Upsweeps appendix

Polity Upsweeps and Collapses Inventory Appendix

Institute for Research on World-Systems

irows.ucr.edu

v. 10-4-12

Appendix to “Polity Scale Shifts Since the Bronze Age: An Inventory of Upsweeps and Collapses”

Hiroko Inoue, Alexis Álvarez, Kirk Lawrence, Anthony Roberts,

E.N. Anderson and Christopher Chase-Dunn

Institute for Research on World-Systems, University of California-Riverside

Riverside, CA 92521-0419 USA

This research was supported by National Science Foundation Grant #: NSF-HSD SES-0527720.

Paper published in the International Journal of Comparative Sociology

Appendix Table of Contents

Excel data files of polity sizes: Mesopotamia, Egypt, Central System, South Asia, East Asia

Excel data file with polities upsweeps and downsweeps counts

Table A1: List of Sweeps and comparison of two methods of calculation

Table A2: Cycles and Sweep Rates Over Time

Figure A1: Largest polities in the Central and East Asian PMNs

Subsample error estimation tables: Robustness Analysis of Non-Uniform Time-Series

Figures from the paper

Table A1: List of Sweeps and comparison of two counting methods

The two methods:

(1) Compared with smallest / largest: comparing with largest/smallest of the peaks of the preceding three cycles

(2) Compared with the average: comparing with the average of the peaks of the preceding three cycles

Mesopotamia

2800 BCE to 1500 BCE

Upsweep

Year	size	Polity name	comp w largest	comp w average
-2400	0.05	Lagash	1 (3471%)	1 (3471%)
-2250	0.8	Akkadia	1 (1500%)	1 (3013%)
-1450	0.3	Mitanni	--- (20%)	1 (100%)
Sweep total			2	3
Ups total			6	6

Down sweep

year	size	Polity name	comp w smallest	comp w average
-2150	0	Akkadia	--- (3233%)	1 (-34%)
Sweep total			0	1
Downs total			5	5

Egypt

2850 BCE to 1500 BCE

Upsweep

Year	Size	Polity name	comp w largest	comp w average
-2400	0.4	5th Dynasty	1 (300%)	1 (300%)
-1850	0.5	12th Dynasty	--- (25%)	1 (100%)
-1650	0.65	Hyksos	--- (30%)	1 (95%)
Sweeps total			1	3
Ups total			4	4

Downsweep

Year	Size	Polity name	comp w smallest	comp w average
-1630	0.08	Hyksos	--- (60%)	1 (-55%)
Sweeps total			0	1
Downs total			3	3

Central PMN

-1500 BCE to 1991AD

Upsweep

Year	Size	Polity name	comp w largest	comp w average
-1450	1	18th Dynasty	--- (25%)	1 (114%)
-670	1.4	Assyria	1 (40%)	1 (87%)
-480	5.4	Achaemenid Persia	1 (286%)	1 (440%)
117	5	Rome	--- (28%)	1 (105%)
750	11.1	Abbasids	1 (122%)	1 (178%)
1294	29.4	Mongol/Yuan	1 (654%)	1 (902%)
1936	34.5	Britain	1 (245%)	1 (505%)
Sweeps total			5	7
Ups total			27	27

Downsweep

Year	Size	Polity name	comp w smallest	comp w average
-200	0.5	Parthia	1 (-80%)	1 (-84%)
-150	0.65	Rome	--- (30%)	1 (-68%)
882	1.3	Kiev	1 (-38%)	1 (-74%)
962	2.05	Samanid	--- (58%)	1 (-55%)
972	2.1	Fatimid-Ayyubid-Mameluk	--- (62%)	1 (-54%)
1200	1	Fatimid-Ayyubid-Mameluk	--- (- 29%)	1 (-45%)
1478	1.24	Timur	1 (-50%)	1 (-69%)
Sweeps total			3	7
Downs total			27	27

East Asia

1300 BCE to 1830 AD

Upsweep

Year	Size	Polity name	comp w largest	comp w average
-1050	1.25	Western Zhou (Chou)	1 (178%)	1 (178%)
-176	4.03	Xiongnu (Hsiung-nu)	1 (222%)	1 (374%)
-50	6	Western Han	1 (49%)	1 (214%)
100	6.5	Eastern (Later) Han	--- (8%)	1 (73%)
624	1.5	E. Turks	1 (33%)	1 (35%)
660	4.9	Tang (T'ang)	--- (23%)	1 (34%)
1294	29.4	Mongol/Yuan	1 (635%)	1 (809%)
1790	14.7	Qing	--- (-50%)	1 (33%)
Sweeps total			5	8
Ups total			17	17

