Tag Archives: world population

Nonlinearity in Growth, Decay and Human Mortality

Nonlinearity in Growth, Decay and Human Mortality

Processes of Growth and Decay abound in natural and economic systems. Growth processes determine biological structure and pattern formation, selection of species or ideas, the outcome of economic competition and of savings in financial portfolios. In this post we will examine a few different types of quantitative growth / decay and their qualitatively different outcomes.


In the media we often hear about nonlinear, exponential, or explosive growth as popular references to seemingly unstoppable increases. Buzzwords like “tipping point” or “singularity” appear on book titles and web sites. Mathematical models can help analytical understanding of such dynamic processes, while visualization can support a more intuitive understanding.

Let’s look three different growth processes: Linear, exponential, and hyperbolic (rows below) by specifically considering three different quantities (columns below):
The absolute amount (as a function of time),
the absolute rate of increase (derivative of that function), and
the relative rate of increase (relative to the amount)

Amounts, Rates, and Relative Rates of three growth processes: Linear, Exponential, Hyperbolic

Linear growth (blue lines) is the result of a constant rate or increment per time interval. The relative rate (size of increment in relation to existing quantity) is decreasing to zero.

Exponential growth (red lines) is the result of a linearly growing rate or increment per time interval. The relative rate is a constant. Think accrual of savings with fixed interest rate. Urban legend has it that Albert Einstein once declared compound interest – an exponential growth process – to be “the most powerful force in the universe”. Our intuition is ill-suited to deal properly with exponential effects, and in many ways it seems hard to conceive of even faster growth processes. However, even with exponential growth it takes an infinite time to reach an infinitely large amount.

Hyperbolic growth (brown lines) is the result of a quadratically growing rate. In this type of growth even the relative rate is increasing. This can be caused by auto-catalytic effects, in other words, the larger the amount, the larger the growth of the rate. As a result, such growth leads to infinite values at a finite value of t – also called a discontinuity or singularity.

When multiple entities grow and compete for limited resources, their growth will determine the outcome as a distribution of the resource as follows:

  • Linear growth leads to coexistence of all competitors; their ratios determined by their linear growth rates.
  • Exponential growth leads to reversible selection of a winner (with the highest relative growth rate). Reversible since a competitor with a higher relative growth rate will win, regardless of when it enters the competition.
  • Hyperbolic growth leads to irreversible selection of a winner (first to dominate). Irreversible since the relative growth rate of the dominant competitor dwarfs that of any newcomer.

Such processes have been studied in detail in biology (population dynamics, genetics, etc.) It’s straightforward to imagine the combination of random fluctuations, exponential (or faster) growth and ‘Winner-take-all’ selection as the main driving processes of self-organized pattern formation in biology, such as in leopard spots or zebra stripes, all the way to the complex structure-formation process of morphogenesis and embryology.

Yet such processes tend to also occur in economics. For example, the competition for PC operating system platforms was won by Microsoft’s Windows due to the strong advantages of incumbents (applications, tools, developers, ecosystem, etc.) Similar effects can be seen with social networks, where competitors (like FaceBook) become disproportionately stronger as a result of the size of their network. I suspect that it also plays a central role in the evolution of inequality, which can be viewed as the dynamic formation of structure (viewed as the unequal allocation of wealth across a population).

Two popular technology concepts owe their existence to nonlinear growth processes:

  • Exponential Growth: The empirical Moore’s Law states that computer power doubles every 18 months or so (similar for storage capacity, transistors on chips and network bandwidth). This allows us to forecast fairly accurately when machines will have certain capacities which seem unimaginable only a few decades earlier. For example, computer power increases by a factor of 1000 in only 15 years, or a million-fold in 30 years or the span of just one human generation!
  • Hyperbolic Growth: Futurist Ray Kurzweil has observed that the doubling period of many aspects of our knowledge society is shrinking. From this observation of an “ever-accelerating rate of technological change” he concludes in his latest book that “The Singularity Is Near“, with profound technological and philosophical implications.

