# A Map of Science in Wikipedia

Puyu Yang  Institute for Logic, Language and Computation (ILLC)
University of Amsterdam, The Netherlands.  Giovanni Colavizza Please address
correspondence to<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Institute for Logic,
Language and Computation (ILLC)
University of Amsterdam, The Netherlands.

###### Abstract

In recent decades, the rapid growth of Internet adoption is offering
opportunities for convenient and inexpensive access to scientific information.
Wikipedia, one of the largest encyclopedias worldwide, has become a reference
in this respect, and has attracted widespread attention from scholars.
However, a clear understanding of the scientific sources underpinning
Wikipedia’s contents remains elusive. In this work, we rely on an open dataset
of citations from Wikipedia to map the relationship between Wikipedia articles
and scientific journal articles. We find that most journal articles cited from
Wikipedia belong to STEM fields, in particular biology and medicine ($47.6$%
of citations; $46.1$% of cited articles). Furthermore, Wikipedia’s biographies
play an important role in connecting STEM fields with the humanities,
especially history. These results contribute to our understanding of
Wikipedia’s reliance on scientific sources, and its role as knowledge broker
to the public.

## Introduction

Wikipedia is the largest, free and collaborative encyclopedia to date. Its
importance cannot be overstated as Wikipedia provides reliable access to
information worldwide. Wikipedia editors use primary and secondary sources in
support of the statements they make in Wikipedia. These sources are quoted,
cited and added to the list of references in any Wikipedia article. While
previous work has focused on the contents of Wikipedia and its collaborative
editing, it is only recently that scholars have begun to systematically
investigate Wikipedia’s sources.

Developing a better understanding of the scientific sources Wikipedia relies
on is important to assess its coverage, reliability and representation of
human knowledge, including in view of informing its future development.
Expanding upon previous work (Teplitskiy et al.,, 2017; Torres-Salinas et
al.,, 2019; Colavizza,, 2020; Arroyo-Machado et al.,, 2020), we pose here the
following two descriptive research questions (RQs):

  1. 1.

RQ1: Which scientific sources are cited from Wikipedia and what are their
characteristics?

  2. 2.

RQ2: Which areas of Wikipedia rely on scientific sources and how do they
relate?

In order to contribute towards answering them, in the present contribution we
focus on citations given from Wikipedia articles to scientific journal
articles. We rely on a recently published dataset as a source of such citation
data (Singh et al.,, 2021). Firstly, we provide a descriptive overview of the
journal articles cited from Wikipedia (RQ1); secondly, we make use of network
analysis and study the bibliographic coupling network of Wikipedia articles,
respectively the co-citation network of journal articles (RQ2).

## Previous work

#### References in Wikipedia

Wikipedia is a tertiary source which strives to provide reliable contents in a
neutral way (Mesgari et al.,, 2015). To this end, Wikipedia editors follow
established standards, guidelines and workflows to expand Wikipedia articles,
and add references to them (Kaffee and Elsahar,, 2021). The bulk of the
referencing activity seems to occur when an article has reached a certain
level of maturity and number of edits. Furthermore, references “also tend to
be contributed by editors who have contributed more frequently and more
substantially to an article, suggesting that a subset of more qualified or
committed editors may exist for each article” (Chen and Roth,, 2012). Despite
this approach, Wikipedia’s contents vary greatly in quality, including across
languages (Roy et al.,, 2021).

The automatic improvement of Wikipedia’s contents is an area of active
research. While bots already patrol and improve the quality of Wikipedia’s
references (Zagorova et al.,, 2021), recent work also focused on automatically
flagging sentences in need for a citation (Redi et al.,, 2019) and on
assessing a source’s reliability (Lewoniewski et al.,, 2020).

#### Using Wikipedia

Given its broad scope in contents, the usage of Wikipedia varies greatly too:
“for instance, we observe long and fast-paced page sequences across topics for
users who are bored or exploring randomly, whereas those using Wikipedia for
work or school spend more time on individual articles focused on topics such
as science”(Singer et al.,, 2017). The usage of Wikipedia is even more
significant when considering countries with varied languages and socio-
economic characteristics (Lemmerich et al.,, 2019). Wikipedia also fulfills a
specific role within the broader Web: it serves as a stepping stone between
search engines and third-party websites (Piccardi et al., 2021b, ). The
complementarity of Wikipedia and search engines, such as Google, is
particularly significant for scientific information seeking (Mesgari et al.,,
2015).

Previous studies have explored the use and usability of Wikipedia as a source
of biomedical information. A randomized controlled trial focused on placing
evidence of the effects of treatments (in this case, for schizophrenia) within
Wikipedia pages finds no significant changes in the full-text accesses of the
treated pages, but an effect in their altmetric scores (Adams et al.,, 2020).
The readability of the most viewed Wikipedia articles on diseases is of
varying quality, with many articles still too difficult to read for a general
readership (Brezar and Heilman,, 2019). More generally, while Wikipedia is a
prominent health information source in terms of views and visibility (e.g.,
often top in Google searches), the study of its impact in this respect is
still too limited to draw any general conclusion (Smith,, 2020).

A distinct set of recent studies explored the use of references in Wikipedia,
and in particular references to external sources. Work on WikiProject Medicine
shows that its readers appear to use links to external sources to verify and
authorize Wikipedia content, rather than to examine the sources themselves
(Maggio et al.,, 2020). A Wikipedia-wide study of engagement with external
references found that users click on them only very rarely (once for every 300
page views, on average) (Piccardi et al.,, 2020). Crucially, users more often
look for more when reading lower quality and shorter articles, which possibly
do not contain what they seek (Piccardi et al., 2021a, ). These findings
further underline the importance of providing high-quality contents within
Wikipedia itself.

#### Science and Wikipedia

Wikipedia strives to convey information grounded in scientific results. It
thus provides visibility to scientific research, and is considered an
altmetric source in this respect (Sugimoto et al.,, 2017), yet the influence
goes both ways. In fact, previous work has established that being cited from
Wikipedia can increase the citation impact of an article (Thompson and
Hanley,, 2018). Understanding and monitoring which scientific results underpin
Wikipedia’s contents, and why, is therefore of critical importance.

Recent work is gradually improving our understanding of the matter. The open
release of datasets of citations from Wikipedia to its sources is helping in
broadening access to the essential data for tackling the question at hand
(Halfaker et al.,, 2018; Zagorova et al.,, 2021; Singh et al.,, 2021).
Furthermore, several previous studies have been able to rely on altmetric
data. Some trends clearly emerge from this literature. Journal articles cited
from Wikipedia are more likely than average published in high-impact journals
(e.g., by impact factor), and in open access (Nielsen,, 2007; Teplitskiy et
al.,, 2017). Articles cited from Wikipedia are ‘uncited’ and untested by
subsequent studies in rates proportional to the rest of the scientific
literature, nevertheless they also receive a higher rate of supporting
citations (Nicholson et al.,, 2021).

Wikipedia’s capacity to rapidly and reliably integrate novel scientific
results to respond to ongoing public events or crises has also been assessed.
The COVID-19 pandemic provides a recent example (Colavizza,, 2020). Most
notably, the areas of Wikipedia where the editors are highly organized and
include domain experts, for example several WikiProjects, appear to fare
better in this respect. Indeed, the scope of expert involvement in editing
Wikipedia is substantial. A recent study found that approximately 10%–30% of
Wikipedia’s contributors have substantial subject-matter expertise in the
topics that they edit (Yarovoy et al.,, 2020).

Recent results by Arroyo-Machado et al., (2020), extending previous work by
the same team (Torres-Salinas et al.,, 2019), directly relate to our study.
The authors perform a co-citation analysis of Wikipedia’s sources, relying on
a dataset of “847,512 references made by 193,802 Wikipedia articles to 598,746
scientific articles belonging to 14,149 journals indexed in Scopus.” They use
Altmetrics data to retrieve Wikipedia citations to journal articles. Their
study the co-citation network of journals and Scopus main field categories, as
referenced by Wikipedia. The most significant scientific domains cited by
Wikipedia include Medicine, Biochemistry, Genetics, and Molecular Biology.
They confirm that the most important journals are multidisciplinary, and
include prominent venues such as Nature, Science, PNAS, and that articles from
high-impact factor journals are more likely to be cited from Wikipedia.
Lastly, they find that only 13.44% of Wikipedia citations are to open access
journals. Directly expanding upon Arroyo-Machado et al., (2020), we use here a
larger and more recent dataset, considering the more granular level of
analysis of journal articles, and we deepen the analysis by comparing the
bibliographic coupling network of Wikipedia articles with the co-citation
network of journal articles.

In conclusion, it is also worth noting that understanding how Wikipedia is
structured according to how it uses scientific publications complements work
relying on its internal link network. For example, previous results have found
that different scientific domains possess distinct internal link network
organizations, with modular structures for Biology and Medicine, but a sparse
structure for Mathematics and a dense core for Physics (Silva et al.,, 2011).

## Data and methods

We assemble data from a variety of sources to perform our study.

#### Wikipedia Citations

We use Wikipedia Citations as our main dataset (Singh et al.,, 2021). It
consists of more than 29M citations extracted from the over 6M articles
composing the English Wikipedia as of May 2020. In Wikipedia Citations, each
citation is automatically classified as being to a book, journal article or
Web content. Approximately 2.5M citations are classified as to a journal
article, of which 1,705,085 are equipped with a DOI, either from Wikipedia
itself or retrieved from Crossref. These citations to journal articles come
from 405,358 distinct Wikipedia article pages and refer to 1,157,571 distinct
DOIs. Citations to journal articles clearly comprise a relatively limited
share of all citations contained in Wikipedia, and thus likely serve a
specific purpose (Singh et al.,, 2021). A large share of these citations are
given to articles published over the past 20 years, and the most cited
journals include Nature, Science, the Journal of Biological Chemistry and
PNAS. We use this set of citations in what follows.

#### ORES Topics, WikiProjects, Dimensions

In order to perform our analysis, we enrich the Wikipedia Citations dataset
with other sources of information. Firstly, we equip Wikipedia articles with
information about their topics and the WikiProject they belong to, if any. The
topics of Wikipedia articles are retrieved using the ORES Web
service11endnote: 1We made use of the second API at the following address:
https://wiki-topic.toolforge.org/#lang-agnostic-model., which exposes a topic
model of Wikipedia trained using Language-Agnostic Topic Classification (LATC)
(Johnson et al.,, 2021) and assigns each Wikipedia article to a taxonomy
rooted into the four categories of: Geography, Culture, History and Society,
and STEM.22endnote:
2https://www.mediawiki.org/wiki/ORES/Articletopic#Taxonomy. Through the ORES
API, we could extract topics for Wikipedia articles covering 99.7% of the
citations in Wikipedia Citations.

Further, we equip Wikipedia articles with information on their WikiProject. “A
WikiProject is a group of contributors who want to work together as a team to
improve Wikipedia”33endnote:
3https://en.wikipedia.org/wiki/Wikipedia:WikiProject., for example to focus on
a specific topic area such as WikiProject Mathematics or WikiProject India, or
curate a specific aspect of the encyclopedia, for instance WikiProject
Disambiguation. The English Wikipedia currently includes over 2,000
WikiProjects. Using public data (Johnson and Halfaker,, 2020), we could equip
with WikiProjects information Wikipedia articles comprising 96.3% of the
citations in Wikipedia Citations.

Finally, we use Dimensions (Herzog et al.,, 2020) to retrieve metadata for all
the journal articles with a DOI cited from Wikipedia. While no ideal single
bibliographic data source exists yet, Dimensions provides for broad source
coverage and relies in substantial part on the open Crossref repository,
making it a meaningful choice for our study (Visser et al.,, 2021). What is
more, we are further interested in an article’s Field of Research (FOR)
classification.44endnote: 4The Fields of Research classification follows the
research areas defined in the Australian and New Zealand Standard Research
Classification (ANZSRC). See:
https://app.dimensions.ai/browse/categories/publication/for. The Fields of
Research are organized into hierarchies, with divisions (top, largest), groups
and fields (bottom, smallest). Dimensions exposes divisions as major fields,
and groups as minor fields; we adopt this naming convention in what follows.
By querying the Dimensions’ API we were able to match 96% of all the unique
DOIs from Wikipedia Citations. All these data were retrieved in June 2021.

