Applying Zipf's Law

 I asked Gemini the following:

Can you compile a table of the top ten Indonesian cities and their populations ranked from highest to lowest. Use this table to determine how closely populations and rank conform to Zipf’s Law.

Here was its response:

Rank	City	Actual  Population	Zipf's Law Predicted Population	Difference
1	Jakarta	10,684,946	10,684,946	0
2	Surabaya	2,921,996	5,342,473	-2,420,477
3	Bekasi	2,644,058	3,561,649	-917,591
4	Bandung	2,528,160	2,671,236	-143,076
5	Medan	2,494,512	2,136,989	+357,523
6	Depok	2,163,635	1,780,824	+382,811
7	Tangerang	1,963,970	1,526,421	+437,549
8	Palembang	1,718,440	1,335,618	+382,822
9	Semarang	1,702,379	1,187,216	+515,163
10	Makassar	1,474,393	1,068,495	+405,898

Population data reflects mid-2024/2025 official estimates.

Zipf's Law posits that the population of the n-th ranked city in a given country should be roughly equal to the population of the largest city divided by n. When applied to Indonesia's top ten cities, the distribution reveals a notable divergence from the theoretical model:

The Primate City Effect 

Jakarta acts as a classic "primate city," meaning its population is disproportionately massive compared to the rest of the urban hierarchy. Because Jakarta is so large, Zipf's Law predicts that the second-largest city (Surabaya) should have over 5.3 million residents. In reality, Surabaya falls nearly 2.4 million people short of this mark, creating a sheer drop-off at the top of the distribution rather than a smooth proportional curve.

Mid-Rank Convergence 

The urban distribution briefly aligns with Zipf's Law around the fourth rank. Bandung's actual population of 2.52 million closely brushes against the model's prediction of 2.67 million. This is the only point in the top ten where the theoretical distribution and actual settlement sizes tightly converge.

Overperformance in the Lower Ranks 

From the fifth rank (Medan) downward, Indonesian cities consistently exceed the population sizes predicted by the model. Rather than trailing off rapidly as the fractions get smaller, cities ranked six through ten maintain robust populations between 1.4 and 2.1 million, resulting in a "fat tail" on the data curve.

The Metropolitan Distortion 

A fundamental geographical factor skews this list: three of the top ten cities—Bekasi, Depok, and Tangerang—are not independent regional hubs but rather satellite cities directly bordering Jakarta within the greater Jabodetabek metropolitan area. Their massive growth inflates the middle ranks of the hierarchy, concentrating immense population density into a single contiguous megalopolis rather than distributing it evenly across the archipelago.

Ultimately, Indonesia's urban population distribution does not closely conform to Zipf's Law. The sheer dominance of Jakarta creates an initial deficit for the secondary cities, while the massive sprawl of Jakarta's satellite cities props up the lower ranks, resulting in an uneven urban hierarchy shaped more by metropolitan gravity than strict mathematical proportionality.

I then asked:

Can you repeat this analysis for Australian cities.

Here was the response:

Rank	City (Significant Urban Area)	Actual Population	Zipf's Law Predicted Population	Difference
1	Melbourne	5,103,528	5,103,528	0
2	Sydney	5,041,275	2,551,764	+2,489,511
3	Brisbane	2,622,585	1,701,176	+921,409
4	Perth	2,289,366	1,275,882	+1,013,484
5	Adelaide	1,426,803	1,020,705	+406,098
6	Gold Coast–Tweed Heads	735,213	850,588	-115,375
7	Newcastle–Maitland	526,515	729,075	-202,560
8	Canberra–Queanbeyan	503,402	637,941	-134,539
9	Sunshine Coast	407,859	567,058	-159,199
10	Central Coast	348,435	510,352	-161,917

Population data reflects Australian Bureau of Statistics (ABS) 2023 Estimated Resident Population for Significant Urban Areas.

When applied to Australia's top ten cities, the distribution reveals a stark contrast to both the theoretical Zipf's Law model and the primate city effect seen in Indonesia.

