The NetFACS package is a tool that allows users to analyse and plot their facial signal data, generated using the Facial Action Coding System (FACS), in a statistical meaningful way using bootstrapping and network analysis. FACS data, based on the cooccurrence of different elements representing facial muscle movements (called ‘Action Units’ or AUs), have some rather awkward properties: single AUs are binary (present/absent), they are part of larger expressions and combined with each other, their use can be nonnormally distributed, individuals can provide multiple datapoints, data are often autocorrelated etc (see Mielke et al., 2021 for more information). NetFACS allows users to overcome some of these problems and present their results in a form that is easy to communicate to other researchers.
This is a tutorial and overview over which functions are currently available in the NetFACS package and which questions we can potentially already answer. While we will use mainly Facial Action Coding System examples throughout (meaning ‘elements’ in the functions are Action Units, or AU), the functions can be applied to any cooccurrence data, so other questions regarding communication are also possible. This allows researchers to directly compare the complexity of different communication systems, e.g. gestures and facial expressions in nonhuman primates.
As the name indicates, NetFACS is based on the use of networks for FACS data. Not all analyses require an understanding of networks: at its most basic, NetFACS can simply represent the probabilities that Action Units occur in a context or not, or contrast AU use between two conditions. Networks become interesting once we talk about the connections between Action Units, or the connections between AUs and contexts that goes past the pure occurrence. All analyses are based on different concepts of probability, which are explained below. Probabilities are used to quantify the strength of connection between AUs, and between AUs and contexts. The third pillar of the package, next to networks and probabilities, are two different resampling methods: permutations and bootstrapping. These are used to determine what occurrence of AUs would be expected under random circumstances or if the data were taken from a different population. We will explain their use further down, but it’s important to understand the distinction. In permutations, existing data are shuffled to create the ‘random’ probability that two AUs cooccur, for example. In bootstraps, subsets of data from a different context are selected over and over again for the same purpose.
NetFACS is based mainly on different probabilities: unconditional probability is the simply the proportion of occurrence; \(P(A \cap B) = \frac{number of occurrences A+B}{number of possible occurrences A+B}\)). The unconditional probability of a combination AB is not directional, i.e. it is the same for AB as it is for BA: (\(P(A \cap B)\) == \(P(B \cap A)\)). We use the unconditional probability mainly to express how often an AU or a combination of AUs occurs at all, either in the whole data set or in a specific context. Conditional probability, on the other hand, is the probability of an event A given an event B; \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)). Thus, it tells us not how often A occurs at all, but how often A occurs in the subset of cases where we know that B is present. Therefore, the conditional probabilities can differ depending on which direction we are looking at (\(P(A \mid B)\) != \(P(B \mid A)\)). For example, in the English language, the letter ‘h’ is often accompanied by the letter ‘s’, but the reverse is not true: most cases of ‘s’ do not involve ‘h’. We can explore the conditional probability between two elements, or between an element/combination and a context. Probabilities are used as they are intuitive for most people, in contrast to many more opaque association indices that are often used in social network analysis. It also means that we can tie in other analytical tools that are very directly based on probabilities, such as Bayesian models and Markov Chains.
Let’s start by installing the package and loading the library.
# install NetFACS from CRAN
install.packages("NetFACS")
# read library
library(NetFACS)
Alternatively, you can install the latest development version of NetFACS from GitHub with:
# install NetFACS from GitHub
::install_github("NetFACS/NetFACS") devtools
Currently, there are two practice data sets in the package: one is the Extended CohnKanade Database for posed FACS data (all basic AUs, 7 emotions) (Lucey et al. 2010), where cooccurrence is defined as Action Units happening in the same facial expression, and one is the Communist Manifesto in French, German, and English, where cooccurrence is defined as letters happening in the same word. These can be accessed the following way:
data("letternet") # this is the Manifesto #
data("emotions_set") # this is the CK Database #
Important for the NetFACS package at the moment is that the basic data used to calculate probabilities are stored in a data frame with each event as a row (this can be a communication sequence, a facial expression, a frame in a video, or a word, for example), and each element as a column (for example, Action Units in FACS data or letters for words). Elements get a 0 if they are absent in this event, and a 1 if they are present. It’s usually good to have a second data frame of the same length that contains additional information about the condition, duration, etc. In this tutorial we will use the Extended CohnKanade dataset as an example.