### Methods

For our study we make a comparison between two undirected networks in turn
extracted from the directed citation network of Wikipedia to journal articles:
the co-citation network among scientific journal articles, and the
bibliographic coupling network among Wikipedia articles. Using the conceptual
framework of Costas et al., (2021), we can say that the co-citation network is
made of co-Wikipedia citations whereby two journal articles are cited by the
same Wikipedia article(s), and the bibliographic coupling network is composed
of Wikipedia articles connected when they cite the same journal articles. Both
networks are clustered using the Leiden algorithm (Traag et al.,, 2019), and
their modular structures are compared. In this way, we aim to clarify which
areas of Wikipedia rely on scientific sources. What is more, we seek to show
how similar clusters of Wikipedia articles (bibliographic coupling) and
similar clusters of co-cited scientific articles (co-citation) are internally
related (RQ2).

#### Citation networks

While constructing the co-citation network, we remove nodes (journal articles)
that are cited only once from Wikipedia, as they would be isolated in the co-
citation network since they are never co-cited. This results in a network of
1,050,686 nodes (91% of 1,157,571) and 17,916,861 edges. Similarly, we remove
nodes (Wikipedia articles) that cite only one journal article, as they would
be isolated in the bibliographic network. This gives a network of 257,452
nodes (64% of 405,358) and 27,473,262 edges. This bibliographic coupling
network is thus not only denser, but it also does not include a higher share
of isolated nodes than the co-citation network. Nodes in both networks are
equipped with relevant metadata: WikiProject and ORES topics for the
bibliographic coupling network, Fields of Research for the co-citation
network. Furthermore, in both networks every edge is weighted according to how
many times any two nodes are cited together (co-citation) or jointly cite the
same items (bibliographic coupling).

#### Network clustering

In order to equip our networks with a clustering solution, we use the Leiden
algorithm (Traag et al.,, 2019), a popular choice that is fast and provides
specific guarantees over the resulting clustering.55endnote: 5For our analyses
we relied on igraph 0.9.6 and leidenalg 0.8.3. The Leiden algorithm uses a
resolution (hyper)parameter to control the balance between the number of
clusters and their size. A higher resolution parameter leads to more, smaller
clusters. We find a reasonable value for the resolution parameter empirically,
by inspecting the number of clusters at varying values of the parameter.
Figure 1(a) shows the number of clusters at the varying size of the resolution
parameter for the co-citation network, and Figure 1(b) does the same for the
bibliographic coupling network. We avoid choosing extreme values, and settle
for reasonable elbows which can be found, in both cases, at a value of the
resolution parameter of $1e-4$.

(a) Co-citation.

(b) Bibliographic coupling.

Figure 1: Number of clusters at varying values of the resolution parameter.

#### Supernetworks

In order to further coarsen our networks, we also construct ‘supernetworks’ of
clusters. In a supernetwork, each node is a cluster of nodes from an
underlying network. A supernetwork is therefore a network of clusters. A
supernetwork is often easier to visualize and inspect, as it contains a much
smaller number of nodes than the original network. We construct supernetworks
by weighting each node with the number of nodes that a cluster contains, and
by weighting each edge by summing the weights of the edges between the nodes
of any two clusters. The use of aggregated metadata from the nodes of a
cluster is detailed below, when discussing results. In order to identify and
focus on the largest and most representative clusters, we further trim the
supernetworks at a given cluster size threshold. We do so empirically, by
inspecting the cumulative share of nodes that would be included in a
supernetwork, at varying cluster size thresholds. Results are shown in Figure
2(a) (co-citation) and Figure 2(b) (bibliographic coupling). In both cases,
reasonable cutoffs can be found at elbows corresponding to a cumulative share
of included nodes of $.7$. This cutoff corresponds in turn to cluster sizes of
98 (co-citation) and 48 (bibliographic coupling), below which clusters are
removed from the supernetworks.

(a) Co-citation.

(b) Bibliographic coupling.

Figure 2: Cumulative share of nodes included in the supernetworks, per cluster
size threshold. Clusters are ordered from largest (left) to smallest (right).

## Results

We organise our results into three parts, addressing RQ1 in the first, and RQ2
in the second and third sections. First, we provide an overview of the
scientific sources cited from Wikipedia. Next we analyse how Wikipedia
articles are organised accordingly using bibliographic coupling networks.
Lastly, we analyse how journal articles cited from Wikipedia are organised in
turn, using co-citation networks.

### Science in Wikipedia

Field of Research (major field) | Citations | Journal articles
---|---|---
06 Biological Sciences | 440,029.8(25.8%) | 255,382.0(22.1%)
11 Medical and Health Sciences | 371,304.7(21.8%) | 278,289.3(24.0%)
04 Earth Sciences | 69,835.7(4.1%) | 42,865.6(3.7%)
02 Physical Sciences | 69,678.3(4.1%) | 42,405.1(3.7%)
03 Chemical Sciences | 67,931.0(4.0%) | 53,396.7(4.6%)
17 Psychology and Cognitive Sciences | 64,092.0(3.8%) | 48,072.9(4.2%)
21 History and Archaeology | 63,292.9(3.7%) | 40,855.7(3.5%)
16 Studies in Human Society | 57,414.7(3.4%) | 40,640.1(3.5%)
09 Engineering | 42,994.5(2.5%) | 34,626.0(3.0%)
01 Mathematical Sciences | 41,408.8(2.4%) | 31,755.2(2.7%)
08 Information and Computing Sciences | 33,286.9(2.0%) | 25,300.0(2.2%)
20 Language, Communication and Culture | 31,969.4(1.9%) | 23,256.9(2.0%)
05 Environmental Sciences | 22,993.5(1.3%) | 15,069.7(1.3%)
22 Philosophy and Religious Studies | 21,448.8(1.3%) | 15,654.1(1.4%)
14 Economics | 182,30.9(1.1%) | 13,720.5(1.2%)
07 Agricultural and Veterinary Sciences | 15,525.6(0.9%) | 12,146.2(1.0%)
15 Commerce, Management, Tourism and Services | 12,691.6(0.7%) | 10,146.7(0.9%)
13 Education | 10,534.9(0.6%) | 8,714.2(0.8%)
19 Studies in Creative Arts and Writing | 10,304.9(0.6%) | 8,085.8(0.7%)
18 Law and Legal Studies | 9,971.6(0.6%) | 7,189.6(0.6%)
10 Technology | 5,740.8(0.3%) | 4,433.4(0.4%)
12 Built Environment and Design | 2,886.5(0.2%) | 2,274.6(0.2%)
Missing | 221,517.0(13.0%) | 143,291.0(12.4%)
Total | 1,705,085(100%) | 1,157,571(100%)
Table 1: Number of citations and of journal articles per Field of Research
(major fields), using fractional counting.

#### Fields of Research

Table 1 shows the number of citations and the number of journal articles cited
from Wikipedia per Dimension’s major Field of Research. Given that journal
articles (and, later on, Wikipedia articles) are our units of analysis, we use
fractional counting to account for an article belonging to multiple categories
(Perianes-Rodriguez et al.,, 2016), unless otherwise specified. It can be seen
that almost half of citations are given to articles in the fields of
biological, medical and health sciences, which in turn make up about 46% of
all journal articles cited from Wikipedia. More in general, STEM fields make
up for a large part of both citations and articles, while the most represented
areas outside of STEM are history and sociology. Looking at minor Fields of
Research, in Table 2 (only the top 10 are shown), we can appreciate how the
most cited fields include genetics and cell biology, evolutionary biology and
ecology in biology; clinical science and public health, psychology and
neurosciences in medicine; history in the humanities. These fields capture
over a fifth of all citations from Wikipedia.

Field of Research (minor fields) | Fractional counting | Unique dois number
---|---|---
0604 Genetics | 76,299.8(4.5%) | 39,393.3(3.4%)
0601 Biochemistry and Cell Biology | 71,505.5(4.2%) | 45,098.7(3.9%)
1103 Clinical Sciences | 49,352.5(2.9%) | 38,700.8(3.3%)
1117 Public Health and Health Services | 32,066.8(1.9%) | 23,282.4(2.0%)
1701 Psychology | 27,375.5(1.6%) | 20,623.2(1.8%)
1109 Neurosciences | 25,030.1(1.5%) | 19,117.9(1.7%)
2103 Historical Studies | 24,134.6(1.4%) | 15,873.0(1.4%)
0403 Geology | 23,114.6(1.4%) | 13,464.6(1.2%)
0602 Ecology | 23,033.6(1.4%) | 14,862.2(1.3%)
0603 Evolutionary Biology | 18,246.4(1.1%) | 8,667.2(0.7%)
Table 2: Number of citations and of journal articles per Field of Research
(minor fields), using fractional counting.

#### Citation counts

We assess the distribution of (journal article) citations given to articles
cited from Wikipedia, using data from Dimensions. There is a wide variation in
citations counts, with many articles with no or few citations, and some with a
high number. The maximum number of received citations is 214,886 and the
minimum is 0, with a mean at 189 and a median at 33. When considering recent
citations (over the past two years), the maximum is 34,845, the minimum is 0,
with a mean at 36.4 and a median at 5. Furthermore, 60% of articles cited from
Wikipedia received fewer than 10 citations in the past two years. Articles
cited fewer than 100 times account for 70% of the total cited articles, and
only about 3% of articles are cited 1,000 times or more. It is worth noting
that 10.4% of journal articles are never cited and 22.5% have no recent
citations either, according to Dimensions. Finally, we note that 75,248 (4.4%)
articles were missing this information from Dimensions.

#### Most cited articles and journals

Among the top journal articles by a number of received citations (See
Appendix, Section Top-10 most cited journal articles), only two are open
access, six are authored by scholars based in the United States, and only one
is published after 2000. The most cited article is “Cleavage of Structural
Proteins during the Assembly of the Head of Bacteriophage T4”, published in
Nature in 1970, and cited 214,886 times according to Dimensions. Instead, the
most cited article in terms of recent citations (past two years) is “Deep
Residual Learning for Image Recognition”, published at the 2016 Conference on
Computer Vision and Pattern Recognition, with 34,845 recent citations. The top
citing Wikipedia articles, instead, frequently include reference or survey
articles. For example, the series “this year in” is often a source of highly
citing Wikipedia articles (e.g., “2018 in paleontology” includes 580 citations
to journal articles). In Table 3, we list the most cited journals in
Wikipedia. Nature, PNAS and Science top the list, confirming findings from
previous work (Arroyo-Machado et al.,, 2020).

Journal name | Citations
---|---
Nature | 37,287
PNAS | 31,801
Science | 26,903
Journal of Biological Chemistry | 24,518
PLOS ONE | 12,997
Zootaxa | 10,006
Cell | 9,318
Genome Research | 8,961
The Astrophysical Journal | 8,882
Astronomy & Astrophysics | 7,292
Table 3: Most cited journals

#### Age of articles and open access

In Figure 8, we show the distribution of the publication years for journal
articles cited from Wikipedia. The trend is clear: most cited articles were
published in the past two decades. We also assess the open access availability
of these journal articles using Dimensions data. We find that 686,952 (41%)
articles are available in some form of open access, while a majority number of
942,885 (55%) remain closed access, and for 75,248 (4%) this information is
missing. This constitutes a considerably higher fraction of cited OA articles
than what previous work has found (Arroyo-Machado et al.,, 2020).

