Discontinuities in citation relations among journals: self-organized criticality as a model of scientific revolutions and change
Discontinuities in citation relations among journals: self
Loet Leydesdorff 0 1 2
Caroline S. Wagner 0 1 2
Lutz Bornmann 0 1 2
0 Division for Science and Innovation Studies, Administrative Headquarters of the Max Planck Society , Hofgartenstr. 8, 80539 Munich , Germany
1 John Glenn College of Public Affairs, The Ohio State University , Columbus, OH 43210 , USA
2 Amsterdam School of Communication Research (ASCoR), University of Amsterdam , PO Box 15793, 1001 NG Amsterdam , The Netherlands
Using 3-year moving averages of the complete Journal Citation Reports 1994-2016 of the Science Citation Index and the Social Sciences Citation Index (combined), we analyze links between citing and cited journals in terms of (1) whether discontinuities among the networks of consecutive years have occurred; (2) are these discontinuities relatively isolated or networked? (3) Can these discontinuities be used as indicators of novelty, change, and innovation in the sciences? We examine each of the N2 links among the N journals across the years. We find power-laws for the top 10,000 instances of change, which we suggest interpreting in terms of ''self-organized criticality'': co-evolutions of avalanches in aggregated citation relations and meta-stable states in the knowledge base can be expected to drive the sciences towards the edges of chaos. The flux of journal-journal citations in new manuscripts may generate an avalanche in the metastable networks, but one can expect the effects to remain local (for example, within a specialty). The avalanches can be of any size; they reorient the relevant citation environments by inducing a rewrite of history in the affected partitions.
With the complete Journal Citation Reports during the period 1994–2016 as data, we
address the question of change and stability in the sciences at the level of the (n2)
aggregated citation links between (n) journals using entropy statistics. Entropy statistics
allows for dynamic analysis and aggregation. This operationalization enables us to relate
micro-developments in the data to theorizing about the sciences in terms of distributed
(Price 1976; cf. Kuhn 1962)
. Our results suggest that the dynamics can be
explained by considering Bak et al.’s (1987) model of ‘‘self-organized criticality’’: the
knowledge base can be considered as a large set of meta-stable constructs which are
continuously disturbed by new knowledge claims bringing also new citation relations.
‘‘Avalanches’’ of variable size can then be expected (Kuhn et al. 2014). The effects,
however, are local; the meta-stable regions operate in parallel. The overall system remains
tending towards meta-stability at ‘‘the edge of chaos’’ because of the ongoing flux of new
manuscripts creating and rewriting journal–journal relations in terms of citations at
(Zitt et al. 2005)
Aggregated journal–journal citation relations provide a possible baseline for studying
the structural dynamics of the sciences
(Studer and Chubin 1980, p. 269)
. Citations and
cocitations at the paper level have a high turnover in fields with a research front such as
biomedicine, but less so in other fields
(Garfield 1979a; Price 1970)
. Repertoires of words
and co-words can be ‘‘translated’’ among fields with flexible interpretations
(Callon et al.
studied self-organization in key words in physics, interpreted as
evolving ‘‘paradigms,’’ showing preferential attachment along a power-law distribution.
However, scholarly journals are established institutions providing quality control by peer
review. This ‘‘journal literature’’ can be considered as the core of scientific literature
; new disciplinary developments can be expected to lead to new journals
Journals play a crucial and institutionalized role in the validation of knowledge claims
and in the incorporation of new knowledge into the archive of science. Given their role in
the codification of knowledge, journals can be considered as an organizing layer of the
scientific literature. Not incidentally, the Science Citation Index (SCI) and its derivates (the
Social Sciences Citation Index (SSCI) and the Arts and Humanities Citation Index (AHCI))
were defined in terms of specific journal selections
(Garfield 1972, 1979b)
, as is Scopus,
the main competitor of the SCI since 2004.
The gate-keeping role of journals is nowadays debated
(Brembs et al.2013; Kling and
Callahan 2003; cf. Zsindely et al. 1982)
. The internet revolution may have changed the
landscape. Google Scholar (since 2004) collects articles on a case-by-case basis by
crawling the web. Consequently, Google Scholar does not delineate the universe of
scholarly documents (
Martin–Martin et al. 2018
). Furthermore, with the introduction of
PLOS ONE in 2006, new journals have emerged that deliberately abstain from disciplinary
criteria in the peer-review process in favor of a focus on novelty in terms of methods and
data. As a consequence, journals may have lost some of their exclusiveness and perhaps
precision in maintaining borders among disciplines
(Harzing and Alakangas 2016)
and Klavans (2011)
Klavans and Boyack (2017)
, for example, argue that ‘‘direct citation’’
at the article level
(Waltman and van Eck 2012)
has become an organizer of the literature
more strongly than journals or other possible groupings of citations.
