Your grade ten math teacher probably wrote this several times on your tests: SIMPLIFY. And, for much of science, that’s part of the work: SIMPLIFY. The universe can be broken down into smaller and smaller chunks in an attempt to find its most basic level and functions. But what do you do when that doesn’t work? Complex systems that defy reduction are all around us, from the elaborate workings of an ant colony—which could never be predicted from the physiology of a single ant—to fluctuations in the financial system that can send ripples around the globe. When broken into their constituent pieces, examined and put back together, such systems do not behave as expected. The sum of the parts does not equal the whole.

Surprisingly, the best way to analyze these decidedly large-scale systems may be by exploiting techniques first developed not in biology or economics, but in microscopic physics. Raissa D’Souza, a complexity scientist at UC Davis and an external professor at the Santa Fe Institute, is applying lessons learned from studying how physical systems go through phase transitions—becoming magnetized, for instance—to try to predict when everyday networks will go through potentially catastrophic changes. Her work has implications for the spread of disease, sustaining the energy infrastructure, the financial health of countries, and for the way we connect with our friends in online communities.

While completing her PhD in statistical physics at MIT in the 1990s, D’Souza became fascinated with complex systems and the behavior patterns that emerge from them. Since she did not know of anyone who specialized in the subject, she went to the library and searched the entire Boston area for someone who did, before finding Norm Margolus, who it turned out was handily also at MIT and with whom she studied pattern formation and computing in natural systems. D’Souza’s background in statistical physics introduced her to the prototypical phase transition. It considers a collection of atoms, each with a magnetic moment, that could either line-up with each other—so that the overall system becomes magnetized—or remain in a disordered mess. There is a tension in this case: on the one hand, the atoms want to line-up, lowering the system’s energy; on the other hand, the laws of thermodynamics tell us that systems prefer to move to a state of increasing disorder, mathematically expressed as having a higher entropy. It was first discovered experimentally that the outcome depends on temperature. At high temperatures entropy rules, the atoms remain disordered and the system does not become magnetized. But below some critical temperature, the system undergoes a phase transition and the atoms align.

That sounds simple enough, but the phase transitions that change a system’s behaviour so profoundly are often unpredictable, especially if you only study the system in its smallest components. How for instance, could you predict what the critical temperature would be, in theory, if you only focus your attention down onto one isolated atom? Instead, you’ve got to see the big picture. And sometimes that picture is very big.

The Power of Networking

Taking a step back, D’Souza sees everything as being interconnected. Her job is to work out when linked objects or entities will go through profound phase transitions, which could lead to a negative (or sometimes positive) outcome. For instance, the United States power grid was once a collection of small isolated grids, made of a few powerplants run by some municipality or corporation. Then, local grids were connected to create state-wide and regional grids that remained largely independent. Distinct regions were then interconnected to allow power transfer in emergency situations. But, with deregulation, those interconnections now transfer massive amounts of power bought and sold on power auction markets each day. As D’Souza points out, this interdependence has changed networks in ways that were originally never intended, leading to unforeseen bad consequences. The U.S. power grid has grown to a point where it has seemingly encountered a phase transition and now occasionally suffers from large cascading blackouts, where a problem in one area can drag many areas down. Worse, a failure in one network can actually drag down many different networks. So a failure in the power grid can cause a failure in the telecommunications grid which causes a failure in the transportation grid and the impact keeps rippling through time and space.

Speaking at FQXi’s Setting Time Aright meeting, D’Souza discussed the conception of time that emerges from considering examples of interconnected networks in terms of complexity theory:

"In the last 3-4 years, I’ve been working to take the ideas from single networks, such as the structure of the network and the dynamics happening on top of the network substrate, and extending it to this bigger context where we have networks of networks," says D’Souza. "I’ve been trying to understand what it does to emergent properties like phase transitions and what it means for the vulnerability of these systems. So, if there is a small ripple in one layer can it go into other layers and how long does it take?"

Understanding how networks interconnect and evolve has huge implications for public health, for instance. D’Souza cites the example of pandemics, where infection rates have changed drastically over time based on advancements in our transportation networks. In the Middle Ages, the bubonic plague took years to spread across Europe, for example; by contrast the Spanish flu pandemic of 1918 killed over 20 million people across the globe, taking only a matter of weeks or months to spread. But now, with the arrival of mass air travel, it only takes hours for SARS, swine flu or bird flu to reach new countries.

Pinpointing the critical point of a phase transition is not easy in the world of networked networks, but part of D’Souza’s work has been to find a generalised model, or set of equations that will apply to many different examples, not just the power grid or the transportation network. In February 2012, D’Souza and colleagues published a paper in the Proceedings of the National Academies of Sciences (PNAS) in which they analysed such a universal model and predicted where the optimal level of connection and interdependence would be—and that, ultimately, too much connectivity would be detrimental.