In many cases, empirical growth observations and measurements can be compared with mathematical models to either verify or falsify hypothesis about the underlying mechanisms controlling the growth processes. For example, world population growth has been tracked closely. To understand the strong increase of world population as a whole over the last hundred years or so one needs to look at the drivers (birth and mortality rates) and their key influencing factors (medical advances, agriculture). Many countries still have high birth rates, while medical advances and better farming methods have driven down the mortality rates. As a result, population has grown exponentially for many decades. (See also the wonderful 2min video visualization of this concept linked to from the previous post on “7 Billion“.) Short of increasing the mortality rate, it is evident that population stabilization (i.e. reduction of growth to zero) can only be achieved by reducing the birth rate. This in turn influences the policy debates, for example to empower women so they have less children (better education and economic prospects, access to contraception, etc.). Here is a graphic on world population growth rates:

Population growth rates in percent (source: Wikipedia, 2011 estimates)

Compare this to the World maps showing population age structure in the Global Trends 2025 post. There is a strong correlation between how old a population is and how high the birth rates are. (Note Africa standing out in both graphs.)


Conversely one can study processes of decay or decline, again with qualitatively different outcomes for given rates of decline such as linear or exponential. One interesting, mathematically inspired analysis related to decay processes comes from the ‘Gravity and Levity’ Blog in the post “Your body wasn’t built to last: a lesson from human mortality rates“. The article starts out with the observation that our likelihood of dying say in the next year doubles every 8 years. Since the mortality rate is increasing exponentially, the likelihood of survival is decreasing super-exponentially. The empirical data matches the rates forecast by the Gompertz Law of mortality almost perfectly.

Death and Survival Probability in the US (Source: Wolfram Alpha)

If the death rate were to grow exponentially – i.e. with a fixed increase per time interval – the resulting survival probability would follow an exponential distribution. If, however, the death rate is growing super-exponentially – i.e. with a doubling per fixed time interval – the survival probability follows a Gompertz distribution.

Lets look at a table similar to the above, this time contrasting three decay processes (rows below): Linear, Exponential, Super-Exponential. (Again we consider the amount, absolute rate and relative rate (columns below) as follows (constants chosen to match initial condition F[0] = 1):

Amounts, Rates, and Relative Rates of three decay processes: Linear, Exponential, Super-Exponential

The linear decay (blue lines) is characterized by a constant rate and reaches zero at a time proportional to the initial amount, at which the relative rate has a discontinuity.

The exponential decay (red lines) is characterized by a constant relative rate and thus leads to a steady, but long-lasting decay (like radio-active decay).

The super-exponential decay (brown lines) leads to the amount following a Gompertz distribution (matching the shape of the US survival probability chart above). For a while the decay rate remains very small near zero. Then it ramps up quickly and leads to a steep decline in the amount, which in turn reduces the rate down as well. The relative rate keeps growing exponentially.

The above linked article goes on to analyze two hypotheses on dominant causes of human death: The single lightning bolt and the accumulated lightning bolt model. If the major causes of death were singular or cumulative accidents (like lightning bolts or murders), the resulting survival probability curves would have a much longer tail. In other words, we would see at least some percentage of human beings living to ages beyond 130 or even 150 years. Since such cases are practically never observed, the underlying process must be different and the lightning bolt model is not able to explain human mortality.

Instead, a so called “cops and criminals” model is proposed based upon biochemical processes in the human body. “Cops” are cells who patrol the body and eliminate bad mutations (“criminals”) which when unchecked can lead to death. From the above post:

 The language of “cops and criminals” lends itself very easily to a discussion of the immune system fighting infection and random mutation.  Particularly heartening is the fact that rates of cancer incidence also follow the Gompertz law, doubling every 8 years or so.  Maybe something in the immune system is degrading over time, becoming worse at finding and destroying mutated and potentially dangerous cells.

