# Detecting and Gauging Impact on Wikipedia Page Views

Xiaoxi Chelsy Xie<EMAIL_ADDRESS>Wikimedia Foundation ,  Isaac Johnson
<EMAIL_ADDRESS>Wikimedia Foundation and  Anne Gomez<EMAIL_ADDRESS>Wikimedia Foundation

(2019)

###### Abstract.

Understanding how various external campaigns or events affect readership on
Wikipedia is important to efforts aimed at improving awareness and access to
its content. In this paper, we consider how to build time-series models aimed
at predicting page views on Wikipedia with the goal of detecting whether there
are significant changes to the existing trends. We test these models on two
different events: a video campaign aimed at increasing awareness of Hindi
Wikipedia in India and the page preview feature roll-out—a means of accessing
Wikipedia content without actually visiting the pages—on English and German
Wikipedia. Our models effectively estimate the impact of page preview roll-
out, but do not detect a significant change following the video campaign in
India. We also discuss the utility of other geographies or language editions
for predicting page views from a given area on a given language edition.

wikipedia; bayesian structural time series; page views; causal inference

††journalyear: 2019††copyright: iw3c2w3††conference: Companion Proceedings of
the 2019 World Wide Web Conference; May 13–17, 2019; San Francisco, CA,
USA††booktitle: Companion Proceedings of the 2019 World Wide Web Conference
(WWW ’19 Companion), May 13–17, 2019, San Francisco, CA, USA††doi:
10.1145/3308560.3316751††isbn: 978-1-4503-6675-5/19/05††ccs: Information
systems Web mining††ccs: Human-centered computing Empirical studies in
collaborative and social computing

##  1\. Introduction

Wikipedia is the fifth-most-visited website worldwide (Alexa, 2018) at 190
billion page views in 2018 alone (Erhart, 2019) and is turned to by readers
for all sorts of reasons ranging from simple curiosity to fact-checking to
making a personal decision (Singer et al., 2017). Despite its success, there
are still many regions in the world where it is relatively unknown (Gill and
McCune, 2018) or access is blocked (Clark et al., 2017). In an attempt to
improve access and awareness worldwide, the Wikimedia Foundation (WMF) has
conducted various campaigns and efforts aimed at improving access to Wikipedia
in various regions.111https://meta.wikimedia.org/wiki/New_Readers

Many researchers have sought to estimate the impact of external events on
Wikipedia page views. This has included the effect of posting Wikipedia
articles on external websites (Vincent et al., 2018; Moyer et al., 2015),
privacy concerns on viewing of sensitive Wikipedia articles (Penney, 2016),
and censorship (Zhang and Zhu, 2011). Conversely, much research has also
sought to use Wikipedia page views as a predictor—i.e. _nowcasting_ or
_forecasting_ (Priedhorsky et al., 2017)—of external entities such as the
stock market (Moat et al., 2013), box office returns for movies (Mestyán et
al., 2013), and disease incidence (Priedhorsky et al., 2017).

Evaluating the impact, however, of a given campaign or external event can be
difficult. Daily page views to Wikipedia projects can be quite noisy, being
affected by weekly, seasonal, holiday-related trends (ten Thij et al., 2012).
A change in the volume of page views following an awareness campaign could
also easily be the result of an unrelated event—e.g., a celebrity marriage or
World Cup game (Erhart, 2019). To account for these challenges, many
researchers rely on some form of regression discontinuity design that focuses
on changes between a short time period (e.g., two weeks) before and after an
event (e.g., (Vincent et al., 2018; Zhang and Zhu, 2011; Moyer et al., 2015)).
While powerful, this approach is limited to studying short-term effects and
does not naturally lend itself to the task of nowcasting or forecasting.

In this paper, we explore the utility of Bayesian structural time series
(BSTS) models for evaluating the impact of external events. BSTS models are
designed to predict a given time series based on historical data, seasonality
components, and additional control time series. They naturally incorporate
uncertainty and the resulting forecast can then serve as a counterfactual—i.e.
be compared against the actual time series following a given intervention to
determine whether there is evidence that the intervention increased or
decreased the magnitude of the time series. We test the BSTS model in two
scenarios: the page preview roll-out in German and English Wikipedia as well
as a video campaign in India designed to increase awareness about Hindi
Wikipedia.

Our contributions are as follows:

  * •

Page Previews: using the roll-out of page previews on the German and English
Wikipedia, we demonstrate that our BSTS model can effectively detect changes
in page views given predictive control series.

  * •

Impact of video campaigns in India: applying our BSTS model to online and TV
awareness campaigns in India, we do not find evidence of increased page views
as a result of the online or TV campaigns.

  * •

Correlations across languages and regions: we evaluate the predictive power of
page-view time-series between pairs of Wikipedia language editions and
regions. We find evidence that page view trends are unique to a given country
and language edition and that control series ideally originate from the same
language edition followed by same country to be a useful predictor.

##  2\. Related Work

In this work, we draw methods from the time series prediction literature and
motivation from the literature that has sought to understand the impact of
external events on Wikipedia activity.

###  2.1. Time Series Modeling

The goal of the time series modeling that we employ in this paper is to
understand whether a specific intervention significantly impacts a given
metric for which we have temporal data—e.g., whether the roll-out of a new
feature causes a change in daily traffic. There are many approaches to time
series modeling that span from quite simple to much more complex in accordance
with how many assumptions they make. Common to these models, however, is that
their validity depends on the model being able to make direct comparisons
between the time series prior to an intervention and the time series following
the intervention (Angrist and Jörn-Steffen, 2009). Threats to this validity
come from a variety of sources that may affect time series independent of the
intervention being studied: seasonality effects such as natural variation by
day of week or month of year, unaccounted external events such as holidays or
changes in the size of the underlying population. The likelihood that these
events affect the time series increases as the time period being studied
increases. A strong model, then, incorporates covariates that can control for
seasonality, holidays, and other external factors that might affect the time
series. A strong model also effectively represents its own uncertainty about
predictions when there are insufficiently strong controls in place.

The core distinguishing features of approaches are 1) whether they include a
control time series, and, 2) whether they directly compare metrics before and
after the intervention or predict the time series after the intervention and
compare this counterfactual prediction with the actual data. A control time
series is a time series that is highly correlated with the “treated” times
series, but, importantly, is known to not be affected by the intervention. The
value of a control time series is that it helps to ensure that if seasonality
or another event affects the treated time series, this effect is not conflated
with the impact of the intervention because the effect should also be present
in the control time series. The value of comparing the post-intervention time
series with a counterfactual, as opposed to just the values from prior to the
intervention, is greater flexibility to changing conditions. A model that
produces the counterfactual can take into account more data about the time
series prior to the intervention and, therefore, better account for shifts in
covariates that might occur following the intervention. This is especially
important when considering the long-term effects of an intervention. For these
models, if the actual time series falls outside of the bounds of the
counterfactual time series, this provides evidence that the intervention had a
significant impact.

There are many considerations for how to build a robust time series model in
order to produce the counterfactual predictions. Primarily, some models that
use static regression to produce the counterfactual predictions falsely assume
the data to be independent and identically distributed (i.i.d), which would
result in an underestimation of the uncertainty (Bertrand et al., 2002).
Secondarily, to help avoid over-fitting, we need to choose appropriate control
time series. Castle et al. (Castle et al., 2009) reviews and compares 21
methods for variable selection, including significance testing (e.g., forward
and backward step-wise regression) and information criteria (e.g., AIC, BIC).
Other popular model selection algorithms in time-series forecasting includes
principal component and factor models, and penalized regression models (e.g.,
Lasso, ridge regression). However, these techniques force us to use a fixed
set of selected variables, or do not account for the uncertainty in variable
selection. Lastly, in order to gauge the uncertainty of the impact, we need to
account for various sources of uncertainty in the model. Besides the
uncertainty in variable selection and auto-correlation mentioned above, we
also want to account for uncertainties in the historical relationships between
treated and control time series, as well as uncertainties in seasonality and
other components in the model.

###  2.2. Impact of External Events on Wikipedia

A number of papers have considered the challenge of establishing how an
external event has affected dynamics within Wikipedia. Vincent et al. (Vincent
et al., 2018) and Moyer et al. (Moyer et al., 2015) take what is known as a
_interrupted time series_ (ITS) approach to examine how posts on Reddit that
include links to Wikipedia articles affect page view traffic on Wikipedia.
Both model a Reddit post with a Wikipedia link as a “shock” to that Wikipedia
article and compare the mean number of page views in a short period of time
before and after the post to determine whether there is a significant
difference. Zhang and Zhu (Zhang and Zhu, 2011) take a similar approach for
the rate of contributions to Chinese Wikipedia from outside editors before and
after a block on mainland China. Zhang and Zhu also seek to control for
seasonal trends by examining the same time period in prior years. These
approaches build on the assumption that there should be no difference in the
expected page views between the pre- and post-intervention periods, and
therefore any difference in page views can be causally tied to the Reddit post
or block. As discussed above, these are assumptions that may hold true for
short time-spans like one week, given that they do not happen to coincide with
major holidays or events. This assumption is increasingly tenuous, however, as
more long-term trends are considered.

