes with a higher product will be more attractive. This, in turn, meant that the idea of timing, which was considered central to the idea of preferred connectivity, had to be rethought. The result of this reconsideration yielded a substantial change in the model of scale-free networks within competitive environments. It moved fitness to the forefront since Google showed that the presence of an early entrant does not de facto mean that a latecomer in such circumstances would find itself in an inferior position in terms of the number of links. In fact, Barabasi and Albert conclude that a node’s fitness connectivity product is all important within a competitive network landscape (2002, p. 97). Whereas node connectivity within a scale-free model would normally be expected to increase as a square root of time, node fitness indicates that the determinant (?) of how fast a node acquires links is different for each node (2002, p. 97). Consequently, longevity no longer determines the speed at which nodes acquire links. This, then, explains why Google, which exhibited a much higher degree of fitness connectivity product by virtue of its superior search capability, quickly outpaced every competitor in the field. 11 The Google example holds implications for other scale-free networks in competitive environments. Other competition-oriented networks might include the rate at which companies establish new affiliations with other businesses, or the rate at which a new actor competing for work with other actors lands parts. A relatively new business, for example, may possess advantages in terms of efficiency, marketing or employee morale that heighten its fitness product and make it more attractive than other companies that can claim greater longevity in the marketplace. A new actor may be perceived as better looking, funnier or more adept in a wide range of roles than competing actors. Fitness is the key underlying factor. That being the case, it becomes necessary to proceed from the proposition that all nodes in a highly competitive network are different due to the presence of newer nodes acquiring links at a higher rate than nodes with more seniority. Other applicable scenarios might include the introduction of a new Web site or the introduction of a new electronic communications and social network “tool,” such as a new version of Apple’s iPhone. The Internet is a highly volatile environment, with change in social network technology constituting a virtual constant. Nevertheless, it remains a prime example of the kind of power-law distribution that has been used, for instance, to show that a low percentage of individuals control a high percentage of the wealth in the United States (known as the “80/20 rule”). The standard equation for this rule is that the value x is proportional to 1/x? where the exponent is positive (Levene 2010, p. 353). This approach has been utilized to show that 80 percent of links on the Internet are directed to roughly 15 percent of the universe of Web sites (2010, p. 353). Power-law distribution can also be used to show that the vast majority of Web sites are small while only a small percentage are very large, and that while small sites may, on a particular day, receive a large number of visits, an extremely large site (such as Google) will receive vast numbers of hits on a daily basis (2010, p. 353). Scale-free properties can be observed in other social network phenomena arising from the 12 worldwide web, such as file swapping. Power-law distribution provides an illuminating perspective on the characteristics of online file-swapping repositories of music, academic research papers and other popular (and practical) media. For instance, the intersection of users and links on a distribution graph will show that download requests tend to overlap a given threshold within a set time frame. The power-law distribution approach has proven quite popular in a relatively short time span (in fact, it might be said to have mirrored the rapidity of Google’s success), and has armed researchers with a ready scale model from which to proceed in the theoretical pursuit of determining the nature of the Internet itself. 8. Criticism And yet some scientists maintain that their colleagues have been too quick in turning to what amounts to little more than an unproven, though promising, model. In an editorial, published in Internet Mathematics, Michael Mitzenmacher contends that power law research, of the kind popularized by the likes of Barabasi and Albert, must evolve from an observational modeling paradigm into the realm of scientific validation. Mitzenmacher uses the web as an example of the kind of situation he decries. “For the web graph, the primary model of study has been variations of preferential attachment: the more links a web page has the more links it is likely to obtain in the future” (Mitzenmacher 2005, p. 527). Yet there has been very little in the way of “systemic or theoretical work” that can validate the assertions of those who postulate power law behavior in web-based scenarios (2005, p. 527). One problem is that a lack of validated experimentation leaves open the possibility that theories other than preferential attachment may be of equal or superior value. Another problem with a modeling approach is its comparative lack of credibility in the absence of empirical data. Mitzenmacher argues that it is difficult to trust “the underlying behavior and therefore how the behavior might change over time. It is not enough to plot data and demonstrate a power law, allowing one to say things about current behavior” (Mitzenmacher 2005, p. 527). The point of modeling, 13 then, is that it provides a base from which one may reliably predict the behavior of a network. Without validated evidence, there is no basis from which one may begin to determine, for instance, whether the distribution of web pages will look the same in a year, or in two years (2005, p. 528). Mitzenmacher does not refute the assertion that the web exhibits power law distribution, only that “the methodology needs to be re-examined” (2005, p. 528). On a human level, validation of power law distribution and