Unfortunately, the full complexity of human biology does not lend itself readily to cartoons about cops and criminals.  There are a lot of difficult questions for anyone who tries to put together a serious theory of human aging.  Who are the criminals and who are the cops that kill them?  What is the “incubation time” for a criminal, and why does it give “him” enough strength to fight off the immune response?  Why is the police force dwindling over time?  For that matter, what kind of “clock” does your body have that measures time at all?

There have been attempts to describe DNA degradation (through the shortening of your telomeres or through methylation) as an increase in “criminals” that slowly overwhelm the body’s DNA-repair mechanisms, but nothing has come of it so far.  I can only hope that someday some brilliant biologist will be charmed by the simplistic physicist’s language of cops and criminals and provide us with real insight into why we age the way we do.

A web calculator for death and survival probability based on Gompertz Law can be found here.

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Posted by on January 12, 2012 in Medical, Scientific, Socioeconomic


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Global Trends 2025

Global Trends 2025

If you like to do some big-picture thinking, here is a document put together by the National Intelligence Council and titled “Global Trends”. It is published every five years to analyze trends and forecast likely scenarios of worldwide development fifteen years into the future. The most recent is called “Global Trends 2025” and was published in November 2008. It’s a 120 page document which can be downloaded for free in PDF format here.

To get a feel for the content, here are the chapter headers:

  1. The Globalizing Economy
  2. The Demographics of Discord
  3. The New Players
  4. Scarcity in the Midst of Plenty?
  5. Growing Potential for Conflict
  6. Will the International System Be Up to the Challenges?
  7. Power-Sharing in a Multipolar World

From the NIC Global Trends 2025 project website:

Some of our preliminary assessments are highlighted below:

  • The whole international system—as constructed following WWII—will be revolutionized. Not only will new players—Brazil, Russia, India and China— have a seat at the international high table, they will bring new stakes and rules of the game.
  • The unprecedented transfer of wealth roughly from West to East now under way will continue for the foreseeable future.
  • Unprecedented economic growth, coupled with 1.5 billion more people, will put pressure on resources—particularly energy, food, and water—raising the specter of scarcities emerging as demand outstrips supply.
  • The potential for conflict will increase owing partly to political turbulence in parts of the greater Middle East.

As interesting as the topic may be, from a data visualization perspective the report is somewhat underwhelming. I counted just 5 maps and 5 charts in the entire document. The maps are interesting, such as the following on World Age Structure:

World Age Structure 2005

World Age Structure 2025 (Projected)

These maps show the different age of countries’ populations by geographical region. The Northern countries have less young people, and the aging trend is particularly strong for Eastern Europe and Japan. In 2025 almost all of the countries with very young population will be in Sub-Saharan Africa and the Arab Peninsula. Population growth will slow as a result; there will be approximately 8 billion people alive in 2025, 1 billion more than the 7 billion today.

In this day and age one is spoiled by interactive charts such as the Bubble-Charts of Gapminder’s Trendalyzer. Wouldn’t it be nice to have an interactive chart where you could set the Age intervals and perhaps filter in various ways (geographic regions, GDP, population, etc.) and then see the dynamic change of such colored world-maps over time? How much more insight would this convey about the changing demographics and relative sizes of age cohorts? Or perhaps display interactive population pyramids such as those found here by Jorge Camoes?

Another somewhat misguided ‘graphical angle’ are the slightly rotated graphics on the chapter headers. For example, Chapter 2 starts with this useful color-coded map of the Youth in countries of the Middle East. But why rotate it slightly and make the fonts less readable?

Youth in the Middle East (from Global Trends 2025 report)

I don’t want to be too critical; it’s just that reports put together with so much systematic research and focusing on long-range, international trends should employ more state-of-the-art visualizations, in particular interactive charts rather than just pages and pages of static text…


Posted by on January 4, 2012 in Industrial, Socioeconomic


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World Cartogram of Mobile Phone Adoption

World Cartogram of Mobile Phone Adoption

Under the slogan “Our Changing World”, FedEx has developed a website with various cartograms showing world-wide socio-economic changes based on publicly available data from sources such as World Bank, UNESCO, World Health Organization and others.