Penney (Penney, 2016) also starts with an ITS model to understand the impact
of the Edward Snowden revelations on page views to “terrorism-related”
Wikipedia articles. Notably, because Penney examines a much longer time-period
comprising 32 months, their analysis also includes a “control time series”
that are security-related articles that are similar in content but less likely
to be affected by the Snowden revelations. This approach is often referred to
as difference-in-differences (DD) and is similar to how we construct the BSTS
models considered in this work, but the BSTS models directly incorporate the
concept of a control series as a core component of the models and captures the
uncertainty of the relationship between the treated series and the control
series. This makes for a much more explicit and robust means of controlling
for additional external effects that may otherwise be conflated with the
treatment under study.

##  3\. Methods

We use a single time series model architecture, described below, and apply it
to two different events. Each event involves an external event that led to a
potential shift in page views. We describe each event alongside its results.

###  3.1. Bayesian Structural Time Series Model

In this work, we use Bayesian structural time series (BSTS) model (Scott and
Varian, 2013b). Per the components discussed in §2.1, these models can
incorporate control covariates and time series, generate counterfactual
predictions of page views for the post-intervention period assuming that the
intervention did not take place, and naturally model their own uncertainty
about these counterfactuals. We can then compare the counterfactual
predictions and actual page views to quantify the causal impact of the
intervention.

BSTS models combine three statistical methods into an integrated architecture
(Scott and Varian, 2013a):

  * •

A structural time series model for trend and seasonality, estimated using
Kalman filters;

  * •

Spike and slab regression for variable selection;

  * •

Bayesian model averaging for the final prediction.

Structural Time Series Model: Under different assumptions, a very large class
of models can be expressed in the form of structural time series models,
including all ARIMA models (Harvey, 1991). This flexibility allows BSTS models
to accommodate multiple sources of variations, including trends, seasonality,
and latent evolutions of the treated series that cannot be explained by known
trends or events. Specifically, a structural time series model (e.g. Eq. 1)
decomposes the time series into four components: a level ($\mu_{t}$), a local
trend or slope ($\delta_{t}$), seasonal effects ($\tau_{t}$) and error terms.
The model described here adds a regression component
($\beta^{T}\mathbf{x}_{t}$) to incorporate the control time series and other
covariates. It is a stochastic generalization of the constant-trend regression
model (e.g. $y_{t}=\mu+\delta t+\beta^{T}\mathbf{x}_{t}+\epsilon_{t}$), where
the level $\mu_{t}$ and slope $\delta_{t}$ parameters each follow a random
walk model instead of a constant. This allows for greater flexibility in the
trends expressed within the model.

(1) $\displaystyle y_{t}$
$\displaystyle=\mu_{t}+\tau_{t}+\beta^{T}\mathbf{x}_{t}+\epsilon_{t},\epsilon_{t}\sim
N(0,\sigma_{\epsilon}^{2})$ $\displaystyle\mu_{t}$
$\displaystyle=\mu_{t-1}+\delta_{t-1}+u_{t},u_{t}\sim N(0,\sigma_{u}^{2})$
$\displaystyle\delta_{t}$ $\displaystyle=\delta_{t-1}+v_{t},v_{t}\sim
N(0,\sigma_{v}^{2})$ $\displaystyle\tau_{t}$
$\displaystyle=-\sum_{s=1}^{S-1}\tau_{t-s}+w_{t},w_{t}\sim
N(0,\sigma_{w}^{2})$

Spike and Slab: There are often many potential control series but including
them all would likely lead to over-fitting and very complex models. A spike-
and-slab prior over coefficients (George and McCulloch, 1997; Madigan and
Raftery, 1994) is designed to solve this challenge. The spike part controls
the probability of whether a given variable would be chosen for the model—i.e.
having a non-zero coefficient. The slab part shrinks the non-zero coefficients
toward prior expectations which is often zero. Upon observing data, Bayes’
theorem updates the inclusion probability of each coefficient. Then when
sampling from the posterior distribution of a regression model, many of the
simulated regression coefficients will be exactly zero (Scott, 2017).

Bayesian Model Averaging: To generate counterfactual predictions, the
procedure uses the Markov chain Monte Carlo (MCMC) algorithm to draw samples
from the parameter’s posterior distribution and then combine that with the
available data to yield a distribution of the counterfactual predictions. The
model can then compute the difference between the actual values of a treated
series in the post-intervention period and the distribution of counterfactual
samples to yield an estimate of the distribution of the impact (Brodersen et
al., 2015). Because the structural time series model, spike-and-slab
regression and model averaging all have natural Bayesian interpretations, BSTS
is able to account for various sources of uncertainties using MCMC. This
allows us to gauge confidence in the magnitude of causal impact and estimate
the posterior probability that the causal impact is non-existent.

Figure 1. Hindi Wikipedia daily externally-referred page views from the top 10
states with the most page views in India. The black (top highlighted) series
is the treated series: page views from Madhya Pradesh. The red (bottom
highlighted) series is one of the most predictive control series according to
the model in §4.2: page views from the neighboring state of Rajasthan. The
blue dashed line is a covariate: interpolated number of internet subscribers
in thousands from Madhya Pradesh. The intervention under study (online video
campaign) started on 3 April 2018.

We use the _BSTS_ 222https://cran.r-project.org/package=bsts and
_CausalImpact_ 333https://cran.r-project.org/package=CausalImpact R packages
to fit the BSTS models. The following parameters comprise each BSTS model.
Figure 1 illustrates the parameters of the model for Hindi online video
campaign in §4.2.

  * •

Treated Time Series: this is the time series under study—e.g., daily
internally-referred page views to English Wikipedia over the period of several
months.

  * •

Intervention: an event that occurred on a specific date during the study
period that is believed to have affected the treated time series—e.g., the
introduction of a new feature onto Wikipedia that might change the daily
number of page views. Model validation is conducted entirely on data prior to
the intervention.

  * •

Pre-Intervention Period: time period from the first data point of the treated
series to the day before the intervention. For each model in this work, we
explore four different pre-intervention period length using grid-search: 12
weeks, 18 weeks, 183 days and 400 days.

  * •

Post-Intervention Period: time period following the intervention for which the
impact is being estimated—e.g., daily page views for the six weeks following
the roll-out of a new feature. For each model in this work, we set this to 6
weeks.

  * •

Covariates: additional variables that help explain the treated time
series—e.g., total population online in a country. This also includes the
control series described below.

  * •

Control Series: a time series that is predictive of the treated time series
prior to the intervention, but that is not be impacted by the
intervention—e.g., daily page views for a similar Wikipedia edition for which
the feature was not rolled out. The authors of the _CausalImpact_ library we
use for estimating the models suggest using 3-50
covariates.444https://stats.stackexchange.com/questions/162930/causalimpact-
should-i-use-more-than-one-control/163554#163554 Thus, for some models in this
work where we have hundreds of control series—e.g., many combinations of
different regions and language editions—we use correlation and dynamic time
warping (DTW) (Toni, 2009) algorithms with pre-intervention data to prescreen
and trim the list of control series before feeding them into the BSTS model.

  * •

Seasonality: weekly and seasonal trends, or holiday effects that did not get
captured by the control series. For each model in this work, we include
features for day-of-week and month. We also include major holidays for the
regions under study as described in the Results section.

  * •

Trend Model: the architecture for the model that predicts how the treated time
series evolves. For each model in this work, we explore the following types of
trend components using grid-search: local level, local linear, semi-local
linear, and static intercept term.

##  4\. Results

Below we describe the context for two events on which we tested our BSTS model
and their results.

###  4.1. Wikipedia Page Previews

Beginning in 2014, Wikipedia began exploring a new feature that would allow
for page previews for the desktop version of the site. When a user moused over
a link, a card would appear with an image and part of the first paragraph from
the article that the link pointed to (see the Wikimedia Blog
post555https://blog.wikimedia.org/2018/05/09/page-previews-documentation/ for
more information and an example of a page preview on Wikimedia
Commons666https://commons.wikimedia.org/w/index.php?curid=47213242). This
allows users to preview the article content without clicking on it (and
thereby recording a page view). As a result of this ability to explore
Wikipedia content without actually visiting the pages, it was expected that
page views would actually drop with the roll-out of this feature.