#### Citation flows

In conclusion of this first overview section, we analyse the flow of citations
from Wikipedia to journal articles, on the one hand, grouping Wikipedia
articles by ORES topics and WikiProjects, and on the other hand, grouping
journal articles by major FOR categories. We show the river plots of citations
from Wikipedia articles by ORES topics in Figure 3 and by (top-10)
WikiProjects in Figure 4. The flow of citations from STEM Wikipedia articles
confirms the importance of the biological, medical and health sciences in
Wikipedia, while other topics are more evenly distributed across fields of
research. When considering WikiProjects, we only kept the top-10 by
(fractional) number of Wikipedia articles, which have a strong STEM focus.
Partially as a consequence, the flow of citations again favours the
biological, medical and health sciences, but also distributes across other
STEM fields of research. It is worth highlighting the role of the WikiProject
Biography, which focuses on the biographies of notable persons. This project
spans across fields of research, for example by covering the lives of
prominent scientists, and connects them with historical fields in turn.

Figure 3: Citation flow from ORES topics to major fields of research. Figure
4: Citation flow from the top-10 WikiProjects to major fields of research.

### Wikipedia from a science perspective

#### Bibliographic coupling network

The bibliographic coupling network of Wikipedia articles includes 257,452
nodes and 27,473,262 edges, with a density of 0.0008. The corresponding
supernetwork has 31,642 nodes (clusters) and 9,158 edges, for a density of
0.000018. We further aggregate information about ORES topics and WikiProjects
to the supernetwork using fractional counting. In Figure 9 we show the
distribution of the cluster sizes of the underlying bibliographic coupling
network, distinguishing between isolated and connected clusters. A large
number of small, isolated clusters exists, while at higher cluster sizes the
clusters become increasingly connected.

Figure 5: Distribution of ORES topics in the 10 largest clusters of the
bibliographic coupling network of Wikipedia articles.

#### Largest clusters

Next, we zoom-in the largest clusters of this network, showing their ORES
topic distribution in Figure 5, and their distribution over WikiProjects in
Figure 6. The combination of topics and projects is helpful in uncovering what
a cluster is about. For example, the largest cluster is devoted to biographies
and has a relatively balanced topic distribution across geography, history and
society, culture. Importantly though, its top WikiProjects include military
history, United States and football, suggesting a certain skew in this
respect. Most clusters are, instead, specialized in specific STEM fields. For
example, cluster 2 is focused on molecular and cell biology, and cluster 9 on
entomology (spiders, in particular). The dominance of STEM in ‘journal-
article-citing’ Wikipedia is further confirmed. Exceptions include history and
biographies, and archaeology/physical anthropology (cluster 7).

Figure 6: Distribution of WikiProjects in the 10 largest clusters of the
bibliographic coupling network of Wikipedia articles.

#### Network visualization

We conclude by a visual comparison of the bibliographic coupling supernetwork
using the same layout but different coloring according to its modularity class
(Figure 11), ORES topics (Figure 12), and WikiProjects (Figure 13). These
visualizations confirm two patterns we have previously highlighted: a) the
important role of biographies in connecting this part of Wikipedia (first
visualization); b) the systematic importance of STEM (second visualization).
Furthermore, we can also appreciate the apparently effective role played by
WikiProjects in helping editors coordinate for the curation of specialized
knowledge in a coherent whole (third visualization).

### Science from a Wikipedia perspective

#### Co-citation network

The co-citation network of journal articles cited from Wikipedia has 1,050,686
nodes and 17,916,861 edges, with a density of around 0.00003. The resulting
supernetwork has 71,983 nodes (clusters) and 23,668 edges, and a density of
0.00001. The distribution of small to large cluster sizes and the fraction of
isolated clusters is, consequently, even more pronounced than for the
bibliographic coupling network, as it can be seen from Figure 10.

Figure 7: Distribution of major Fields of Research in the 10 largest clusters
of the co-citation network of journal articles. The categories in each bar are
ordered by color (in the legend), from left to right, top to bottom. For
example, cluster one top major fields of research include (in order of
appearance in the legend): Biological Sciences, Biochemistry and Cell Biology,
Genetics, Medical and Health Sciences, Other.

#### Largest clusters

In Figure 7 we show the distribution of the top major Fields of Research per
top-10 cluster in the co-citation network. This result strongly confirms the
dominance of biology and medicine as the two top fields cited from Wikipedia.
Virtually all top clusters include a dominant or at least sizable fraction of
citations to these two fields, with the exception of the second cluster which
contains a sizeable share of contributions in earth sciences, geology, and
evolutionary biology, and the third one, focused on physics and astronomy.

#### Network visualization

Also for the co-citation supernetwork we visualize it using the same layout
and different coloring, respectively by modularity class (Figure 14) and top
major Field of Research (Figure 15). The results are consistent with what we
previously discussed. Furthermore, the two top clusters, which from Figure 7
we know being roughly focused on cell biology and genetics (cluster 1) and
physical anthropology (cluster 2), group differently by modularity, with
medicine, psychology and clinical sciences showing a stronger connection to
the former than the latter.

## Discussion

We provided a map of science in Wikipedia highlighting clear trends, while
rising some further questions. Firstly, our first research question asked
which scientific sources are cited from Wikipedia and what are their
characteristics. The dominant role of STEM literature clearly emerges, with
biology and medicine as the top fields, followed by earth sciences, physics
and astronomy. Furthermore, a sizeable fraction of this literature is well
cited and published in notorious venues. These results confirm previous
studies conducted on smaller datasets (Colavizza,, 2020) or at the journal
level of analysis (Arroyo-Machado et al.,, 2020).

We were able to delve deeper as well, in view of our second research question
asking which areas of Wikipedia rely on scientific sources and how do they
relate. Firstly, the seemingly marginal role of journal articles from non-STEM
fields is attenuated by the connecting role of biography articles in
Wikipedia, which effectively bridge history, geography and culture with STEM
topics. Secondly, the bibliographic coupling network of Wikipedia articles is
not only smaller, but also better connected than the co-citation network of
journal articles cited from Wikipedia. This might showcase the consolidating
encyclopedic role of Wikipedia, as well as the positive impact of WikiProjects
as a means to coordinate editorial efforts.

A set of new questions emerge from our work. To begin with, on the
characteristics of journal articles cited from Wikipedia: we have found that
on average these are well cited outside of Wikipedia too, yet many remain
poorly cited or even not cited at all – for example, 70% of journal articles
cited from Wikipedia received fewer than 100 citations from other journal
articles at the time of the study. Furthermore, the fraction of Open Access
articles cited from Wikipedia appears relatively high even though not
dominant. Lastly, the age of such journal articles is mostly distributed
within the past 20 years, possibly hinting at a chronological debt in
Wikipedia: several citations might have been current when most of Wikipedia
articles were created, but might no longer be up to date. Wikipedia is in this
respect a ‘slow altmetric indicator’, which takes time to accumulate (Fang and
Costas,, 2020). Secondly, our network analysis finds interesting clusters of
articles which warrant further study. How the central role of the largest
fields of research (biology, medicine) articulates with the networks’
‘periphery’, in particular, remains an open question.

Some of the limitations of our study constitute possible directions for future
work as well. The most important one is that we only considered citations to
journal articles. Adding other cited sources, such as books and Web contents,
would complement the map which we provided here. Secondly, the dataset we used
constitutes a snapshot of Wikipedia at a certain point in time: a study of
citations including time information would provide for a clearer picture of
the dynamics of negotiation and consolidation of knowledge in Wikipedia.
Lastly, Wikipedia’s internal structure can also be mapped using information
such as the internal link structure or the textual similarity of the articles.
A comparison of the citations networks we studies here with these would
further enrich the map of Wikipedia.

## Conclusion

In this study, we mapped the organization of Wikipedia according to its use of
scientific journal articles via citations. We made use of a recent dataset of
citations from Wikipedia, and relied on network analysis techniques, in
particular network clustering. We were able to show that Wikipedia heavily
relies on scientific contents from biology, medicine and a handful of other
STEM fields, including physics and earth sciences. Journal articles cited from
Wikipedia are, on average, well-cited, published in notorious journals such as
Nature or Science, and have been published over the past 20 years. While non-
STEM fields are only marginally represented in journal articles cited from
Wikipedia, they play an important connecting role via Wikipedia’s biographies.
This is but an example of how Wikipedia is able to interconnect knowledge
across scientific fields and also with other non-scientific topics. In this
respect, the most interesting future work which awaits us is the extension of
this map of science in Wikipedia to include books, Web contents and all other
sources cited in Wikipedia.

### Code and data availability statement

The code to replicate our work is made available online:
https://github.com/alsowbdxa/Code-of-Science-in-Wikipedia. The Wikipedia
Citations dataset is openly available (Singh et al.,, 2021), while access to
Dimensions can be requested through their portal. All other supporting
datasets we used are openly available and referenced from the Data and Methods
section.

### Acknowledgements

We acknowledge the support of Dimensions in making their data available.

###### Notes

  1. 1 We made use of the second API at the following address: https://wiki-topic.toolforge.org/#lang-agnostic-model.
  2. 2 https://www.mediawiki.org/wiki/ORES/Articletopic#Taxonomy.
  3. 3 https://en.wikipedia.org/wiki/Wikipedia:WikiProject.
  4. 4 The Fields of Research classification follows the research areas defined in the Australian and New Zealand Standard Research Classification (ANZSRC). See: https://app.dimensions.ai/browse/categories/publication/for.
  5. 5 For our analyses we relied on igraph 0.9.6 and leidenalg 0.8.3.

## Appendix

### Top-10 most cited journal articles

  1. 1.

Laemmli U. K. (1970) “Cleavage of Structural Proteins during the Assembly of
the Head of Bacteriophage T4”. Nature. Open access: closed. Research
organization country: United Kingdom. Number of citations: 214,886. Number of
recent citations: 6,111.

  2. 2.

Bradford M. M. (1976) “A rapid and sensitive method for the quantitation of
microgram quantities of protein utilizing the principle of protein-dye
binding”. Analytical Biochemistry. Open access: closed. Research organization
country: United States. Number of citations: 193,330. Number of recent
citations: 16,578.

  3. 3.

Perdew J. P., Burke K., Ernzerhof M. (1996) “Generalized Gradient
Approximation Made Simple”. Physical Review Letters. Open access: closed.
Research organization country: United States. Number of citations: 99,164.
Number of recent citations: 27,949.

  4. 4.

G.M. Sheldrick. (2007) “A short history of SHELX”.  Acta Crystallographica
Section A: Foundations and advances. Open access: open. Research organization
country: Germany. Number of citations: 72,560. Number of recent citations:
7,355.

  5. 5.

Axel D. Becke. (1993) “Density-functional thermochemistry. III. The role of
exact exchange”. The Journal of Chemical Physics. Open access: closed.
Research organization country: Canada. Number of citations: 69,187. Number of
recent citations: 8,871.

  6. 6.

Chengteh Lee, Weitao Yang, Robert G. Parr. (1988) “Development of the Colle-
Salvetti correlation-energy formula into a functional of the electron
density”. Physical Review B. Open access: closed. Research organization
country: United States. Number of citations: 66,421. Number of recent
citations: 8,872.

  7. 7.

Marshal F. Folstein, Susan E. Folstein, Paul R. McHugh. (1975) ““Mini-mental
state” A practical method for grading the cognitive state of patients for the
clinician”. Journal of Psychiatric Research. Open access: closed. Research
organization country: United States. Number of citations: 64,625. Number of
recent citations: 8,428.

  8. 8.

Stephen F. Altschul, Warren Gish, Webb Miller, Eugene W. Myers, David J.
Lipman. (1990) “Basic local alignment search tool”. Journal of Molecular
Biology. Open access: closed. Research organization country: United States.
Number of citations: 63,340. Number of recent citations: 10,717.

  9. 9.

F. Sanger, S. Nicklen, A. R. Coulson. (1977) “DNA sequencing with chain-
terminating inhibitors”. PNAS. Open access: open. Research organization
country: United Kingdom. Number of citations: 58,637. Number of recent
citations: 1,046.

  10. 10.

Piotr Chomczynski, Nicoletta Sacchi. (1987) “Single-step method of RNA
isolation by acid guanidinium thiocyanate-phenol-chloroform extraction”.
Analytical Biochemistry. Open access: closed. Research organization country:
United States. Number of citations: 56,286. Number of recent citations: 993.

