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Network representation and complex systems

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Abstract

In this article, network science is discussed from a methodological perspective, and two central theses are defended. The first is that network science exploits the very properties that make a system complex. Rather than using idealization techniques to strip those properties away, as is standard practice in other areas of science, network science brings them to the fore, and uses them to furnish new forms of explanation. The second thesis is that network representations are particularly helpful in explaining the properties of non-decomposable systems. Where part-whole decomposition is not possible, network science provides a much-needed alternative method of compressing information about the behavior of complex systems, and does so without succumbing to problems associated with combinatorial explosion. The article concludes with a comparison between the uses of network representation analyzed in the main discussion, and an entirely distinct use of network representation that has recently been discussed in connection with mechanistic modeling.

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Notes

  1. Canonical sources for these developments can be found, respectively, in Rual et al. (2005), Christakis and Fowler (2007), Pastor-Satorras and Vespignani (2001), and Supekar et al. (2008).

  2. If we interpret “network science” loosely, its history is much longer. Sociologists have applied small networks to social phenomena since the work of Moreno in the 1930s (Easley and Kleinberg 2010), and operations researchers made use of graph theory in WWII (Cunningham et al. 1984). The very large-scale work described here, however, requires computational power that has only been available more recently. It should also be noted that the more recent history of network science has had a much broader impact than early work in the field. In 2008, the Thompson Reuters publication “Science Watch” ranked the Watts and Strogatz paper as the 6th most cited paper in the history of physics. On August 9th, 2014, the Thomson Reuters Web of Knowledge (apps.webofknowledge.com) shows that the paper has received 10,308 citations.

  3. The term “long-range” refers to path-length in graph-theoretic, rather than physical space. The idea can be expressed intuitively in terms of a social network. In a graph of my social network, the set of all my friends are represented as being one unit removed from me. The set of all the friends of my friends (who are not also friends with me directly) are represented as being two units removed from me, and so on. A long-range connection is one that directly connects two nodes that would otherwise be separated by a long path, relative to the global average.

  4. The ring-format of the lattice was chosen for the computational convenience of periodic boundary conditions, and is not meant to have any representational significance.

  5. They also studied how the structure of networks influences the interactions of models of rational agents in a Prisoner’s dilemma scenario. Although the results of this initial work also led to a huge amount of subsequent study, I focus on the epidemiological aspect of their work to avoid the special complications associated with models of rational decision-making.

  6. The form of these equations is slightly more complex than the one actually implemented by Watts and Strogatz. I have chosen to write this version of the model both because it is the most widely recognized version of the SIR model, and also because it allows me to define a biological parameter in Section 3.5, where I argue that the network epidemiology is reasonably autonomous from certain biological details. It is also worth noting that I am following epidemiological practice by referring to the model parameters as “rates.” In fact they are just proportions and do not literally have a temporal component.

  7. Some texts use the term “population structure” to describe the disease topology of biological population. I prefer the term “contact structure” because the set of relevant connections—instances of physical proximity between two organisms that make transmission biologically possible—will be different for each population/disease pairing. “Contact structure” does a better job of avoiding the misleading implication that there is a single structure associated with each population.

  8. Specifically, it is about \(ln(n)/ln(k)\), where \(n\) is the number of nodes in the graph, and \(k\) is the number of initially connected neighbors.

  9. I do not mean to presume that all causal explanations are mechanistic. I discuss causal explanation in this section, and take up a comparison with mechanistic explanation in Sect. 6.

  10. Jones (2014) presents a similar view. However, the focus of that article is on the way that diagrammatic representations facilitate inferences about counterfactual structure. In large networks, very few interesting properties are detectable by means of visual inspection. What makes the representations useful is rather the fact that they can be studied algorithmically.

  11. It should be noted that Huneman is sensitive to the fact that topology and graph theory are distinct mathematical disciplines, but chooses to use the term “topological” in a broad sense that covers both disciplines.