In this study, we use aggregated journal–journal citation relations as units of analysis
instead of journals. Each relation specifically combines two journals. The journal–journal
citation relation is a link in the networks of which journals are the nodes. The citation
relations among the 10,000 ? journals contained in the Journal Citation Reports (JCR) of
the Science Citation Index and the Social Sciences Citation Index can be organized as a
matrix of (10,000 ?)2 cells, each representing a unique relation between a citing and a
cited journal. These (valued) relations can change over time to the extent that they can
disappear or emerge; that is, turn from a zero into a positive value larger than or equal to
From case-study research, we know that specific journal–journal relations can catalyze
structural changes in the journal networks over time. For example,
shown how the introduction of the Scandinavian Journal of Work Environment and
Health—owned by the Swedish trade-unions—triggered a clustering in the (relatively
small) network of journals about occupational health around 1978
den Besselaar and Leydesdorff (1996) found the merging of three journals (Artificial
Intelligence, AI Magazine, and IEEE Expert)2 into a citation community decisive for the
take-off of artificial intelligence as a specialism since 1988. Major disciplinary
developments such as the development of nanoscience and technology in recent decades can also
be analyzed and visualized in terms of journal citation relations
(e.g., Leydesdorff and
Schank 2008; Rosvall and Bergstrom 2010; cf. Klavans and Boyack 2009)
. However, new
developments at the level of specialties and disciplines cannot be indicated unequivocally
in terms of new journals, since journals often emerge within existing fields, and then do not
indicate change but growth and stability within disciplinary boundaries
(Leydesdorff et al.
Journal names can also be changed indicating a change in focus or, for example,
reflecting a shift from a national to an international orientation. Each year, the two JCRs
(of the SCI and SSCI) provide lists of the changes in the previous year.3 Note that changes
in journal names and/or journal citation relations do not indicate novelty at the level of
(Uzzi et al. 2013; Wang et al. 2017)
. From an evolutionary perspective,
atypical citation patterns in papers provide the variation, whereas journal structures can be
considered as selection environments. Changes at the aggregated level are more structural
and occur more slowly than those at the level of articles containing knowledge claims. We
will analyze the asymmetrical journal–journal citation matrix from the ‘‘citing’’ side which
follows the research front, while ‘‘cited’’ represents the archive that can be expected to
develop at a lower speed
(Leydesdorff 1995; Small 1978; Small and Griffith 1974)
In sum, we compare cells within citation matrices of the aggregated journal–journal
citation relations during the period 1994-2016 taking journal name changes into account.
1994 is the first year that the JCRs were available as CD-Roms (replacing earlier paper and
microfiche versions) and 2016 is the last year for which this data is available. The
comparison is done using (1) dynamic measures such as
Kullback and Leibler’s (1951)
divergence measure (KL), which enables us to compare the relative frequency distribution
ex ante as an expectation of the distribution ex post; (2) Theil’s (1972) measure of
improving on or worsening the prediction using KL divergences; and (3) Leydesdorff’s
(1991) measure of critical path-dependency based on Theil’s measure. (The analytical
relations are elaborated in the Appendices.)
1 Citation relations between journals are counted in the JCRs as the sum of unique citation relations among
(Garfield 1979a, 1979b)
2 The journal name was changed to IEEE Intelligent Systems in 1997.
3 For analytical reasons, one may wish to include only journals present during all the years under study or,
in other words, a ‘‘fixed journal set’’
(Narin, 1976; cf. Leydesdorff, 1988)
The use of algorithms that focus on the dynamics represents an improvement over
comparing statics maps year-by-year consecutively
(e.g., Rosvall and Bergstrom 2010)
The dynamics is more than the resulting difference between two states. Because we focus
on journal–journal links, those that pass the statistical tests can be recomposed into
networks and analyzed and visualized by standard programs such as Pajek and VOSviewer.