### Figures

Figure 8: Publication year of journal articles cited from Wikipedia, from 2000
to 2020. Figure 9: Distribution of clusters by size in the bibliographic
coupling network of Wikipedia articles. Figure 10: Distribution of clusters by
size in the co-citation network of journal articles. Figure 11: Bibliographic
coupling supernetwork coloured by modularity class. Figure 12: Bibliographic
coupling supernetwork coloured by top ORES topic within a node/cluster.
Compare with Figure 5. Figure 13: Bibliographic coupling supernetwork coloured
by top WikiProject within a node/cluster. Compare with figure 6. Figure 14:
Co-citation supernetwork coloured by modularity class. Figure 15: Co-citation
supernetwork coloured by top major Field of Research. Compare with Figure 7.

## References

  * Adams et al., (2020) Adams, C. E., Montgomery, A. A., Aburrow, T., Bloomfield, S., Briley, P. M., Carew, E., Chatterjee-Woolman, S., Feddah, G., Friedel, J., Gibbard, J., et al. (2020).  Adding evidence of the effects of treatments into relevant Wikipedia pages: A randomised trial.  BMJ open, 10(2):e033655.
  * Arroyo-Machado et al., (2020) Arroyo-Machado, W., Torres-Salinas, D., Herrera-Viedma, E., and Romero-Frías, E. (2020).  Science through Wikipedia: A novel representation of open knowledge through co-citation networks.  PloS one, 15(2):e0228713.
  * Brezar and Heilman, (2019) Brezar, A. and Heilman, J. (2019).  Readability of English Wikipedia’s health information over time.  WikiJournal of Medicine, 6(1):1–6.
  * Chen and Roth, (2012) Chen, C.-C. and Roth, C. (2012).  Citation Needed: The Dynamics of Referencing in Wikipedia.  In Proceedings of the Eighth Annual International Symposium on Wikis and Open Collaboration, WikiSym ’12, New York, NY, USA. Association for Computing Machinery.
  * Colavizza, (2020) Colavizza, G. (2020).  COVID-19 research in Wikipedia.  Quantitative Science Studies, 1(4):1349–1380.
  * Costas et al., (2021) Costas, R., Rijcke, S., and Marres, N. (2021).  “Heterogeneous couplings”: Operationalizing network perspectives to study science-society interactions through social media metrics.  Journal of the Association for Information Science and Technology, 72(5):595–610.
  * Fang and Costas, (2020) Fang, Z. and Costas, R. (2020).  Studying the accumulation velocity of altmetric data tracked by Altmetric.com.  Scientometrics, 123(2):1077–1101.
  * Halfaker et al., (2018) Halfaker, A., Mansurov, B., Redi, M., and Taraborelli, D. (2018).  Citations with identifiers in Wikipedia.
  * Herzog et al., (2020) Herzog, C., Hook, D., and Konkiel, S. (2020).  Dimensions: Bringing down barriers between scientometricians and data.  Quantitative Science Studies, 1(1):387–395.
  * Johnson et al., (2021) Johnson, I., Gerlach, M., and Sáez-Trumper, D. (2021).  Language-agnostic Topic Classification for Wikipedia.  In Companion Proceedings of the Web Conference 2021, pages 594–601.
  * Johnson and Halfaker, (2020) Johnson, I. and Halfaker, A. (2020).  Wikipedia Articles and Associated WikiProject Templates.  Figshare.
  * Kaffee and Elsahar, (2021) Kaffee, L.-A. and Elsahar, H. (2021).  References in Wikipedia: The Editors’ Perspective.  In Companion Proceedings of the Web Conference 2021, pages 535–538.
  * Lemmerich et al., (2019) Lemmerich, F., Sáez-Trumper, D., West, R., and Zia, L. (2019).  Why the World Reads Wikipedia: Beyond English Speakers.  In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining, WSDM ’19, page 618–626, New York, NY, USA. Association for Computing Machinery.
  * Lewoniewski et al., (2020) Lewoniewski, W., Węcel, K., and Abramowicz, W. (2020).  Modeling Popularity and Reliability of Sources in Multilingual Wikipedia.  Information, 11(5):263.
  * Maggio et al., (2020) Maggio, L. A., Steinberg, R. M., Piccardi, T., and Willinsky, J. M. (2020).  Meta-Research: Reader engagement with medical content on Wikipedia.  Elife, 9:e52426.
  * Mesgari et al., (2015) Mesgari, M., Okoli, C., Mehdi, M., Nielsen, F. Å., and Lanamäki, A. (2015).  “The sum of all human knowledge”: A systematic review of scholarly research on the content of Wikipedia.  Journal of the Association for Information Science and Technology, 66(2):219–245.
  * Nicholson et al., (2021) Nicholson, J. M., Uppala, A., Sieber, M., Grabitz, P., Mordaunt, M., and Rife, S. C. (2021).  Measuring the quality of scientific references in Wikipedia: an analysis of more than 115M citations to over 800,000 scientific articles.  The FEBS Journal, 288(14):4242–4248.
  * Nielsen, (2007) Nielsen, F. Å. (2007).  Scientific citations in Wikipedia.  arXiv preprint arXiv:0705.2106.
  * Perianes-Rodriguez et al., (2016) Perianes-Rodriguez, A., Waltman, L., and van Eck, N. J. (2016).  Constructing bibliometric networks: A comparison between full and fractional counting.  Journal of Informetrics, 10(4):1178–1195.
  * (20) Piccardi, T., Gerlach, M., Arora, A., and West, R. (2021a).  A Large-Scale Characterization of How Readers Browse Wikipedia.
  * Piccardi et al., (2020) Piccardi, T., Redi, M., Colavizza, G., and West, R. (2020).  Quantifying engagement with citations on Wikipedia.  In Proceedings of The Web Conference, pages 2365–2376.
  * (22) Piccardi, T., Redi, M., Colavizza, G., and West, R. (2021b).  On the Value of Wikipedia as a Gateway to the Web.  In Proceedings of the Web Conference 2021, WWW ’21, page 249–260, New York, NY, USA. Association for Computing Machinery.
  * Redi et al., (2019) Redi, M., Fetahu, B., Morgan, J., and Taraborelli, D. (2019).  Citation Needed: A Taxonomy and Algorithmic Assessment of Wikipedia’s Verifiability.  In The World Wide Web Conference, WWW ’19, page 1567–1578, New York, NY, USA. Association for Computing Machinery.
  * Roy et al., (2021) Roy, D., Bhatia, S., and Jain, P. (2021).  Information asymmetry in Wikipedia across different languages: A statistical analysis.  Journal of the Association for Information Science and Technology.
  * Silva et al., (2011) Silva, F., Viana, M., Travençolo, B., and da F. Costa, L. (2011).  Investigating relationships within and between category networks in Wikipedia.  Journal of Informetrics, 5(3):431–438.
  * Singer et al., (2017) Singer, P., Lemmerich, F., West, R., Zia, L., Wulczyn, E., Strohmaier, M., and Leskovec, J. (2017).  Why We Read Wikipedia.  In Proceedings of the 26th International Conference on World Wide Web, WWW ’17, page 1591–1600, Republic and Canton of Geneva, CHE. International World Wide Web Conferences Steering Committee.
  * Singh et al., (2021) Singh, H., West, R., and Colavizza, G. (2021).  Wikipedia citations: A comprehensive data set of citations with identifiers extracted from English Wikipedia.  Quantitative Science Studies, 2(1):1–19.
  * Smith, (2020) Smith, D. A. (2020).  Situating Wikipedia as a health information resource in various contexts: A scoping review.  PloS one, 15(2):e0228786.
  * Sugimoto et al., (2017) Sugimoto, C. R., Work, S., Larivière, V., and Haustein, S. (2017).  Scholarly use of social media and altmetrics: A review of the literature.  Journal of the Association for Information Science and technology, 68(9):2037–2062.
  * Teplitskiy et al., (2017) Teplitskiy, M., Lu, G., and Duede, E. (2017).  Amplifying the impact of open access: Wikipedia and the diffusion of science.  Journal of the Association for Information Science and Technology, 68(9):2116–2127.
  * Thompson and Hanley, (2018) Thompson, N. and Hanley, D. (2018).  Science is shaped by Wikipedia: Evidence from a randomized control trial.  MIT Sloan Research Paper, (5238-17).
  * Torres-Salinas et al., (2019) Torres-Salinas, D., Romero-Frías, E., and Arroyo-Machado, W. (2019).  Mapping the backbone of the Humanities through the eyes of Wikipedia.  Journal of Informetrics, 13(3):793–803.
  * Traag et al., (2019) Traag, V. A., Waltman, L., and Van Eck, N. J. (2019).  From Louvain to Leiden: guaranteeing well-connected communities.  Scientific reports, 9(1):1–12.
  * Visser et al., (2021) Visser, M., van Eck, N. J., and Waltman, L. (2021).  Large-scale comparison of bibliographic data sources: Scopus, Web of Science, Dimensions, Crossref, and Microsoft Academic.  Quantitative Science Studies, 2(1):20–41.
  * Yarovoy et al., (2020) Yarovoy, A., Nagar, Y., Minkov, E., and Arazy, O. (2020).  Assessing the Contribution of Subject-matter Experts to Wikipedia.  ACM Transactions on Social Computing, 3(4):1–36.
  * Zagorova et al., (2021) Zagorova, O., Ulloa, R., Weller, K., and Flöck, F. (2021).  "I updated the <ref>": The evolution of references in the English Wikipedia and the implications for altmetrics.  Quantitative Science Studies, pages 1–27.


# On Stability and Convergence of Distributed Filters

Sayed Pouria Talebi, Stefan Werner, Vijay Gupta, and Yih-Fang Huang Sayed
Pouria Talebi and Stefan Werner are with the Department of Electronic Systems,
Faculty of Information Technology and Electrical Engineering, Norwegian
University of Science and Technology, Trondheim NO-7491 Norway, E-mail:
<EMAIL_ADDRESS>
Vijay Gupta and Yih-Fang Huang are with the Department of Electrical
Engineering, University of Notre Dame, Notre Dame, IN 46556 USA,
<EMAIL_ADDRESS>
The work of Sayed Pouria Talebi and Stefan Werner was supported in part by
Norwegian Research Council.

###### Abstract

Recent years have bore witness to the proliferation of distributed filtering
techniques, where a collection of agents communicating over an ad-hoc network
aim to collaboratively estimate and track the state of a system. These
techniques form the enabling technology of modern multi-agent systems and have
gained great importance in the engineering community. Although most
distributed filtering techniques come with a set of stability and convergence
criteria, the conditions imposed are found to be unnecessarily restrictive.
The paradigm of stability and convergence in distributed filtering is revised
in this manuscript. Accordingly, a general distributed filter is constructed
and its estimation error dynamics is formulated. The conducted analysis
demonstrates that conditions for achieving stable filtering operations are the
same as those required in the centralized filtering setting. Finally, the
concepts are demonstrated in a Kalman filtering framework and validated using
simulation examples.

###### Index Terms:

Stability of linear systems, observers for linear systems, Kalman filtering,
sensor fusion, sensor networks.

##  I Introduction

In recent years, large-scale multi-agent systems that can interact over a
network, such as sensor networks, have started to emerge in a wide range of
engineering applications including, power system monitoring[1, 2], multi-
vehicle coordination [3, 4], as well as collaborative information processing
and decision making [5, 6, 7]. The common thread among most of these
applications is the requirement for a robust underlying information processing
and filtering framework. Although optimal filtering solutions are attainable
through the use of a centralized coordinator, these so-called centralized
approaches are vulnerable to failure of the centralized coordinator and
require complex communication protocols [8, 9, 10]. Therefore, one of the most
fundamental problems in this area has become that of designing distributed
filtering techniques. Distributed filtering techniques aim to enable agents of
the network to estimate and track the state of a system or learn a set of
parameters in a fashion that limits communication to the neighbourhood of each
agent and would not require a centralized coordinator.