Cartograms visualize a particular metric by adjusting a country’s size corresponding to that metric. It leaves country neighborhood relationships (which we blogged about here) intact, but inflates or deflates countries, often dramatically so. Here is a series of three cartograms showing the adoption of mobile phones in the years 1995, 2000, and 2008. Size of each country is proportional to the density of mobile phones (average # mobile phones per 100 people).

Mobile Phone Density 1995

Mobile Phone Density 2000

Mobile Phone Density 2008

From the Topic Info on the Mobile Phone Presence display:

In 1996, mobile phones were a Nordic phenomenon. A Swede was twice as likely as an American to own one, and five times as likely as a German. Skip forward four years and the picture changed radically. Mobile phone usage boomed ten-fold across Europe; most European nations caught up with their northern neighbours. Eight years later. Africa suddenly loomed large. Mobile-phone penetration in same emerging economies now outstrips that of the developed world; Algeria tops the US. In most countries, mobile phone use is now ubiquitous. Lacking a mobile phone is more striking today than possessing one.

Indeed, it’s hard to find a country with very small mobile phone presence – and then to pinpoint it on the cartogram. One country I found was Cuba: While most countries in the Americas have between 50-100, Cuba has only 3 mobile phones per 100 people.

A few months ago Nathan Yau covered this topic on his FlowingData Blog here. As he already suggested, there are many more data to explore on FedEx’s website, so check it out for yourself here.

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Posted by on November 20, 2011 in Industrial, Scientific, Socioeconomic


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7 Billion

7 Billion

World population has just reached 7 Billion this week. Exploring the growth of population and related aspects such as consumption, land use, urbanization etc. lends itself very well to data visualization. In this context, the National Geographic Society has released a free iPad app called “7 Billion” together with its Special Series: 7 Billion website.

The iPad app features some interesting charts under the heading “The Shape Of Seven Billion”. These visualizations come in the form of cartograms, a type of map that ignores a country’s true physical size and scales the size according to other data. Here they show population (current 2011 vs. 1960, when world population was around 3 Billion).

Population Cartogram 2011 (Source: National Geographic iPad App 7 Billion)

The position of countries is roughly preserved, the size is proportionate to the country population, and the color legend shows the amount of growth since 1960. The strongest growth (red, more than 300%) happened in Africa and the Middle East. Europe, Russia and Japan had the least amount of growth (blue, under 50%). India and China are by far the most populous countries, with India growing faster than China.

Another interesting cartogram illustrates consumption (as measured in Gross Domestic Product, GDP). Here the reference year is 1980 and is shown first in black & white:

Consumption Chart 1980 (Source: National Geographic, iPad App 7-Billion)

Compare this to the current Consumption or GDP distribution as of 2011:

World Consumption Chart 2011 (Source: National Geographic iPad App 7 Billion)

The size of the countries here is proportionate to their GDP (in constant international dollars using purchase power parity rates). The color scale has red (more than $40,000 per capita) and blue (less than $3,000 per capita) on both ends of the spectrum. While the United States is clearly dominating this picture, Europe has about the same size and China isn’t far behind. However, China has had the world’s largest GDP increase of 1,506% since 1980 (~15 fold increase), whereas the GDP of the U.S. grew by 119% (a bit more than doubled) during the same period of time.

Ideally on would be able to see this cartogram animated over time with sizes of countries shrinking or growing and changing colors over time, similar to the Bubble Charts we looked at earlier on this Blog.

There are many other interesting charts in this interactive eBook style app. For example, here is a chart showing the population growth over time – a good visualization of the power of exponential growth.

World Population Growth and Projection (Source: National Geographic 7 Billion iPad App)

One graphic aims at explaining the main drivers behind the explosive growth over the last two centuries after relatively slow growth for millennia – the improvements in health care and resulting drop in death rate led to a period of far greater birth rates than death rates.