Links may be connected into large components. However, incidental changes in links
between two nodes in a single year can be the effect of chance processes and should
therefore not be considered as equally valid indicators of change
(Bianconi et al. 2014; de
Nooy and Leydesdorff 2015)
The Science Citation Index has existed since 1964,4 but the first edition of the Journal
Citation Reports dates from 1975. The Social Sciences Citation Index was first published in
1973, and extended with Journal Citation Reports in 1978
(Garfield 1979b, at p. 16;
. At the time, JCRs were made available in print and since approximately
1990 also on microfiches. Electronic versions on CD-Rom were distributed between 1994
and 2008. In 2009, this modality was abandoned in favor of downloadable files on the
internet. (The internet version at WoS goes back to 1997.) However, the complete
electronically available series between 1994 and 2016 is organized similarly (using MS
Access), albeit with some reorganization and further extension in 2001.
Journal coverage has been expanded over the years, with a deliberate discontinuity in
2008-2009 when the database was ‘‘regionally expanded’’ given its traditional
underrepresentation of Eastern-European journals
(Testa 2010, 2011)
and perhaps also in response
to competition from Scopus, which was introduced in 2004 by Elsevier. Scopus covers
many more journals than WoS (Leydesdorff et al. 2016).
On issuance of the JCRs 2016 in November 2017, one of us reorganized the JCR data for
SCI and SSCI combined during the period 1994–2016 into a single standard format in
order to obtain a cube of data containing 23 yearly slices. For each year, one can construct
a matrix of N journals in the columns citing the same N journals in the rows. The number of
journals N grows from 5765 journals in 1994 to 11,487 in 2016 (Fig. 1). Note that not all
journals covered are also processed from the citing side; for example, 189 of the journals in
2016 (1.6%) were not processed for citing; articles in some journals (e.g., The Scientist) do
not have reference lists. In our design, such cases lead to an empty column (zeros) in the
matrix, but in the final analysis, we will use only values above a threshold. The zeros can
thus be considered as missing values.
Figure 1 shows the growth of the SCI and SSCI in terms of the numbers of journals
covered during the period under study; Fig. 2 adds the numbers of citations stored in the
two databases. The difference between SCI and SSCI in the volume of citation is striking.
Whereas the contribution of SSCI to the combined set grows from 24.3% of the journals in
1994 to 28.2% in 2016, the citations in the SSCI contribute only 4.6% to the combined set
in 1994 and 6.8% in 2016. The overlap in terms of journals grows from 150 (2.6%) in 1994
4 An experimental version of the SCI—the Genetics Citation Index—was available for 1961
to 633 (5.5%) in 2016. However, we count these journals only once. Note that the regional
expansion of the journal set in 2009 (in Fig. 1) did not significantly affect the time series of
the cumulative citations in Fig. 2.
Changes in journal names
Each JCR (in both databases) contains a file with name changes, mergers, and splittings of
journals. We use the latest name abbreviation and backtrack from the most recent year of
change adding this name to all previous years. For example, the name of the Journal of
Zhejiang University Science C: Computers and Electronics was changed to Frontiers of
Information Technology and Electronic Engineering in 2015. We use the abbreviated
journal name (FRONT INFORM TECH EL) to follow this journal from its first inclusion
into the database in 2011. These uniform labels are needed for comparisons over time.
However, we were not able to backtrack journal names in the case of journal splittings.
In such cases, it is not possible to tell which name expresses the line of inheritance in terms
of content. Furthermore, journals can be split into more than two new journals. For
example, the journal Biochimica Biophysica Acta (BBA; established in 1947) was split into
nine titles as sections of BBA. BBA—Molecular and Cell Biology of Lipids commenced in
1998 and is itself a continuation of BBA—Lipids and Lipid Metabolism (1965–1998) and
BBA—Specialized Section on Lipids and Related Subjects (1963–1964). In other words,
BBA can nowadays be considered as a family of (Elsevier) journals.
In summary, we organize the data of the JCRs (SCI and SSCI combined) into a matrix
for each year during the period 1994–2016 with unique identifiers for each cell; that is, the
citing and cited journal name abbreviations. The matrix is asymmetrical (‘‘citing’’ as
column variables and ‘‘cited’’ as row variables), but equally sized in both directions (that
is, 1-mode). The 23 matrices were thereafter transformed into 21 matrices (1996–2016)
with 3-year moving averages. Since each evaluation requires 3 years (t as the a posteriori
year, t - 1 for the revision of the prediction, and at t - 2 as the a priori distribution), we
have 19 observations for each cell (1998–2016). In the final step, we incorporate only
values larger than ten (as the aggregate for 3 years) in order to suppress the possible noise
effects of small values.