Seminal works in distributed filtering include [11, 12, 9, 13, 14, 15], which
have since created a vibrant research area (see, e.g., [16, 17, 18, 19, 20,
21, 5, 22, 23, 24] and references therein). In general, these distributed
filtering techniques are accompanied with stability analysis. However, the
assumptions imposed for guaranteeing stable behaviour are, in the authors
opinion, restrictive. For instant, distributed Kalman filtering techniques
require the state vector process to be fully observable to each agent via its
observation information only or in conjunction with the observation
information shared from its neighbours [14, 21, 1]. More recent contributions
in this field achieve less strict observability criteria, which nonetheless
result in the introduction of other requirements that mainly manifest themself
as conditions on how information flows throughout the network [25, 26, 19].
Despite exhibiting less complex behaviour, the made statements also extend to
distributed filters based on gradient descent, e.g., the least mean square
(LMS) algorithm [13, 27].

In what follows, a generalized distributed filtering formulation is
constructed and its behaviour is analysed. The obtained results establish that
the stability criteria of the considered distributed filter are, in essence,
the same as that of its centralized dual. More importantly, due to the
generalized setting of the constructed framework, the concept can be extended
to a wide range of filtering applications. Finally, the introduced concepts
are verified in a Kalman filter setting using simulation examples, showing
that in addition to modern techniques [10, 15], traditional distributed Kalman
filters [14, 9, 21] also converge under more relaxed conditions than initially
reported.

Mathematical Notations: Scalars, column vectors, and matrices are denoted by
lowercase, bold lowercase, and bold uppercase letters. Inequality symbols
refer to generalized matrix inequalities. Remainder of the nomenclature is
organized as follows:

$\mathbf{1}$

column vector of appropriate size with unit entries

$\mathbf{I}$

identity matrix of appropriate size

$(\cdot)^{\mathsf{T}}$

transpose operator

$|\cdot|$

cardinality operator

$\mathsf{E}\left\\{\cdot\right\\}$

statistical expectation operator

$\mathsf{p}\left\\{\cdot\right\\}$

spectral radius operator

$\mathsf{vec}\left\\{\cdot\right\\}$

vectorization operator

$\mathsf{diag}\left\\{\cdot\right\\}$

constructs a block-diagonal matrix from its entries

$\mathsf{col}\left\\{\cdot\right\\}$

constructs a block-column matrix from its entries

$\otimes$

Kronecker product

$\delta(\cdot)$

Kronecker delta function

##  II Preliminaries & Background

###  II-A Network Model

Following the conventional setting [14, 11], the networked agents are modelled
as the connected graph $\mathcal{G}=\\{\mathcal{N},\mathcal{E}\\}$ with the
node set $\mathcal{N}$ representing the agents and their bidirectional
communication links represented by the edge set $\mathcal{E}$. The
neighbourhood of node $l$ is the set of agents that agent $l$ receives
information from, which includes agent $l$ itself, and is denoted by
$\mathcal{N}_{l}$. Finally, the cardinality of a node set is the number of
nodes that it contains. For example, $|\mathcal{N}_{l}|$ denotes the number of
nodes in the neighbourhood of node $l$.

###  II-B Distributed Filtering Paradigm

Consider the state vector sequence $\\{\mathbf{x}_{n},n=1,2,\ldots\\}$
described through the linear dynamics

$\mathbf{x}_{n+1}=\mathbf{A}\mathbf{x}_{n}+\mathbf{v}_{n}$ (1)

with $\mathbf{A}$ denoting the state evolution matrix and $\mathbf{v}_{n}$
representing the state evolution noise at time $n$. The state vector is
observed at node $l$, so that we have

$\mathbf{y}_{l,n}=\mathbf{H}_{l,n}\mathbf{x}_{n}+\mathbf{w}_{l,n}$ (2)

where at time instant $n$ and node $l$, $\mathbf{y}_{l,n}$ is the observation
and $\mathbf{H}_{l,n}$ is the observation matrix, while $\mathbf{w}_{l,n}$
denotes the observation noise.

######  Assumption 1.

In keeping with the conventional distributed filtering framework [14, 11, 9,
10], all noise sequences are assumed to be stationary zero-mean Gaussian
processes, with covariance matrix

$\mathsf{E}\left\\{\begin{bmatrix}\mathbf{v}_{n}\\\
\mathbf{w}_{l,n}\end{bmatrix}\begin{bmatrix}\mathbf{v}^{\mathsf{T}}_{m}&\mathbf{w}^{\mathsf{T}}_{i,m}\end{bmatrix}\right\\}=\begin{bmatrix}\boldsymbol{\Sigma}_{\mathbf{v}}&\mathbf{0}\\\
\mathbf{0}&\boldsymbol{\Sigma}_{\mathbf{w}_{l}}\delta(l-i)\end{bmatrix}\delta(n-m).$

A centralized filtering operation can be achieved through the following
operations [10, 11]

$\hat{\mathbf{x}}_{n+1}=\left(\mathbf{I}-\mathbf{G}\mathbf{H}_{\mathrm{col}}\right)\hat{\mathbf{x}}_{n}+\mathbf{G}\mathbf{y}_{\mathrm{col},n}$
(3)

where $\mathbf{G}$ is an adaptation gain, whereas

$\mathbf{H}_{\mathrm{col}}=\mathsf{col}\left\\{\mathbf{H}_{l}:\forall
l\in\mathcal{N}\right\\}\hskip 5.69046pt\text{and}\hskip
5.69046pt\mathbf{y}_{\mathrm{col},n}=\mathsf{col}\left\\{\mathbf{y}_{l,n}:\forall
l\in\mathcal{N}\right\\}.$

Importantly, if $\\{\mathbf{A},\mathbf{H}_{\mathrm{col}}\\}$ is detectable and
$\\{\mathbf{A},\boldsymbol{\Sigma}^{\frac{1}{2}}_{\mathbf{v}}\\}$ is
stabilizable; then, the algebraic Riccati recursion

$\mathbf{M}^{{}^{-1}}_{n+1}=\left(\mathbf{A}\mathbf{M}_{n}\mathbf{A}^{\mathsf{T}}+\boldsymbol{\Sigma}_{\mathbf{v}}\right)^{{}^{-1}}+\mathbf{H}^{\mathsf{T}}_{\mathrm{col}}\mathbf{R}^{-1}\mathbf{H}_{\mathrm{col}}$
(4)

where $\mathbf{R}$ is a block-diagonal weighting matrix given as

$\mathbf{R}=\mathsf{diag}\left\\{\mathbf{R}_{l}:l=1,\ldots,|\mathbf{N}|\right\\}\hskip
3.41418pt\text{so~{}that}\hskip 3.41418pt\forall l:\mathbf{R}_{l}\geq 0$

converges to a unique stabilizing solution $\mathbf{M}$, so that

$\forall\mathbf{M}_{1}>0:\hskip
3.41418pt\lim_{n\rightarrow\infty}\mathbf{M}_{n}=\mathbf{M}$

for which the adaptation gain
$\mathbf{G}=\mathbf{M}\mathbf{H}^{\mathsf{T}}_{\mathrm{col}}\mathbf{R}^{-1}$
makes the estimation error of the filtering operations in (3) into a globally
stable linear system [28, 29].

The aim in distributed filtering is to enable each node in the network to
track the state vector sequence in (1) via observations available to the agent
as described in (2) and cooperation achievable through communication only with
its neighbouring nodes. In the sequel, a general framework for achieving this
goal is introduced. Then, its behaviour is rigorously analysed, establishing
its stable operating criteria. Finally, the results are shown to be applicable
to a number of seminal distributed filtering approaches.

##  III Distributed Filtering Framework

###  III-A Distributed Filtering Operations

To achieve a distributed filtering operation, and without loss of generality,
consider the following operations at each node

$\forall l\in\mathcal{N}:\left\\{\begin{aligned}
\boldsymbol{\phi}_{l,n+1}=&\left(\mathbf{I}-\mathbf{G}_{l}\mathbf{H}_{l}\right)\mathbf{A}\hat{\mathbf{x}}_{l,n}+\mathbf{G}_{l}\mathbf{y}_{l,n}\\\
\hat{\mathbf{x}}_{l,n+1}=&\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\boldsymbol{\phi}_{i,n+1}\end{aligned}\right.$ (5)

where node $l$ implements a local filtering operation using adaptation gain
$\mathbf{G}_{l}$; then, combines its intermediate estimate,
$\boldsymbol{\phi}_{l,n}$, with those of its neighbours using the combination
weights $\\{c_{l,i}:\forall l,i\in\mathcal{N}\\}$ to arrive at the final
estimate $\hat{\mathbf{x}}_{l,n}$.

######  Remark 1.

The filtering operation in (5) is designed to encompass the seminal works in
distributed Kalman filtering [14, 18, 10]. In addition, it is straightforward
to simplify the operation in (5) to describe distributed filters based on
gradient descent [21, 13, 27].

######  Assumption 2.

In keeping with the diffusion and consensus information fusion setting [30,
31, 32, 21], the combination weights in (5) are chosen so that

$\forall l\in\mathcal{N}:\sum_{\forall i\in\mathcal{N}_{l}}c_{l,i}=1.$

Thus, matrix $\mathbf{C}$ with $l^{\text{th}}$ row and $i^{\text{th}}$ column
element

$\mathbf{C}^{\\{l,i\\}}=\left\\{\begin{matrix}c_{l,i}&{}&\text{if}\hskip
3.41418pti\in\mathcal{N}_{l}\\\ 0&{}&\text{if}\hskip
3.41418pti\notin\mathcal{N}_{l}\end{matrix}\right.$

would be right-stochastic and primitive [21].

###  III-B Stability and Convergence Analysis

The filtering operations in (5) are combined to give

$\hat{\mathbf{x}}_{l,n+1}=\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\Big{(}\left(\mathbf{I}-\mathbf{G}_{i}\mathbf{H}_{i}\right)\mathbf{A}\hat{\mathbf{x}}_{i,n}+\mathbf{G}_{i}\mathbf{y}_{i,n}\Big{)}.$
(6)

Now, subtracting $\mathbf{x}_{l,n+1}$ from both sides of (6) yields

$\displaystyle\boldsymbol{\epsilon}_{l,n+1}=$
$\displaystyle\mathbf{x}_{n+1}-\hat{\mathbf{x}}_{l,n+1}$ (7) $\displaystyle=$
$\displaystyle\mathbf{x}_{n+1}-\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\Big{(}\left(\mathbf{I}-\mathbf{G}_{i}\mathbf{H}_{i}\right)\mathbf{A}\hat{\mathbf{x}}_{i,n}+\mathbf{G}_{i}\mathbf{y}_{i,n}\Big{)}.$

Substituting $\mathbf{x}_{n+1}$ from (1) and $\mathbf{y}_{l,n}$ from (2) into
(7) gives the regressive error expression

$\displaystyle\boldsymbol{\epsilon}_{l,n+1}=$ $\displaystyle\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\left(\mathbf{I}-\mathbf{G}_{i}\mathbf{H}_{i}\right)\mathbf{A}\boldsymbol{\epsilon}_{i,n}-\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\mathbf{G}_{i}\mathbf{w}_{i,n}$
$\displaystyle+\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\left(\mathbf{I}-\mathbf{G}_{i}\mathbf{H}_{i}\right)\mathbf{v}_{n}.$
(8)

The expression in (8) shows how the performance of node $l$ is linked to that
of its neighbours, which are in turn, linked to their respective neighbours.
Therefore, an over all analysis is only possible if the entire network is
considered as unified filter. To this end, the network-wide error vector is
defined as

$\mathbfcal{E}_{n}=\mathsf{col}\left\\{\boldsymbol{\epsilon}_{l,n}:\forall
l\in\mathcal{N}\right\\}.$ (9)

After some mathematical manipulation, replacing the rows of
$\mathbfcal{E}_{n}$ in (9) with expressions in (8) gives

$\mathbfcal{E}_{n+1}=\mathbfcal{C}\Big{(}\mathbfcal{F}\mathbfcal{E}_{n}+\mathbfcal{P}\left(\mathbf{1}\otimes\mathbf{v}_{n}\right)-\mathbfcal{G}\mathbfcal{W}_{n}\Big{)}$
(10)

where $\mathbfcal{C}=\mathbf{C}\otimes\mathbf{I}$,
$\mathbfcal{G}=\text{diag}\\{\mathbf{G}_{l}:\forall l\in\mathcal{N}\\}$, and

$\displaystyle\mathbfcal{F}=$
$\displaystyle\mathsf{diag}\left\\{\left(\mathbf{I}-\mathbf{G}_{l}\mathbf{H}_{l}\right)\mathbf{A}:\forall
l\in\mathcal{N}\right\\}$ (11) $\displaystyle\mathbfcal{P}=$
$\displaystyle\mathsf{diag}\left\\{\left(\mathbf{I}-\mathbf{G}_{l}\mathbf{H}_{l}\right):\forall
l\in\mathcal{N}\right\\}$ (12) $\displaystyle\mathbfcal{W}_{n}=$
$\displaystyle\mathsf{col}\left\\{\mathbf{w}_{l,n}:\forall
l\in\mathcal{N}\right\\}.$ (13)

######  Theorem 1.

If $\\{\mathbf{A},\mathbf{H}_{\mathrm{col}}\\}$ is detectable and
$\\{\mathbf{A},\boldsymbol{\Sigma}^{\frac{1}{2}}_{\mathbf{v}}\\}$ is
stabilizable; then, there exists a matrix set $\\{\mathbf{M}_{l}:\forall
l\in\mathcal{N}\\}$ so that

$\forall
l\in\mathcal{N}:\mathbf{G}_{l}=\mathbf{M}_{l}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}$
(14)

makes the estimation error of the distributed filtering operations in (5)
globally stable.