In late 2017, the feature was finally rolled out to a proportion of anonymous
users on German (de-wiki) and English (en-wiki) Wikipedia in a series of A/B
tests. These tests were analyzed and it was determined that page preview
feature led to a drop of approximately 4% in page views across these Wikipedia
communities.777https://www.mediawiki.org/wiki/Page_Previews/2017-18_A/B_Tests
The full deployment of page previews to all anonymous users of these
communities occurred on April 11 and 17 respectively. These A/B tests present
us with an opportunity to explore the power of our BSTS models because they
experimentally determined the expected effect on overall page views from the
full deployment of page previews. Specifically, from the A/B tests starting on
December 21 2017, in which 1.5% and 4% of anonymous users on English and
German Wikipedias by default had access to page preview functionality
respectively, we expect our BSTS model to detect:

  * •

de-wiki: a 3.0% decrease in page views after April 11
2018.888https://phabricator.wikimedia.org/T191966

  * •

en-wiki: a 4.7% decrease in page views after April 17
2018.999https://phabricator.wikimedia.org/T191101

####  4.1.1. Model Parameters

For each model, we set the pre-intervention period to be 400 days and the
post-intervention period to be 6 weeks. For English Wikipedia, this means that
the time series starts on 13 March 2017 and includes daily page views data
through 28 May 2018, with the intervention occurring on 17 April 2018. For
German Wikipedia, this means that the time series starts on 7 March 2017 and
includes daily page views data through 22 May 2018, with the intervention
occurring on 11 April 2018. Alongside day-of-week and monthly seasonality, we
also include the following holidays: Christmas and New Year’s. For trend
modeling, we choose a static intercept term for German Wikipedia—i.e. we
expect the trend of the time series to be soaked up by the regression
component, and a local level model for English Wikipedia—i.e. the trend will
be predicted around the weighted average values of recent observations.

For the control series, we rely on the assumption that while page previews
should impact the internally-referred page views—i.e. page views as a result
of navigating from one Wikipedia page to another—there is no reason that the
previews would impact externally-referred page views—i.e. page views that
result from someone navigating from a search engine or other, non-Wikimedia
website—or direct page views without referrer. Specifically, we select the
daily internally-referred page views from en-wiki or de-wiki as the treated
time series. For our control time series, we include the daily externally-
referred page views and direct page views from the same Wikipedia language
edition as the treated time series under study (i.e. en-wiki or de-wiki), as
well as daily externally-referred page views from the other top-20 largest
Wikipedia editions (e.g. Russian and Spanish Wikipedia).

####  4.1.2. Results

We find that our time series models for both English and German Wikipedia are
quite accurate. The validation statistics associated with the model provide an
indication of how effective the model was at predicting the pre-intervention
time series. With 10-fold cross validation and prediction evaluated on 6 weeks
of daily page views (from the end of the pre-intervention period), the holdout
mean absolute percentage error (MAPE) of the English Wikipedia model is 2.54%,
and the holdout MAPE of the German Wikipedia model is 3.92%.

Figure 2. Impact of the page preview feature on 6 weeks of English Wikipedia
internally-referred page views. Vertical dashed line represents the date of
the roll-out (17 April 2018). Shaded areas indicate 95% credible intervals.
The first panel shows the data (black solid line) and counterfactual
prediction (blue dashed line) for the post-intervention period. The second
panel sums the difference between observed data and counterfactual
predictions—i.e. point-wise causal effect as estimated by the model, resulting
in a plot of the cumulative effect of the intervention. The figures of point-
wise causal effect are removed in this paper for space consideration.

Turning to the estimate of the causal impact of the page preview roll-out, the
treated time series along with counterfactual estimates from model for en-wiki
are shown in Figure 2. Recall that the early A/B tests indicated that there
would be a 4.7% decrease in page views for en-wiki. Our BSTS model, using the
externally-referred and direct page views as control series, estimates a 3.0%
decrease and correctly determines that no impact—i.e. 0% change—falls outside
of the 95% credible interval [1.9%, 3.9%], indicating that the roll-out
resulted in a significant change in page views. The most predictive control
series in this model is the search engine referred page views on English
Wikipedia with an average standardized coefficient of 0.65—i.e. when search
engine referred page views change 1 standard deviation, we expect to see
internally-referred page views change 0.65 standard deviation, and the
posterior inclusion probability—i.e. the probability of this coefficient being
different from zero—is 100%.

We see analogous results for de-wiki: the BSTS model estimated a 2.6% decrease
in page views with a 95% credible interval of [1.9%, 3.4%], in line with the
3.0% decrease that had been determined via A/B testing. Similarly, the most
predictive control series in this model is the search engine referred page
views on German Wikipedia with an average standardized coefficient of 0.95 and
a posterior inclusion probability of 100%.

###  4.2. Hindi Video Campaign

In India, only 33% of Hindi internet users have heard of Wikipedia (Gill and
McCune, 2018) and, while there are 120,000 Wikipedia articles in Hindi, many
people do not know that Hindi content exists. Meanwhile, internet access is
growing 20%+ per year across
India101010http://www.internetlivestats.com/internet-users/india/ and Hindi
online content consumption is growing 94% per
year111111https://economictimes.indiatimes.com/tech/internet/hindi-content-
consumption-on-internet-growing-at-94-google/articleshow/48528347.cms. In July
2017, the Wikimedia Foundation and the Hindi Wikimedians User Group began
collaborating to reach “New Readers” in India through production and promotion
of an online
video.121212https://meta.wikimedia.org/wiki/New_Readers/Raising_Awareness_in_India
The goal is to increase awareness and drive new usage of Wikipedia among Hindi
speaking internet users.

To explain and promote Hindi Wikipedia (hi-wiki), the Wikimedia Foundation
launched the video campaign on 3 April 2018. The
Ektara131313https://commons.wikimedia.org/wiki/File:Wikipedia_-
_Ektara_(English_subtitles).webm video was promoted on YouTube and Facebook
targeting Hindi internet users in Madhya Pradesh, many of whom who had not
heard of Wikipedia. The online promotion ran for three weeks and the video
gathered 2.61 million views. This was followed by a second push over TV during
a major Cricket event (on DD Sports during the Indian Premier League finals)
on 27 May 2018 to the whole country, which reached 1.37 million
viewers.141414TV data was collected by Eurodata TV via BARC in India.

####  4.2.1. Model Parameters

The pre-intervention period is 400 days and the post-intervention period is 6
weeks. Alongside day-of-week and monthly seasonality, we also include the
following major Hindu holidays: Diwali, Raksha Bandhan, Holi, Dussehra, and
New Year.151515These Hindu festivals are picked because of their relative big
impact on the treated time series. Their dates of each year are obtained from
https://www.officeholidays.com/countries/india/index.php Local level model and
a static intercept term are chosen as the trend for online campaign model and
TV campaign model respectively.

For the evaluation of the impact of the online campaign, we set the treated
time series to be daily externally-referred page views to hi-wiki from the
Indian state of Madhya Pradesh (as determined by IP geolocation) because the
promotion of the online video campaign was targeted at Madhya Pradesh. We
select the externally-referred page views because it is a good indicator of
the general brand awareness. The time series starts on 27 February 2017 and
includes daily page view data through 14 May 2018, where the intervention
occurred on 3 April 2018. For the control series, we use daily hi-wiki page
views, as well as page views to other popular Wikipedia language editions and
Wikimedia projects,161616We selected the top 10 Wikimedia projects in India
with the most page views, and Wikipedia of major Indian languages spoken by
more than 4% of the population, according to 2011 census of India. from the
rest of India by states. We also included the daily number of internet
subscribers in Madhya Pradesh as a covariate, which is linearly interpolated
from a quarterly series reported by Telecom Regulatory Authority of
India.171717https://www.trai.gov.in/release-publication/reports/performance-
indicators-reports

For the evaluation of the impact of the TV campaign, we set the treated time
series to be daily externally-referred hi-wiki page views from the entire
country of India because there was no state-specific targeting of the
campaign. The time series starts on 22 April 2017 and includes daily page view
data through 7 July 2018, where the intervention occurred on 27 May 2018. For
the control series, we use daily hi-wiki page views and page views to other
popular Wikipedia language editions and Wikimedia projects from other
countries181818Countries that contribute more than 5% of Hindi Wikipedia page
views, countries whose official language is Hindi, and other nearby
countries.. Additionally, we included the daily number of internet subscribers
in India as a covariate, which is linearly interpolated from a quarterly
series reported by Telecom Regulatory Authority of India.

####  4.2.2. Results

Figure 3. Impact of the online campaign in 6 weeks on Hindi Wikipedia
externally-referred page views from Madhya Pradesh. Vertical dashed line
represents the start date of the campaign 3 April 2018. Figure 4. Impact of
the TV campaign in 6 weeks on Hindi Wikipedia externally-referred page views
from all of India. Vertical dashed line represents the date of the campaign 27
May 2018.

For both the online and TV campaigns for Hindi Wikipedia, we do not detect a
significant change in page views. As we discuss below and in §5.1, this is
likely a combination of factors: low impact and imprecise control series. The
results from the BSTS models for the online campaign are in Figure 3 and TV
campaign in Figure 4.

First we examine the results for the online video campaign that was targeted
at the state of Madhya Pradesh. As before, we performed a 10-fold cross
validation with the pre-intervention time series and predict 6-week’s daily
page views in each fold. The average holdout MAPE is 7.6%.

As Figure 3 indicates, no significant impact on page views was detected
following the intervention. While there does appear to be a downward trend in
page views, zero change is still within the 95% credible interval. Page views
to hi-wiki from the states of Rajasthan and Chhattisgarh, both of which border
Madhya Pradesh, are the most predictive control series in the model, with
average standardized coefficients of 0.23 and 0.13 respectively. The posterior
probabilities that their coefficients are different from zero are greater than
95%.

