CHAPTER 1 INTRODUCTION
1.1 C OMPLEX N ETWORK T OPOLOGY
Many systems in nature and technology consist of large numbers of highly interconnected dynamical units [2, 16]. Examples include coupled biological and chemical systems, neural networks, social interaction, and the Internet. An initial approach to capturing the global properties of such systems is to model them as graphs whose nodes represent dynamical units (e.g., neurons in the brain or
individuals in a social system) and whose links represent interactions between units.
Of course, this is a very strong approximation that requires translating interactions
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between dynamical units (generally dependent on temporal, spatial, and many other details) into simple binary numbers designating the existence or lack of links between two corresponding nodes. Such approximations provide simple yet informative representations of whole systems. The development of powerful and reliable data analysis tools represent better mechanisms for exploring the topological properties of multiple networked systems, thus supporting topological analyses of interactions in a diverse range of systems (e.g., communication, social, and biological). These efforts reveal that despite inherent differences, most real networks have the same topological properties [1, 5]. The most significant are the small-world effect, degree scale-free distributions, correlations, and clustering.
1.1.1 Randomness
The first non-regular network model [17, 18] was introduced by Paul Erdös and Alfred Rényi in the late 1950s [19]. In this dissertation I will variously refer to this as the random model, the Erdös-Rényi model, or the ER model. The ER model of a random network starts with N nodes and connections between pairs of nodes at a p probability, resulting in graphs with approximately N(N–1)/2 randomly placed links (Fig. 2, part Aa). Node degrees follow a Poisson distribution (Fig. 2, part Ab),
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indicating that most nodes have approximately the same number of links (close to the average degree <k>). The tail (high k region) of the P(k) degree distribution decreases exponentially, indicating the rarity of nodes that significantly deviate from the
average.
1.1.2 Small-world property
This property was first investigated in the 1960s in a social context, as part of a series
of experiments designed by Milgram [20, 21] to estimate the number of steps in acquaintance chains. In his first experiment, Milgram asked randomly selected people
in Nebraska to send letters that would eventually arrive at the home of an individual living in Boston, identified only by his name, occupation, and city of residence. The
step-by-step letters could only be sent to individuals that the current sender knew by first name, and who were presumably closer to the final recipient. Milgram kept track of the paths followed by the letters and of the demographic characteristics of their handlers. At the time of these experiments, the commonly held belief was that it would take hundreds of steps for letters to reach their final destination, but Milgram found that the number of links needed to reach the targeted individual was six. Dodds et al. [22] have recently replicated Milgram‘s experiment using e-mail, completing
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enough connecting chains so as to allow for a thorough statistical characterization.
The small-world property has been observed in a variety of real networks (including biological and technological [2, 4, 23]), and is now an accepted mathematical property in some network models (e.g., random graphs).
In 1998, Watts and Strogatz [21] proposed a new model for explaining small path lengths and large clustering coefficients that are independent of network size—two properties shared by many real networks. According to their model, the first step is to construct a network with a one-dimensional ring lattice of N nodes (or d-dimensional regular lattice) in which each node is wired to its neighbors up to kth nearest neighbor.
Such regular lattices have high average path lengths. Decreasing those lengths requires the rewiring of each link with a p probability to another randomly picked node—a process that establishes long-range connections. A small-world network displays characteristics of a regular lattice for very small p values and an ER network for very large p values, meaning that small-world networks lie somewhere between order and randomness. Average path length in a small-world network is expressed as
)
This function is a constant for u << 1, and behaves as ln(u)/u for u >> 1. Accordingly,
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the clustering coefficient for small-world networks is CSW ∝ (1 − p)3.
Small-world networks share some properties with a number of real networks.
However, their degree distribution has a pronounced peak at <k> = K and
exponentially decaying wings for large k, thereby distinguishing them from the power law degree distributions of networks such as the WWW, the Internet, and many social networks.
1.1.3 Scale-free distributions
Many scale-free networks are characterized by a power-law degree distribution [24] in which the probability that a node has k links follows P(k) ~ k–γ, where γ is the degree exponent. The probability that a node is highly connected is statistically more
significant than in a random graph(see Fig. 2, part Ba), with network properties often determined by a relatively small number of highly connected nodes known as hubs. In the Barabási–Albert scale-free model network model [24], a node with M links is added to the network at each time point and connects to an already existing node I
with probability
I kI/
J kJ, where k is the degree of node I and J the index denoting the sum over network nodes. The network generated by this growth process has a power-law degree distribution characterized by the degree exponent γ = 3, a8
distribution represented by a straight line on a log–log plot (see Fig. 2, part Bb). The network created using the Barabási–Albert model [24, 25] does not have an inherent
modularity, meaning that C(k) is independent of k. Scale-free networks with degree exponents 2<γ<3(a range that is observed in most biological and non-biological
networks) are ultra-small, with average path lengths that follow l ~ log log N. This is significantly shorter than log N, which is characteristic of random small-world networks [21].
Fig. 2. The comparison between the random network and the scale-free network.