From comparative statics to dynamic analysis
There are both theoretical and methodological reasons for not using the differences
between maps of consecutive years as indicators of change. Cluster analysis and similar
techniques (e.g., community finding) are static analyses. Furthermore, community-finding
algorithms often begin with a random seed, so that it may even be difficult to reproduce the
cluster structure in the same year. When one compares the results of static analyses
yearon-year by subtraction, one loses control of whether one is measuring substantive change
or the choice of another sub-optimum by the clustering algorithm. One risks confounding
the dynamic analysis with the development of error in the measurement as a consequence
of the model.
For example, Fig. 3 shows an alluvial diagram based on the 3-year moving average
matrices of journal–journal relations in 2006, 2011, and 2016. It illustrates the problems:
(1) journals and journal groups may be positioned differently from year to year because of
different (sub)optimalizations in the delineation of clusters, and (2) the emergence of PLOS
ONE in 2006, and even more importantly the clustering by RSC Advances
the Royal Society of Chemistry in 2011)
. Unlike multi-disciplinary journals such as
Science and Nature, these new journals are oriented towards publishing a large volume of
The emergence of these ‘‘mega-journals’’ has made the clustering of journals an
unreliable basis for assessing longitudinal developments using comparative statics. The
ratios of contributions by different specialties or disciplines within these journals may
change from year to year for reasons very different from substantive ones. This relatively
new development adds to the methodological point that comparative-static clustering
cannot provide a basis for longitudinal inferences in the multi-variate case. Thus, one needs
dynamic measures for the longitudinal evaluation.
Change and discontinuity
Baur and Schank’s (2008)
dynamic extension of multi-dimensional scaling
Leydesdorff and Schank (2008)
elaborated a dynamic version of visone for
bibliometric network analysis. However, the size of the networks here under study is
computationally beyond the capacity of these routines.
Information theory enables us to study longitudinal developments first at the level of
cells and then to aggregate, since the Shannon-formulae are based on using Rs. There are
virtually no size limitations for the decomposition of large networks since the evaluation is
on a cell-by-cell basis. Note that
Rosvall and Bergstrom’s (2010)
information theory for the decomposition and is therefore equally fit for handling (virtually
unlimited) large sets. The limits are set only by computer hardware limitations such as
available memory and disk size.
The dynamic extension of Shannon’s (1948) definition of the information content of a
distribution [H = Pi pi log2 pi] is provided by
Kullback and Leibler’s (1951)
measure I = Ri qi log2 (qi/pi). where I measures the expected information of the message
that the prior distribution (Ri pi) has turned into the posterior distribution (Ri qi). (When
the two-base of the logarithm is used, I is expressed in bits of information.) Theil (1972, at
pp. 59 f.) has proven that I is necessarily equal to or larger than zero. However, the
nonnegative aggregated value for I allows for local entropy-changes as contributions which
are negative. Note furthermore that I is asymmetrical in p and q: the information content of
a change along the arrow of time is different from one in the reverse (backward) direction.
The prior distribution can also be considered as a prediction of the posterior one. In the
case of a perfect prediction, I = 0 and the two distributions are similar. If the prediction is
imperfect, it can be improved by a distribution at an in-between moment of time (Fig. 4).
This improvement of the prediction of the a posteriori probability distribution (Ri qi) on the
basis of an in-between probability distribution (Ri pi’) compared with the original
prediction (Ri pi) can be formulated as follows
(Theil 1972, at p. 77)
i qi log2ðqi=piÞ
i qi log2ðqi=p0iÞ
Iðq : pÞ
Iðq : p0Þ ¼
i qi log2ðp0i=piÞ
If I(q:p) [ I(q:p’) ? I(p’:p), the path via the revision (in Fig. 4) is a more efficient
channel for the communication between sender and receiver in terms of bits of information
than their direct link. Contrary to the geometry of Fig. 4, the sum of the information
distances via the intermediate station is then shorter than the direct information path
between the sender and the receiver.
