While we continue to write on many diverse topics related to complex phenomena and socio-economic systems, some of the topics are related to some of the core concepts in this research direction. We would like to use this page to briefly introduce some of them. A full list of topics can be found here.

Agent-based modeling

Some of the systems we study are so complex that they cannot be easily studied using traditional reductionist approach. The systems are special, because interactions between the constituent parts of these systems play crucial role in the observed dynamics. These systems can not be understood as a simple sum of their parts.

Human body is one of the examples of such systems. You can hardly understood what it means to be human by studying our hearts and brains. These studies will also provide only minor hints to how human body works. In order to understand the inner workings one needs to see how each organ interacts with other organs. Though it is worth to note that human body is at the same time perfect and slightly inapt example of a complex system. It is so, because that each internal organ has it’s own differing function and thus somewhat different structure.

Individuals, e.g., within financial markets in this sense are different. At least from the bird’s eye view they appear to be identical as they all perform similar function while participating in global exchange of money and goods. It is thought that essential properties of time series from financial markets could be explained by studying complex interactions of somewhat similar and somewhat different individuals within financial markets. Evidently these ideas can be applied while studying other socio-economic systems.

It is possible to explore the socio-economic interactions through the lens of agent-based models. In these models we replace the individuals from the real life socio-economic systems with abstract entities called agents (hence the term for this framework). These agents act based on predefined rules and sometimes a more complex behavioral patterns emerge.

Physicists study agent-based models to understand the ways complexity emerge. While social scientists are tempted to test their behavioral theories in this artificial playground.

Cellular automata

One could see Cellular automata as precursors of agent-based models. Here we also have generalized objects, cells, which behave according to very simple rules. Usually these cells are arranged in a 1D or 2D grids and react to the states of their neighbors. Despite these limitations Cellular automatons exhibit rich and complex behavior.

Rich enough for Stephen Wolfram to write a book "A New Kind of Science" and to propose an idea that nature is simplistic and apparent complexity is just impression of the observer. Within the book Wolfram rediscovers so-called simple programs, which were previously known as cellular automata. Those programs, as the nature itself, as proposed by Wolfram, are by definition very simple, but their evolution might be very complex and in some cases apparently random. In this sense cellular automata appear to be very similar to stochastic processes.

But there is one significant difference – cellular automata are most usually deterministic! Those simple programs could have no random noise within them (usually only input), but they would still be able to evolve chaotically. In this sense cellular automata seem to behave very similarly to dynamical systems exhibiting dynamical chaos. Though let us remind you that cellular automata are by definition simple programs in contrast to complex mathematics behind the dynamical description of systems.

Cellular automatons are fascinating topic, because they are excellent examples how simple rules can lead to dynamics reminiscent of solutions of very sophisticated dynamical equations.

Network models

Human society is not a grid of people. Nor it is a hierarchical ladder. But how does its structure look like? In the middle of XIX century Auguste Comte proposed to analyze society in micro level, where each individual would be represented some object, e.g., agent, and its relation to the other objects. After nearly half of century ideas of Comte found their place in the first works about social network theory, mostly by G. Simmel and J. Moreno. In his works J. Moreno widely used sociograms. These useful charts are now known as networks. Why are they useful? Well as J. Moreno wrote:

with these charts, we have the opportunity to grip the myriad of networks of human relations and at the same time, view any part or portion which we desire to relate to or distinguish.

Theory here once again looks very promising, but the problem is that we will never know precisely how actual social networks look like. There are varying reasons for it – it is impossible to survey every human being, while polling only random ones will give no useful information about network structure at all, finally social networks are constantly changing at every moment. Though we can gather some limited data, which could give insights towards how social networks are forming and by using models we could create social networks with similar properties to real social networks. Currently network modeling is very active research field with wide applications in varying scientific fields – from epidemiology (prevention and control of diseases), sociology (selection of individuals for polls) and marketing theory (how to efficiently promote new products?).

Stochastic models

All earlier topics cover the modeling of the microscopic details of human behavior. Macroscopic dynamics in these models emerge from microscopic rules, but in some applications simple description of macroscopic dynamics is more useful than a sophisticated microscopic models.

Usually macroscopic dynamics of complex systems could potentially be described using Stochastic Calculus. Yet most of the complex systems are well described by non-linear equations. Thus those systems (and their models) exhibit dynamical chaos – minor error in initial conditions causes system evolution to deviate from initial predictions. Typical example of such unpredictability is found in weather forecast – due to Butterfly effect even modern supercomputers can give credible week-long predictions of weather. But there is set of non-linear systems within which even minor impact from outside, or let it be error in initial conditions, can cause major and momentary change in observed behavior. Typical example of this kind of system might be financial markets.

It is convenient, in case of sensitive systems, to assume those impacts to be random. In such case it is said that system has stochastic nature. Thus models of such system are called stochastic models. Those models are usually implemented using Stochastic Calculus, namely stochastic differential (usually Langevin) equations.

Dynamical chaos

As we have mentioned non-linear stochastic models are in a certain sense similar to sophisticated dynamical systems, which exhibit dynamical chaos. But what dynamical chaos actually is?

Physicists love mathematics. Especially and most usually differential calculus. Mathematical description of any system using differential calculus is usually referenced as dynamical description. Thus in this case equations themselves are called dynamical equations. Purely theoretically it appears that if we know precise form dynamical equations we know everything about the system, we can make precise predictions of its evolution.

The problem is that this impression is only theoretical. In practice one must take measurements of reality. As we all know it measurements are just approximation – every measurement has some error related to it, no matter how small or apparently insignificant it appears. Measurement bias in the linear systems may play no role at all – system evolves as predicted with small or no corrections at all. But most of the interesting systems are not linear! In fact most of them are strongly non-linear. Non-linearity in dynamical equations may cause accumulating differences between solutions with apparently insignificant variation in initial conditions. Thus over the time in non-linear system prediction of evolution and observed evolution may start to disagree by far more than just by primary measurement bias. This behavior is known as dynamical chaos.

Most widely known example of dynamical chaos is the Butterfly effect – butterfly flapping its wings may cause hurricane on next week, but thousand kilometers away. In this section of Physics of Risk blog we aim to show that even smallest bias of initial conditions, of reality itself, might be the reason for unexpected results and chaos.


Last bit of complexity seems to be unrelated to all over bits we have discussed previously. Yet this first impression is wrong - usually models of complex systems and complex systems themselves exhibit fractal-like properties.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line ...

were the words using which B. Mandelbrot attempted to lure his readers into the fractal geometry of nature. This of course is an obvious truth, but the message behind it is truly fascinating. So what is fractal and what significance does it have?

Word fractal originated from Latin word fractus (en. “composed of pieces”). Thus fractal ought to be composed of smaller parts. And thus the most fascinating property of fractals lies within it, as each smaller part fully or partially resembles every other smaller and larger parts. Mathematicians call it self-similarity and it’s inherent property of any fractal.

Self-similarity of fractals seems to be appealing to both – scientists and people far from science. Former group are interested in fractals due to purely scientific reasons – fractals give philosophical insight into the underlying rules of nature. Some of philosophical ideas resemble ideology behind the cellular automata, while some relate to the ideas of non-linearity within dynamical chaos. Mesmerizing beauty of fractals also attracts ordinary people, some of who see fractals as a form of art.

Fractals will mesmerize you with the beauty, both visual and philosophical.