The basis for the whole package so far is the netfacs
function. The function creates a list for all possible combination of
elements that has the information on their observed probability of
occurrence in the chosen data set. It also tests whether this observed
probability is significantly different from an expected or null
distribution, and gives information about the shape of that null
distribution. The user defines whether this null distribution is the
expected random distribution of elements (e.g., do AU1 and AU2 occur
together more than expected by chance) based on permutations of the raw
data that control for the overall probability of single elements
occurring and the number of elements per event, or whether we are
comparing two conditions (e.g., do AU1 and AU2 occur together more in
sad than in angry faces) with one condition providing the null
condition. In the latter case, bootstraps of the null condition are used
to establish the null distribution. Bootstraps can be done by randomly
selecting events, or by randomly selecting on a higherlevel variable,
e.g. subject, by including that variable in the ‘random.level’ parameter
of the function call. For direct comparisons, one can include ‘control
variables’ into the bootstrap: for example, if individual sex is
included, then the bootstrap will select the null condition on the same
ratio as the test condition, so if 1/3 of participants in the test
condition are males, then 1/3 of cases in each random selection will
also be male. This prevents biases due to sampling differences between
test and null condition.
It is important to note that combination of elements here does not mean that ONLY those elements were active, but that they were among the active elements. So, the combination of AU1 + 2 is counted when the overall facial expression shows AU1+2, but also AU1+2+5, AU1+2+5+10, etc. Conditional probabilities will be based on the number of times either of the two elements were present.
These are the function parameters and what they do:
data
: this is the data matrix that contains each event
as a row and each element as a column, with 0 and 1 denoting whether an
element was active at allcondition
: this denotes the condition (e.g., the
emotion, language, gender …) that we want to find out about. The input
should be a vector of the same length as data with the
condition each event belongs to, to allow the algorithm to assign each
event. If no condition is set, then all data points are compared against
random distribution. If a condition is set, the user has to define a
test condition: one of the levels of the condition
vector that we are interested in.test.condition
: marks the condition that we want to
find out about.null.condition
: marks the comparison level. If NULL,
then the test condition is compared with all other data points
(e.g. anger against all other emotions). If it is set to a value (e.g.,
‘happy’), then only the test and null condition are compared with each
other (e.g., happy and angry faces).duration
: when analysing videos with FACS, each event
could differ in duration; for example, some events contain 5 frames,
while another contains 10 frames. If this information is available, FACS
will weight the results accordingly. Here, you should enter a vector of
the same length as the data set that defines the duration of each event.
It doesn’t matter if you put seconds or frames as duration, the function
will determine the smallest value and compare all other values to
this.ran.trials
: determines how many permutations or
bootstraps are performed. The larger, the better and more accurate, but
things take longer. 1000 iterations is usually good.control
: Here, you would enter a list of control
variables. So, for example, to control for gender and place of origin,
you would enter ‘list(gender, place.of.origin)’; if these were
available. This way, in the bootstraps, he algorithm will approximate
the distribution of these control parameters in the test and null
condition. So, if there are 10 males and 20 females in the angry faces,
the null condition will select a ratio of 1 to 2 as well, to prevent
difference to be driven by differences in the data set. Note that you
need enough data to include control variables: if there are no female
faces in one of the conditions, the algorithm will fail at adjusting the
distribution.random.level
: Should be a vector of the same length as
the data matrix that assigns each event a higherlevel identifier, such
as individual or video. For the bootstraps, this works similar to a
random effect in that it controls for the fact that some data points
belong to the same hierachical level. So, if an individual provides
several data points, or data points come from the same video, noting
this in the random.level parameter means that random selection
occurs on the level of the video or individual rather than each event
separately. Thus, the probability space of the null distribution is not
biased by one individual or video.combination.size
: If null, the algorithm will pop out
all possible combinations. If we don’t care for the higherorder
combinations (e.g. combinations of 5 elements), we can speed the
analysis up by limiting this to 2 or 3.tail
: Should pvalues be upper.tail (the
observed value is larger than the expected values) or left.tail
(the observed value is smaller than the expected values)?use_parallel
: Should processing be done in
parallel?n_cores
: How many cores should be used for parallel
processing?Here, I will give an example where we compare the facial expression of anger against all other facial expression. We assume that every individual provided one case and there is no information about gender etc, so we do not use any controls:
# here, we test whether any Action Units and combinations appear more frequently than expected under one condition than under another.