Next we turn to the results for the country-wide TV campaign. The average
holdout MAPE from the cross validation is 10.2%. Figure 4 shows the results
from the BSTS model. As the graphs indicate, no significant impact on page
views was detected in the first 3 weeks following the intervention. There was
a bump in the 4th week after the campaign, but it is most likely to be the
result of an unknown event. Overall, we did not detect significant impact in 6
weeks. The number of internet subscribers in India, hi-wiki page views from
the United States, Bengali Wikipedia page views from Bangladesh and English
Wiktionary pageviews from Nepal are the most predictive control series in the
model, with an average standardized coefficient of 0.84, 0.35, 0.14 and 0.16
respectively. The posterior probabilities that their coefficients are
different from zero are greater than 95%.

We did not include the page views to other Wikipedia language editions that
also were geolocated to India—e.g., page views to en-wiki from India—in the
set of control series because it may violate the independence assumption. Most
people in India are multilingual, so if our brand awareness was affected by
the campaign, the impact would likely be revealed on page views of other
Wikipedia language editions from the target region as well. After seeing the
relatively high MAPE (10.2%) of the TV campaign model, we tried to include the
page views to other Wikipedia language editions and other Wikimedia projects
from India into the model to see if they help. The average holdout MAPE
decreased to 8.5%, but that model also does not detect significant impact.

##  5\. Discussion

###  5.1. Choosing a Control Time Series

| Treated Series (lang; region) | Control Series (lang; region) | Avg MAPE |  | Avg MAPE
---
(No Control Series)
Model 1 | hi-wiki page views; Madhya Pradesh | hi-wiki page views; other states of India | 7.54% | 11.54%
Model 2 | hi-wiki page views; Madhya Pradesh | other wikis page views; Madhya Pradesh | 9.31% | 11.54%
Model 3 | hi-wiki page views; all of India | hi-wiki page views; other countries | 7.22% | 7.92%
Model 4 | hi-wiki page views; all of India | other wikis page views; all of India | 9.14% | 7.92%
Table 1. Comparison of the predictive power of four sets of control time
series for the Hindi Wikipedia campaign, of which the treated series and the
control series either share the same geographic region or the same Wikipedia
language edition.

A predictive control series is one of the most important aspects of a BSTS
model and also the part of the model that is often most difficult to choose.
From the page preview roll-out analysis (§4.1), we see that the BSTS model,
with a well-chosen control time series, can effectively estimate the impact of
a given external event. In that case, externally-referred and internally-
referred page views for the same project are highly correlated, but only
internally-referred page views were believed to be impacted by the page
preview roll-out. For both German and English Wikipedia, the estimated causal
impact was slightly conservative—i.e. lower in magnitude than expected based
on the A/B tests—but still quite close to the expected impact. We had less
success selecting an effective control series for the Hindi video campaign.
Furthermore, the fact that the online campaign model for Madhya Pradesh had
less error than the TV campaign model for all of India raises questions about
how factors like geography or language edition affect the predictive power of
a control series for Wikipedia.

To better understand the power of different types of control series, we tested
four additional control time series models for the Hindi analysis. All are
trained on 400 days of daily externally-referred page views and evaluated via
10-fold cross validation on 6 weeks of page views prior to the intervention (3
April 2018). All models include day-of-week and monthly seasonality, holiday
effects, local level trends, and a set of control time series but no further
covariates. The control series for each model and its respective validation
error are shown in Table 1.

We see that the control time series that are from the same language edition as
the treated time series (Models 1 and 3) have a consistently lower error than
the models that are from different language edition but the same geographic
region (Models 2 and 4). This indicates that language edition plays a more
important role than geographic region in the page view trends on Wikipedia.
Comparing Model 1 and 3 where the control and treated series are from the same
language edition, page views between states within the same country are more
predictive of each other (adding these controls into the model decrease the
MAPE from 11.54% to 7.54%) than page views between different countries (adding
them into the model only decrease the MAPE from 7.92% to 7.22%), which
indicates that while language appears to be most important, country borders
are still a highly salient aspect of page view trends on Wikipedia. Further
research would be needed to understand how these effects play out in other
language communities and the inter-relatedness of different countries and
language pairs.

##  6\. Future Work and Limitations

While this work has a number of limitations, as we lay out below, we believe
it lays the groundwork for exploring more standardized methods of predicting
trends such as page views on Wikipedia with the goal of understanding the
effect of external events. Limitations for this work largely relate to
temporal evolution of impact, data pre-processing, prior distributions of
parameters in BSTS, and the need for additional experiments.

In this work, we focus only on the cumulative effect by the end of the post-
intervention period—its existence and magnitude—without discussing the
temporal evolution of an impact. In practice, how an effect evolves over time,
especially its onset and decay structure, is often a valuable question as
well. The point-wise effect from BSTS reflects the temporal evolution and
future implementation should consider analyzing this result.

Small volume Wikipedia editions such as hi-wiki are more sensitive to
undetected bot behavior, which can cause anomalies in page-view data.
Anomalies in the prediction or post-intervention period would increase the
error rate of validation, or the model might detect an impact that is
unrelated to the known intervention. When the number of control series is very
large, removing outliers manually is not feasible and thus requires a robust
algorithm to detect and adjust outliers while preserving those known
“outliers” such as holiday effects. It is possible that further pre-processing
would also provide benefits—e.g., including more holidays, removing seasonal
patterns in predictors before fitting the model, more extensive grid search
for parameters like the length of pre-intervention period.

We were expecting that spike-and-slab prior in the BSTS would prevent over-
fitting by forcing the coefficients of poor predictors to zero, so we would at
least have predictions not worse than that of a model which only contains the
historical information of the treated series itself. Contrary to this
expectation, Model 4 from Table 1 (predicting hi-wiki page views in India)
shows that including control series from other Wikipedia language editions and
other Wikimedia projects within India actually added noise to the prediction.
To solve this problem, we can further tune the hyper-parameter that controls
the expected model size—the expected number of coefficients that are different
from zero—so that when most of the predictors do not have enough predictive
power, we can lower the expected model size and force more coefficients to be
zero (we set the expected model size to be 10% of total number of controls in
the model but not greater than 5 in this work).

Finally, future work should continue to explore these models in more contexts.
This would hopefully provide guidance for how to select control time
series—e.g., which pairs of regions and language editions (or even other
Wikimedia projects) are predictive, what is the best way to split Wikipedias
into control and treatment articles, how to take advantage of more granular
information as with the internal/external referrer information. This would
also hopefully provide guidance for how to set priors in BSTS models—e.g., a
prior likelihood of relationships between treatment and control time series or
prior standard deviation of the Gaussian random walk of the trend models
(conservatively, we use a non-informative prior for the former and 0.01 for
the latter). In future analyses, we can increase the prior inclusion
probabilities for control series that are likely to be correlated with the
treated series, and increase the prior standard deviation for the trend models
if we believe the volatility of residuals would be large after regressing out
known predictors.

###### Acknowledgements.

We would like to thank Tilman Bayer, Mikhail Popov, Leila Zia, Kate Zimmerman
and Jon Katz for guidance and support as well as our anonymous reviewers for
their feedback.

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Multiplicative Automorphisms of Incidence Algebras

Evgenii Kaigorodov111Gorno-Altaisk State University, e-mail:
<EMAIL_ADDRESS>Piotr Krylov222National Research Tomsk State
University, e-mail<EMAIL_ADDRESS>Askar Tuganbaev333National Research
University MPEI; Lomonosov Moscow State University; e-mail:
<EMAIL_ADDRESS>
Abstract. Let $I(X,R)$ be the incidence algebra of the preordered set $X$ over
the ring $R$. In the case of a finite connected partially ordered set $X$, we
prove that the subgroup of inner multiplicative automorphisms is a direct
factor of the group of multiplicative automorphisms of the algebra $I(X,R)$.
As a consequence, we obtain several matching criteria of the subgroup of inner
multiplicative automorphisms with the group of multiplicative automorphisms.

Key words: incidence algebra, multiplicative automorphism.

MSC2020 database 16W20, 16S50

The study is supported by Russian Scientific Foundation.

##  1 Introduction

The incidence algebras of preordered (in particular, partially ordered) sets
represent an important and very characteristic example of rings of functions.
In this paper, we study multiplicative automorphisms of the incidence algebra
$I(X,R)$ of a preordered set $X$ over the ring $R$. Automorphisms of incidence
algebras have long attracted the attention of specialists. For information on
this subject, you can refer to the book [12] and the papers [3], [4], [6],
[7]. Multiplicative automorphisms are interesting in themselves. They play an
essential role in the study of arbitrary automorphisms of the algebra
$I(X,R)$; moreover, they act as one of the types of its standard automorphisms
(see [4], [6], [12]). In addition, multiplicative automorphisms are used to
describe involutions of the algebra $I(X,R)$ (see [1], [2], [4]).

Incidence algebras were defined by Rota in his familiar paper [10] as a tool
for solving some problems of combinatorics and, above all, for studying
generalizations of the Möbius inversion formula in number theory in a unified
way. Over time, incidence algebras themselves, regardless of their
applications in combinatorics and other areas of mathematics, have turned out
to be a meaningful algebraic object. Many works are dedicated to them,
including the book [12].