Population Growth as Function of Birth Rate minus Death Rate

An interesting visualization idea has been published in a video by NPR using buckets for each continents and visualizing birth rate as water drops into the bucket and death rates as drops out of the bucket. It is obvious that when more water is dropping in on the top (births) than dropping out at the bottom (deaths), then the buckets fill up.

As a final example, consider this chart visualizing our even faster growing environmental impact: Since there is not just the Population size, but at least two other factors – Affluence and Technology – the multiplicative impact is growing even faster. With the use of three dimensions and the formula I = P * A * T this yields a simple but effective illustration.

Multiplicative Human Impact through Population, Affluence and Technology

Of course a short Blog post can’t do justice to all aspects of an app or eBook. There is a lot more to this app than shown here. But I hope you got an impression as to how interactive graphics can help communicate abstract and quantitative ideas in a more intuitive way.


Posted by on November 4, 2011 in Socioeconomic


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Visual Human Development Index

Alex Simoes, MIT Media Lab student working with Professor Cesar Hidalgo, developed a graphical representation of the Human Development Index (HDI). The so-called HDI trees are based on data published in the United Nations 2010 edition of the Human Development Report. The interactive version on their website allows for comparisons between two countries, or between two years of one country.

Human Development Index – HDI Tree Representation

From Hidalgo’s website:

The HDI Tree aggregates data in the Human Development Index graphically instead of numerically. A long standing criticism of the Human Development Index is that, because it averages indicators of Income, Health and Education, it is possible for countries to obtain the same score with different combinations of indicators. This creates the possibility of substituting Education for Health, Health for Income or Income for Education.

The HDI tree deals with the numerical aggregation problem by using a graphical representation in which the total value of a country’s HDI is presented together with that of its components and subcomponents. This way it is possible to see immediately the contribution of each dimension to the value of a country’s HDI.

Moreover, the HDI tree represents an alternative way of branding the idea of Human Development and communicating its message graphically to a wide audience. For more on the HDI tree, see the original report or this summary document.

Inevitably, there are times when one wishes to collapse multiple dimensions or factors into one numerical score. However, one loses the details underlying the score. Such tree-like visual representations of aggregate information can be used for compound measurements used in business, such as the Balanced Scorecard.

Note: Hidalgo’s gallery features many more interesting projects, such as Disease Network Data visualizing disease associations or the Product Space visualizing economic capabilities of countries based on their trading activities.

Addendum: I did some more research on this and found a great summary on the HDI tree posted under the title “Visualizing Human Development” at One particularly interesting chart is a summary of 35 African nations, showing their respective HDI tree for both 1970 and 2005.

From the original summary paper “A Visual HDI” by C. Hidalgo:

The Development Tree also facilitates searching and comparing features over large volumes of data. For example, consider Figure [above], a chart in which the HDI trees of 35 African Nations are shown for both 1975 and 2005. This figure shows information on 420 numerical values (35 countries x 2 years x 6 values). In this chart, however, there are several observations that are easy to spot despite the large amount of information being presented. For instance, it is relatively easy to find out what are the countries in the set with higher levels of development. Algeria, Botswana, Libya, Mauritius, Morocco, South Africa and Tunisia in this case. Moreover, their increases are also rather conspicuous. Also, the lopsidedness of some nations also becomes conspicuous, as it can be seen in the examples of Botswana, South Africa and Swaziland, regarding the life dimension, and that of Libya in 1970, regarding high Income, or of Congo DRC in 2005, regarding low income.

Again, I can easily picture applications of this visual representation of an aggregate score in a typical business environment. Consider an internal ranking of employees based on an aggregation of several orthogonal dimensions such as skill, teamwork, communication, innovation and business savvy. You could look at a dozen of these employees and their respective visual aggregate tree scores to spot trends, outliers, and relative strengths. Another example is the Balanced Scorecard approach mentioned above. Suppose you are aggregating measures about Finance, Schedule, Quality, Innovation, and People into the score of an Engineering organization. Then you could picture the tree for aggregate performance of this business unit over time (quarters or years) to spot trends.


Posted by on July 6, 2011 in Socioeconomic


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