# load data
< emotions_set[[1]]
au.data # this is the basic data frame, with each video represented by a row, and each Action Unit represented by a column
< emotions_set[[2]]
au.info # this is the additional information about the emotions etc
# We remove AU 25 here, because it is not informative, and all AUs that have 2 or fewer occurrences, because not meaningful interpretation is available for them
< au.data[, setdiff(colnames(au.data), "25")]
au.data < au.data[, colSums(au.data) > 2]
au.data
# create netfacs object for angry faces
< netfacs(
angry.face data = au.data, # this is the data matrix
condition = au.info$emotion, # info about condition of each case
test.condition = "anger", # condition we are interested in
null.condition = NULL, # null condition (test against all other emotions)
duration = NULL, # we could add duration information for videos
ran.trials = 1000, # number of randomizations. The larger the better
control = NULL, # control variables, e.g. gender, ethnicity etc.
random.level = NULL, # Works like a random effect.
combination.size = 4, # limit the analysis to make things faster,
tail = "upper.tail", # should the pvalue reflect twotailed or right/left tailed testing?
use_parallel = TRUE, # use parallel processing
n_cores = NULL # number of cores for parallel processing
)
combination  combination.size  count  expected.prob  observed.prob  effect.size  pvalue  prob.increase 

4  1  40  0.291  0.889  0.598  0.000  3.051 
17  1  39  0.266  0.867  0.601  0.000  3.257 
23  1  36  0.025  0.800  0.775  0.000  32.469 
24  1  33  0.036  0.733  0.698  0.000  20.525 
7  1  32  0.167  0.711  0.544  0.000  4.260 
6  1  8  0.309  0.178  0.132  1.000  0.575 
14  1  7  0.089  0.156  0.067  0.000  1.751 
5  1  6  0.307  0.133  0.174  1.000  0.434 
18  1  4  0.007  0.089  0.082  0.000  12.546 
15  1  3  0.106  0.067  0.040  1.000  0.627 
9  1  3  0.206  0.067  0.139  1.000  0.324 
10  1  2  0.046  0.044  0.002  0.588  0.967 
12  1  1  0.281  0.022  0.259  1.000  0.079 
16  1  1  0.039  0.022  0.017  0.977  0.566 
4_17  2  36  0.195  0.800  0.605  0.000  4.098 
17_23  2  31  0.014  0.689  0.675  0.000  48.604 
4_23  2  31  0.014  0.689  0.675  0.000  48.604 
4_24  2  31  0.000  0.689  0.689  0.000  NA 
17_24  2  28  0.014  0.622  0.608  0.000  43.075 
7_17  2  28  0.093  0.622  0.530  0.000  6.724 
Now we have the angry.face object that contains all the information we need to make networks and claims about which Action Units make up a facial expression.