Downsweep

Year	Size	Polity name	comp w smallest	comp w average
-1700	0.4	Xia (Hsia)	---	---
-600	0.05	Western Zhou (Chou)	1 (-88%)	1 (-88%)
221	1.5	Wu	--- (2900%)	1 (-58%)
316	2	Earlier Zhao (Chao)	--- (33%)	1 (-41%)
502	1.3	Huns / Tuoba	--- (-13%)	1 (-35%)
907	0.8	Tang (T'ang)- Jin (Chin) - Song (Sung)	1 (-74%)	1 (-77%)
947	0.5	Liao (Kitan)	1 (-38%)	1 (-84%)
Sweeps total			3	6
Downs total			17 (Qing’s starting down is counted)	17 (Qing’s starting down is counted)

South Asia

-420 BCE to 1008 AD

Upsweep

Year	Size	Polity name	comp w largest	comp w average
-260	3.72	Mauryan	---	1 (---)
Sweeps total			0	1
Ups total			5	5

Downsweep

Year	Size	Polity name	comp w smallest	comp w average
-130	1.9	Mauryan	---	1 (---)
330	0.97	Kusanas	1 (-49%)	1 (-49%)
600	0.05	Gupta	1 (-95%)	1 (-97%)
1050	0.2	Rastrakutas	--- (300%)	1 (-72%)
Sweeps total			2	4
Downs total			5	5

Two different counting methods compared

Upsweeps and downsweeps represent the major changes in the scale of polities that form 33% increase or decrease of polity size from the previous level. A peak is any data point that is preceded and followed by data points that are smaller than it is. A trough is any data point that is preceded and followed by data points that are larger than it it. Each peak or trough data point is compared with preceding three peaks or troughs to determine if that data point constitutes an upsweep or downsweep. But this comparison can be done in several possible ways. In determining this, we applied the following two methods.

(1) Comparison of each peak or trough data point with the average of preceding three peak or trough data points.

In order to determine whether or not a data point is a downsweep, the trough data point is compared with the average of preceding three low points (troughs). In order to determine whether or not at peak data point is an upsweep, the data point is compared with the average of preceding three peaks.

(2) Comparison of each peak or trough data point with the smallest (for downsweep) or largest (for upsweep) among preceding three peak or trough data points.

In order to determine whether or not a data point is a downsweep, the trough data point is compared with the smallest among preceding three trough data points; in order to determine whether or not a data point is an upsweep, the peak data point is compared with the largest among preceding three peak data points.

We decided to use the method (1) for determining the upsweeps and downsweeps in our paper. The cutting point that forms upsweep or downsweep is 33% growth or decline from the previous reference data. The method (2) is presented for comparative purposes to see how frequently to two methods converge or diverge.

Example:

The following is an example of cases from East Asia about how we determined the upsweeps or downsweeps. Method 1 from this example was applied to all five regions of our study.

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Figure 1. Growth and decline of largest empire sizes in square megameters in East Asia between 1900 BCE and 860 CE

Figure 1 shows the growth and decline of the largest empires in East Asia data from 1900 BCE to 860 CE. The red circles represent the trough data points used to determine if Wu (220 AD) formed a downsweep (presented in the left side of the graph); the yellow circles represent the peak data points used to determine if Tang, Jin, or Song (660AD) made an upsweep (presented in the right side of the graph)

Upsweeps

In order to determine whether the data point peak of Tang, Jing, or Song (660AD) forms an upsweep or not, we compare it with the three previous peaks ( Eastern Turks (624AD), GokTurk, 557AD) and Rouran (405AD) [yellow circles in the graph].

Both methods 1 and 2 are applied for comparison.

Method (1)

The rate of increase of the peak of Tang, Jing, or Song (660) from the average of the peaks of the Eastern Turks, the GokTurks and Rouran is calculated. The sizes of the thee data points are: Tang, Jing, Song =4.9, Eastern Turks= 4, GokTurks=3,and Rouran=2.8. Thus, the increase rate is:

It shows a 50% increase. Since it is more than 33%, we decide Tang, Jing, Song (660) forms an upward sweep.

