Abstract
Complex systems are characterized by processes that exhibit feedback, nonlinearity, heterogeneity, and path dependencies, and accurately modeling such systems is becoming increasing important. To help realize the potential of complex systems modeling we need new methods that are capable of capturing the dynamical properties of such processes across disciplines and modeling frameworks. This chapter presents a portion of the methodology development that includes formal and domain-agnostic definitions of phenomena related to tipping points, criticality, robustness, and sustainability. For each included concept I provide a probabilistic definition based on a Markov model generated from time-series data in a specific way. These rigorous mathematical definitions clearly distinguish multiple distinct dynamical properties related to each concept, and they also function as measures of these properties. Though only a small portion of the methodology’s capabilities, theorems, and applications can be included in this treatment, it does include all the foundational material necessary to apply the methodology.
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- 1.
According to Google’s Ngram viewer, the term “tipping point” appeared in English-language books (within Google’s scanned corpus) steadily between 0.000002 and 0.000004% between 1976 and 2000 (although it saw a spike to 0.00001% in 1975). From 2000 to 2007 its usage increased steadily from 0.000004 to 0.000029%—more than a seven-fold increase.
- 2.
As a rough measure of the increased prevalence of robustness considerations, Google’s Ngram viewer reports that the term “robustness” increases in usage in English-language books from 0.00002% in 1956 to more than 0.0002% in 2006 (with a slight drop since then)—a ten-fold increase. “Sustainable” quite remarkably goes from nearly 0.0 to 0.0014% over the same period (also dropping recently).
- 3.
This will be true of the Markov representation even if the data is event-driven. The time interval is a resolution parameter of the Markov representation building process. Time-homogeneity can be relaxed with the appropriate modifications to the measures and algorithms.
- 4.
As is defined formally later, stability refers to a tendency to self-transition.
- 5.
Recall that many of the things that can be included as aspects of states are not numeric parameters, and so what counts as a “value” for that aspect is meant to be interpreted broadly.
- 6.
The value equals the iterated sum of the previous transition’s criticality and the product of the transition criticality with the previous transition’s criticality’s complement.
- 7.
In this case it does not matter whether the sum is limited to span over neighbors or all the vertices because \(P_{ij} = 0\) for \(S_j\) that are not neighbors. This convention will be used throughout—including cases where limiting an operation to neighbors matters.
- 8.
The phrase “punctuated equilibrium” is in quotes here because I will use it in a more formal sense in Sect. 6.4, and though the variable changes also have the feature of being stable for long periods of time between abrupt changes, the internal dynamics within an equivalence class for these properties probably do not satisfy equilibrium-like conditions.
- 9.
There must be at least one attractor per system, but that attractor may be an orbit consisting of every state in the system.
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Bramson, A. (2019). Formal Measures of Dynamical Properties: Tipping Points, Robustness, and Sustainability. In: Carmichael, T., Collins, A., Hadžikadić, M. (eds) Complex Adaptive Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-20309-2_5
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