Package overview: netropy
Installation
You can install the released version of netropy from CRAN with:
install.packages("netropy")The development version from GitHub with:
# install.packages("devtools")
devtools::install_github("termehs/netropy")Statistical Entropy Analysis of Network Data
Multivariate entropy analysis is a general statistical method for analyzing and finding dependence structure in data consisting of repeated observations of variables with a common domain and with discrete finite range spaces. Only nominal scale is required for each variable, so only the size of the variable’s range space is important but not its actual values. Variables on ordinal or numerical scales, even continuous numerical scales, can be used, but they should be aggregated so that their ranges match the number of available repeated observations. By investigating the frequencies of occurrences of joint variable outcomes, complicated dependence structures, partial independence and conditional independence as well as redundancies and functional dependence can be found.
This package introduces these entropy tools in the context of network data. Brief description of various functions implemented in the package are given in the following but more details are provided in the package vignettes and the references listed.
library('netropy')Loading Internal Data
The different entropy tools are explained and illustrated by exploring data from a network study of a corporate law firm, which has previously been analysed by several authors (link). The data set is included in the package as a list with objects representing adjacency matrices for each of the three networks advice (directed), friendship (directed) and co-work (undirected), together with a data frame comprising 8 attributes on each of the 71 lawyers.
To load the data, extract each object and assign the correct names to them:
data(lawdata)
adj.advice <- lawdata[[1]]
adj.friend <- lawdata[[2]]
adj.cowork <-lawdata[[3]]
df.att <- lawdata[[4]]Variable Domains and Data Editing
A requirement for the applicability of these entropy tools is the specification of discrete variables with finite range spaces on the same domain: either node attributes/vertex variables, edges/dyad variables or triad variables. These can be either observed or transformed as shown in the following using the above example data set.
We have 8 vertex variables with 71 observations, two of which (years
and age) are numerical and needs categorization based on their
cumulative distributions. This categorization is in details described in
the vignette “variable domains and data editing”. Here we just show the
new dataframe created (note that variable senior is omitted as it only
comprises unique values and that we edit all variable to start from 0):
att.var <-
data.frame(
status = df.att$status-1,
gender = df.att$gender,
office = df.att$office-1,
years = ifelse(df.att$years <= 3,0,
ifelse(df.att$years <= 13,1,2)),
age = ifelse(df.att$age <= 35,0,
ifelse(df.att$age <= 45,1,2)),
practice = df.att$practice,
lawschool= df.att$lawschool-1
)
head(att.var)
#> status gender office years age practice lawschool
#> 1 0 1 0 2 2 1 0
#> 2 0 1 0 2 2 0 0
#> 3 0 1 1 1 2 1 0
#> 4 0 1 0 2 2 0 2
#> 5 0 1 1 2 2 1 1
#> 6 0 1 1 2 2 1 0These vertex variables can be transformed into dyad variables by using
the function get_dyad_var(). Observed node attributes in the dataframe
att_var are then transformed into pairs of individual attributes. For
example, status with binary outcomes is transformed into dyads having
4 possible outcomes $(0,0), (0,1), (1,0), (1,1)$:
dyad.status <- get_dyad_var(att.var$status, type = 'att')
dyad.gender <- get_dyad_var(att.var$gender, type = 'att')
dyad.office <- get_dyad_var(att.var$office, type = 'att')
dyad.years <- get_dyad_var(att.var$years, type = 'att')
dyad.age <- get_dyad_var(att.var$age, type = 'att')
dyad.practice <- get_dyad_var(att.var$practice, type = 'att')
dyad.lawschool <- get_dyad_var(att.var$lawschool, type = 'att')Similarly, dyad variables can be created based on observed ties. For the
undirected edges, we use indicator variables read directly from the
adjacency matrix for the dyad in question, while for the directed ones
(advice and friendship) we have pairs of indicators representing
sending and receiving ties with 4 possible outcomes :
dyad.cwk <- get_dyad_var(adj.cowork, type = 'tie')
dyad.adv <- get_dyad_var(adj.advice, type = 'tie')
dyad.frn <- get_dyad_var(adj.friend, type = 'tie')All 10 dyad variables are merged into one data frame for subsequent entropy analysis:
dyad.var <-
data.frame(cbind(status = dyad.status$var,
gender = dyad.gender$var,
office = dyad.office$var,
years = dyad.years$var,
age = dyad.age$var,
practice = dyad.practice$var,
lawschool = dyad.lawschool$var,
cowork = dyad.cwk$var,
advice = dyad.adv$var,
friend = dyad.frn$var)
)
head(dyad.var)
#> status gender office years age practice lawschool cowork advice friend
#> 1 3 3 0 8 8 1 0 0 3 2
#> 2 3 3 3 5 8 3 0 0 0 0
#> 3 3 3 3 5 8 2 0 0 1 0
#> 4 3 3 0 8 8 1 6 0 1 2
#> 5 3 3 0 8 8 0 6 0 1 1
#> 6 3 3 1 7 8 1 6 0 1 1A similar function get_triad_var() is implemented for transforming
vertex variables and different relation types into triad variables. This
is described in more detail in the vignette “variable domains and data
editing”.