Method (2)

The rate of increase for the Tang, Jing, Song (660) peak is compared with the largest size among the three preceding peaks. This is Eastern Turk with size 4. Thus the increase rate is:

By rounding, it is 23 % increase, and it is less than 33%. Using method (2), we do not categorize Tang, Jing, Song (660) as an upsweep.

Downsweeps

In order to determine whether or not Wu (220 AD) formed a downsweep we compared its size with the previous three troughs. These include the Western Han (10AD), the Xiongnu (110BCE), and the Chu (600BCE) [red circles in the graph].

Method (1)

The rate of decline of the Wu trough is compared with the average of the sizes of Western Han, Xiongnu, and Chu troughs. The sizes are: Wu=1.5, Western Han=4.7, Xiongnu=4 and Chu=.05. Thus the decline rate is:

By rounding, it is a 49% decline. Comparing with the average of preceding three points, Wu shows a downsweep with more than 33% change. Thus, we determine Wu should be categorized as a downsweep.

Method (2)

The rate of decline of the Wu trough is compared with the smallest size among the three preceding troughs. This is Chu=.05. The decline rate is:

This shows 2900 % increase from Chu to Wu because the smallest of the previous three (Chu) is smaller than Wu. Thus by this method Wu would not be considered a downsweep.

We prefer Method 1 because it is less vulnerable to single deviations of the kind shown in the example of Wu.

Table A2: Cycles and Sweep Rates Over Time

Method:

The total length of the period for each region is divided by 5 equal intervals (phase1 to 5). The number of cycles and sweeps in 5 regions are counted in the 5 intervals.

Mesopotamia

-2800~-1500

(1300 yrs)

	interval 260 years	# of cycle	# of upsweeps	# of down sweeps
phase1	- 2800 ~ -2540	0.5	0	0
phase2	-2540 ~ - 2280	1	1	0
phase3	-2280 ~ - 2020	1	1	1
phase4	-2020 ~ - 1760	2	0	0
phase5	- 1760 ~ - 1500	1.5	1	0

Egypt

-2850~ - 1500

(1350 yrs)

	interval 270 years	# of cycle	# of upsweeps	# of down sweeps
phase1	- 2850 ~ -2580	0.5	0	0
phase2	-2580 ~ - 2310	0.5	1	0
phase3	-2310 ~ - 2040	0.5	0	0
phase4	-2040 ~ - 1770	0.5	1	0
phase5	-1770 ~ -1500	2	1	1

Central PMN

-1500~2000

(3500 yrs)	interval 700 years	# of cycle	# of upsweeps	# of down sweeps
phase1	-1500 ~ -800	2.5	1	0
phase2	-800 ~ -100	8.5	2	2
phase3	-100 ~ 600	2	1	0
phase4	600 ~ 1300	10	1	4
phase5	1300 ~ 2000	4	2	1

East Asia

-1300~1830

(3130 yrs)

	interval 626 years	# of cycle	# of upsweeps	# of down sweeps
phase1	-1300 ~ -674	1	1	0
phase2	-674 ~ -48	1.5	2	1
phase3	-48 ~ 578	3.5	1	3
phase4	578 ~ 1204	8	2	2
phase5	1204 ~ 1830	3	2	0

South Asia

-420~1008

(1428 yrs)

	interval 285.6 years	# of cycle	# of upsweeps	# of down sweeps
phase1	-420 ~ -134.4	1	1	0
phase2	-134.4 ~ 151.2	0.5	0	1
phase3	151.2 ~ 436.8	1	0	1
phase4	436.8 ~ 722.4	1.5	0	1
phase5	722.4 ~ 1008	1	0	1

Figure A1: Largest polities in the Central and East Asian PMNs

Description: Description: Description: eastwestemps

Subsample error estimation tables: Robustness Analysis of Non-Uniform Time-Series

Two important problems for our analysis are:

1. large differences in the number of largest polity size estimates across regions, and

2. the non-uniformity of the time intervals between estimates for each region.

