The physics of citation growth

Giacomin, Alan J.; Zatloukal, Martin; Kanso, Mona A.; Phan-Thien, Nhan

doi:10.2478/jdis-2025-0004

Figures & Tables

Scatter-plot of growth penalty versus y over the period 1999-2023.

Distribution of available fractional annual citation growth, y, for the fluids journal tri-cluster over the period 1999-2023. Taken from Columns 7 of Table 2 through Table 4. Gaussian curve from Eq. (12) with μy ≅ 2.724 and σy ≅ 0.267. Histogram bin-width, Δy = 0.1.

Growth penalty versus x from Eq. (5) over the period 1999-2023: μy ≅ 2.724 (black), y = 1 (cyan) and y = 1.5 (magenta). y = 5 (blue), y = 10, (green), y = ∞ (red).

Scatter-plot of growth penalty versus Ur over the period 1999-2023.

Distribution of ratio of twice the growth-adjusted impact factor to three-year running average immediacy index, Ur, for the fluids journal tri-cluster over the period 1999-2023 which is nearly Gaussian. Curve from Eq. (19) with μUr ≅ 10.23 and σUr ≅ 1.70. Eq. (18) overwrites Eq. (19) within a line width. Histogram bin-width, ΔUr = 1.

Annual fractional citation growth versus Ur from Eq. (17) with small and large Ur asymptotes y ≈ Ur (leftmost) and y≈Ur−12y \approx \sqrt {{\rm{Ur}}} - {1 \over 2} (rightmost). Operating point to the three fluids journal cluster for 56 journal years is near the large asymptote Ur, in the nearly linear region.

Comparison of annual fractional citation growth distribution from Eq. (12) versus Ur distribution from Eq. (17). Unit areas under each distribution. Histogram bin-widths, Δy = 0.1 (left) and ΔUr = 1 (right).

Annual fractional citation growth distribution of Figure 2 [from Eq. (12)] projected geometrically (horizontal dashed lines) onto the curve of Eq. (17) projected orthogonally in turn (vertical dashed lines) onto the Ur distribution of Figure 5 [from Eq. (18)]. Since y is both Gaussian and nearly linear with Ur near the operating point, the projected Ur distribution is nearly Gaussian.

Test of the running average approximation of Eq. (20), showing confidence interval on the regression proportionality:. Regression employed 212 ordered pairs of the 56 journal years of the three fluids journal cluster.

Illustration of one urgency metric Ug as a function of the other, Ur, from Eq. (28). The large Ur asymptote of Eq. (29) is also illustrated, along with the operating point (10.23,48.00) of the three fluids journal cluster. Eqs. (28) and (29) agree within a pen-width. Graphical projection method of Figure 8 used to project the Ug distribution from the Ur distribution of Figure 5.

Distribution of ratio of four times the growth-adjusted CiteScore to three-year running average immediacy index, Ug, for the fluids journal tri-cluster over the period 1999-2023 which is nearly Gaussian. Curve obtained graphically in Figure 10 obeys Eq. (19) with μUg ≅ 48.00 and σUg ≅ 9.59. Histogram bin-width, ΔUg = 4.

Region y > 0 (crosshatched in orange) is where the CiteScore is anticompetitive too. Subregion y > 1 (crosshatched in blue) is where impact factor is anticompetitive. All 56 journal years of the three fluids journal cluster are well within both anticompetitive zones: POF, JFM and PRF.

Growth penalty versus x from Eq. (5) over the period 1999-2023, y = 1 (cyan) and y = 0.60 (magenta). y = 0.33 (blue), y = 0.14 (green), y = 0 (red).