As we can see in the last table already, there is a lot of information in this object:
combination
: the name of the combinationcombination.size
: how many elements make up the
combinationcount
: how often it occurs in the test conditionobserved.prob
: Probability that the combination occurs
in a frame of that conditionexpected.prob
: Expected probability of the combination
if the data set was drawn from the null condition; mean of all
randomisations.effect.size
: Difference between the observed
probability and the mean expected probability.pvalue
: how many random probabilities were more extreme
than the observed probability (larger for right.tail, smaller
for left.tail)probability.increase
: how many times more likely is the
combination in this condition than the null condition?As this table is very large in many instances, we can extract results more easily while already cleaning the table a bit using the ‘netfacs_extract’ function:
# extract angry face information for the first level (single elements)
< netfacs_extract(
anger.aus netfacs.data = angry.face,
combination.size = 1, # only looking at combinations with 1 element (here, Action Units)
min.count = 1, # minimum number of times that the combination should occur
min.prob = 0, # minimum observed probability of the combination
significance = 0.01
# significance level we are interested in )
condition  combination  combination.size  count  expected.prob  observed.prob  effect.size  pvalue  prob.increase 

anger  23  1  36  0.025  0.800  0.775  0  32.469 
anger  24  1  33  0.036  0.733  0.698  0  20.525 
anger  17  1  39  0.266  0.867  0.601  0  3.257 
anger  4  1  40  0.291  0.889  0.598  0  3.051 
anger  7  1  32  0.167  0.711  0.544  0  4.260 
anger  18  1  4  0.007  0.089  0.082  0  12.546 
anger  14  1  7  0.089  0.156  0.067  0  1.751 
The results show that in angry faces, the Action Units 4, 7, 14, 17, 18, 23, and 24 are significantly more common than would be expected given the information we have from the other emotions. AU23, for example, occurs 36 times, which is 80% of all ‘angry’ faces. Expected would be 2.5%, so we have an increase in probability of 32 times.
We can also plot this. In element.plot, the yaxis is based on the logtransformed change in probability between the observed and all randomised distributions, and the xaxis contains each element. Almost all plots that are included in the package at this point are ggplot based, so the user can subsequently change most parameters to their own liking.
# create plot showing the importance of each AU for the angry faces
element.plot(netfacs.data = angry.face)
In the distribution.plot function, we see the distribution of the null probabilities and where the observed values fall in comparison. It does this independently for each element. Here, we look at AU4 (which occurs more than expected) and AU9 (which occurs less than expected).
# create plot showing the distribution of the null probabilities and how the observed probability compares
distribution.plot(netfacs.data = angry.face)$"4"
distribution.plot(netfacs.data = angry.face)$"9"
This already gives us an important insight into the structure of facial expressions, by clarifying how strongly certain facial muscles are connected to specific contexts. What we can do now is look at higherorder combinations of Action Units. Let’s look at combinations of 3 Action Units. As there are many combinations that only occur once or twice, and these are probably not very important, let’s set the minimum count that we consider meaningful to 5. It is important to note that, given the large number of possible higherorder combinations (e.g., 20 elements create over 1100 combinations of 3, almost 5000 combinations of 4 etc), most of these will not have meaningful information unless the data set is quite large.
# extract information for threeelementcombinations in angry faces
< netfacs_extract(
anger.aus3 netfacs.data = angry.face,
combination.size = 3, # only looking at combinations with 3 elements (here, Action Units)
min.count = 5, # minimum number of times that the combination should occur
min.prob = 0, # minimum observed probability of the combination
significance = 0.01 # significance level we are interested in
)
condition  combination  combination.size  count  expected.prob  observed.prob  effect.size  pvalue  prob.increase 

anger  4_17_23  3  28  0.014  0.622  0.608  0  43.9 
anger  4_17_24  3  27  0.000  0.600  0.600  0  NA 
anger  4_23_24  3  23  0.000  0.511  0.511  0  NA 
anger  4_7_24  3  23  0.000  0.511  0.511  0  NA 
anger  7_17_23  3  22  0.000  0.489  0.489  0  NA 
anger  7_17_24  3  22  0.000  0.489  0.489  0  NA 
Here, we see that for example combination AU4_17_23 appears 28 times (62% of all angry faces), which is 44 times more than we would have expected. Many of the combinations only occur in the angry condition, and they get assigned an ‘increased probability’ value of NA. That many of these higherorder combinations contain very similar combinations (e.g. AU4_17_23 and AU4_17_24) can mean two things: either, they are significant because all these elements declare anger and they are combined randomly; or they are all part of one standard higherorder combination (e.g. AU4_17_23_24) that occurs over and over again. By looking at the different levels (3rd order, 4th order etc), we can resolve this question.