Bronze Age estimates from Mesopotamia, Egypt and East Asia were farther apart in time, and we had fewer estimates overall for Mesopotamia and Egypt. This could have meant that we were missing important changes in the sizes of largest polities. We know that there is a missing estimate for the Uruk polity in Mesopotamia.

We performed a secondary analysis to address the critical issue of having a non-uniform time-series when estimating the average time to upswings, upsweeps, downswings and downsweeps (events). We measured the number of years per event by multiplying the number of time points per event by several mean interval lengths. This allows us to simulate a uniform time-series for our estimates while accounting for uncertainty of our sample. By controlling for the unevenness and comparing the results of this analysis to the reported results, we can better determine whether the non-uniformity in our time-series influences the estimates for the average number of years between events.

Table 1 shows the variation in the time intervals between estimates and in the total number of estimates for each region. The smallest sample of estimates is for the Mesopotamia region, with 20 estimates. The largest sample of estimates is for the Central PMN with 123 estimates. The mean interval length between estimates for the regions ranges from 20 years (Egypt) to 72.5 years (Mesopotamia). Across all regions, the average difference (standard deviation) in the interval lengths ranged from 28 years (Central System) to 65 years (Mesopotamia). Given the variation in the number of estimates and the variation in interval lengths, it is possible that the number of events that we found and the intervals between the events were heavily influenced by the structure of the data. Therefore, it is important to estimate possible differences in results when adjusting both issues.

A critical assumption underlying the analysis is that the average length of intervals between observations in the time-series is a function of some unobserved stochastic process. Given this assumption, we constructed an average interval length for each region and a confidence interval for the average length. For example, in Mesopotamia, the average interval between estimates is 72.5 years, where intervals ranged from 10 years to 300 years. Assuming that the average interval length is a product of our ‘sample’ of intervals, in the ‘population’ of intervals, we would expect that the average interval length should be between 44 to 101 years for Egypt. Therefore, for calculating the average number of years to an event, we use the average, lower limit average, and upper limit average interval lengths were used.

Table 1. Estimate Intervals in Regional Time-Series

	*Mesopotamia*	*Egypt*	*Central PMN*	*East Asia*	*South Asia*
N	20	117	123	91	24
*Avg* *Interval*	72.5	20	29	38	64
*Min Interval*	10	450	1	0	4
*Max Interval*	300	117	140	400	280
*SD Interval*	65	34	28	57	66
*SE Interval*	15	50	2.5	6	13
*95% LCL*	44	71	24	27	38
*95% UCL*	101	183	34	50	90
Note: N = Total Number of Estimates.

Tables 2 and 3 show the results from re-estimating the duration of intervals between estimates based on the average interval length for each region. The time between estimates was calculated by multiplying the number of estimates per event by the average interval length, the 95% lower limit of average interval length, and the upper limit average. The analysis provides an average year per event and an interval estimate for the average year per event. According to the results in Tables 2 and 3, the unevenness of estimates across region has little effect on the estimated duration of intervals between events. Comparing Tables 2 and 3 to the results presented in the paper, we see that there are few significant difference between the point estimates (average years per event) and the interval estimates.

Table 2. Duration Analysis of Upswings and Upsweeps.
	*Estimates per Upsweep*	*Estimates per Upswing*	*Average Years per Upsweep*	*Average Years per Upswing*	*Interval Estimate for Average Year per Upsweep*		*Interval Estimate for Average Year per Upswing*
*Mesopotamia*	7	3	483	242	294	673	147	337
*Egypt*	4	3	467	350	201	732	151	549
*Central PMN*	18	5	503	130	416	589	108	153
*East Asia*	11	5	436	205	304	569	143	268
*South Asia*	24	5	1534	307	901	2167	180	433

Table 3. Duration Analysis of Downswings and Downsweeps.
	*Estimates per Downsweep*	*Estimates per Downswing*	*Average Years per Downsweep*	*Average Years per Downswing*	*Interval Estimate for Average Years per Downsweep*		*Interval Estimate for Average Years per Downswing*
*Mesopotamia*	20	4	1450	290	881	2019	176	404
*Egypt*	12	3	1400	350	604	2196	151	549
*Central PMN*	18	5	503	130	416	589	108	153
*East Asia*	15	5	582	205	405	758	143	268
*South Asia*	6	5	383	307	225	542	180	433