Variables with Dimensions_

Name	Symbol	Calculation	Dimensions
fractional biennial publication growth, Eq. (32)	X	2p3p1+p2{{2{p_3}} \over {{p_1} + {p_2}}}	dimensionless
fractional annual citation growth, Eq. (7)	Y	12[ (x+1)2+4IFII¯(2+x)−(x+1) ]{1 \over 2}\left[ {\sqrt {{{(x + 1)}^2} + 4{{{\rm{IF}}} \over {\overline {{\rm{II}}} }}(2 + x)} - (x + 1)} \right]	dimensionless
fractional annual publication growth	x	ratio of citable items in the current and previous year, p₃/p₂	dimensionless
growth penalty, Eq. (5)	IF¯(n3)\overline {{\rm{IF}}} \left( {{n_3}} \right)	2(y+1+x)(y+1)(2+x){{2(y + 1 + x)} \over {(y + 1)(2 + x)}}	dimensionless
growth-adjusted impact factor, Eq. (13)	GIF	IF(n3)IF¯(n3)¯{{{\mathop{\rm IF}\nolimits} \left( {{n_3}} \right)} \over {\overline {\overline {{\mathop{\rm IF}\nolimits} } \left( {{n_3}} \right)} }}	citations/publication
immediacy index	II	Web of Science	citations/publication
impact factor	IF	Web of Science	citations/publication
journal urgency, Eq. (15)	Ur	2GIF(n3)II¯2{{{\mathop{\rm GIF}\nolimits} \left( {{n_3}} \right)} \over {\overline {{\rm{II}}} }}	dimensionless
mean value of variable i, Eqs. (12) and (19)	μ_i		i
number of articles and reviews (citable items)	p	Web of Science	number of publications
number of articles and reviews (citable items) in year n_i	p_i	Web of Science	number of publications
standard deviation of variable i	σ_i		i
three-year mean immediacy index	II¯\overline {{\rm{II}}}	arithmetic mean of II for the current and two previous years	citations/publication
year	n_i	Web of Science	year

Growth-adjusted impact factors of Physics of Fluids_

n_i	p	IF	II	II¯\overline {{\rm{II}}}	x	y	IF¯(n3)\overline {{\rm{IF}}} \left( {{n_3}} \right)	GIF	GIF/IF	Ur
1997	369	1.630	0.298	-	-	-	-	-	-	-
1998	314	1.348	0.274	-	-0.1491	-	-	-	-	-
1999	358	1.420	0.302	0.2913	0.1401	2.7098	0.9698	1.464	1.0310	10.051
2000	305	1.442	0.334	0.3033	-0.1480	2.5718	1.0352	1.393	0.9660	9.186
2001	346	1.799	0.298	0.3113	0.1344	2.9904	0.9686	1.857	1.0322	11.931
2002	412	1.697	0.374	0.3353	0.1908	2.7873	0.9589	1.770	1.0430	10.558
2003	375	1.566	0.307	0.3263	-0.0898	2.6067	1.0209	1.534	0.9796	9.402
2004	435	1.761	0.313	0.3313	0.1600	2.8577	0.9643	1.826	1.0369	11.023
2005	575	1.728	0.303	0.3077	0.3218	3.0100	0.9305	1.857	1.0747	12.07
2006	503	1.697	0.250	0.2887	-0.1252	2.9110	1.0326	1.643	0.9682	11.382
2007	567	1.780	0.217	0.2567	0.1272	3.3182	0.9679	1.839	1.0331	14.328
2008	485	1.738	0.241	0.2360	-0.1446	3.2934	1.0416	1.669	0.9603	14.144
2009	487	1.638	0.300	0.2527	0.0041	3.1370	0.9989	1.640	1.0012	12.98
2010	445	1.722	0.267	0.2693	-0.0862	3.0710	1.0229	1.683	0.9774	12.499
2011	488	1.926	0.355	0.3073	0.0966	3.1179	0.9763	1.973	1.0244	12.841
2012	499	1.942	0.319	0.3137	0.0225	3.0639	0.9944	1.953	1.0057	12.451
2013	587	2.040	0.349	0.3410	0.1764	3.0678	0.9588	2.128	1.0431	12.481
2014	604	2.031	0.434	0.3673	0.0290	2.8743	0.9931	2.045	1.0069	11.135
2015	553	2.017	0.374	0.3857	-0.0844	2.7402	1.0205	1.976	0.9797	10.246
2016	554	2.232	0.444	0.4173	0.0018	2.8094	0.9996	2.233	1.0004	10.702
2017	617	2.279	0.415	0.4110	0.1137	2.9117	0.9737	2.341	1.0272	11.392
2018	659	2.627	0.724	0.5277	0.0681	2.7187	0.9848	2.668	1.0156	10.112
2019	859	3.514	0.984	0.7077	0.3035	2.7925	0.9377	3.747	1.0663	10.589
2020	1212	3.521	1.116	0.9413	0.4109	2.3793	0.9304	3.784	1.0747	8.04
2021	1970	4.980	1.325	1.1417	0.6254	2.6676	0.8917	5.585	1.1215	9.784
2022	2203	4.6	1.1	1.18	0.12	2.37	0.98	4.7	1.02	8
2023	3008	4.1	1.6	1.34	0.37	2.09	0.94	4.4	1.07	6.6