The in-between year can also be considered as an auxiliary station in the signal
transmission from the sender to the receiver. If the auxiliary station boosts the signal from
the sender to the receiver, the system loses its history because what happened before the
rewrite no longer matters. In other words, the generation of a negative entropy indicates a
discontinuity. The Kullback–Leibler divergences can thus be used to analyze critical or
path-dependent transitions in a set of sequential events
(Frenken and Leydesdorff 2000;
Leydesdorff 1991, 1995, at p. 341)
Whereas a path dependency is generated historically if I(q:p’) ? I(p’:p) \ I(q:p) in the
forward direction, Lucio-Arias and
noted that the forward arrow of
time models diffusion from a sender to a receiver, whereas the backward arrow assumes
the perspective of a receiver looking backward (Rousseau et al. in preparation). The
backward perspective of the citing receiver provides meaning to the events from the
perspective of hindsight. Using the same notation, the inequality can analogously be
formulated as: I(p:p’) ? I(p’:q) \ I(p:q).
In summary, let us define an indicator
U ¼ Iðq : p0Þ þ Iðp0 : pÞ
Iðq : pÞ
in which q indicates posterior, p prior, and p’ revision of the prediction. If U \ 0, the
transition is critical in the forward direction. In other words, negative entropy is generated
along the arrow of time. Analogously:
V ¼ Iðp : p0Þ þ Iðp0 : qÞ
Iðp : qÞ
indicates a critical transition against the arrow of time (that is, from the perspective of
hindsight). The signs in the equations are chosen so that the reasoning remains consistent
with (Shannon-type) information theory: a negative entropy indicates the irregularity
which needs to be explained. A mathematical elaboration of these equations is provided
below in Appendices 1 and 2.
Since I(p:q) is unequal to I(q:p), one can expect the path dependencies with the arrow of
time (representing diffusion) to be different from the ones against the arrow of time
(representing codification). While we are interested in the significance of discontinuities
for codification more than diffusion, we focus the presentation on the results with the
reverse arrows. However, the differences are often small.
We first explore the data for 2016 (as the most recent year) and then extend the analysis to
the full set 1994–2016. In 2016, the domain is based on the (11,487 2 =) 131,951,169
possible relations among 11,487 journals. Of these cells, 3,020,242 (2.3%) have a value
larger than zero in the JCR, containing in total 50,030,365 citation relations. On average,
this is 16.6 per cell. However, citation distributions are very skewed: 97.7% of the cells are
In accordance with the findings of de Nooy and Leydesdorff (2015), we found so much
volatility when comparing the data for 2016 with the preceding years 2014 and 2015 that
we decided to use 3-year moving aggregates. A 3-year moving aggregate or average,
however, is only reliable for relations present during each of three consecutive years (t,
t - 1, and t - 2). In 2016, this is the case for 1,387,423 relations (45.9%) containing
122,778,368 citation relations aggregated over the 3 years 2014–2016.5 The average cell
value is now (122,778,368/3)/1,387,432 = 29.5. We further reduce noise by using a
threshold of above ten in each of the three consecutive years for the assessment of critical
transitions and improvements of the prediction. This leads to a file of 844,476 unique
5 This is derived as follows: (122,778,368/3 =) 40,926,123 relations on average, or 81.8% of the original
journal–journal citation relations6 that will be used in the analysis of 2016 data, and
analogously in the other years.
In addition to the N of links for each year, Fig. 5 shows the longitudinal development of the
numbers of observations for which the predictions are improved and path-dependencies are
generated during the period 1998-2016. Some of the lines are virtually coinciding: the
improvement of the prediction in the forward direction almost coincides with the absence
of a path-dependency in both the forward and backward directions (U [ 0 or V [ 0). This
is the case in on average 43.0% of the observations (SD = 1.8%).7 Secondly,
path-dependency in the forward direction coincides with path-dependency in the backward
direction and improvement in the backward prediction (57.0% of the observations). We
had not expected this coincidence; but we derive in the Appendices why this is the case for
The patterns are repeated from year to year. Note that critical transitions are the rule
(57%) more than the exception. The values of the critical transitions (in bits), however,
vary widely in each year (Fig. 6).