Section 2 of the given paper contains some definitions and auxiliary results.
In Section 3, we first give several general facts on multiplicative
automorphisms. The main result of the section and the whole paper is Theorem
3.6 which states that if $X$ is a finite connected partially ordered set, then
the subgroup of inner multiplicative automorphisms is a direct factor in the
group of multiplicative automorphisms of the algebra $I(X,R)$. As a
consequence of this theorem, several conditions have been found under which
this multiplicative automorphism is inner.

We only consider associative rings with non-zero identity element. For some
ring (or an algebra) $S$, $U(S)$ is its group of invertible elements, $C(S)$
is its center, $M(n,\,S)$ is the ring of all square $n$-order matrices with
values in $S$, $M(m\\!\times\\!n,\,S)$ is the group of all rectangular
matrices of order $m\\!\times\\!n$ with values in $S$. Further,
$\mathrm{Aut}\,S$ is the automorphism group of the algebra $S$,
$\mathrm{In}(\mathrm{Aut}\,S)$ is a subgroup of its inner automorphisms. The
semidirect product of two groups $G$ and $H$ is denoted by $G\leftthreetimes
H$.

##  2 Preliminaries

We briefly outline some initial information about preordered sets (one can get
acquainted with them in more detail in the book [12]).

Let $X$ be an arbitrary set and let $\leqslant$ be a reflexive transitive
relation on $X$. In such a case, the system $\langle X,\leqslant\rangle$ is
called a preordered set and $\leqslant$ is a preorder on $X$. If the relation
$\leqslant$ is also antisymmetric, then $\langle X,\leqslant\rangle$ is a
partially ordered set.

Further, we assume that $\langle X,\leqslant\rangle$ is a preordered set. For
any two elements $x,\,y\in X$, we denote by $[x,\,y]$ the subset $\\{z\in
X\;|\;x\leqslant z\leqslant y\\}$. It is called an interval in $X$. An
interval of the form $[x,x]$ is denoted by $[x]$. We note two useful
properties of intervals.

(a) For any $y,\,z\in[x]$, we have the equality $[y,\,z]=[x]$;

(b) If $x<y$, then $s<t$ for arbitrary elements $s\in[x]$ and $t\in[y]$.

We define a binary relation $\sim$ on $X$ by setting $x\sim y$
$\Leftrightarrow$ $x\leqslant y$ and $y\leqslant x$. It is clear that $\sim$
is an equivalence relation on $X$. The corresponding equivalence classes are
of the form $[x]$ for every $x\in X$. It follows from the property $(b)$ that
the pre-order relation $\leqslant$ is agreed with the equivalence relation
$\sim$. Consequently, the induced relation $\leqslant$ appears on the factor
set $\overline{X}=X/\\!\\!\sim$ and $\langle\overline{X},\leqslant\rangle$ is
a partially ordered set.

With the preordered set $X$ and the partially ordered set $\overline{X}$, we
can associate a directed graph (we do not take into account the loops that
arise in this case). It is more convenient to proceed from the set
$\overline{X}$. When necessary, we consider $\overline{X}$ as a simple graph
related to a directed graph $\overline{X}$. In this case, we use standard
notions of graph theory such that a semi-path and its length, spanning tree,
cycle and cyclomatic number.

We agree that all intervals in $X$ are finite. In such a case, $X$ is called a
locally finite preordered set.

The length of the interval in a locally finite partially ordered set is called
the greatest of lengths of chains in this interval.

We agree in the future to denote by $x$ elements of the partially ordered set
$\overline{X}$, i.e., the equivalence classes of the form $[x]$. In other
words, we use some representative of the class $[x]$ to represent $[x]$. This
arrangement is correct and should not lead to confusion. In a specific
situation, it is always clear which elements of the set ($\overline{X}$ or
$X$) we are talking about.

Further, let the letter $R$ denote an algebra over some commutative ring $T$.
However, the ring $T$ itself is hardly used explicitly.

An incidence algebra is some function ring. Let $\langle X,\leqslant\rangle$
be an arbitrary locally finite preordered set. We set $I(X,R)=\\{f\colon
X\\!\\!\times\\!\\!X\to R\;|\;f(x,\,y)=0,\,\text{ if }\,x\not\leqslant y\\}$.
Functions are added pointwise. The product of functions $f$ and $g$ from
$I(X,R)$ is given by the formula

$(fg)(x,\,y)=\sum_{x\leqslant z\leqslant y}f(x,\,z)\cdot g(z,\,y)$ $None$

for any $x,\,y\in X$. Since $X$ is a locally finite set, we can write $z\in X$
instead of $x\leqslant z\leqslant y$ in $(1)$. For any $t\in T$ and $x,\,y\in
X$, we also assume $(tf)(x,\,y)=tf(x,\,y)$. As a result, we obtain a
$T$-algebra $I(X,R)$ called the incidence algebra or the ring incidence of the
preordered set $X$ over the ring $R$. In what follows, we denote by $A$ the
specific algebra $I(X,R)$.

We define some special functions in $I(X,R)$. For $x\in X$, we set
$e_{[x]}(t,t)=1$ for all $t\in[x]$ and $e_{[x]}(z,\,y)=0$ for remaining pairs
$(z,\,y)$. The system $\\{e_{[x]}\;|\;x\in X\\}$ consists of pairwise
orthogonal idempotents which are central in $L$ (the ring $L$ is defined in
the next paragraph). In accordance with the convention of denoting the class
$[x]$ by some of its representatives, we will write $e_{x}$ instead of
$e_{[x]}$.

We define a subring $L$ and an ideal $M$ in $A$. We set $L=\\{f\in
A\;|\;f(x,\,y)=0,\,\text{ if }\,x\not\sim y\\}$ and $M=\\{f\in
A\;|\;f(x,\,y)=0,\,\text{ if }\,x\sim y\\}$. We have the direct sum $A=L\oplus
M$ of $T$-modules, i.e., the ring $A$ is a splitting extension of the ideal
$M$ with the use of the subring $L$. The ideal $M$ is a natural $L$-$L$\-
bimodule. In addition, $M$ is a non-unital algebra.

The ideal $M$ is contained in the Jacobson radical of the algebra $A$.
Consequently, the element $1+d$ invertible in $A$ for every $d\in M$ [12,
Theorem 1.2.3].

Let we have an arbitrary interval $[x]$. We denote by $R_{[x]}$ the set of
functions $f\in A$ such that $f(z,\,y)=0$ if $z\not\sim x$ or $y\not\sim x$.
Similar to the case of idempotents $e_{x}$, we write $R_{x}$ instead of
$R_{[x]}$. There are the equalities $R_{x}=e_{x}Ae_{x}=e_{x}Le_{x}.$ We
conclude that $R_{x}$ is a ring with identity element $e_{x}$. If we pass to
restrictions of functions in $A$ to $[x]\\!\times\\![x]$, then, in fact,
$R_{x}$ is an algebra of all functions $[x]\\!\times\\![x]\to R$ with
pointwise addition and product of convolution type as in $(1)$. We take some
numbering of the interval $[x]\colon[x]=\\{x_{1},\ldots,x_{n}\\}$. After this,
if the function $f\in R_{x}$ is associated with the matrix $(f(x_{i},x_{j}))$,
then we obtain an algebra isomorphism $R_{x}\cong M(n,R)$. Now we take two
distinct intervals $[x]$, $[y]$ and set

$M_{xy}=\\{f\in A\;|\;f(s,t)=0,\,\text{ if }\,s\not\sim x\,\text{ or
}\,t\not\sim y\\}.$

Then $M_{xy}=e_{x}Ae_{y}$ and, therefore, $M_{xy}$ is an
$R_{x}$-$R_{y}$-bimodule. In addition, the equality $M_{xy}=e_{x}Me_{y}$ is
true.

We clarify that $M_{xy}=0$ for $x\not\leqslant y$. For $x<y$, there exists the
canonical isomorphism

$M_{xy}\cong M(n\\!\times\\!m,R),\quad
n=\big{|}[x]\big{|},\;m=\big{|}[y]\big{|}$

with respect to above isomorphisms $R_{x}\cong M(n,R)$, $R_{y}\cong M(m,R)$.
After identifications of all algebras $R_{x}$ with $M(n,R)$ and bimodules
$M_{xy}$ with $M(n\\!\times\\!m,R)$, it becomes clear that ring actions
$R_{x}$ and $R_{y}$ on $M_{xy}$ are ordinary multiplications of matrices. It
is clear also that $M_{xy}$ is an $L$-$L$-bimodule. The action $L$ on $M_{xy}$
reduces to the action $R_{x}$ on the left and $R_{y}$ on the right.

We note once again that by the indices in $R_{x}$ and $M_{xy}$ we mean $[x]$
and $[x][y]$, respectively (see above).

