Predicting the tipping point of complex systems

While preparing soup, have you ever wondered at what point does a pot of boiling water start evaporating?

Now, we know from basic physics that water can be of two states- liquid or gas.

But is it possible to determine the exact interaction and energy between the billions of water molecules that converts liquid to vapor?

Century-old theoretical physics dictates that we can reduce these billions of interactions and parameters to a very few observables. In this case basically just one -temperature. We can at all times, figure out the state of water when we measure the temperature.

But what about other complex systems such as the weather patterns, ecology, disease states etc.? These systems interact via complicated networks, with many members in each system intricately linked through one another. Is it possible to quantify the state of such complicated systems?

That is precisely the focus of a new study by a team of researchers, including an Israeli applied mathematician at the Bar-Ilan University, Dr. Baruch Barzel, one of the lead authors.

Drawing comparisons [to other systems] similar to water's 'desired' state of the liquid form versus an 'undesired state' of an evaporated form, Barzel says, "A system can be in a desirable state, such as bees for example which pollinate flowers and thrive. Or that bees could be in an undesirable state as is happening over the past decade with the decline in bee population."

Although we are presented with examples of systems residing in either of two states, sometimes the number of states could exceed two. How to reconcile the observations then?

"In order to quantify which one of the states the system is going to be in- meaning either on it's way to extinction or thriving, you need to measure many parameters," remarks Barzel.

System resilience

The team of researchers led by senior scientist, Dr. Albert-Laszlo Barabasi, at the Northeastern University has investigated a central tenet of any complex system behavior- how does a system with inherently many parameters respond to perturbation?

There are several forms of perturbation. For instance, weather patterns are disturbed in light of global warming. In another instance, an ecosystem may be affected if a particular species faces extinction or a new species is introduced in it.

Given such versatile scenarios with a rich repertoire of interaction within systems, in this study, the researchers have addressed a very pertinent problem- Can the systems withstand such perturbations or are they vulnerable? In other words, how 'resilient' is a system and at what stage does it breakdown?

"Metaphorically speaking, what we were looking for is an analogy for 'temperature' in these systems; Basically what are the relevant number(s) that can quantify the state of an complex system and take the millions of parameters that we would need to know and crush them all down into a single parameter that determines- is this system going to be liquid? That's good. Or is it going to be gas?" remarks Barzel.

But real life systems are seldom as straightforward as the two physical states of water. The bigger challenge here is not just analyzing the states of such multi-dimensional systems but also quantifying it. That's where mathematics comes to the rescue! The researchers used mathematical equations to construct models and map the original complex system onto a single dimensional system to accurately predict the system's resilience.

Independent experts have hailed the work of Dr. Barabasi and his team as a very important and useful discovery within and outside the exciting new field of network science. The current work has defied popular belief that differences between biochemical, ecological and other networks (modeled using certain equations) are too large for any single solution to fit so many types.

Dr. Robert May, former Chief Scientific Adviser to the UK government and currently a professor of Zoology at the Oxford University emphasizes the broad impact of such predictive studies, "There has, over the past several decades, been much research on the way natural ecosystems can be threatened by human disturbances.  Likewise, the recent bad behaviour of some major banking systems raises important questions about stability and complexity. Although not explicitly addressed by these authors, their work is in many ways paralleled by efforts to understand how banking systems might be made more resilient and thus better able to handle the kinds of problems we have seen over the past few years in that area."

"Barzel's work claims to bridge these differences by providing an incredibly broad solution to a very important and difficult problem. If experiments back up Barzel's theory, it will be recognized as one of the most important and useful discoveries," says Dr. Neo Martinez, Associate Professor of Ecology and Evolutionary Biology at University of Arizona, not directly associated with this study.

To illustrate the practical adaptability of their theoretical model, the researchers used it to study the resilience of multidimensional technological system- power grids.

The scientists used the loss of power in the Northeastern United States, in the wake of snowstorms every year as a motivation to test their results on the resilience of transformers.

Dr. Jianxi Gao, another lead author at Northeastern University expresses his amazement on the versatility of their model, "That this theory can be applied to AC power systems to predict blackouts, surprises me very much, because AC power system is more complex than the ecological and biological systems."

An independent expert, Dr. Shlomo Havlin, Professor of Physics at Bar Ilan University and not a part of this study, highlights the benefits of such predictive studies for complex systems, which could be highly resilient yet quite fragile, "This paper helps us understand three crucial aspects: 1. What are the different states or phases of a complex system? For instance, the power grid can be at a functioning state or at blackout. 2. What are the precise transition points? e.g., what is the critical load beyond which the power system risks collapse.  3. What structural characteristics enhance a system's resilience, namely what changes in design would make the power grid more reliable? These are very important findings, for which we lacked a theoretical understanding until now."

Tipping point

The scientists posit that systems could be very similar. The moment that they cross the tipping point (in the water example, when water ceases to be in its liquid state and transforms into the vaporized state), it might take some time for the system to die out.

"For instance, it might take an epidemic a year to spread to a global scale or about 24 hours before a cell becomes pathogenic," says Barzel.

But there is always time for intervention and the current study is a call for it!

Now, what comes next? As Barzel puts it, "In the pot of water, it's clear. You turn down the gas. But how do you do it in the case of Zika or Ebola?"

In other words, 'how' to efficiently intervene so as to stem a disease from spreading, cross the tipping point and develop into a global epidemic?

In order to address such pressing issues several research teams are trying to identify and study early warning signals for systems - in the case of boiling water those are the bubbles.

Such studies should serve as a wake up call in identifying emerging problems across various platforms from biology to infrastructure and formulating intervention strategies to create resilient systems.

However, resilience could be a double-edged sword. "Cancer is a resilient condition able to withstand many pressures. In such cases, we need to impact resilience in the negative direction!" remarks Barzel.

The scientists are optimistic however that sometime in the near future, the stock market or flu forecast will be done accurately with the aid of mathematical models akin to the one used in this study. And there will come a day, when such predictions will not be futuristic events any more! We will depend heavily on those similar to our absolute reliance on weather forecast nowadays.

Future applications

So what's next as far as some of the exciting real life applications of this model is concerned? "Network science and applications to brain- that's the upcoming decade. If I am allowed to be a prophet and not a scientist for a second," remarks Barzel and adds that there are some caveats to it.

Ecosystems for instance, can benefit tremendously from similar studies. Martinez expresses the importance of ecology and its studies on networks that determine whether ecosystems continue to support human life with the food and oxygen we need, "These networks range from the networks of pollinating insects that fertilize our crops to networks of feeding interactions among species that move carbon in our atmosphere and store it in soils and fossil fuels."

Ecologists are always keen to explore the dynamic behavior of these networks. And the only way to 'predict' the success or the failure of these networks and their impact on the environment such as global warming is via theoretical models and statistical methods.

"If it passes the tests, Barzel's theory would have significantly advanced ecology and have helped us predict and hopefully better manage the resilience of ecosystems ranging from orchards to fisheries," adds Martinez.

According to May, an understanding of the network characteristics is becoming increasingly important as humanity's impacts on natural systems grow- such as ocean acidity and/or global warming.

As evidenced from this study a theoretical model seems to be a successful predictor of a system's resilience and collapse, yet sometimes its implications could be quite profound.

"Infrastructure just as much it seems to be a matter of lifestyle, in many cases it is life or death. There are fatalities in the Northeast every year due to power outage during harsh winter storms. It's more than just the matter of a quality of life sometimes, it's sometimes just a matter of life!" reflects Barzel.

Image credits: (Anthony Rossbach)


Edited: May 12, 2016 (Included comments by Dr. Havlin and Dr. May)

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