The number of observations in 2016 is 844,476, of which 770,170 (91.2%) have values
between - 0.1 and ? 0.1 millibits. This large segment is represented in Fig. 6 as a flat line
along the x-coordinate. At both ends, however, the critical transitions can have much larger
absolute values. Plotting these values log–log for the top 10,000 on either end provides two
power-law-type distributions (Fig. 7) with an excellent fit
(r2[ .99; b = - .998; p \ .001;
using the SPSS routine for curve estimation; cf. Clauset et al. 2009)
We added the equations to the figures to show the exponents, which are on the order of
0.7. For a scale-free network, this exponent has to be larger than one
(e.g., Broido and
. The distribution also fits more than .99 to a non-scale-free distribution such
as the log-normal or Weibull distributions. The interpretation of this finding is therefore not
trivial. We tested the distributions also using curve estimation in SPSS (v.22), since r2 does
not explain the variation in log–log curves. However,
Clauset et al. (2009)
more sophisticated test for power-law distributions.
One possible interpretation of the curves might be that the fit marks the signature of
self-organized criticality or 1/f-noise. For explaining self-organized criticality, Bak and
Chen (1991, at pp. 26f.) used the example of a pile of sand on which one grain of sand is
dropped regularly: ‘‘Now and then, when the slope becomes too steep somewhere on the
pile, the grains slide down, causing a small avalanche. […] When a grain of sand is added
to a pile in the critical state, it can start an avalanche of any size, including a ‘catastrophic’
event. But most of the time, the grain will fall so that no avalanche occurs.’’ Even the
largest avalanches involve only a small proportion of the grains in the pile, and therefore
even catastrophic avalanches cannot cause the slope of the pile to deviate significantly
from the critical slope.
In other words, the effects of an avalanche are local and do not affect the overall
structure of the pile. The system remains in a critical state so that one can expect
avalanches to remain equally possible. ‘‘Even though sand is added to the pile at a uniform
rate, the amount of sand flowing off the pile varies greatly over time.’’ In contrast to white
6 This is 28.0% of the original 3,020,242 non-zero relations and 60.9% of the 1,387,423 cells with
threeyear moving averages.
7 The improvements have a larger standard deviation (3.0%) than the path-dependencies (1.8%).
noise, 1/f noise suggests that the dynamics of the system are strongly influenced by past
events. The pile has a history of construction and reorganizations over time. Self-organized
criticality can also be studied by using a cellular automaton for the simulation.
Self-organized criticality as a model of the development of journal literature
The fit with a power-law in Fig. 7 suggests self-organized criticality (SoC) in the system of
journal–journal citation relations. New knowledge claims in manuscripts continuously
generate journal–journal citation relations potentially leading to an equivalent of
‘‘avalanches’’ of reconstructed inter-journal relations. These avalanches can occur anywhere;
their effects may be very different; but the consequences are local, that is, within the
discipline or specialty. The self-organized criticality remains globally available for new
critical transitions in the full range from minor, but more frequently occurring changes, to
rare but disruptive ones. The comparison with earthquakes provides another metaphor.
This model of self-organized criticality differs from the Kuhnian model of normal
science versus revolutionary science as phases in paradigm transitions
(Kuhn 1962; Marx
and Bornmann 2013; van den Daele and Weingart 1975)
. The flux of manuscripts with
knowledge claims contain references to other journals which can be compared with the
grains of sand that hit the sand pile or, in this case, the knowledge base as a construct. The
effect can be an avalanche of any size depending on the state of the system at that specific
place and time. The selection environments determine the size of the avalanches more than
the intrinsic qualities of the knowledge claims providing the variation.
Unlike grains of sand, however, one expects knowledge claims to be related. Bak and
(Bak and Chen 1991; Bak et al. 1987)
worked in their physical experiments
with actual sand grains of uniform granularity. Our ‘‘grains’’ are dropped on a sand pile,
but they are of different granularity in that they may be impure, containing, for example,
lumps of clay. Golyk (s. d.) compared Bak’s model with Zipf’s Law, which states that in
literary texts, the frequency of a word is inversely proportional to its rank in the frequency
table, given a large sample of words used. As against sand grains, such texts have
complicated non-local correlations such as syntax and cognitive structures (e.g. references), yet
the accumulation leads similarly to log–log lines (Price, 1976).
For self-organized criticality to occur, a large number of unrelated meta-stable
configurations is needed. Golyk concluded that both ‘‘models with local interactions (such as
BTW) as well as models with non-local (literary texts) correlations may lead to power-law
distributions’’ (at p. 4). However, the theory of self-organized criticality has hitherto been
rather phenomenological. There is no strict criterion for the value of the exponent, such as
one finds in the case of preferential attachment leading to power-law distributions where
one uses 2 \ a \ 3 as a criterion for scale-freeness. Bak et al. (1987, at p. 383) report
values of the exponent as low as .42 in studies of SOC. As noted, SOC can be simulated
using a cellular automaton as a grid. The exponent is also determined by the dimensionality
of the model, and perhaps by the different objects of study such as earthquakes, water
droplets on surfaces, human brains, etc.