The product $\\!\\!\\!\prod\limits_{x,\,y\in X}\\!\\!M_{xy}$ is an
$L$-$L$-bimodule. Namely, if $f\in L$,
$(g_{xy})\in\\!\\!\\!\prod\limits_{x,\,y\in X}\\!\\!M_{xy}$, then
$f(g_{xy})=(f_{x}g_{xy})\,\text{ and }\,(g_{xy})f=(g_{xy}f_{y})$ where
$f_{x}=e_{x}fe_{x}$ and $f_{y}=e_{y}fe_{y}$.

In the bimodule $\\!\\!\\!\prod\limits_{x,\,y\in X}\\!\\!M_{xy}$, we define a
multiplication through the formula $(g_{xy})\cdot(h_{xy})=(d_{xy}),$ where
$d_{xy}=\\!\\!\\!\sum\limits_{x\leqslant z\leqslant y}\\!\\!g_{xz}\cdot
h_{zy}$. After which this bimodule becomes a (non-unital) algebra.

Proposition 2.1. There exist canonical algebra isomorphisms and also
$L$-$L$-bimodule isomorphisms $L\cong\\!\prod\limits_{x\in X}R_{x}$, and also
an algebra isomorphism $M\cong\\!\\!\\!\prod\limits_{x,\,y\in X}\\!\\!M_{xy}$.

Proof. We define a mapping $\omega\colon L\to\\!\\!\prod\limits_{x\in
X}\\!R_{x}$ by setting $\omega(f)=(f_{x})$ for every $f\in L$, where
$f_{x}=e_{x}fe_{x}$. Then $\omega$ is an algebra isomorphism.

The mapping $\varepsilon\colon M\to\\!\\!\\!\prod\limits_{x,\,y\in
X}\\!\\!M_{xy}$, $\varepsilon(g)=(g_{xy})$, where $g_{xy}=e_{x}ge_{y}$, is an
$L$-$L$-bimodule isomorphism and an algebra isomorphism. $\square$

In what follows, we don’t distinguish objects corresponding to each other
under isomorphisms $\omega$ and $\varepsilon$.

Remark 2.2. We can consider an algebra $A$ as a function algebra and as an
abstract ring presented as a splitting extension $L\oplus M$. These two
approaches can be called a ”functional approach” and an ”abstract approach”;
of course, they are equivalent. We use the both of these approaches.

Every invertible element $v$ of the algebra $A$ provides an automorphism
$\mu_{v}$ of $A$ with the use of the following rule: $\mu_{v}(a)=v^{-1}av$,
$a\in A$. Such an automorphism is called an inner automorphism defined by the
element $v$. All inner automorphisms form a normal subgroup
$\mathrm{In}(\mathrm{Aut}\,A)$ of the group $\mathrm{Aut}\,A$. We denote by
$\mathrm{In_{1}}(\mathrm{Aut}\,A)$ (correspondingly,
$\mathrm{In_{0}}(\mathrm{Aut}\,A)$) the subgroup of inner automorphisms of the
algebra $A$ defined by invertible elements of the form $1+d$, $d\in M$
(correspondingly, by invertible elements of the algebra $L$). The first
subgroup is normal in $\mathrm{Aut}\,A$. By the use of the semidirect
decomposition $U(A)=(1+M)\leftthreetimes U(L)$, we can verify that the
decomposition
$\mathrm{In}(\mathrm{Aut}\,A)=\mathrm{In_{1}}(\mathrm{Aut}\,A)\leftthreetimes\mathrm{In_{0}}(\mathrm{Aut}\,A)$
exists (also see [9, Section 4]).

##  3 Multiplicative Automorphisms

In this section, we consider the ring $L$ and the bimodules $M$, $M_{xy}$
defined above. It follows from Remark 2.2 that $A=I(X,R)=L\oplus M$. In fact,
we do not impose any conditions on the preordered set $X$, the ring $R$, and
the algebra $A$ which is equal to $I(X,R)$.

Let $\varphi$ be an arbitrary automorphism of the algebra $A$. Based on the
$T$-module direct sum $A=L\oplus M$, the automorphism $\varphi$ (as well as
any $T$-endomorphism) can be associated in the usual way with a $2$-order
square matrix $\begin{pmatrix}\alpha&\gamma\\\ \delta&\beta\end{pmatrix}$. We
will only deal with matrices of the form $\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix}$. If the automorphism $\varphi$ corresponds to such a
matrix, then $\beta$ is an automorphism of the (non-unital) algebra $M$ and an
automorphism of the $L$-$L$-bimodule $M$. Conversely, let some mapping
$\beta\colon M\to M$ satisfy the above properties. We obtain an automorphism
$\varphi$ of the algebra $A$ by setting $\varphi(a+c)=\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix}\begin{pmatrix}a\\\ c\end{pmatrix}=a+\beta(c)$ for any
$a\in L$, $c\in M$. All these assertions are directly verified. Their validity
also follows from more general considerations in Section 3 of of the paper
[9]. In what follows, we not distinguish automorphisms of the algebra $A$ and
the matrices corresponding to them.

Automorphisms of the algebra $A$ of the form $\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix}$ are said to be multiplicative. All multiplicative
automorphisms form a normal subgroup in $\mathrm{Aut}\,A$. We denote it by
$\mathrm{Mult}\,A$.

Inner automorphisms of the algebra $A$, corresponding to invertible central
elements of the ring $L$, are said to be fractional. We denote by
$\mathrm{Mult_{0}}\,A$ the subgroup of all fractional automorphisms. In the
book [12], the definitions of multiplicative and fractional automorphisms are
seemingly different definitions from those given here (there is a text on this
topic at the end of the section). The following result establishes a
connection between groups $\mathrm{Mult}\,A$ and $\mathrm{Mult_{0}}\,A$.

Proposition 3.1. There are equalities

$\mathrm{Mult}\,A\cap\text{In}(\text{Aut}\,A)=\mathrm{Mult}\,A\cap\text{In}_{0}(\text{Aut}\,A)=\mathrm{Mult_{0}}\,A.$

Proof. We verify a non-obvious inclusion
$\mathrm{Mult}\,A\cap\text{In(Aut}\,A)\subseteq\mathrm{Mult_{0}}\,A$. We take
some automorphism $\mu=\mu_{1}\mu_{0}$, where $\mu\in\mathrm{Mult}\,A$,
$\mu_{1}\in\text{In}_{1}(\text{Aut}\,A)$,
$\mu_{0}\in\text{In}_{0}(\text{Aut}\,A)$. Let $\mu_{1}$ be defined by the
element $1+d$, $d\in M$, and let $\mu_{0}$ be defined by the element $v\in
U(L)$. Then $\mu$ is defined by the element $v+vd$, and $\mu^{-1}$ is defined
by the element $v^{-1}+d^{\prime}$, where $d^{\prime}\in M$.

For any element $a\in L$, we have

$\mu(a)=v^{-1}av+(v^{-1}avd+d^{\prime}av+d^{\prime}avd).$

Since $\mu\in\mathrm{Mult}\,A$, we have that $v^{-1}av=a$ and $av=va$.
Therefore, $v\in C(L)$. The expression in parentheses is zero. In particular,
$v^{-1}e_{x}vd+d^{\prime}e_{x}v+d^{\prime}e_{x}vd=0$,
$e_{x}v^{-1}e_{x}vd+e_{x}d^{\prime}e_{x}v+e_{x}d^{\prime}e_{x}vd=e_{x}d=0$ for
every $x\in X$. Therefore, $d=0$, whence we have $\mu_{1}=1$ and
$\mu=\mu_{0}$. Since $v$ is an invertible central element,
$\mu\in\mathrm{Mult_{0}}\,A$. $\square$

It can be concluded that fractional automorphisms coincide with inner
multiplicative automorphisms (see also Proposition 3.4).

In this section, we consider various information about groups
$\mathrm{Mult}\,A$ and $\mathrm{Mult_{0}}\,A$. If $X$ is a finite connected
set, then the subgroup $\mathrm{Mult_{0}}\,A$ can be calculated exactly
(Corollary 3.7) and it is a direct factor of the group $\mathrm{Mult}\,A$
(Theorem 3.6).

The following fact is standard and essentially known (see also [9, Proposition
13.2]).

Proposition 3.2. Let $P=M(n,R)$, $Q=M(m,R)$ and $V=M(n\\!\times\\!m,R)$.
Endomorphisms (correspondingly, automorphisms) of the $P$-$Q$-bimodule $V$
coincide with multiplications by central elements (correspondingly, invertible
central elements) of the ring $R$.

Let we have a multiplicative automorphism $\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix}$. For every element $c\in M_{xy}$, we have the equalities
$\beta(c)=\beta(e_{x}ce_{y})=e_{x}\beta(c)e_{y}$. Consequently, $\beta$ leaves
in place every bimodule $M_{xy}$. We denote by $\beta_{xy}$ the restriction
$\beta|_{M_{xy}}$. Since $\beta$ is an automorphism of the algebra $M$, the
equality

$\beta_{xy}(ab)=\beta_{xz}(a)\beta_{zy}(b)$ $None$

is true for any $x,\,z,\,y\in X$ with $x<z<y$ and all $a\in M_{xz}$, $b\in
M_{zy}$ . Due to the fact that $\beta$ is an automorphism of the
$L$-$L$-bimodule $M$ at the same time, we have that $\beta_{xy}$ is an
automorphism of the $R_{x}$-$R_{y}$-bimodule $M_{xy}$ for any $x,\,y$ with
$x<y$.