###### Proof:

Take the matrix recursions

$\displaystyle\mathbf{S}_{l,n+1}=$
$\displaystyle\left(\mathbf{A}\mathbf{M}_{l,n}\mathbf{A}^{\mathsf{T}}+\boldsymbol{\Sigma}_{\mathbf{v}}\right)^{{}^{-1}}+\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l}$
(15a) $\displaystyle\mathbf{M}^{{}^{-1}}_{l,n+1}=$ $\displaystyle\sum_{\forall
l\in\mathcal{N}_{l}}c_{l,i}\mathbf{S}_{i,n+1}$ (15b)

which can be carried out by the agents in a distributed fashion. Comparing
(15) to (4), and assuming, without loss of generality, that $\forall
l\in\mathcal{N}:0<\mathbf{M}_{1}\leq\mathbf{M}_{l,1}$, it can be concluded
that $\forall l\in\mathcal{N}:0<\mathbf{M}_{n}\leq\mathbf{M}_{l,n}$.
Proceeding on this basis, it follows algebraically that for $\forall
l\in\mathcal{N}$:

$\left(\mathbf{I}-\mathbf{M}_{l,n}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l}\right)\leq\left(\mathbf{I}-\mathbf{M}_{n}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l}\right).$
(16)

Moreover, the assumption that $\\{\mathbf{A},\mathbf{H}_{\mathrm{col}}\\}$ is
detectable and
$\\{\mathbf{A},\boldsymbol{\Sigma}^{\frac{1}{2}}_{\mathbf{v}}\\}$ is
stabilizable, from the Riccati recursions in (4), we have

$\mathsf{p}\left\\{\left(\mathbf{I}-\mathbf{M}_{n}\sum_{\forall
l\in\mathcal{N}}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l}\right)\mathbf{A}\right\\}<1.$
(17)

Thus, from (17) and (16) it follows that,
$(\mathbf{I}-\mathbf{M}_{l,n}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l})\mathbf{A}$,
would be stable on modes detectable at node $l$. Subsequently, $\sum_{\forall
l\in\mathcal{N}_{l}}c_{l,i}\left(\mathbf{I}-\mathbf{M}_{i,n}\mathbf{H}^{\mathsf{T}}_{i}\mathbf{R}^{{}^{-1}}_{i}\mathbf{H}_{i}\right)\mathbf{A}$,
would be stable on modes detectable to nodes in $\mathcal{N}_{l}$.

From Assumption 2, recall that $\mathbf{C}$ is primitive, and hence
$\mathbfcal{C}$ is block primitive, that is, there exits $m$ so that
$\mathbfcal{C}^{m}=\mathbf{C}^{m}\otimes\mathbf{I}$ consists of identity
matrices scaled by positive non-zero real-valued numbers. Given the block
diagonal structure of $\mathbfcal{F}$, extending the statements in the
previous paragraph, there exists $k$ for which
$\left(\mathbfcal{C}\mathbfcal{F}\right)^{k}$ is a block matrix, where each
block consists of an appropriate combination of the matrix set

$\left\\{\left(\left(\mathbf{I}-\mathbf{M}_{l,n}\mathbf{H}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}_{l}\mathbf{H}_{l}\right)\mathbf{A}\right)^{i}:\forall
l\in\mathcal{N},i=1,\ldots,k\right\\}$

so that $\mathsf{p}\left\\{(\mathbfcal{C}\mathbfcal{F})^{k}\right\\}<1$.
Therefore, the statistical expectation of any norm, including the second-order
norm, of the estimation error sequence $\\{\mathbfcal{E}_{n+k}:n=1,2,\dots\\}$
would become convergent, which concludes the proof. ∎

######  Remark 2.

Note that Theorem 1 broadens the scope of our previous results in [10] and
presents the least restrictive convergence criteria for an all-inclusive class
of distributed filters, essentially equating convergence criteria of
centralized and distributed filtering techniques.

######  Theorem 2.

If $\\{\mathbf{A},\mathbf{H}_{\mathrm{col}}\\}$ is detectable and
$\\{\mathbf{A},\boldsymbol{\Sigma}^{\frac{1}{2}}_{\mathbf{v}}\\}$ is
stabilizable; then, the gain matrices resulting from (14) and (15) make the
estimates of the introduced distributed filtering operations in (5) globally
asymptotically unbiased.

###### Proof:

From the recursive expression of the estimation error in (10) and conditions
set on state evolution and observation noise in Assumption 1, we have

$\mathsf{E}\left\\{\mathbfcal{E}_{n}\right\\}=\left(\mathbfcal{C}\mathbfcal{F}\right)^{n}\mathsf{E}\left\\{\mathbfcal{E}_{1}\right\\}.$
(18)

Furthermore, in the proof of Theorem 1, it was demonstrated that under the
made assumptions, there exits a $k$ for which
$\left(\mathbfcal{C}\mathbfcal{F}\right)^{k}$ is a contracting operator, and
therefore,

$\text{as}\hskip 3.41418ptn\rightarrow\infty\hskip 3.41418pt\text{then}\hskip
3.41418pt\left(\mathbfcal{C}\mathbfcal{F}\right)^{n}\rightarrow\mathbf{0}.$
(19)

As a direct result of (18) and (19), we have

$\lim_{n\rightarrow\infty}\mathsf{E}\left\\{\mathbfcal{E}_{n}\right\\}=\mathbf{0}$

which concludes the proof. ∎

To provide a more practical perspective, a reformulation of (15) is
considered, so that

$\mathbf{M}^{{}^{-1}}_{l,n}=\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\mathbf{S}_{i,n}=\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\mathbf{P}^{{}^{-1}}_{i,n}+\bar{\mathbf{H}}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}\bar{\mathbf{H}}_{l}$
(20)

where $\forall
l\in\mathcal{N}:\mathbf{P}_{l,n}=\mathbf{A}\mathbf{M}_{l,n-1}\mathbf{A}^{\mathsf{T}}+\boldsymbol{\Sigma}_{\mathbf{v}}$,
while

$\bar{\mathbf{H}}_{l}=\left[\sqrt{\mathbf{C}^{\\{l,1\\}}}\mathbf{H}^{\mathsf{T}}_{1},\ldots,\sqrt{\mathbf{C}^{\\{l,|\mathcal{N}|\\}}}\mathbf{H}^{\mathsf{T}}_{|\mathcal{N}|}\right]^{\mathsf{T}}.$

Now, using (20) and some mathematical manipulation, the operations in (15) are
rearranged as

$\displaystyle\mathbf{P}_{l,n}=$
$\displaystyle\mathbf{A}\left(\mathbf{Z}^{{}^{-1}}_{l,n-1}+\bar{\mathbf{H}}^{\mathsf{T}}_{l}\mathbf{R}^{{}^{-1}}\bar{\mathbf{H}}_{l}\right)^{{}^{-1}}\mathbf{A}^{\mathsf{T}}+\boldsymbol{\Sigma}_{\mathbf{v}}$
(21a) $\displaystyle\mathbf{Z}^{{}^{-1}}_{l,n}=$ $\displaystyle\sum_{\forall
i\in\mathcal{N}_{l}}c_{l,i}\mathbf{P}^{{}^{-1}}_{i,n}.$ (21b)

From (21a), it is clear that $\mathbf{P}_{l,n}$ is dual of the a posteriori
estimate error covariance matrix of a Kalman filter with a priori estimate
error covariance of $\mathbf{Z}_{l,n-1}$ using observations
$\\{\mathbf{y}_{i,n}:\forall i\in\mathcal{N}_{l}\\}$. Following on the same
line, from (21b), $\mathbf{Z}_{l,n}$ becomes the dual of the post estimate
error covariance of a Kalman filter using observations
$\\{\mathbf{y}_{i,n}:\forall i\in\mathcal{N}_{l_{2}}\\}$, where
$\mathcal{N}_{l_{k}}$ is the $k$-hop neighbourhood of node $l$. Thus, through
iteration, $\mathbf{Z}_{l,n+m}$ becomes the dual of the a posteriori estimate
error covariance of a Kalman filter using observations

$\bigcup\limits^{m}_{t=0}\\{\mathbf{y}_{i,n+k}:\forall
i\in\mathcal{N}_{l_{t+1}},k=0,\ldots,m-t\\}.$ (22)

If $\\{\mathbf{A},\mathbf{H}_{\mathrm{col}}\\}$ is detectable and
$\\{\mathbf{A},\boldsymbol{\Sigma}^{\frac{1}{2}}_{\mathbf{v}}\\}$ is
stabilizable; then, there exists $m$ for which observation set in (22) is
sufficient for tracking the state vector, and thus,
$\\{\mathbf{M}_{l,n}:\forall l\in\mathcal{N}\\}$ converges to stabilizing
solutions.

It is prudent to note that the Riccati style recursions in (15) have their
roots in the decentralized controllers in [10, 15]. The structure in (15) is
specifically designed to stabilize controllers with the dynamics in (10),
which results in its versatile application in this context. However, this
manuscript, further eases convergence criteria and removes the restriction for
using the consensus framework for information fusion, accommodating for both
diffusion and consensus. Traditional distributed Kalman filters inspired from
[14, 9] that do not use (15) and are reliant on local observability
conditions, will experience unstable modes in their corresponding
$\mathbf{M}_{l}$ matrices. However, even in these cases, if the error dynamic
follows (10) and the gain follows (14), the overall filtering operation will
converge. Note that this might take some skilful coding legerdemain to ensure
unstable modes do not result in computation errors, which fall beyond the
scope of this manuscript. On a final note, the distributed filter in (5) can
be simplified to the distributed filters based on the LMS [13, 27, 21] and
even a wide range of gradient-based approaches for nonlinear filters, in which
case, the concepts introduced in this manuscript would hold true.

##  IV Simulation Examples

The problem of tracking a target in two dimensions is considered [14, 10]. In
this case, the state evolution function is

$\mathbf{x}_{n}=\begin{bmatrix}1&0&\Delta T&0\\\ 0&1&0&\Delta T\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}\mathbf{x}_{n-1}+\begin{bmatrix}\frac{1}{2}(\Delta
T)^{2}&0\\\ 0&\frac{1}{2}(\Delta T)^{2}\\\ \Delta T&0\\\ 0&\Delta
T\end{bmatrix}\mathbf{v}_{n}\\\ $

where
$\mathsf{E}\left\\{\mathbf{v}_{n}\mathbf{v}^{\mathsf{T}}_{n}\right\\}=0.01\times\mathbf{I}$
and $\Delta T=$$0.04$ s representing the sampling interval. The network shown
in Fig. 1 was used for simulations. Excluding the node with one link sitting
on the top left, all nodes observed movements of the target on the horizontal
axis solely. The mentioned node, on the other hand, could only observe moments
on the vertical axis, presenting a worst-case scenario for distributed
filtering. The observation noise covariance for all nodes was $0.16$ which was
used as the weight factors $\mathbf{R}_{l}$.