Univariate, Bivariate and Trivariate Entropies
The function entropy_bivar() computes the bivariate entropies of all
pairs of variables in the dataframe. The output is given as an upper
triangular matrix with cells giving the bivariate entropies of row and
column variables. The diagonal thus gives the univariate entropies for
each variable in the dataframe:
H2 <- entropy_bivar(dyad.var)
H2
#> status gender office years age practice lawschool cowork advice
#> status 1.493 2.868 3.640 3.370 3.912 3.453 4.363 2.092 2.687
#> gender NA 1.547 3.758 3.939 4.274 3.506 4.439 2.158 2.785
#> office NA NA 2.239 4.828 4.901 4.154 5.058 2.792 3.388
#> years NA NA NA 2.671 4.857 4.582 5.422 3.268 3.868
#> age NA NA NA NA 2.801 4.743 5.347 3.411 4.028
#> practice NA NA NA NA NA 1.962 4.880 2.530 3.127
#> lawschool NA NA NA NA NA NA 2.953 3.567 4.186
#> cowork NA NA NA NA NA NA NA 0.615 1.687
#> advice NA NA NA NA NA NA NA NA 1.248
#> friend NA NA NA NA NA NA NA NA NA
#> friend
#> status 2.324
#> gender 2.415
#> office 3.044
#> years 3.483
#> age 3.637
#> practice 2.831
#> lawschool 3.812
#> cowork 1.456
#> advice 1.953
#> friend 0.881Bivariate entropies can be used to detect redundant variables that
should be omitted from the dataframe for further analysis. This occurs
when the univariate entropy for a variable is equal to the bivariate
entropies for pairs including that variable. As seen above, the
dataframe dyad.var has no redundant variables. This can also be
checked using the function redundancy() which yields a binary matrix
as output indicating which row and column variables are hold the same
information:
redundancy(dyad.var)
#> no redundant variables
#> NULLMore examples of using the function redundancy() is given in the
vignette “univariate bivariate and trivariate entropies”.
Trivariate entropies can be computed using the function
entropy_trivar() which returns a dataframe with the first three
columns representing possible triples of variables V1,V2, and V3
from the dataframe in question, and their entropies H(V1,V2,V3) as the
fourth column. We illustrated this on the dataframe dyad.var:
H3 <- entropy_trivar(dyad.var)
head(H3, 10) # view first 10 rows of dataframe
#> V1 V2 V3 H(V1,V2,V3)
#> 1 status gender office 4.938
#> 2 status gender years 4.609
#> 3 status gender age 5.129
#> 4 status gender practice 4.810
#> 5 status gender lawschool 5.664
#> 6 status gender cowork 3.464
#> 7 status gender advice 4.048
#> 8 status gender friend 3.685
#> 9 status office years 5.321
#> 10 status office age 5.721Joint Entropy and Association Graphs
Joint entropies is a non-negative measure of association among pairs of variables. It is equal to 0 if and only if two variables are completely independent of each other.
The function joint_entropy() computes the joint entropies between all
pairs of variables in a given dataframe and returns a list consisting of
the upper triangular joint entropy matrix (univariate entropies in the
diagonal) and a dataframe giving the frequency distributions of unique
joint entropy values. A function argument specifies the precision given
in number of decimals for which the frequency distribution of unique
entropy values is created (default is 3). Applying the function on the
dataframe dyad.var with two decimals:
J <- joint_entropy(dyad.var, 2)
J$matrix
#> status gender office years age practice lawschool cowork advice
#> status 1.49 0.17 0.09 0.79 0.38 0.00 0.08 0.02 0.05
#> gender NA 1.55 0.03 0.28 0.07 0.00 0.06 0.00 0.01
#> office NA NA 2.24 0.08 0.14 0.05 0.13 0.06 0.10
#> years NA NA NA 2.67 0.61 0.05 0.20 0.02 0.05
#> age NA NA NA NA 2.80 0.02 0.41 0.01 0.02
#> practice NA NA NA NA NA 1.96 0.04 0.05 0.08
#> lawschool NA NA NA NA NA NA 2.95 0.00 0.01
#> cowork NA NA NA NA NA NA NA 0.62 0.18
#> advice NA NA NA NA NA NA NA NA 1.25
#> friend NA NA NA NA NA NA NA NA NA
#> friend
#> status 0.05
#> gender 0.01
#> office 0.08
#> years 0.07
#> age 0.05
#> practice 0.01
#> lawschool 0.02
#> cowork 0.04
#> advice 0.18
#> friend 0.88
J$freq
#> j #(J = j) #(J >= j)
#> 1 0.79 1 1
#> 2 0.61 1 2
#> 3 0.41 1 3
#> 4 0.38 1 4
#> 5 0.28 1 5
#> 6 0.2 1 6
#> 7 0.18 2 8
#> 8 0.17 1 9
#> 9 0.14 1 10
#> 10 0.13 1 11
#> 11 0.1 1 12
#> 12 0.09 1 13
#> 13 0.08 4 17
#> 14 0.07 2 19
#> 15 0.06 2 21
#> 16 0.05 7 28
#> 17 0.04 2 30
#> 18 0.03 1 31
#> 19 0.02 5 36
#> 20 0.01 5 41
#> 21 0 4 45As seen, the strongest association is between the variables status and
years with joint entropy values of 0.79. We have independence (joint
entropy value of 0) between two pairs of variables:
(status,practice), (practise,gender), (cowork,gender),and
(cowork,lawschool).