Further, it follows from Proposition 3.2 that there exists an element
$c_{xy}\in C(U(R))$ with property $\beta_{xy}(g)=c_{xy}g$, $g\in M_{xy}$.
Taking into account the equality $M_{xz}M_{zy}=M_{xy}$, we obtain from $(2)$
the equality

$c_{xy}=c_{xz}c_{zy},$ $None$

where $x<z<y$. Thus, the automorphism $\psi\in\mathrm{Mult}\,A$ can be
associated with the system of invertible central elements $c_{xy}$ ($x,\,y\in
X$, $x<y$) of the ring $R$. Conversely, every system of elements $\\{c_{xy}\in
C(U(R))\;|\;x<y\\}$, satisfying to equality $(3)$, provides a multiplicative
automorphism $\psi$. Namely, for the element $g=(g_{xy})\in M$, we set
$\psi(g)=(c_{xy}g_{xy})$. In this case we have to take into account the rule
of multiplication of elements in $M$ (see Section 2).

We formulate everything just stated above as follows.

Proposition 3.3.

1. There is a one-to-one correspondence between multiplicative automorphisms and systems of elements $\\{c_{xy}\in C(U(R))\;|\;x<y\\}$ such that $c_{xy}=c_{xz}c_{zy}$ for all $x,\,z,\,y\in X$ with $x<z<y$.

2. The group $\mathrm{Mult}\,A$ is embedded in the product of $\mathcal{M}$ copies of the group $C(U(R))$, where $\mathcal{M}=\big{|}\\{(x,\,y)\;|\;x<y\\}\big{|}$.

A more formal point of view on the situation under consideration is as
follows. For every pair $(x,\,y)$ of elements in $X$ with $x<y$, we take a
copy $U_{xy}$ of the group $C(U(R))$. Then the group $\mathrm{Mult}\,A$ is
isomorphic to a subgroup of the product $\prod\limits_{x<y}\\!U_{xy}$,
consisting of vectors $(u_{xy})$, the elements of which satisfy the equality
$u_{xy}=u_{xz}u_{zy}$ if $x<z<y$.

What can be said about fractional automorphisms, i.e. on automorphisms from
the subgroup $\mathrm{Mult_{0}}\,A$? Let a fractional automorphism
$\psi=\begin{pmatrix}1&0\\\ 0&\beta\end{pmatrix}$ be defined by a invertible
central element $v=(v_{x})\in L$, where $v_{x}\in C(U(R))$. For any $x$, $y$
with $x<y$ and every $g\in M_{xy}$, we have the equalities

$\beta(g)=v^{-1}gv=v_{x}^{-1}v_{y}g.$

Consequently, the system of elements $\\{c_{xy}\;|\;x<y\\}$ from Proposition
3.3, corresponding to the automorphism $\psi$, consists of all elements of the
form $v_{x}^{-1}v_{y}$, where $x<y$.

We have the following result.

Proposition 3.4. Let a multiplicative automorphism $\psi$ correspond to a
system of elements $c_{xy}$ $(x,\,y\in X,\;x<y)$ satisfying equality $(2)$.
The following assertions are equivalent.

1) $\psi$ is a fractional automorphism.

2) $\psi$ is an inner automorphism.

3) There exists an element $v=(v_{x})\in L$ with $v_{x}\in C(U(R))$ such that
is true the equality $c_{xy}=v_{x}^{-1}v_{y}$ for all $x,\,y$ $(x<y)$.

Everywhere else we assume that $X$ is a finite connected set. For technical
convenience, we also assume that $X$ is partially ordered.

Note that the incidence algebra $I(X,R)$ for a finite set $X$ is sometimes
called the ring of structural matrices. You can get acquainted with such rings
in the papers [5], [11]. Structural matrix rings are one type of a large class
of formal matrix rings. The book [8] is dedicated to the latter.

Let we have an arbitrary set $\\{c_{xy}\;|\;x,\,y\in X,\;x<y\\}$ of central
invertible elements of the ring $R$. Based on this set, we assign weights to
the edges and the half-paths of the graph $X$ (see [3]). If $(x,\,y)$ is an
edge, then we set $w(x,\,y)=c_{xy}$ for $x<y$ and $w(x,\,y)=c_{yx}^{-1}$ for
$x>y$. For a semi-path $P=z_{1}z_{2}\ldots z_{k+1}$ in $X$, we set

$w(P)=\prod_{i=1}^{k}w(z_{i},z_{i+1}).$

If $\psi$ is some multiplicative automorphism, then we obtain weights of edges
and semi-paths of the graph $X$. They are induced by the system
$\\{c_{xy}\;|\;x<y\\}$ corresponding to the automorphism $\psi$ in the sense
of Proposition 3.3. Let $\psi$ be a fractional automorphism (i.e., inner
multiplicative automorphism; see Proposition 3.4) which is defined by an
invertible element $v=(v_{x})$. It is easy to verify that for any two elements
$x,\,y\in X$, the weight of any semi-path from $x$ to $y$ is equal to
$v_{x}^{-1}v_{y}$ (thus, it does not depend on specific semi-path).

We fix some spanning tree $T$ of the graph $X$. For every edge $(x,\,y)$ in
$T$ with $x<y$, we take an element $c_{xy}\in C(U(R))$. Similar to what was
done above, we use the elements $c_{xy}$ to assign the weight $w$ to edges and
semi-paths in $T$. Let $(x,\,y)$ be an edge in $X$ which is not contained in
$T$ and $x<y$. We set $c_{xy}=w(P)$, where $P$ is a (unique) semi-path in $T$
from $x$ to $y$. As a result, we obtain a system of elements $\\{c_{xy}\in
C(U(R))\;|\;x,\,y\in X,\,x<y\\}$ and weights of edges and semi-paths in $X$.

Lemma 3.5. The constructed system of elements $\\{c_{xy}\;|\;x<y\\}$ satisfies
equalities $(3)$. Consequently, it induces a multiplicative automorphism
$\psi$ in the sense of Proposition 3.3.

Proof. Let $x,\,z,\,y$ be elements with $x<z<y$. We denote by $P_{xz}$,
$P_{zy}$ and $P_{xy}$ semi-paths in $T$ from $x$ to $z$, from $z$ to $y$ and
from $x$ to $y$, correspondingly. Since semi-pathes in $T$ are unique, we have
$P_{xy}=P_{xz}P_{zy}$. Therefore, $w(P_{xy})=w(P_{xz})\cdot w(P_{zy})$ and
$c_{xy}=c_{xz}c_{zy}$. It follows from Proposition 3.3 that the system
$\\{c_{xy}\;|\;x<y\\}$ corresponds to a quite definite automorphism
$\psi\in\mathrm{Mult}\,A$. $\square$

Since $X$ is a partially ordered set, the bimodules $M_{xy}$ are copies of the
ring $R$ in our case. We can define analogues of matrix units. Let $x,\,y\in
X$ and $x<y$. We denote by $e_{xy}$ a function $X\times X\to R$ such that
$e_{xy}(s,t)=1$ if $s=x$, $t=y$ and $e_{xy}(s,t)=0$ for all remaining pairs
$(s,t)$. The functions $e_{xy}$ satisfy the following property: if $x<z<y$,
then $e_{xz}e_{zy}=e_{xy}$.

Note that we have chosen the spanning tree $T$ of the graph $X$. We denote by
$\mathrm{Mult_{1}}\,A$ the subgroup in $\mathrm{Mult}\,A$ consisting of
automorphisms $\psi$ such that $\psi(e_{xy})=e_{xy}$ for all edges $(x,\,y)\in
T$ with $x<y$. Equivalently, $c_{xy}=1$ for the specified edges $(x,\,y)$.
Here $\\{c_{xy}\in C(U(R))\;|\;x<y\\}$ is the system from Proposition 3.3. We
also note that this proposition implies that the group $\mathrm{Mult}\,A$ is
commutative.

Theorem 3.6. There exists a group direct decomposition
$\mathrm{Mult}\,A=\mathrm{Mult_{1}}\,A\\!\times\\!\mathrm{Mult_{0}}\,A$.

Proof. Let $\psi\in\mathrm{Mult}\,A$ and let $\\{c_{xy}\in C(U(R))\;|\;x<y\\}$
be the corresponding system of elements from Proposition 3.3. For every vertex
in $X$, we now define some central invertible element.