There are drawbacks to basing your mathematical analyses on equations inspired by mathematical physics that are usually used to analyse the collective behavior of atoms and molecules, however. Such statistical equations usually work by considering a collection of around 10²⁶ atoms (that’s 10 followed by 26 zeros, Avogradro’s number). By contrast, even the biggest real-world networks today only get up to about a billion (10⁹), which makes it difficult to take theoretical predictions from the equations and apply them directly to real-world networks. Nonetheless, independent network scientists aiming to forecast financial crises have found intriguing evidence that backs D’Souza’s theoretical predictions about interdependence and phase transitions.

Financial Contagion

Soon after D’Souza’s PNAS paper appeared, Stefano Battiston, at ETH Zurich, and colleagues published an independent study in the Journal of Economic Dynamics and Control that investigated the dominant view in finance that diversification is good. The idea is that it is better to spread your money around, so that even if one investment goes bad, you will still win overall. However, Battiston’s group found that diversification may not be the best strategy. In fact, they calculated that it could actually spread "financial contagion."

What Battiston’s group realized was that a naturally occurring financial mechanism known as trend reinforcement was enough to push a single failure through the entire system. Trend reinforcement works through a rating body that evaluates an entity’s performance. In the financial case that Battiston’s group evaluated, when the market was disappointed by a company’s returns, they downgraded that company, which resulted in additional selling, which caused the company to underperform and disappoint the rating body again. This negative cycle and penalization altered the probability of the company doing well and magnified the initial shock. Furthermore, they found that if the shock became big enough, then a phase transition would occur, as D’Souza hypothesizes, allowing the shock to travel through the entire system.

"There are some benefits in diversification and connections of course," says Battiston, "but there are serious threats that come from connecting everything in a single system that behaves in synchrony because recovering from a complete collapse is obviously very costly, not just economically but socially as well."

Extending their tentacles beyond the financial world, networks can also help expose the way politicians and nation-states act. Zeev Maoz, a political scientist at UC Davis and a distinguished fellow at the Interdisciplinary Center in Herzliya, Israel, has found that geopolitical networks have significant spillover to other networks, for instance security and trade. Importantly, Maoz has also shown that nations are not connected equally; often smaller states are connected through more central players. So, you get a situation where there are a few major players each with a large cadre of states connected on their periphery, and this can be destabilizing.

"The uneven structure is a cause of potential instability because if everyone is connected to a few major partners and the major powers experience a shock then everyone suffers," says Maoz. Unfortunately, there aren’t any levers that can help mitigate a shock like that because of the nature of connectivity, he explains. Take for instance Greece, which is dependent on Germany and France and the United States. If a shock because of the recession hits the big players, then Greece suffers more than Germany, France, or the USA, because Greece is dependent on them and does not have trading partners of its own.

Complex Conceptions of Time

All these studies converge on one conclusion: complex systems are, fittingly, even more complex than first thought. Complexity theorists have long accepted that you cannot just look at components and understand the whole system—their discipline is based on that assumption, after all. But now complexity scientists have learned that you cannot even look at a single system and understand it without the context of all the other systems it interacts with. "So we’re at the point where we can begin to analyze systems of systems," says D’Souza, each evolving on its own timescale, with feedbacks between these systems. Take, for instance, online social networks that evolve much faster than say social norms or transportation networks. Users of Facebook or Twitter typically develop a web of "friends" or "followers" that extends well beyond the number of people they would have time to physically meet up with and interact with face-to-face, says D’Souza: "How do we characterize time in these disparate systems?"

At first sight, the ability of online social networks to bring people around the world closer together and shrink the time that it takes to interact may seem like an unambiguously positive thing. But even social networks are vulnerable to phase transitions, so D’Souza urges caution: At some point that connectivity might backfire and potentially cause the network to collapse. "Maybe we will find that Facebook becomes a great big mush and isn’t interesting anymore because there is no way to differentiate who is a true friend and who is someone that you used to know 20 years ago and you’re just overwhelmed with information," D’Souza says. "That could be one way that a network like Facebook could fail. It could break down if it got so dense that you couldn’t distinguish meaningful information from noise anymore."

And your number of Facebook friends is only going to increase, according to D’Souza. In fact, she believes that to be almost a rule in thinking about networks of networks. "I firmly believe networks become more interdependent in time," says D’Souza. "We see the global economy becoming more interdependent. We see Facebook making everyone more interconnected. We’re relying increasingly on technologies like the Internet and communications networks, for instance, the smart-grid, a cyber-physical system. All these networks that used to operate more independently are now becoming more interconnected, and to me that is really a signature of time."