Growth-adjusted impact factors of Journal of Fluid Mechanics_

n_i	p	IF	II	II¯\overline {{\rm{II}}}	x	^y	IF¯(n3)\overline {{\rm{IF}}} \left( {{n_3}} \right)	GIF	GIF/IF	Ur
1997	370	1.609	0.370	-	-	-	-	-	-	-
1998	334	1.672	0.338	-	-0.0973	-	-	-	-	-
1999	340	1.686	0.226	0.3113	0.0180	2.8359	0.9957	1.693	1.0042	10.877
2000	321	1.601	0.374	0.3127	-0.0559	2.7180	1.0133	1.580	0.9869	10.106
2001	389	1.912	0.357	0.3190	0.2118	3.0852	0.9511	2.010	1.0513	12.602
2002	398	1.882	0.369	0.3667	0.0231	2.7511	0.9947	1.892	1.0053	10.319
2003	360	1.811	0.339	0.3550	-0.0955	2.6974	1.0230	1.770	0.9774	9.972
2004	378	1.853	0.384	0.3640	0.0500	2.7478	0.9886	1.874	1.0113	10.297
2005	401	2.061	0.446	0.3897	0.0608	2.8133	0.9860	2.090	1.0141	10.726
2006	487	2.022	0.472	0.4340	0.2145	2.6617	0.9560	2.115	1.0460	9.747
2007	501	2.026	0.413	0.4437	0.0287	2.5724	0.9938	2.039	1.0064	9.191
2008	450	2.315	0.422	0.4357	-0.1018	2.7583	1.0251	2.258	0.9754	10.365
2009	450	2.283	0.433	0.4227	0	2.8245	1	2.283	1	10.802
2010	524	2.457	0.534	0.4630	0.1644	2.8565	0.9634	2.550	1.0379	11.015
2011	565	2.459	0.550	0.5057	0.0782	2.6852	0.9828	2.502	1.0175	9.895
2012	557	2.183	0.445	0.5097	-0.0142	2.4648	1.0030	2.176	0.9968	8.538
2013	679	2.294	0.594	0.5297	0.2190	2.5498	0.9569	2.397	1.0449	9.05
2014	663	2.383	0.670	0.5697	-0.0236	2.4282	1.0050	2.371	0.9950	8.324
2015	669	2.514	0.526	0.5967	0.0090	2.4483	0.9981	2.519	1.0020	8.443
2016	739	2.821	0.591	0.5957	0.1046	2.6526	0.9775	2.886	1.0230	9.689
2017	806	2.893	0.627	0.5813	0.0907	2.7261	0.9799	2.952	1.0204	10.157
2018	993	3.137	0.708	0.6420	0.2320	2.7434	0.9516	3.297	1.0510	10.271
2019	975	3.333	0.833	0.7227	-0.0181	2.5719	1.0040	3.320	0.9961	9.188
2020	973	3.627	1.095	0.8787	-0.0021	2.4158	1.0004	3.626	0.9997	8.253
2021	1360	4.245	1.151	1.0263	0.3977	2.5270	0.9282	4.573	1.0773	8.912
2022	1078	3.7	0.9	1.05	-0.21	2.15	1.04	3.6	0.97	6.9
2023	1056	3.6	0.9	0.98	-0.02	2.25	1	3.6	1	7.3

Growth-adjusted impact factors of Physical Review Fluids_

n_i	p	IF	II	II¯\overline {{\rm{II}}}	x	^y	IF¯(n3)\overline {{\rm{IF}}} \left( {{n_3}} \right)	GIF	GIF/IF	Ur
2016	234	N/A	0.188	-	-	-	-	-	-	-
2017	471	2.021	0.452	-	1.0128	-	-	-	-	-
2018	484	2.442	0.576	0.4053	0.0276	3.0190	0.9932	2.459	1.0070	12.13
2019	546	2.512	0.443	0.4903	0.1281	2.7858	0.9716	2.585	1.0291	10.55
2020	583	2.537	0.590	0.5363	0.0678	2.6389	0.9852	2.575	1.0150	9.603
2021	575	2.895	0.706	0.5797	-0.0137	2.6947	1.0032	2.886	0.9969	9.957
2022	538	2.7	0.6	0.63	-0.06	2.45	1.01	2.7	1	8.6
2023	431	2.5	0.8	0.7	-0.2	2.17	1.04	2.4	0.96	6.9

The physics of citation growth

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Figure 13.

Variables with Dimensions_

Growth-adjusted impact factors of Physics of Fluids_

Growth-adjusted impact factors of Journal of Fluid Mechanics_

Growth-adjusted impact factors of Physical Review Fluids_

Paradigm

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