One of the oft-cited features of complex systems is their ability to adapt to environmental changes and shocks. This is often contrasted with human-made engineered systems which are typically specialized and optimized to work in only the limited conditions for which they were designed. The point being underlined in these discussions is that complex systems are self-organizing (and often self-perpetuating) and thus their behaviors are contingent on inputs in ways that purpose-built systems are typically not. What these discussions often leave out is the crucial fragility that many complex systems exhibit to specific inputs and disruptions.

I am not saying that those claiming that complex systems are often robust are incorrect, only that 1) a system can be robust to some shocks and vulnerable to others, 2) robustness is neither sufficient nor necessary for complexity, and 3) these claims (typically) use a loose and overly broad meaning of robustness to make them seem valid. I'll address these one at a time.

For the robust/vulnerable dichotomy we can take the human body as clear example of a complex system. It is extremely robust to certain insults, like the removal of a limb or a lung, which are large and disruptive, but can be dealt with. On the other hand, there are some specific regions in the brain and heart where the removal of just a cubic millimeter of tissue results in the immediate death of the individual. Given an average body mass of 70kg, that's just 1/70,000,000 of the total volume. That sounds like a fragile system to me. We can contrast that to a society of 70 million people that can survive the loss of any, and perhaps any quarter, of its population.

We can say that a system is robust with respect to some property (like being alive or perpetuating) when that system has that property over a large proportion of changes; i.e. the overall likelihood that the system will have that property is high. And we can say that a system is vulnerable to a particular change if the likelihood of loosing that property is high for such changes (and fragile if it vulnerable to at least one change). And from these definitions it is clear that a system can be both robust and fragile (with respect to some property). However, this does not mean that every system is equally robust and vulnerable…or robust and vulnerable through the same sorts of changes. Differences in robustness and vulnerability characteristics can be used to establish equivalence classes of system dynamics and thus allow us to categorize systems (or at least models of systems) by how they change over time.

Instead of offering exact criteria for systems to be complex, the style has shifted to listing some common features of complex systems, and features of systems that make them complex. Robustness is typically on these lists, but vulnerability is typically not. Part of the reason is political; practitioners of complexity science want to make it sound good. A better reason is that it's the complexity of the system that makes it robust, but systems are vulnerable where the complexity fades. For example, brains and hearts and eyes are very specific, optimized organs that are more like designed systems and less like self-organizing systems than other tissues, and that's why they suffer from single-point failures.

This is merely an unsatisfactory ad hoc rationalization to justify the connection between complexity and robustness; i.e. it's a political explanation. The truth is that at the current state of the art, robustness applies to some complex and some non complex systems, and fails to apply to others. Until some specific mechanistic connection between the complexity of systems (measured somehow) and their robustness and vulnerability characteristics is established in a general way, the claims are mere rhetoric. And furthermore, once this work is done, I'm sure we will find that robustness is neither a necessary nor sufficient feature for systems to be complex (or vice versa). It is a contingent property, and working out when systems are and are not robust will be interesting and difficult research.

Finally, the claims relating robustness to complexity often make unfair use of the lack of specificity of the two terms. My research on formalizing robustness has revealed that there are many distinct meanings that, though related, indicate very specific dynamics. A property can be maintained without being lost, regained if lost, obtained if not present, or present over a large proportion of possible futures (among others). Saying that a system is robust, without specifying the way in which it is robust, offers little to no insight into the relationship of that property and system dynamics.

Complexity is even more vague, and identifying what characteristics make systems complex is fraught with confusion. Something like: collections of at least an intermediate number of components that interact in such a way that self-organization and/or adaptation is possible in some situations. If that's the kind of thing that people mean by "complex system", then certainly it is folly to claim that such systems will be robust in general, or that robust systems will be complex in the sense of satisfying that loose definition. My guess is that people who consent (or acquiesce) to claims that complex systems are robust will retract their support when expressed in these more detailed (but equally vague) terms.

So it seems that under closer inspection complex systems are no more characterized by their robustness than their fragility, contrary to many unfounded claims made by practitioners of complex systems modeling, and expounders of complexity theory. This makes our whole field seem unscientific and foolish. For the field to mature we need specific, testable, and falsifiable claims about the relationships of system properties like interaction structure, scale, rates of change, behavior rule types, etc. and properties of system dynamics such as tipping points, robustness, path dependence, and adaptability. I've started on some of this work with my measures of some of these phenomena, but the scale of this project is science-wide. Building and using the appropriate formal tools is an important first step, but it is even more crucial to convince people of the importance of thinking and speaking clearly on these issues.