One question that is of relevance in this context is which element
actually contributes to our decoding of the message that the other one
is trying to send. For example, AU1 and AU2 will both appear on their
own, but also as AU1+2, AU1+2+5, AU1+2+5+26 etc. The question is whether
the elements actually add information about the condition: If I remove
AU1 from AU1+2+5+26, does the resulting combination still convey the
same message? We do this through the specificity measure (i.e., strength
association of the element with the condition in question). First, the
function specificity
calculates the specificity of all
elements to a context, then the function
specificity_increase
goes through all combinations and
calculates the mean specificity with and without an element. The result
tells us which Action Units actually add the information ‘angry!’ and
which ones potentially just appear as part of combinations:
< specificity(angry.face)
spec < specificity_increase(spec) spec.increase
Here, we see that even though AU5 is pretty rare, as it only occurred 6 times, and is in itself not very specific to the context of anger (only 5% of all events containing AU5 occur in anger), adding AU5 to other combinations makes them ‘angry’: the specificity increases by 35%. Combinations that would usually be relatively neutral (for example, wrinkling your forehead) look really angry if you add some eye flashing. Thus, AU5 on itself is not a sign of anger, but when added to other combinations it might have a different function. A smile (AU12) actually decreases the likelihood that a combination is ‘angry’ (specificity increase = 0.09). The specificity increase is available for the first level (single elements) and second level (dyads of elements).

To understand the importance of any element in our communication system, we should look for the ones that combine two aspects: they should be common in the context (i.e., high probability), but they should also be limited to that context (i.e., high specificity). We will come back to that idea when talking about the overlap networks below.
The previous sections all dealt with the unconditional probability of two elements occurring together; for example, we wanted to know whether Action Units 1 and 2 occur together more often than expected. However, these two elements will sometimes occur together and sometimes on their own, and this might be important from a communication point of view: for example, if AU1 is always active when AU2 is there, and vice versa, it is reasonable to assume that these two elements are actually used as one functional unit. These two elements would have respective conditional probabilities of 1, or close to it. Alternatively, two elements could have conditional probabilities of 0; this could indicate that sender use either one or the other. For example, in facial expressions, AU26 (jaw drop) and AU27 (mouth stretch) denote different levels of opening the mouth, and they can never happen at the same time, which is a relevant aspect of the structure of the FACS. The conditional probabilities of two elements can be highly asymmetric: AU2 can always be associated with AU1, but AU1 can be more common and happen outside of this combination. This case will usually indicate that one element is either way more common than the other, or that the second one functions to change the meaning of the other. For example, consonants in English, when spoken, include a vowel, so the conditional probability of hearing the vowel when the consonant is used is 1; however, the same vowel is shared between many consonants. The last alternative is the the two elements are not related to each other and randomly cooccur across events.
In NetFACS, we can get information on the conditional probabilities
using the conditional_probabilities
function:
< conditional_probabilities(angry.face) conditional.probs
element_A  element_B  combination  count  probability_A  probability_B  probability_AandB  probability_AgivenB  probability_BgivenA 

4  17  4_17  36  0.89  0.87  0.80  0.92  0.90 
17  4  17_4  36  0.87  0.89  0.80  0.90  0.92 
4  23  4_23  31  0.89  0.80  0.69  0.86  0.78 
4  24  4_24  31  0.89  0.73  0.69  0.94  0.78 
17  23  17_23  31  0.87  0.80  0.69  0.86  0.79 
23  4  23_4  31  0.80  0.89  0.69  0.78  0.86 
5  23  5_23  6  0.13  0.80  0.13  0.17  1.00 
6  17  6_17  6  0.18  0.87  0.13  0.15  0.75 
17  5  17_5  6  0.87  0.13  0.13  1.00  0.15 
23  5  23_5  6  0.80  0.13  0.13  1.00  0.17 
17  6  17_6  6  0.87  0.18  0.13  0.75  0.15 
4  5  4_5  5  0.89  0.13  0.11  0.83  0.12 
7  14  7_14  5  0.71  0.16  0.11  0.71  0.16 
The above table shows us the two elements (e.g., 5 an 23), how often they occur together (‘count’), the probability of each of them to occur across events P(A) and P(B); their unconditional probability to occur together P(A+B), and the probability of A occurring when B is also present. So for the AU 5 and 23, AU23 is much more common in angry faces (80% of videos) than AU5 (13% of videos). They also occur both together in 13% of cases, which means that in all cases where AU5 was observed, AU23 was also observed, but only 17% of events that have AU23 also have AU5 present. In this case, it appears as if AU5 is an ‘addition’ to AU23 or facial expressions including AU23. If we look at AU17 and AU23, they appear to be part of a fixed expression: their conditional probabilities are around 80% each way, indicating that they both occur together most of the time. Below, we will also show what the network visualisation of this function looks like.
