Abstract
This chapter provides a summary of relevant insights into complex networks theory. It is the foundation for the development of network theoretical hypotheses and the respective network methodology. In the first step, a brief overview on the background of complex networks research is provided, before relevant structural and locations properties of networks are presented. The network measures are used to illustrate insights into the structure and on the dynamics of networks. At the end of this chapter, research hypotheses are developed concerning the open research questions on network network-centric valuation in software markets. They are challenged with the complex networks diffusion simulator that is developed in the following chapter.
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Notes
- 1.
Complex networks are networks with nontrivial network characteristics that differ from those of simple networks such as lattice rings.
- 2.
Accordingly, complex customer networks are customer networks investigated from a complex network research perspective.
- 3.
- 4.
Please note the difference between the social network theory and complex networks research. While the social network theory analyzes the characteristics of individual nodes in small networks, complex networks research investigates statistical properties of networks in order to observe large-scale structural and locational properties, as well as the large-scale network dynamics.
- 5.
- 6.
- 7.
This terminology is typically used by computer scientists, while mathematicians speak of graphs that are defined by a set of nodes connected through edges (Simon 1962). The etymological origin of the word roots back to the Indogerman word “net” meaning “being tied” (Duden 1989). Complex networks have non-trivial topological features, i.e. features that do not occur in simple networks. The complexity of a system is related to the amount of information that is necessary to describe its behavior. The most fundamental subunit of a network is a vertex, which is also termed site in physics, node in computer science, and actor or agent in sociology. In contrast, an edge is the connection between two nodes, which is also termed bond in physics, link in computer science, or tie in sociology. Links are coined directed if they have an orientation, whereas bidirectional links are termed undirected. A set of connected describes a graph. Depending on the context, expressions may be used interchangeably throughout the subsequent research.
- 8.
A nonlinear state space model is a mathematical representation of a nonlinear dynamical system that accounts for the state of the system, as well as its past and future development. In other words, the state space is a set of all states a system can be found in. If the state space and the transition paths between the states are known it is possible to follow the dynamics of the system and to treat its behavior not only deterministically, but also in a probabilistic manner, which allows one to calculate and to understand time dependent physical properties. These are frequently determined as weighted averages over the whole state space, e.g. according to the mean-field approximation.
- 9.
Please note the difference between the network connectivity, i.e. connectivity defined for the whole network, and the general connectivity, i.e. the connectivity of a node. Please confer Sect. 10.2.5.
- 10.
Some authors use this term also for the average geodesic distance in a graph, although strictly the two quantities are quite distinct.
- 11.
It is important to note that the term clustering is probably misleading as it has also another meaning. A traditional method for extracting community structure from a network is called cluster analysis with a different connotation (Everitt 1974).
- 12.
The largest component is frequently equated with the giant component, although both are only the same in the limit of very large networks.
- 13.
There is much more that could be said about prototypical network topologies, but a complete survey of al1 the material is beyond the scope of this paper. Please confer Barabasi (2002) and Newman et al. (2006) for further details.
- 14.
Please note that from a mathematical topological perspective a circle is a one-dimensional lattice.
- 15.
The Szemerdi regularity lemma states that if a property exists on a random graph this property exists on nearly all other network topologies. It is a derivative of the observation that as every large finite undirected graph can be approximated by a set of structured and pseudo-random parts (Komlós et al. 2002).
- 16.
Please note that in a Small-World Network Analysis the development of the clustering coefficient of a network is compared to the development of its characteristic path lengths for varying rewiring probabilities β (Watts 1999). Moreover, it is important to note that such investigations are not limited to small-world graphs, but can be conducted also for other network topologies in order to investigate vital differences among network topologies.
- 17.
Please note that the rewiring probability is also termed beta. This is the name of the respective variable in the complex networks adoption and diffusion simulator.
- 18.
Please confer (Newman 2000) for further mathematical or physical details on both methods.
- 19.
In other words, a network is termed scale-free if the probability that a node selected uniformly at random has a certain number of links follows a mathematical function called a powerlaw. The origin of the term refers to any functional form f(x) that is unchanged to within a multiplicative factor under a rescaling of the independent variable x. In essence, this implies powerlaw forms, which are the sole solutions to f(ax)=bf(x). For this reason, the terms powerlaw and scale-free are used as synonyms in the following.
- 20.
Please note that this expression is misleading as there is no inherent threshold above which a node can be viewed as a hub. Otherwise, the network would not have a scale-free distribution.
- 21.
In statistics, the YuleSimon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon which is the result of this Yule process.
- 22.
Please confer also the Sect. 10.4.
- 23.
This mechanism is sometimes also termed “the rich getting richer”.
- 24.
There is much more that could be said about the evolution of complex networks, but a complete survey of al1 the material is beyond the scope of this paper. Please confer Barabasi (2002) and Newman (2003b) for further details.
- 25.
Please confer (Albert et al. 1999) for further information on the generalization of the preferential attachment model.
- 26.
Please note that these concepts are not mutually exclusive and collectively exhaustive as they can complement each other in the investigations of diffusion dynamics, e.g. it is possible to investigate phase transitions of a diffusion process which is modeled with an SIR model.
- 27.
Please note that there is much more that could be said about processes on networks, but a complete survey of all the material is beyond the scope of this paper. Please confer (Strogatz 2001; Albert 2001; Newman 2003b; Newman et al. 2006; Kemper 2006) for further details on processes on complex networks.
- 28.
Please note that although the more common word is infectious the standard term among epidemiologists is infective.
- 29.
While some authors also use the term removed implying the possibility that people may die of the disease and are removed from the infective pool, others, studying reaction diffusion processes use the term refractory (Strogatz 1994).
- 30.
Please confer Sect. 10.5.2 for further details.
- 31.
The epidemic threshold exists if a non-zero fraction of the population is infected in the limit of large networks.
- 32.
Please note that this critical mass is different from the definition of an economic break-even of a company.
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Kemper, A. (2010). Complex Networks Theory. In: Valuation of Network Effects in Software Markets. Contributions to Management Science. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2367-7_10
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