scale-free networks could help determine, for instance, whether the true essence of life lies in the make-up of individual genes or in the interrelated connectivity of genes (Bogust 2006, p. 147). In another example, critics argue that experimentation and validation could help further research into the long-term survivability of scale-free networks and their resistance to catastrophic chain reactions when a link is broken. As such, a more empirical approach could yield alternatives that serve as better predictors of such an occurrence. One theory that could benefit from a renewed emphasis on experimentation and validation is the notion of boundedness in complex systems. This model postulates the stability of complex systems which remain so only in a situation where the energy built up by any break in a solid structure remains permanently bounded (Kaleva 1990, p. 2). Thus, the breaking of links and even the failure of nodes would not necessarily signal the destruction of a complex system. The problem is that despite the many variations proposed for power law distribution, none have been disproven to any significant degree. Some critics have argued that it is not enough to simply argue that a power law is present solely because one’s favored mechanism is at work (Power Law Distribution 2012). As well, statistical physicists argue that too seldom do scientists test the possibility that they are dealing with a power law versus the possibility that they may be in the presence of stretched exponentials, or log-normals (2012). Other critics insist that phase transitions, the transformation of a given material into another form, exhibit characteristics that appear to behave like power laws but bear scrutiny for the very reason that 14 they so evidently appear in this guise. Perhaps a more common (and human) explanation for the tendency to automatically endow systems with power law distributive properties can be found in the simple desire to explain systems in terms of interesting and complex characteristics. As has been discussed, many naturally occurring social networks (i.e. food webs and friendship networks), and other processes that mimic them (i.e. the worldwide web) do offer pristine examples of scale-free networks. However, other viewpoints could show that there are other models that could, and should, be tested and validated in the interest of trial and error. One of the hallmarks of scientifically testing a hypothesis is the time-honored practice of first invalidating it, then adjusting one’s equation accordingly before testing it again, and again if necessary. This, in the view of many critics, describes the weakness inherent in the widespread assignation of scale-free networks in social networks. 9. Conclusion Network modeling offers a means by which real-world systems may be assessed for connectivity behaviors that can reveal inherent weaknesses and strengths. These resiliencies or vulnerabilities can help theorists better understand the potential outcome of threats to both natural and man-made networks based on the linking predilections of random and scale-free models and, in competitive environments, determine the relative fitness and predict the consequences of introducing new and dynamic nodes. Network modeling enables scientists to construct theoretical scenarios resulting from the introduction of a virus within the delicate balance of a natural ecosystem or from the directed attack of a computer virus. Despite its proven usefulness across a wide spectrum of natural and man-made applications, theoretical network modeling may well benefit from a more deliberately experimental scientific approach, one which seeks to validate the presence of, for instance, a power law distribution within a given system. It is anticipated that further research in this promising field of study will yield robust processes whereby physicists, statisticians and others may determine more conclusively the presence of network connectivity by proving the absence of alternate explanations. 15 References Albert, R. & Barabasi, A.L. (2002). “Statistical Mechanics of Complex Networks.” Reviews of Modern Physics, 74, 57-94. Barabasi, A.L. (2002). Linked: The New Science of Networks. Cambridge, MA: Perseus Publishing. Barabasi, A.L. & Albert, R. (1999). “Emergency of Scaling in Random Networks.” Science. 286. Bogost, I. (2006). Unit Operations: An Approach to Videogame Criticism. Cambridge, MA: MIT Press. Caldarelli, G. (2007). Scale-Free Networks: Complex Webs in Nature and Technology. New York, NY: Oxford University Press. Dehmer, M. & Emmert-Streib, F. (2009). Analysis of Complex Networks: From Biology to Linguistics. Weinheim: Wiley VCH Verlag. Dizikes, P. (2010). “Better Health Through Social Networking.” MIT News. Web. http://web.mit.edu. Golub, B. & Livne, Y., et al. (2010). “Strategic Random Networks: Why Social Networking Technology Matters.” Web. http://www.mendeley.com. Jackson, M.O. & Rogers, B.W. (2004). “Meetings Strangers and Friends of Friends: How Random are Social Networks?” American Economic Review, 1-48. Jones, J.H. & Handcock, M.S. (2002). “Statistical Evidence Tells Tails of Human Sexual Contact.” Seattle, WA: University of Washington. Kaleva, O. (1990). “The Cauchy Problem for Fuzzy Differential Equations.” Fuzzy Sets and Systems, 35(3), 389-396. Levene, M. (2010). An Introduction to Search Engines and Web Navigation. Hoboken, NJ: John Wiley. Liljeros, F., Edling, C.R. & Amaral, L.A.N. (2003). “Sexual Networks: Implications for the Transmission of Sexually Transmitted Infections.” Microbes and Infection, 5, 189-196. 16 Malkevitch, J. (2012). “Complex Networks.” American Mathematical Society. Web. http://www.ams.org. Mitzenmacher, M. (2005). “The Future of Power Law Research.” Internet Mathematics, 2(4), 525-532. Montoya, J.M. & Sol, R.V. (2002). “Small World Food Patterns in Food Webs.” Journal of Theoretical Biology, 214(3), 405-412. Norbert, J. & Cumming, G.S. (2008). Complexity Theory for a Sustainable Future. New York, NY: Columbia University Press. “Power Law Distributions, 1/f Noise, Long-Memory Time Series.” 8 January 2012. Web. http://cscs.umich.edu. Read More