Now that we have our information about the probabilities of different elements and conditions extracted, we can start looking at different networks that potentially contain information about our Action Units that we rarely consider. There are different networks here that are of interest: first, we can make a bipartite network that shows us how different conditions are linked and which Action Units are specific to certain context. To a have high information value, we would expect an Action Unit to have high specificity in one context and that context only. Second, we can use networks to visualise the connections between Action Units. Third, we can use network metrics to identify central Action Units in a condition, and understand the network as a whole better.
Let’s start with the bipartite overlap network: we can visualise
which conditions (in our case, emotions), share Action Units. To make
the process easier, there is a netfacs_multiple
function
that runs the netfacs function for all levels of a condition against all
others.
< netfacs_multiple(
multi.facs data = au.data,
condition = au.info$emotion,
ran.trials = 1000,
combination.size = 2,
use_parallel = TRUE
)# calculate element specificity
< specificity(multi.facs) multi.spec
Now we can make a network where each condition is a node, and they
are connected through the Action Units. We only consider Action Units
that occur at least 3 times in the condition. Let’s also remove Action
Unit 25 because it can be created by different muscles and is therefore
ambiguous. The overlap_network
function creates different
graphs for different questions: the ‘specificity’ graph shows the
conditional probability that we are in a condition given that we observe
the element; the ‘occurrence’ graph shows the conditional probability to
observe an element in a given context:
< overlap_network(
overlap.net
multi.spec,min.prob = 0, # minimum probability of a connection to be included
min.count = 3, # minimum count of cooccurrences for a connection to be included
significance = 0.01, # significance level for combinations to be considered
clusters = FALSE, # should the bipartite network be clustered
plot.bubbles = TRUE,
)
plot(overlap.net$specificity)
plot(overlap.net$occurrence)
Now, we have an easy way to understand our FACS data: Above we see
how specific an AU is to a condition. 84% of all AU12 happen in happy
faces; while all AU27 happen in surprised faces. AU4 is shared between
multiple conditions without being specific to any of them. Below, we
have the occurrence data: AU6 and AU12 occur in almost all ‘happy’
faces, but AU16 is rare (P = 0.06). AU15 occurs in 82% of sad faces, and
22% of contemptuous faces. The underlying data are stored in the
overlap_network
object as well, in case the network is
overwhelmingly full and hard to understand. With the ‘clusters’
parameter, we could explore whether these results create clearly
separated clusters using the igraph ‘fast and greedy’ clustering
algorithm.
Earlier, we explored conditional probability, but we did not yet look at the output graph, which can help us illuminate our data even more. We restrict connections to those that are at least 0.5 to make things easier to understand.
< network_conditional(
conditional.probs netfacs.data = angry.face,
min.prob = 0.5,
min.count = 5,
ignore.element = NULL,
plot.bubbles = TRUE
)
# plot conditional probabilities
$plot conditional.probs
Here, we see a visual representation of the angry face and how the AUs are linked: for example, AU4, 17, 23, and 24 are bigger than the others, indicating that they are more common. Accordingly, AU17 and AU23 happen almost every time when AU6 is active, but AU6 does not happen above our threshold of 0.5 for cases with those AUs active. The common AUs, on the other hand (e.g., AU4 and 23) have reciprocal conditional probability: in the majority of cases when one is active, the other one is also active.
To do all the following calculations, we extract the netfacs object and turn it into a network with specific properties. Here goes:
< netfacs_network(
angry.net netfacs.data = angry.face,
link = "unweighted", # edges are linked for significant results only
significance = 0.01,
min.count = 3, # remove rare elements as they might be random variation
min.prob = 0
)
Now we have our angry.net, which is an igraph
object.