The Twin-Primate Anomaly 

Australia exhibits a bimodal or "twin-primate" distribution at the top of its urban hierarchy. Melbourne and Sydney are virtually identical in size, functioning as a duopoly. Zipf's Law dictates that the second-ranked city should be half the size of the first. Because Sydney is nearly 99% the size of Melbourne, it overshoots the model's prediction by nearly 2.5 million residents, thoroughly breaking the curve at the very first step.

The State Capital Inflation 

Because Melbourne sets a relatively "low" mathematical baseline compared to the combined dominance of Australia's two largest cities, the next tier of state capitals massively overperforms. Brisbane, Perth, and Adelaide operate as vast, isolated economic hubs for enormous geographical states. Without nearby rival cities to siphon off growth, they concentrate their state's populations, exceeding Zipf's predictions by 400,000 to over 1 million residents each.

The Steep Regional Drop-Off 

The distribution flips entirely at the sixth rank. From the Gold Coast down to the Central Coast, every city falls significantly short of the model's predictions. Unlike the state capitals, these cities are secondary regional centers lacking dominant political and economic gravity. Rather than a smooth curve, Australia's hierarchy demonstrates a "missing middle"—a sheer drop of almost 700,000 people between fifth-ranked Adelaide and sixth-ranked Gold Coast.

Corridor Megalopolises 

Geographical proximity heavily skews the lower half of the list. Six of the top ten cities are concentrated within just two massive coastal conurbations: the Sydney-Newcastle-Central Coast web in New South Wales, and the Brisbane-Gold Coast-Sunshine Coast corridor in Queensland. This highlights that Australia's lower-ranked top-ten cities are largely sprawling coastal satellites anchored to the major state capitals, rather than independent urban centers distributed evenly across the continent.

Australia's urban geography rejects strict mathematical proportionality, favoring a highly concentrated model defined by two mega-cities, a few hyper-dominant state capitals, and a steep drop-off into coastal satellite cities.

I then asked:

Clearly neither Indonesia nor Australia has cities that conform to Zipf’s Law. Is there a country that does conform closely to Zipf’s Law?

Here was the response:

Rank	City	Actual Population	Zipf's Law Predicted Population	Difference
1	New York	8,546,038	8,546,038	0
2	Los Angeles	3,863,148	4,273,019	-409,871
3	Chicago	2,727,758	2,848,679	-120,921
4	Houston	2,419,191	2,136,510	+282,681
5	Phoenix	1,677,947	1,709,208	-31,261
6	San Antonio	1,572,360	1,424,340	+148,020
7	Philadelphia	1,567,280	1,220,863	+346,417
8	San Diego	1,410,522	1,068,255	+342,267
9	Dallas	1,336,440	949,560	+386,880
10	Fort Worth	1,049,355	854,604	+194,751

Population data reflects 2026 demographic estimates for city limits.

Top-Tier Convergence 

Unlike the massive primate city distributions or bimodal anomalies seen in other regions, the top of this urban hierarchy aligns tightly with mathematical expectations. Chicago falls short of the prediction by a mere 4%, and Phoenix sits a fraction of a percent away from its exact mathematical target.

The Sunbelt Expansion 

The most notable deviations occur in the lower half of the top ten, driven by the rapid modern expansion of cities in Texas and the Sunbelt. Houston, San Antonio, Dallas, and Fort Worth all overperform their theoretical predictions. This reflects a contemporary migration pattern where expansive, less dense southern cities have grown rapidly, creating a slight bulge in the mid-tier ranks.

Absence of a Primate City 

New York is undeniably the largest settlement, but it does not exert the suffocating gravitational pull of a primate city. Its size is balanced by a robust network of secondary and tertiary economic hubs distributed across a massive continental landmass, allowing cities like Los Angeles and Chicago to scale naturally according to the law's proportions.

The urban geography of the United States conforms remarkably closely to Zipf's Law, serving as the classic mathematical model for proportional urban population distributions.

Top 20 Largest US Metropolitan Areas 2023 This video visualizes the population scale and rankings of the largest urban centers in the United States, providing a clear illustration of how the demographic distribution roughly follows the predicted curve.