We take some root $x_{0}$ in the tree $T$. Let $y$ be some vertex in $T$
adjacent with $x_{0}$. If $x_{0}<y$, then we set $v_{x_{0}}=1$ and
$v_{y}=v_{x_{0}}c_{x_{0}y}$. If $x_{0}>y$, then we set
$v_{y}=v_{x_{0}}c_{yx_{0}}^{-1}$. Then we take some vertex $z$, which is
adjacent with $y$ in $T$, and do the same. Namely, let $v_{z}=v_{y}c_{yz}$ for
$y<z$ and let $v_{z}=v_{y}c_{zy}^{-1}$ for $y>z$. We continue in this manner
until we reach some hanging vertex in $T$. We obtain a semi-path in $T$ from
$x_{0}$ to this hanging vertex. If there are vertices left in $T$ that do not
belong to this semi-path, then let $s$ be the first vertex in the semi-path
with degree exceeding two. Starting from the vertex $s$, we similarly form
another half-path, and so on. As a result, each vertex $x$ in $T$ has an
element $v_{x}\in C(U(R))$ associated with it.

We take an invertible central element $v=(v_{x})$ in $L$. Let $\psi_{0}$ be
the inner automorphism induced by $v$. Then $\psi_{0}\in\mathrm{Mult_{0}}\,A$,
since $v\in C(U(L))$. We set $\psi_{1}=\psi\psi_{0}^{-1}$. If $(x,\,y)\in T$
and $x<y$, then $\psi(e_{xy})=c_{xy}e_{xy}$, by Proposition 3.2, and
$\psi_{0}(e_{xy})=v_{x}^{-1}v_{y}e_{xy}$. By construction of elements $v_{x}$,
the equality $c_{xy}=v_{x}^{-1}v_{y}$ is true. Consequently, $\psi$ and
$\psi_{0}$ act the same on elements $e_{xy}$ with $(x,\,y)\in T$. Therefore,
the inclusion $\psi_{1}\in\mathrm{Mult_{1}}\,A$ is true. Thus,
$\psi=\psi_{1}\psi_{0}$, where $\psi_{1}\in\mathrm{Mult_{1}}\,A$,
$\psi_{0}\in\mathrm{Mult_{0}}\,A$.

It remains to verify that
$\mathrm{Mult_{1}}\,A\cap\mathrm{Mult_{0}}\,A=\langle 1\rangle$. Let
$\xi\in\mathrm{Mult_{1}}\,A\cap\mathrm{Mult_{0}}\,A$. Then
$\xi(e_{xy})=e_{xy}$ for every edge $(x,\,y)\in T$, where $x<y$. In the same
time, the automorphism $\xi$ is defined as an inner automorphism by some
central invertible element $v=(v_{x})$ of the ring $L$. Therefore, we have the
equality $\xi(e_{xy})=v_{x}^{-1}v_{y}e_{xy}$. Consequently,
$v_{x}^{-1}v_{y}=1$ and $v_{x}=v_{y}$. Now let an edge $(x,\,y)$ be not
contained $T$. There exists a semi-path in $T$ from $x$ to $y$. By the use of
this semi-path, we obtain the equality $v_{x}=v_{y}$ from only the above.
Therefore, we obtain the equality $\xi=1$. Therefore,
$\mathrm{Mult_{1}}\,A\cap\mathrm{Mult_{0}}\,A=\langle 1\rangle$ which
completes the proof. $\square$

For a graph $X$, we denote by $m(X)$ and $\lambda(X)$ the number of edges and
the cyclomatic number of $X$, respectively.

Corollary 3.7. There is a group isomorphism

$\mathrm{Mult_{0}}\,A\cong\\!\\!\\!\\!\\!\\!\\!\prod_{m(X)-\lambda(X)}\\!\\!\\!\\!\\!\\!\\!C(U(R)).$

Proof. We denote by the letter $G$ the product in the right part of the
isomorphism. We define a mapping $g\colon\mathrm{Mult}\,A\to G$ as follows.
Let $\psi\in\mathrm{Mult}\,A$ and let $\\{c_{xy}\;|\;x<y\\}$ be the system of
elements from Proposition 3.3 corresponding to $\psi$. The mapping $g$
associates the automorphism $\psi$ with the vector $(c_{xy})$, where
$(x,\,y)\in T$ and $x<y$. It is clear that $g$ is a group homomorphism.
Surjectivity of $g$ follows from Lemma 3.5. It is also clear that
$\text{Ker}\,g=\mathrm{Mult_{1}}\,A$. We have an isomorphism
$\mathrm{Mult}\,A/\mathrm{Mult_{1}}\,A\cong G$. On the other hand, it follows
from Theorem 3.6 that there is an isomorphism
$\mathrm{Mult}\,A/\mathrm{Mult_{1}}\,A\cong\mathrm{Mult_{0}}\,A$. $\square$

Corollary 3.8. The equality $\mathrm{Mult}\,A=\mathrm{Mult_{0}}\,A$ is true if
and only if for any $\psi\in\mathrm{Mult}\,A$, we have $\psi=1$ for any edge
$(x,\,y)\in T$ such that $\psi(e_{xy})=e_{xy}$.

Corollary 3.9 [3, Theorem 5]. A multiplicative automorphism $\psi$ is inner if
and only if for any two elements $x,\,y\in X$, weights of any two semi-paths
from $x$ to $y$ coincide.

Proof. If a multiplicative inner automorphism $\psi$ is defined by the element
$v=(v_{x})$, then the weight of any semi-path from $x$ to $y$ is equal to
$v_{x}^{-1}v_{y}$. This has already been noted above.

For the automorphism $\psi$, we assume that weight conditions for semi-paths
holds. Similar to Theorem 3.6, we write $\psi=\psi_{1}\psi_{0}$. If $(x,\,y)$
is an edge in $T$, then $\psi(e_{xy})=\psi_{0}(e_{xy})$. Now let $(x,\,y)$ be
an edge in $X$ not contained in $T$. Then $\psi(e_{xy})=c_{xy}e_{xy}$ and
$w(x,\,y)=c_{xy}$. If $P$ is a semi-path in $T$ from $x$ to $y$, then
$w(P)=v_{x}^{-1}v_{y}$, where the invertible element $v=(v_{x})$ defines an
inner automorphism $\psi_{0}$. It follows from the assumption that
$c_{xy}=v_{x}^{-1}v_{y}$. Therefore, $\psi$ acts on all elements $e_{xy}$ as
$\psi_{0}$; therefore, $\psi=\psi_{0}$. $\square$

Let $C$ be some simple cycle of the graph $X$ and let $\psi$ be some
multiplicative automorphism. We define the weight $w(C)$ of the cycle $C$ with
respect to $\psi$ as the weight of one of two semi-paths from any vertex of
the cycle to itself. When selecting the opposite route direction of the cycle,
the weight is equal to $w(C)^{-1}$. Therefore, the weight of the cycle $C$ is
one of the elements $w(C)$ and $w(C)^{-1}$.

Let $C_{1},\ldots,C_{k}$ be the fundamental system of cycles associated with
spanning tree $T$ of the graph $X$, where $k=\lambda(X)$.

Corollary 3.10. A multiplicative automorphism $\psi$ is inner if and only if
$w(C_{1})=\ldots=w(C_{k})=1$.

Proof. The necessity of the condition follows from the first paragraph of the
proof of Corollary 3.9.

Now we assume that $w(C_{1})=\ldots=w(C_{k})=1$. We write the multiplicative
automorphism $\psi$ in the form $\psi=\psi_{1}\psi_{0}$ (see the proof of
Corollary 3.9). Let an edge $(x,\,y)$, $x<y$, be not contained in $T$. Then
$(x,\,y)\in C_{i}$ for some $i$. If $P$ is a semi-path in $C_{i}$ from $x$ to
$y$, then we have equalities

$1=w(C_{i})=w(P)w(x,\,y)^{-1},\quad w(x,\,y)=w(P).$

By repeating the argument from the proof of Corollary 3.9, we obtain that
$c_{xy}=v_{x}^{-1}v_{y}$ and $\psi=\psi_{0}$. $\square$

In the book [12], multiplicative and fractional automorphisms are defined with
the use of some special functions in $I(X,R)$, where $R$ is a commutative
ring. These definitions can be also extended to the case of an arbitrary ring
$R$.

A function $m\in I(X,R)$ is said to be multiplicative if for every triple of
elements $x,\,z,\,y$ in $X$ with $x\leqslant z\leqslant y$, we have

$m(x,\,y)\in C(U(R)),\quad m(x,\,y)=m(x,\,z)\cdot m(z,\,y).$

Let $m$ be some multiplicative function in $I(X,R)$. Then the mapping

$\psi_{m}\colon A\to A,\;\psi_{m}(f)=m*f,\;f\in A,$

is an automorphism of the algebra $A$ (here $*$ is the Hadamard product of
functions, i.e., the pointwise product). It is clear that
$\psi_{m}\in\mathrm{Mult}\,A$. Conversely, with the use of some multiplicative
function, every automorphism in $\mathrm{Mult}\,A$ can be obtained this way.

Let $q\colon X\to C(U(R))$ be some mapping. Then a function $m_{q}\in A$ such
that $m_{q}(x,\,y)=q^{-1}(x)\cdot q(y)$ for $x\leqslant y$ and
$m_{q}(x,\,y)=0$ in remaining cases, is multiplicative. In addition, the
corresponding automorphism $\psi_{m_{q}}$ is fractional. All fractional
automorphisms are obtained in such a way.

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