‘igraph’ is the most commonly used package for network analysis in R, so
now we can use all the functions that have been developed for plotting
and analysing networks.
Let’s plot our network for angry faces.
network_plot(
netfacs.graph = angry.net,
title = "angry network",
clusters = FALSE,
plot.bubbles = TRUE,
hide.unconnected = TRUE
)
This looks like a pretty tight cluster of some Action Units that tend
to occur together. AU5 and AU6 are not significantly more common in this
context than expected, but occur in combination with other AUs more than
expected (that’s why they are smaller). This is again relatively
different from the other networks: we can apply the
netfacs_network
function across our conditions to get a
visual representation of how networks differ:
< multiple_netfacs_network(
multi.net
multi.facs,link = "weighted", # network contains edges where significantly connected
significance = 0.01,
min.count = 3, # again remove rare connections
min.prob = 0
)
multiple_network_plot(multi.net)
Here, we can see that the network for anger actually looks pretty complex compared to the other networks: in Contempt, only the combination of AU12 and 14 occurs more often than expected. Also, not all dyads in Anger are connected, indicating that they do not always all occur together. This is very different in Surprise, for example, where all significant AUs are also significantly more likely to cooccur.
One interesting aspect of network analysis is that we might be able to move away from our posed facial expressions and detect unknown underlying patterns. For example, if we would not know which emotion was posed, would we be able to analytically detect the basic emotions because specific Action Units cluster with each other? Let’s create a network based on all data.
<
all.face netfacs(
data = au.data,
condition = NULL,
ran.trials = 1000,
combination.size = 2,
use_parallel = TRUE
)<
all.net netfacs_network(all.face,
min.count = 3,
link = "unweighted")
In this network, dyadic connections between Action Units mean that
they occur more often together than would be expected given their own
likelihood and the number of elements per event. When plotting this
network, we can say clusters = TRUE
. In that case,
igraph
has a community detection algorithm (groups of AUs
that form clusters).
network_plot(
all.net,title = "all network with clusters",
clusters = TRUE,
plot.bubbles = TRUE
)
Modularity is high (0.50; above 0.3 is high). This means that there are clear clusters (AUs that are connected with each other, but not others). First, some AUs just drop out because they do not connect with any of the others (AU10, 14, 16, 18, 26). Then we have a ‘happy’ network (AU6 and AU12), a ‘surprise’ and ‘fear’ network (AU1, 2, 5, 20, 27). Anger, Sadness, and Disgust all overlap in their use of AU4 and 17, which are also the most central AUs of the remaining cluster. AU14, which is the most common AU in contempt, does not fall into either of these clusters. Given the small size of this data set, even without previous knowledge of where the emotions fall, we could assume that there are at least three distinct clusters in the data.
While making graphs is nice, having our FACS data as networks also allows us to calculate the centrality and importance of specific elements in the network. Again, an Action Unit that conclusively describes a specific emotion should be highly central in that network, but not at all central in any other. There are a number of different centrality measures, and one can extract them all at once.
< network_summary(angry.net) net.sum
element  strength  eigenvector  betweenness  transitivity  hub_score  page_rank  modularity  comm.membership  comm.value 

4  0.194  0.914  0.062  0.714  0.914  0.110  0.117  2  0.02 
17  0.222  1.000  0.100  0.643  1.000  0.125  0.114  2  0.02 
7  0.194  0.936  0.044  0.762  0.936  0.109  0.115  2  0.02 
23  0.222  0.936  0.278  0.536  0.936  0.133  0.093  1  0.02 
6  0.111  0.616  0.000  1.000  0.616  0.067  0.062  2  0.02 
14  0.111  0.613  0.000  1.000  0.613  0.066  0.045  2  0.02 
5  0.083  0.461  0.000  1.000  0.461  0.054  0.001  1  0.02 
18  0.139  0.764  0.000  1.000  0.764  0.080  0.020  2  0.02 
9  0.028  0.151  0.000  NaN  0.151  0.027  0.001  3  0.02 
24  0.194  0.936  0.044  0.762  0.936  0.109  0.000  2  0.02 
15  0.000  0.000  0.000  NaN  0.000  0.013  NA  4  0.02 
10  0.000  0.000  0.000  NaN  0.000  0.013  NA  5  0.02 
12  0.000  0.000  0.000  NaN  0.000  0.013  NA  6  0.02 
16  0.000  0.000  0.000  NaN  0.000  0.013  NA  7  0.02 
1  0.000  0.000  0.000  NaN  0.000  0.013  NA  8  0.02 
2  0.000  0.000  0.000  NaN  0.000  0.013  NA  9  0.02 
20  0.000  0.000  0.000  NaN  0.000  0.013  NA  10  0.02 
26  0.000  0.000  0.000  NaN  0.000  0.013  NA  11  0.02 
27  0.000  0.000  0.000  NaN  0.000  0.013  NA  12  0.02 
The different network measures capture different aspects of centrality.
