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Literacy: When Is a Network Model Explanatory?

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Network Analysis Literacy

Part of the book series: Lecture Notes in Social Networks ((LNSN))

Abstract

Network analysis relies strongly on network models for several reasons: they show that a certain structure can be generated by a set of simple rules, they predict certain behaviors and functions, and they can be used as null-models. In this chapter, the question discussed is whether the classic network models are likely to be explanatory network models for complex systems like metabolic networks, the internet , or collaboration networks.

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Notes

  1. 1.

    To follow common usage of terms, the term ‘random graph model’ will be used for random graph families used as a null-model, i.e., where it is expected that a real-world network deviates from randomness. The term ‘network model’ will be used for random graph families used as a model for the generation of a complex network in a complex system.

  2. 2.

    Note that many papers citing Watts and Strogatz’ paper actually treated the small-world model as an explanatory one, but the authors do not seem to have had this in mind when they proposed it.

  3. 3.

    The original statement is often put in a more general setting by referring to “hypotheses” instead of “models”. The statement itself has been made by many authors in various different forms and may never have been stated in any of these forms by Ockham himself. As he applied it in his scientific research, it is today best known as “Ockham’s razor”.

  4. 4.

    See a short video interview with Dyson on this story here: http://www.webofstories.com/play/freeman.dyson/94.

  5. 5.

    While this statement by von Neumann was just a pun with some serious background, in 2010 Mayer et al. have indeed managed to draw an elephant-like shape with complex equations using four parameters and a different but still recognizable elephant-like model based on five parameters, who wiggles its trunk [33].

  6. 6.

    He also calls them network flows or network flow processes.

  7. 7.

    With respect to Weisberg’s classification, the equations of classical mechanics are still useful as a model of the real-world but with a smaller intended scope and new fidelity criteria, that guarantee their usage only in appropriate situations.

  8. 8.

    I think it is folklore that anyone ever really assumed that real-world networks are random, i.e., whether anyone used random graph models as an explanatory model for a real-world phenomenon. E. Fox Keller cites Barabási and Bonabeau with the sentence: “science treated all complex networks as being completely random” [20, as cited by]. This citation stems from an article in Scientific American and is clearly an exaggeration that emphasizes that nobody saw the need to create more elaborate graph models than the random graph model. However, I am not aware of any serious, scientific application of a random graph model as an explanatory model of a real-world complex network and a quote like “For decades, we tacitly assumed that the components of such complex systems as the cell, the society, or the Internet are randomly wired together.” [2, p. 412] has to be clearly rejected.

  9. 9.

    For the computer scientists among the readers: the question is to find the minimum edge cover in an online fashion.

  10. 10.

    It is clear that in essence, all pairs of proteins have to be tested to be sure that all edges have been explored. The question is thus how far one can get with as few baits as possible.

  11. 11.

    It is unclear how pairs of unconnected nodes were treated.

  12. 12.

    Note again that for networks with such small ranges of the degree such as this one, it is not really possible to analyze which of the many right-skewed distributions fits best.

  13. 13.

    The authors thus do not quantify the possibly increased number of hops necessary to transfer a message but only the question of whether the message can still be delivered.

  14. 14.

    Already in 2002, Barabási et al. showed on the example of a dynamic scientific collaboration data set that the average degree can increase over time [5].

  15. 15.

    While this model ‘heals’ the problem and produces more realistic network structures, it needs again to be checked whether this model is explanatory for any given system of interest.

  16. 16.

    See the discussion of whether this data set really captures the evolution of the Internet well in Sect. 10.6.1.

  17. 17.

    Note that statistical tests might not agree with this.

  18. 18.

    Krapivsky et al. state that it is actually difficult to tell the two distributions apart; for that case as for finite networks the difference is small. Still, a linear preferential attachment mechanism can, in general, be more easily explained and that is why any deviation from it raises curiosity and induces new hypotheses.

  19. 19.

    As explained in the paper by Chen et al., the data on which the analysis of both, Jeong et al. [19] and Chen et al. [8], is based, is incomplete. For more information on this, see Sects. 10.3. Another problem is that due to the noisy nature of the data, it needs some care to determine whether a new AS came into the network or whether this AS was just not contained in the last measurement due to outages or other problems. The authors thus defined an AS to be new if it was not contained in any of the preceding data sets.

  20. 20.

    The assumptions are explained by Simon on the example of the word frequency distribution. The first assumption is that a word is taken with a probability linear to the total occurrence of all words with the same frequency sofar (“linear preferential attachment”), the second assumption is the existence of a constant probability for taking a totally new word, and the third makes an assumption on the rate with which words are dropped.

  21. 21.

    Some also changed it to: “no one should be without a beer because they cannot afford one...”.

  22. 22.

    For hard problems like the graph coloring problem there is often a set of so-called benchmark problems on which new solving strategies can be tested. Mainly, these are especially difficult instances of the problem.

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  1. TU Kaiserslautern, FB Computer Science, Graph Theory and Analysis of Complex Networks, Gottlieb-Daimler-Str. 48, 67331, Kaiserslautern, Germany

    Katharina A. Zweig

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Zweig, K.A. (2016). Literacy: When Is a Network Model Explanatory?. In: Network Analysis Literacy. Lecture Notes in Social Networks. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0741-6_12

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  • DOI: https://doi.org/10.1007/978-3-7091-0741-6_12

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