. The original claim that SOC would
(Bak et al. 1987, p. 381)
is not needed for SOC as a phenomenon. Scale-free
networks are rare
(Broido and Clauset 2018; cf. Baraba´si 2018)
, whereas SOC is
abundantly the case in very different systems.
Aggregation of the critical links into networks
Our units of analysis are links, and thus the avalanches also occur in terms of links.
Building links upon links, one can expect the link structures to become meta-stable at
numerous places. A region may be ‘‘poised’’ for change based upon tension
(Foster et al.
where tradition and innovation coexist, but ‘‘tradition-shattering complements to the
tradition-bound activity of normal science’’
(Kuhn 1962, p. 6, 1977)
reconstruct the organization. Thus, an event introducing new links or abandoning old ones
may lead to large, but local shifts at specific places and moments. The non-local
correlations hold the structures otherwise together. Sets of journals may be reoriented without
losing their cohesive structure as a group.
Moreover, one would not expect multiple disruptions of the system to co-occur. The
system operates at the edge of chaos, and thus will respond to perturbations, but an overall
system collapse and revolution would not be expected. This suggests that other parts of the
network will be robust and resistant to change, even with considerable tensions within
them. However, the links that are involved in a reconstruction can again be considered as
parts of a network.
Using the value of U \ –1 mbit as a threshold, for example, 4207 links are carried by
1,633 journals. Of these journals 1532 form a single (large) component (Fig. 8). Among
these journals are not surprisingly multi-disciplinary journals such as PLOS ONE, PNAS,
Science, and Nature. These journals are also part of the citation structures in the
contributing disciplines. However, the content of these journals is not subject-bound and varies
in its disciplinary composition from year to year. Therefore, they seem to change radically
from year to year in terms of the cited knowledge base, but this is an effect of their
versatility in terms of disciplinary orientation. However, further away from the center, one
finds disciplinarily specific groups of journals and titles indicating specialisms. For
example, a cluster focusing on renewable energy sources is positioned on the right side of
Fig. 8. We return to this cluster below.
First, Fig. 9 shows forty other components comprising 101 journals and related with
values of U \ –1 mbit, but not related to the main component. However, the threshold of
- 1 mbit is arbitrary (and set for reasons of presentation); with a lower threshold, more
journals and clusters are involved, and with a higher threshold the main component is
further decomposed. In other words, one can consider the groups as islands where
‘‘avalanches’’ of various sizes are indicated. Our parameter choices (e.g., thresholds) structure
what we can observe of these avalanches, which can be assumed to occur at all scales and
Changes between years
In Fig. 10, we compare the components which contain the journals Renewable Energy and
Renewable and Sustainable Energy Reviews. In 2010 and 2016, both journals were part of
the main component. In 2004—another 6 years earlier—neither journal was involved in
any critical transitions of more than one bit. Using
Blondel et al.’s (2008
) algorithm for
community finding, we can extract a subcluster of 72 journals in 2010 and 85 in 2016. The
overlap is only 23 journals. How are the non-overlapping journals providing different
In 2010 (Fig. 10a), the group of journals representing this specialty is positioned at the
top of the map, but only connected via Renewable and Sustainable Energy Reviews to
journals focusing on ‘‘biomass’’ as the renewable energy resource at that time. In 2016
(Fig. 10b), the group (light blue) is much more central. The focus on biomass has been
replaced with one on construction, environmental pollution, and electrical power systems.
In Fig. 11, the two maps are compared as overlays using dynamic MDS
. The journals with red labels in the upper part show the situation in 2010,
and the blue labels at the bottom half the situation in 2016; in the middle is an overlap.
Over a period of 6 years, the orientation of this field has changed.
Testing self-organized criticality
Since we have 19 observations, we can test whether the power-law in Fig. 5 is reproducible
in other years. Using the test for power laws in SPSS
(cf. Clauset et al. 2009)
, we found this
fit to be significant at the level of p \ .001 in all years. The exponent is -0.639 on average
with a standard deviation of 0.030. Figure 12 shows this for the years 1998, 2004, 2010,
and 2016. In sum, the phenomenon can be reproduced.