Figure 1: Network of $20$ nodes and $40$ links used in simulations.

The distributed Kalman filter in [14], as a representative of classical
techniques, and the distributed Kalman filter in [10], as a representative of
modern techniques, were used to track the state vector. The estimation error
dynamics for both distributed filters is presented in Fig. 2. Note that, as
suggested in Theorem 1 and 2, both distributed Kalman filtering approaches
exhibited error dynamics that converge to zero. This result is specially
intriguing in the case of traditional distributed Kalman filtering techniques
as the scenario considered here does not meet their original convergence
criteria (see [14, 21]).

The range for eigenvalues of matrix set $\\{\mathbf{M}_{l,n}:\forall
l\in\mathcal{N}\\}$ obtained through the recursions in (15) are presented in
Fig. 3. In addition, Fig. 3 includes the range of eigenvalues of comparable
set of matrices obtained through the framework of [14, 9, 21]. Note that as
proven in Theorem 1, the eigenvalues of matrices in (15) converge and
stabilize to a final value, whereas their counterparts from classical
filtering frameworks diverge. It can be shown through straightforward
algebraic manipulations that the divergent modes at each node correspond to
modes unobservable at that node, which in principle, results in these modes
only being updated through the state evolution model. This will not cause a
divergent behaviour, however, as information regarding modes unobservable to
one node will be made available through the diffusion/consensus structure as
long as there exists another node in the network to whom these modes are
observable, which encapsulates the spirit of this manuscript in a more
intuitive manner. It should be noted that when using classical approaches, the
presence of the unobservable modes for each node should be considered in the
implementation phase, as the infinitely growing eigenvalues can cause
computational problems. However, modern approaches, such as [10, 15] and that
presented in this manuscript, deal with this matter through rigorous modelling
and skilful use of mathematics.

Figure 2: Error dynamics for both modern and classical distributed Kalman
filtering techniques. Figure 3: Range of eigenvalues for the matrix set
$\\{\mathbf{M}_{l,n}:\forall l\in\mathcal{N}\\}$ for both modern and classical
distributed Kalman filtering techniques.

##  V Conclusion

The performance of a general class of distributed filters has been considered.
The analysis culminated in two theorems, which indicate that distributed
filters and their centralized duals essentially require the same convergence
criteria. The distributed filters considered were formulated to make the
obtained results applicable in a wide range of distributed filtering
frameworks. The concepts were verified using simulation examples on both
modern and classical distributed Kalman filters, showing convergence is
possible under much looser conditions than previously thought.

## References

  * [1] S. P. Talebi, S. Kanna, and D. P. Mandic, “A distributed quaternion Kalman filter with applications to smart grid and target tracking,” _IEEE Transactions on Signal and Information Processing over Networks_ , vol. 2, no. 4, pp. 477–488, December 2016.
  * [2] N. Kashyap, S. Werner, and Y. Huang, “Decentralized PMU-assisted power system state estimation with reduced interarea communication,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 12, no. 4, pp. 607–616, 2018.
  * [3] R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” _IEEE Transactions on Automatic Control_ , vol. 51, no. 3, pp. 401–420, March 2006.
  * [4] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control,” _IEEE Control Systems Magazine_ , vol. 27, no. 2, pp. 71–82, 2007.
  * [5] J. S. Shamma, _Cooperative control of distributed multi-agent systems_. John Wiley & Sons, 2007.
  * [6] S. Fattahi, G. Fazelnia, J. Lavaei, and M. Arcak, “Transformation of optimal centralized controllers into near-globally optimal static distributed controllers,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 1, pp. 66–80, Jan 2019.
  * [7] W. Han, H. L. Trentelman, Z. Wang, and Y. Shen, “A simple approach to distributed observer design for linear systems,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 1, pp. 329–336, Jan 2019.
  * [8] J. Speyer, “Computation and transmission requirements for a decentralized linear-quadratic-Gaussian control problem,” _IEEE Transactions on Automatic Control_ , vol. 24, no. 2, pp. 266–269, April 1979.
  * [9] R. Olfati-Saber, “Distributed Kalman filtering for sensor networks,” _In Proceedings of IEEE Conference on Decision and Control_ , pp. 5492–5498, December 2007.
  * [10] S. P. Talebi and S. Werner, “Distributed Kalman filtering and control through embedded average consensus information fusion,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 10, pp. 4396–4403, 2019.
  * [11] R. Olfati-Saber, “Distributed Kalman filter with embedded consensus filters,” _In Proceedings of IEEE Conference on Decision and Control_ , pp. 8179–8184, December 2005.
  * [12] F. S. Cattivelli, C. G. Lopes, and A. H. Sayed, “A diffusion RLS scheme for distributed estimation over adaptive networks,” _In Proceeding of IEEE 8th Workshop on Signal Processing Advances in Wireless Communications_ , pp. 1–5, 2007.
  * [13] F. S. Cattivelli and A. H. Sayed, “Diffusion LMS strategies for distributed estimation,” _IEEE Transactions on Signal Processing_ , vol. 58, no. 3, pp. 1035–1048, 2010.
  * [14] F. S. Cattivelli and A. H. Sayed, “Diffusion strategies for distributed Kalman filtering and smoothing,” _IEEE Transactions on Automatic Control_ , vol. 55, no. 9, pp. 2069–2084, September 2010.
  * [15] S. P. Talebi, S. Werner, and D. Mandic, “Quaternion-valued distributed filtering and control,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 10, pp. 4246–4257, 2020.
  * [16] E. Agarwal, S. Sivaranjani, V. Gupta, and P. J. Antsaklis, “Distributed synthesis of local controllers for networked systems with arbitrary interconnection topologies,” _IEEE Transactions on Automatic Control_ , Preprint, 2020.
  * [17] Y. Chen, S. Kar, and J. M. F. Moura, “Resilient distributed estimation: Sensor attacks,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 9, pp. 3772–3779, 2019.
  * [18] R. Olfati-Saber, “Kalman-consensus filter: Optimality, stability, and performance,” _In Proceedings of IEEE Conference on Decision and Control_ , pp. 7036–7042, December 2009.
  * [19] L. Wang and A. S. Morse, “A distributed observer for a time-invariant linear system,” _In Proceedings of American Control Conference_ , pp. 2020–2025, May 2017.
  * [20] O. Hlinka, F. Hlawatsch, and P. M. Djuric, “Distributed particle filtering in agent networks: A survey, classification, and comparison,” _IEEE Signal Processing Magazine_ , vol. 30, no. 1, pp. 61–81, January 2013.
  * [21] A. H. Sayed, “Adaptation, learning, and optimization over networks,” _Foundations and Trends ${}^{\text{\tiny{\textregistered}}}$ in Machine Learning_, vol. 7, no. 4-5, pp. 311–801, 2014.
  * [22] S. Werner, M. Mohammed, Yih-Fang Huang, and V. Koivunen, “Decentralized set-membership adaptive estimation for clustered sensor networks,” pp. 3573–3576, 2008.
  * [23] S. Werner, Y. Huang, M. L. R. de Campos, and V. Koivunen, “Distributed parameter estimation with selective cooperation,” pp. 2849–2852, 2009.
  * [24] D. Zhang, P. Shi, W. Zhang, and L. Yu, “Energy-efficient distributed filtering in sensor networks: A unified switched system approach,” _IEEE Transactions on Cybernetics_ , vol. 47, no. 7, pp. 1618–1629, 2017\.
  * [25] Q. Liu, Z. Wang, X. He, and D. H. Zhou, “On Kalman-consensus filtering with random link failures over sensor networks,” _IEEE Transactions on Automatic Control_ , vol. 63, no. 8, pp. 2701–2708, 2018.
  * [26] G. Battistelli, L. Chisci, G. Mugnai, A. Farina, and A. Graziano, “Consensus-based linear and nonlinear filtering,” _IEEE Transactions on Automatic Control_ , vol. 60, no. 5, pp. 1410–1415, 2015.
  * [27] C. G. Lopes and A. H. Sayed, “Diffusion least-mean squares over adaptive networks: Formulation and performance analysis,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 7, pp. 3122–3136, 2008.
  * [28] T. Kailath, A. H. Sayed, and B. Hassibi, _Linear estimation_. Prentice Hall, 2000.
  * [29] P. Kumar and P. Varaiya, _Stochastic systems: Estimation, identification, and adaptive control_. Prentice Hall, New Jersey, 1986.
  * [30] L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” _Systems & Control Letters_, vol. 53, no. 1, pp. 65–78, 2004.
  * [31] L. Xiao, S. Boyd, and S. Lall, “A scheme for robust distributed sensor fusion based on average consensus,” _In Proceedings of Fourth International Symposium on Information Processing in Sensor Networks_ , pp. 63–70, 2005.
  * [32] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” _Proceedings of the IEEE_ , vol. 95, no. 1, pp. 215–233, 2007.