These results can be illustrated in a association graph using the
function assoc_graph() which returns a ggraph object in which nodes
represent variables and links represent strength of association (thicker
links indicate stronger dependence). To use the function we need to load
the ggraph library and to determine a threshold which the graph drawn
is based on. We set it to 0.15 so that we only visualize the strongest
associations
library(ggraph)
assoc_graph(dyad.var, 0.15)Given this threshold, we see isolated and disconnected nodes
representing independent variables. We note strong dependence between
the three dyadic variables status,years and age, but also a
somewhat strong dependence among the three variables lawschool,
years and age, and the three variables status, years and
gender. The association graph can also be interpreted as a tendency
for relations cowork and friend to be independent conditionally on
relation advice, that is, any dependence between dyad variables
cowork and friend is explained by advice.
A threshold that gives a graph with reasonably many small independent or conditionally independent subsets of variables can be considered to represent a multivariate model for further testing.
More details and examples of joint entropies and association graphs are given in the vignette “joint entropies and association graphs”.
Prediction Power Based on Expected Conditional Entropies
The function prediction_power() computes prediction power when pairs
of variables in a given dataframe are used to predict a third variable
from the same dataframe. The variable to be predicted and the dataframe
in which this variable also is part of is given as input arguments, and
the output is an upper triangular matrix giving the expected conditional
entropies of pairs of row and column variables (denoted $X$ and $Y$) of
the matrix, i.e. EH(Z|X,Y) where $Z$ is the variable to be predicted.
The diagonal gives EH(Z|X) , that is when only one variable as a
predictor. Note that NA’s are in the row and column representing the
variable being predicted.
Assume we are interested in predicting variable status (that is
whether a lawyer in the data set is an associate or partner). This is
done by running the following syntax
prediction_power('status', dyad.var)
#> status gender office years age practice lawschool cowork advice
#> status NA NA NA NA NA NA NA NA NA
#> gender NA 1.375 1.180 0.670 0.855 1.304 1.225 1.306 1.263
#> office NA NA 2.147 0.493 0.820 1.374 1.245 1.373 1.325
#> years NA NA NA 2.265 0.573 0.682 0.554 0.691 0.667
#> age NA NA NA NA 1.877 1.089 0.958 1.087 1.052
#> practice NA NA NA NA NA 2.446 1.388 1.459 1.410
#> lawschool NA NA NA NA NA NA 3.335 1.390 1.337
#> cowork NA NA NA NA NA NA NA 2.419 1.400
#> advice NA NA NA NA NA NA NA NA 2.781
#> friend NA NA NA NA NA NA NA NA NA
#> friend
#> status NA
#> gender 1.270
#> office 1.334
#> years 0.684
#> age 1.058
#> practice 1.427
#> lawschool 1.350
#> cowork 1.411
#> advice 1.407
#> friend 3.408For better readability, the powers of different predictors can be
conveniently compared by using prediction plots that display a color
matrix with rows for $X$ and columns for $Y$ with darker colors in the
cells when we have higher prediction power for $Z$. This is shown for
the prediction of status:
Obviously, the darkest color is obtained when the variable to be
predicted is included among the predictors, and the cells exhibit
prediction power for a single predictor on the diagonal and for two
predictors symmetrically outside the diagonal. Some findings are as
follows: good predictors for status are given by years in
combination with any other variable, and age in combination with any
other variable. The best sole predictor is gender.
More details and examples of expected conditional entropies and prediction power are given in the vignette “prediction power based on expected conditional entropies”.
Divergence Tests of Goodness of Fit
Occurring cliques in association graphs represent connected components
of dependent variables, and by comparing the graphs for different
thresholds, specific structural models of multivariate dependence can be
suggested and tested. The function div_gof() allows such hypothesis
tests for pairwise independence of $X$ and $Y$: $X \bot Y$, and pairwise
independence conditional a third variable $Z$: $X\bot Y|Z$.
To test friend$\bot$ cowork$|$advice, that is whether dyad
variable friend is independent of cowork given advice we use the
function as shown below:
div_gof(dat = dyad.var, var1 = "friend", var2 = "cowork", var_cond = "advice")
#> the specified model of conditional independence cannot be rejected
#> D df(D)
#> 1 0.94 12Not specifying argument var_cond would instead test
friend$\bot$cowork without any conditioning.
References
Parts of the theoretical background is provided in the package vignettes, but for more details, consult the following literature:
Frank, O., & Shafie, T. (2016). Multivariate entropy analysis of network data. Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique, 129(1), 45-63. link