  12. Although I am borrowing the terminology from Carl Craver’s well-known article “When mechanistic models explain,” I do not mean to imply that the small-world model should be counted as a mechanistic. Reasons for resistance are discussed in Sect. 6.

  13. This freedom from Galilean idealization is one reason why network representations can be viewed as epistemologically distinctive. The computational methods used to study them provide a kind of understanding that at least partially breaks free from the processing limitations intrinsic to human cognition.

  14. It is important that many epidemiological models assume constancy of infection time. If the target population had, for example, a bi-modal distribution of infection times in which one cluster of organisms is disposed to become infected quickly and another is disposed to become infected slowly, it is likely that some of the collective dynamical behavior could be explained on the basis of a decompositional strategy.

  15. The mathematical measure is sometimes called betweenness and sometimes called betweenness centrality. I use the term “betweenness” to refer to the mathematical property and the term “road centrality” to refer to an empirical property of a road.

  16. As a matter of fact, Levy and Bechtel mention the Watts and Strogatz model as well as the concept of a small-world, apparently because the concept has played such a major role in helping to popularize network science. But they quickly set aside such “large-scale statistical models” and explicitly restrict their discussion to much smaller and non-statistical models.

  17. Thanks to an anonymous referee for alerting me to this article, which has a lot in common with the discussion in this section. Given the overlap in emphasis, it is worth noting three points of contrast. First, the Silberstein and Chemero article is aimed exclusively at explanations in biology and neuroscience. My aims are much broader. Second, it is my view that network models have the best chance at explaining in cases in which the individuation of nodes is unproblematic and can be relocated to the background (this point is discussed below). In systems neuroscience—which receives the bulk of the attention in the Silberstein and Chemero article - the task of individuation is often quite controversial. This is not to say that examples in the literature are inadequate, but to suggest that there are explanatory hurdles in neuroscience over and above those typically present in the social and epidemiological sciences. Third, Silberstein and Chemero are interested in showing that the style of explanation they see in systems neuroscience is something “more lawlike” than what the mechanists propose (p. 970). In my view, the relation between law-like and mechanistic forms of explanation is not mutually exclusive. The explanatory status of the cases I have described depends neither on a commitment to predictivism nor on the role of nomological necessity. Rather, they are cases in which we have empirical justification that a mathematical model accurately represents a particular empirical phenomenon. The accuracy of the representation justifies certain explanatory inferences. The question of how mathematics matches up with reality is both more general and more difficult than the question of how network representations explain. Good discussions of this topic that conform to the perspective defended here can be found in Pincock (2012) and Humphreys (2014).

  18. In fact, a position like this was anticipated by Bechtel and Richardson two decades ago. Describing a continuum of cases in which the pattern of interaction among elements plays an increasingly greater role in understanding system behavior, they conclude: “If organization becomes even more dominant in explaining the behavior of the system, and we appeal less to different and distinctive functions performed by the components, we reach a point where decomposition and localization have to be surrendered” (Bechtel and Richardson 1993, p. 199).

  19. Baker also argues that one of the dangers associated with network science is the unwarranted reification of networks. The fact that network representations are explanatorily useful does not entail that network-like empirical phenomena constitute a natural kind. This is one reason that I have stressed the epistemic, rather than ontological contribution of network science. In areas such as molecular biology, neuroscience, and perhaps even ecology, it is reasonable to treat networks as real representation-independent features of the empirical world. In many other areas, however, including those discussed in this article, it is far from clear that this realist attitude is appropriate. In any case, arguments for such a view would require premises that are not clearly supported by the present discussion.

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Acknowledgments

This study was funded by the National Science Foundation (award number 1430601). The author would like to thank Dr. Paul Humphreys for providing extensive comments on multiple drafts of this article.

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Correspondence to Charles Rathkopf.

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Rathkopf, C. Network representation and complex systems. Synthese 195, 55–78 (2018). https://doi.org/10.1007/s11229-015-0726-0

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