strength
: how many connections does the AU haveeigenvector
: high if AU is connected with a lot of AUs
that also have a lot of connectionsbetweenness
: number of shortest connections that go
through the AU; does it connect otherwise unconnected elements?transitivity
: how many triads is the element in (triad
== all three elements are connected)hub_score
: similar to eigenvectorpage_rank
: calculates the influence an element has over
all it’s neighboursmodularity
: if modularity is high, the element clearly
clusters with other elementscommunity
membership: elements that have the same
membership cluster with each othercommunity.value
: if this is above 0.3, then there are
clearly distinct clusters in this data setWhile these centrality measures concern the elements within a network, we can also calculate information flow etc within across a network.
< network_summary_graph(angry.net) net.sum.graph
nr.elements  nr.edges  density  transitivity  diameter  degree_centralization  mean_distance 

19  0  0.158  0.729  3  0.287  1.422 
Here, we see that there are 19 elements, 28 connections between them, which means that 16.4% of all possible connections are filled. The graph is highly transitive (== if A and B are connected and A and C are connected, B and C are also connected), so the dyadic connections probably arise out of higherorder combinations. The diameter is the ‘furthest’ connection. The degree centralisation is the mean degree/strength of all elements. Mean distance is the mean number of ‘steps’ to get from one element to the next in the network.
Let’s see what happens if we do this across the different emotions.
< lapply(multi.net, function(x) {
xx network_summary_graph(x)
})< do.call(rbind, xx)
xx < cbind(emotion = names(multi.net), xx) xx
emotion  nr.elements  nr.edges  density  transitivity  diameter  degree_centralization  mean_distance 

anger  19  28  0.164  0.755  0.291  0.281  0.135 
contempt  19  4  0.023  0.600  0.468  0.143  0.271 
disgust  19  14  0.082  0.780  0.571  0.251  0.242 
fear  19  35  0.205  0.561  0.313  0.406  0.175 
happy  19  7  0.041  0.600  0.133  0.181  0.071 
sadness  19  10  0.058  0.778  0.979  0.164  0.567 
surprise  19  17  0.099  0.814  0.160  0.234  0.064 
We see that the density of the different emotion networks differs considerably, but they all have pretty low density (mainly because they all rely on a small set of AUs). Fear and Anger seem to involve more AUs than the others. Some (Anger, Surprise, Disgust) are pretty transitive (AUs all usually appear together), while some (Fear, Happy) are less transitive (indicating that they have different configurations).
One question that might be of interest when we talk about complexity could be whether the number of elements is larger than expected in a context: for example, it might be that happy faces usually use fewer elements than angry faces because there is less information necessary to transmit the message. The ‘netfacs’ function also pops out the observed probabilities that events consist of 1, 2, 3 … elements, and this is tested against the null distribution.
< angry.face$event.size.information
event.size.angry < event.size.plot(netfacs.data = angry.face) size.plot
combination.size  observed.prob  expected.prob  effect.size  pvalue 

1  0.00  0.04  0.04  0.00 
2  0.00  0.26  0.26  0.00 
3  0.04  0.18  0.14  0.00 
4  0.40  0.36  0.04  0.04 
5  0.33  0.12  0.22  0.00 
6  0.20  0.04  0.16  0.00 
7  0.00  0.00  0.00  0.39 
8  0.02  0.01  0.01  0.00 