Summary and conclusions
Using dynamic entropy measures such as Kullback–Leibler’s (1951) divergence, Theil’s
(1972) improvement of the prediction, and Leydesdorff’s (1991) test for critical transitions
as indicators of change over time, we unexpectedly found ‘‘self-organized criticality’’ in
the 10,000 ? most pronounced cases of evolutionary change. It seems to us that the model
of self-organized criticality makes it possible to show regularities that help us to
understand the evolutionary development of the sciences where disciplines are both enabled and
constrained by what is past—or conventional—and what is possible or can be sustained by
the system. While it is impossible to say where change will occur, the fact that it will occur
is expected. Smaller, local events may be fractals of what occurs elsewhere on a larger
scale. However, we do not expect the system to be scale-free, since the scaling coefficients
may vary among disciplines.
The sciences can be understood as developing in terms of interdependent continuities
and discontinuities. The continuities are needed for the accumulation of the ‘‘sand pile’’
into a knowledge base, with the inertia of institutionalized relations maintaining the
relevant structures. The discontinuities provide options for a rewrite
reorganization. These may happen as lightning (‘‘grains of sand’’) striking the ground—in
an unpredictable way—but the effect can be a local ‘‘avalanche’’ of any size and
accordingly a reorganization.
Unlike a sand pile, the knowledge base is a construct comprising specific structures.
Whereas some multidisciplinary journals participate in this reorganization of the
knowledge base every year—because of their multidisciplinary character—large numbers of
journal links are unresponsive to changes in the environment, so that the transitions remain
close to zero in terms of bits of information. The system is both dynamic by chance and
structurally ‘‘frozen,’’ but in different parts of the structure.
Kauffman and Johnson (1991)
described this as a co-evolution to the edge of chaos; the constructed knowledge base is at
many places meta-stable, while at other places, the systems may be temporarily locked and
therefore unable to change.
In addition to this unexpected result, the use of critical transitions enables us to show
how a research front can shift rapidly. In the case of research about renewable energy, for
example, the research focus on ‘‘biomass’’ in 2010 shifted to an orientation towards themes
like electrical power, construction, and clean production in 2016. The journals were present
in 2004, but this research area was not yet involved in the dynamics of discontinuities.
The data allow for raising many more questions, for example, about the dynamics of
disciplinary groups of journals
(e.g., Leydesdorff and Schank 2008; Rosvall and Bergstrom
. It might be interesting to follow up with a study of the social sciences separately.
Another extension would be a further study about codification on the citing side versus the
diffusion of a signal from a source to a set of receivers—as is a common assumption in
citation analysis (cf. Zitt and Small 2008). As noted, what events mean is decided from the
perspective of hindsight using deterministic (!) selection mechanisms that operate on the
forward generation of variation. One obvious limitation of the present study is the
assumption that citations are simple counts, independent of the questions of who is citing,
what is cited, at which place, in which context, etc.
Acknowledgement The authors wish to thank Clarivate for making the JCR data available.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix 1: Elaboration of the inequalities in the case of critical transitions
A transition is critical in the forward direction if:
q log2 q=p [ q log2 q=p0 þ p0 log2 p0=p
Or after elaboration of the logarithms as exponents:
Analogously in the backward direction:
pq [ ðp0Þq
p q [ ðp0Þ q ðp0Þp0 ð pÞ
p p0 q [ ðp0Þp0 q
if ðp0 [ q then [ qÞ OR ðif p0 \ q then p\p0\qÞ
p log2 p=q [ p log2 p=p0 þ p0 log2 p0=q
pp pp p0p0
qp [ ðp0Þp qp0
q p [ ðp0Þ p ðp0Þp0 ðqÞ
q p0 p [ ðp0Þp0 p
ðif p0 [ p then [ pÞ OR ðif p0\p then q\p0\pÞ
These conditions are the same in both directions: when the sequence between the
probabilities increases or decreases monotonically (with or against the arrow of time), the
transition is path-dependent.
Appendix 2: Elaboration of the inequalities in the case of improvements of the prediction
The condition for an improvement of the prediction in the forward direction is:
q log2 p0=
This is true if p0 [ p. In the other case (p0 \ p), the perdiction is worsened.
Mutatis mutandis in the backward direction:
This is true if p0 [ q. In the other case (p0 \ q), the prediction is worsened.
As against the symmetrical conditions for critical revisions, improvement is
asymmetrical in both the backward and the forward direction.
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