* [Aro50] N. Aronszajn “Theory of Reproducing Kernels” In _Transactions of the American Mathematical Society_ 68.3, 1950, pp. 337–404
  * [Arr+24] Paola Arrubarrena et al. “Novelty Detection on Radio Astronomy Data using Signatures” In _Preprint, arXiv 2402.14892_ , 2024
  * [Bag+18] Anthony Bagnall et al. “The UEA multivariate time series classification archive, 2018” In _Preprint, arXiv 1811.00075_ , 2018
  * [BBC20] José R. Berrendero, Beatriz Bueno-Larraz and Antonio Cuevas “On Mahalanobis distance in functional settings” In _J. Mach. Learn. Res._ 21.1 JMLR.org, 2020
  * [BC85] S. Boyd and L. Chua “Fading memory and the problem of approximating nonlinear operators with Volterra series” In _IEEE Transactions on Circuits and Systems_ 32.11, 1985, pp. 1150–1161
  * [BGW24] Francesca Biagini, Lukas Gonon and Niklas Walter “Universal randomised signatures for generative time series modelling” In _Preprint, arXiv 2406.10214_ , 2024
  * [Bog15] V.I. Bogachev “Gaussian Measures”, Mathematical Surveys and Monographs American Mathematical Society, 2015
  * [Bre+00] Markus M. Breunig, Hans-Peter Kriegel, Raymond T. Ng and Jörg Sander “LOF: identifying density-based local outliers” In _Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data_ Dallas, Texas, USA: Association for Computing Machinery, 2000, pp. 93–104
  * [Bro+22] Paul O. Brown et al. “Mahalanobis Distance Based K-Means Clustering” In _Big Data Analytics and Knowledge Discovery_ Cham: Springer International Publishing, 2022, pp. 256–262
  * [Cha+20] Zhipeng Chang, Wenhe Chen, Yuping Gu and Haoyue Xu “Mahalanobis-Taguchi System for Symbolic Interval Data Based on Kernel Mahalanobis Distance” In _IEEE Access_ 8, 2020, pp. 20428–20438
  * [CLS23] Nicola Muça Cirone, Maud Lemercier and Cristopher Salvi “Neural signature kernels as infinite-width-depth-limits of controlled ResNets” In _Proceedings of the 40th International Conference on Machine Learning_ , 2023
  * [CM24] Christa Cuchiero and Janka Möller “Signature Methods in Stochastic Portfolio Theory” In _Preprint, arXiv 2310.02322_ , 2024
  * [Com+23] Enea Monzio Compagnoni et al. “On the effectiveness of Randomized Signatures as Reservoir for Learning Rough Dynamics” In _IJCNN_ , 2023, pp. 1–8
  * [CS01] Felipe Cucker and Stephen Smale “On the mathematical foundations of learning” In _Bulletin of the American Mathematical Society_ 39, 2001, pp. 1–49
  * [CS24] Thomas Cass and Cristopher Salvi “Lecture notes on rough paths and applications to machine learning” In _Preprint, arXiv 2404.06583_ , 2024
  * [CT24] Thomas Cass and William F. Turner “Free probability, path developments and signature kernels as universal scaling limits” In _Preprint, arXiv 2402.12311_ , 2024
  * [Cuc+21] Christa Cuchiero et al. “Expressive Power of Randomized Signature” In _Advances in Neural Information Processing Systems_ , 2021
  * [Cuc+22] Christa Cuchiero et al. “Discrete-time signatures and randomness in reservoir computing” In _IEEE Trans. Neural Netw. Learn. Syst._ 33.11, 2022, pp. 6321–6330
  * [Cut+07] Marco Cuturi, Jean-Philippe Vert, Oystein Birkenes and Tomoko Matsui “A Kernel for Time Series Based on Global Alignments” In _2007 IEEE International Conference on Acoustics, Speech and Signal Processing_ 2, 2007, pp. II-413-II–416
  * [Cut11] Marco Cuturi “Fast global alignment kernels” In _Proceedings of the 28th International Conference on Machine Learning_ Omnipress, 2011, pp. 929–936
  * [CZ07] Felipe Cucker and Ding Xuan Zhou “Learning Theory: An Approximation Theory Viewpoint”, Cambridge Monographs on Applied and Computational Mathematics Cambridge University Press, 2007
  * [Dan+20] Salva Daneshgadeh Çakmakçı, Thomas Kemmerich, Tarem Ahmed and Nazife Baykal “Online DDoS attack detection using Mahalanobis distance and Kernel-based learning algorithm” In _Journal of Network and Computer Applications_ 168, 2020, pp. 102756
  * [DJM00] R. De Maesschalck, D. Jouan-Rimbaud and D.L. Massart “The Mahalanobis distance” In _Chemometrics and Intelligent Laboratory Systems_ 50.1, 2000, pp. 1–18
  * [GGK12] I. Gohberg, S. Goldberg and M. Kaashoek “Basic Classes of Linear Operators” Birkhäuser Basel, 2012
  * [GGO22] Lukas Gonon, Lyudmila Grigoryeva and Juan-Pablo Ortega “Reservoir kernels and Volterra series” In _Preprint, arXiv 2212.14641_ , 2022
  * [GO21] Lyudmila Grigoryeva and Juan-Pablo Ortega “Dimension reduction in recurrent networks by canonicalization” In _Journal of Geometric Mechanics_ 13.4, 2021, pp. 647–677
  * [Hai23] Martin Hairer “An Introduction to Stochastic PDEs”, 2023 URL: https://www.hairer.org/notes/SPDEs.pdf
  * [HKF04] V. Hautamaki, I. Karkkainen and P. Franti “Outlier detection using k-nearest neighbour graph” In _Proceedings of the 17th International Conference on Pattern Recognition_ 3, 2004, pp. 430–433 Vol.3
  * [HXD03] Zengyou He, Xiaofei Xu and Shengchun Deng “Discovering Cluster Based Local Outliers” In _Pattern Recognition Letters_ 24, 2003, pp. 1641–1650
  * [JH04] Herbert Jaeger and Harald Haas “Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication” In _Science_ 304.5667, 2004, pp. 78–80
  * [KO19] Franz J. Kiraly and Harald Oberhauser “Kernels for Sequentially Ordered Data” In _Journal of Machine Learning Research_ 20.31, 2019, pp. 1–45
  * [Kre13] R. Kress “Linear Integral Equations”, Applied Mathematical Sciences Springer New York, 2013
  * [KSZ08] Hans-Peter Kriegel, Matthias Schubert and Arthur Zimek “Angle-based outlier detection in high-dimensional data” In _Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ Las Vegas, Nevada, USA: Association for Computing Machinery, 2008, pp. 444–452
  * [Lah+17] Hajer Lahdhiri et al. “A new fault detection index based on Mahalanobis distance and kernel method” In _The International Journal of Advanced Manufacturing Technology_ 91.5, 2017, pp. 2799–2809
  * [Lax14] P.D. Lax “Functional Analysis”, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts Wiley, 2014
  * [Led01] M. Ledoux “The Concentration of Measure Phenomenon”, Mathematical surveys and monographs American Mathematical Society, 2001
  * [Lif12] Mikhail Lifshits “Lectures on Gaussian Processes”, Graduate Texts in Mathematics Springer Berlin, Heidelberg, 2012
  * [LKK10] Raz Lin, Eliyahu Khalastchi and Gal A. Kaminka “Detecting anomalies in unmanned vehicles using the Mahalanobis distance” In _2010 IEEE International Conference on Robotics and Automation_ , 2010, pp. 3038–3044
  * [LLN22] Hang Lou, Siran Li and Hao Ni “Path Development Network with Finite-dimensional Lie Group Representation” In _Preprint, arXiv 2204.00740_ , 2022
  * [LLN23] Hang Lou, Siran Li and Hao Ni “PCF-GAN: generating sequential data via the characteristic function of measures on the path space” In _Thirty-seventh Conference on Neural Information Processing Systems_ , 2023
  * [LO23] Darrick Lee and Harald Oberhauser “The Signature Kernel” In _Preprint, arXiv 2305.04625_ , 2023
  * [LTZ08] Fei Tony Liu, Kai Ming Ting and Zhi-Hua Zhou “Isolation Forest” In _2008 Eighth IEEE International Conference on Data Mining_ , 2008, pp. 413–422
  * [Lyo98] Terry J. Lyons “Differential equations driven by rough signals.” In _Revista Matemática Iberoamericana_ 14.2, 1998, pp. 215–310
  * [Mah36] Prasanta Chandra Mahalanobis “On the generalized distance in statistics” In _Proceedings of the National Institute of Sciences (Calcutta)_ 2, 1936, pp. 49–55
  * [MNM02] W. Maass, T. Natschläger and H. Markram “Real-time computing without stable states: a new framework for neural computation based on perturbations” In _Neural Computation_ 14, 2002, pp. 2531–2560
  * [Pan+21] Guansong Pang, Chunhua Shen, Longbing Cao and Anton Van Den Hengel “Deep Learning for Anomaly Detection: A Review” In _ACM Comput. Surv._ 54.2 New York, NY, USA: Association for Computing Machinery, 2021
  * [PH09] Elzbieta Pekalska and Bernard Haasdonk “Kernel discriminant analysis for positive definite and indefinite kernels” In _IEEE Trans Pattern Anal Mach Intell_ 31.6, 2009, pp. 1017–1032
  * [PL15] Esdras Joseph Pedro Galeano and Rosa E. Lillo “The Mahalanobis Distance for Functional Data With Applications to Classification” In _Technometrics_ 57.2 Taylor & Francis, 2015, pp. 281–291
  * [RL01] A Ruiz and P E López-de-Teruel “Nonlinear kernel-based statistical pattern analysis” In _IEEE Trans Neural Netw_ 12.1, 2001, pp. 16–32
  * [RS05] J.. Ramsay and B.. Silverman “Functional Data Analysis” Springer, 2005
  * [Ruf+20] Lukas Ruff et al. “Deep Semi-Supervised Anomaly Detection” In _8th International Conference on Learning Representations_ , 2020
  * [Ruf+21] Lukas Ruff et al. “A Unifying Review of Deep and Shallow Anomaly Detection” In _Proceedings of the IEEE_ PP, 2021, pp. 1–40
  * [Rui+21] Alejandro Pasos Ruiz et al. “The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic advances” In _Data Mining and Knowledge Discovery_ 35.2, 2021, pp. 401–449
  * [Sal+21] Cristopher Salvi et al. “The Signature Kernel Is the Solution of a Goursat PDE” In _SIAM Journal on Mathematics of Data Science_ 3.3, 2021, pp. 873–899
  * [Sal+21a] Cristopher Salvi et al. “Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes” In _Advances in Neural Information Processing Systems_ 34 Curran Associates, Inc., 2021, pp. 16635–16647
  * [San83] I W Sandberg “Series expansions for nonlinear systems” In _Circuits, Systems and Signal Processing_ 2.1 Springer, 1983, pp. 77–87
  * [Sch+23] Bernhard Schäfl, Lukas Gruber, Johannes Brandstetter and Sepp Hochreiter “G-Signatures: Global Graph Propagation With Randomized Signatures” In _Preprint, arXiv 2302.08811_ , 2023
  * [Sch09] Alexander J Schölkopf “Learning with kernels: support vector machines, regularization, optimization and beyond”, Adaptive computation and machine learning MIT Press, 2009
  * [SCZ18] Jun Shang, Maoyin Chen and Hanwen Zhang “Fault detection based on augmented kernel Mahalanobis distance for nonlinear dynamic processes” In _Computers & Chemical Engineering_ 109, 2018, pp. 311–321
  * [Sha+23] Zhen Shao et al. “Dimensionless Anomaly Detection on Multivariate Streams with Variance Norm and Path Signature” In _Preprint, arXiv 2006.03487_ , 2023
  * [Shi+01] Hiroshi Shimodaira, Ken-ichi Noma, Mitsuru Nakai and Shigeki Sagayama “Dynamic Time-Alignment Kernel in Support Vector Machine” In _Advances in Neural Information Processing Systems_ 14 MIT Press, 2001
  * [SK20] Hassan Sarmadi and Abbas Karamodin “A novel anomaly detection method based on adaptive Mahalanobis-squared distance and one-class kNN rule for structural health monitoring under environmental effects” In _Mechanical Systems and Signal Processing_ 140, 2020, pp. 106495
  * [SP16] Shashank Singh and Barnabas Poczos “Finite-Sample Analysis of Fixed-k Nearest Neighbor Density Functional Estimators” In _Advances in Neural Information Processing Systems_ 29 Curran Associates, Inc., 2016
  * [SSM98] Bernhard Schölkopf, Alexander Smola and Klaus-Robert Müller “Nonlinear Component Analysis as a Kernel Eigenvalue Problem” In _Neural Computation_ 10.5, 1998, pp. 1299–1319
  * [TO20] Csaba Toth and Harald Oberhauser “Bayesian Learning from Sequential Data using Gaussian Processes with Signature Covariances” In _Proceedings of the 37th International Conference on Machine Learning_ 119, 2020, pp. 9548–9560
  * [TOS23] Csaba Toth, Harald Oberhauser and Zoltan Szabo “Random Fourier Signature Features” In _Preprint, arXiv 2311.12214_ , 2023
  * [Var58] V.. Varadarajan “On the Convergence of Sample Probability Distributions” In _Sankhyā: The Indian Journal of Statistics (1933-1960)_ 19.1/2 Springer, 1958, pp. 23–26
  * [VF11] Ghislain Verdier and Ariane Ferreira “Adaptive Mahalanobis Distance and $k$ -Nearest Neighbor Rule for Fault Detection in Semiconductor Manufacturing” In _IEEE Transactions on Semiconductor Manufacturing_ 24.1, 2011, pp. 59–68
  * [Wan+11] Pa-Chun Wang, Chao-Ton Su, Kun-Huang Chen and Ning-Hung Chen “The application of rough set and Mahalanobis distance to enhance the quality of OSA diagnosis” In _Expert Systems with Applications_ 38.6, 2011, pp. 7828–7836
  * [Wie58] Norbert Wiener “Nonlinear Problems in Random Theory” The Technology Press of MIT, 1958, pp. 131
  * [WYT07] Defeng Wang, Daniel S. Yeung and Eric C.. Tsang “Weighted Mahalanobis Distance Kernels for Support Vector Machines” In _IEEE Transactions on Neural Networks_ 18.5, 2007, pp. 1453–1462
  * [XNZ08] Shiming Xiang, Feiping Nie and Changshui Zhang “Learning a Mahalanobis distance metric for data clustering and classification” In _Pattern recognition_ 41.12 Elsevier, 2008, pp. 3600–3612