Vignettes of the influential

Overview

The influential package contains several functions that could be categorized into four groups according to their purpose:

• Network reconstruction
• Calculation of centrality measures
• Assessment of the association of centrality measures
• Identification of the most influential network nodes (hubs)

library(influential)

Network reconstruction

Two functions have been obtained from the igraph package for the reconstruction of networks.

• From a data frame

  In the data frame the first and second columns should be composed of source and target nodes.

  A sample appropriate data frame is brought below:

This is a co-expression dataset obtained from PMID: 31211495.

MyData <- coexpression.data        # Preparing the data

My_graph <- graph_from_data_frame(d=MyData)        # Reconstructing the graph

If you look at the class of My_graph you should see that it has an igraph class:

class(My_graph)
#> [1] "igraph"

• From an adjacency matrix

A sample appropriate data frame is brought below:

MyData <- coexpression.adjacency        # Preparing the data

My_graph <- graph_from_adjacency_matrix(MyData)        # Reconstructing the graph

• Network vertices

Network vertices (nodes) are required in order to calculate their centrality measures. Thus, before calculation of network centrality measures we need to obtain the name of required network vertices. To this end, we use the V function, which is obtained from the igraph package.

MyData <- coexpression.data        # Preparing the data

My_graph <- graph_from_data_frame(MyData)        # Reconstructing the graph

My_graph_vertices <- V(My_graph)        # Extracting the vertices

Calculation of centrality measures

• Degree centrality

  Degree centrality is the most commonly used local centrality measure which could be calculated via the degree function obtained from the igraph package.

MyData <- coexpression.data        # Preparing the data

My_graph <- graph_from_data_frame(MyData)        # Reconstructing the graph

GraphVertices <- V(My_graph)        # Extracting the vertices

My_graph_degree <- degree(My_graph, v = GraphVertices, normalized = FALSE) # Calculating degree centrality

Degree centrality could be also calculated for directed graphs via specifying the mode parameter.

• Betweenness centrality

  Betweenness centrality, like degree centrality, is one of the most commonly used centrality measures but is representative of the global centrality of a node. This centrality metric could also be calculated using a function obtained from the igraph package.

MyData <- coexpression.data        # Preparing the data

My_graph <- graph_from_data_frame(MyData)        # Reconstructing the graph

GraphVertices <- V(My_graph)        # Extracting the vertices

My_graph_betweenness <- betweenness(My_graph, v = GraphVertices,    # Calculating betweenness centrality
                                    directed = FALSE, normalized = FALSE)

Degree centrality could be also calculated for directed and/or weighted graphs via specifying the directed and weights parameters, respectively.

• Neighborhood connectivity

Neighborhood connectivity is one of the other important centrality measures that reflect the (semi-) local centrality of a node. This centrality measure was first represented in a Science paper in 2002 and is for the first time calculable in R environment via the influential package.

MyData <- coexpression.data        # Preparing the data

My_graph <- graph_from_data_frame(MyData)        # Reconstructing the graph

GraphVertices <- V(My_graph)        # Extracting the vertices

neighrhood.co <- neighborhood.connectivity(graph = My_graph,    # Calculating neighborhood connectivity
                                           vertices = GraphVertices,
                                           mode = "all")

Neighborhood connectivity could be also calculated for directed graphs via specifying the mode parameter.

Assessment of the association of centrality measures

• Conditional probability of deviation from means

The function cond.prob.analysis assesses the conditional probability of deviation of two centrality measures (or any other two continuous variables) from their corresponding means in opposite directions.

MyData <- centrality.measures        # Preparing the data

My.conditional.prob <- cond.prob.analysis(data = MyData,       # Assessing the conditional probability
                                          nodes.colname = "name",
                                          Desired.colname = "BetweennessCentrality",
                                          Condition.colname = "NeighborhoodConnectivity")

print(My.conditional.prob)
#> $ConditionalProbability
#> [1] 55.35386
#> 
#> $ConditionalProbability_split.half.sample
#> [1] 55.16751

• As you can see in the results, the whole data is also randomly splitted into half in order to further test the validity of conditional probability assessments.
• The higher the conditional probability the more two centrality measures behave in contrary manners.

• Nature of association (considering dependent and independent)

The function double.cent.assess could be used to automatically assess both the distribution mode of centrality measures (two continuous variables) and the nature of their association. The analyses done through this formula are as follows:

1. Normality assessment:
   • Variables with lower than 5000 observations: Shapiro-Wilk test
   • Variables with over 5000 observations: Anderson-Darling test
     
     
2. Assessment of non-linear/non-monotonic correlation:
   • Non-linearity assessment: Fitting a generalized additive model (GAM) with integrated smoothness approximations using the mgcv package
     
     
   • Non-monotonicity assessment: Comparing the squared coefficients of the correlation based on Spearman’s rank correlation analysis and ranked regression test with non-linear splines.
     • Squared coefficient of Spearman’s rank correlation > R-squared ranked regression with non-linear splines: Monotonic
     • Squared coefficient of Spearman’s rank correlation < R-squared ranked regression with non-linear splines: Non-monotonic
       
       
3. Dependence assessment:
   • Hoeffding’s independence test: Hoeffding’s test of independence is a test based on the population measure of deviation from independence which computes a D Statistics ranging from -0.5 to 1: Greater D values indicate a higher dependence between variables.
   • Descriptive non-linear non-parametric dependence test: This assessment is based on non-linear non-parametric statistics (NNS) which outputs a dependence value ranging from 0 to 1. For further details please refer to NNS: Nonlinear Nonparametric Statistics: Greater values indicate a higher dependence between variables.
     
     
4. Correlation assessment: As the correlation between most of the centrality measures follows a non-monotonic form, this part of the assessment is done based on the non-linear non-parametric statistics (NNS) which itself calculates the correlation based on partial moments and outputs a correlation value ranging from -1 to 1. For further details please refer to NNS: Nonlinear Nonparametric Statistics.
   
   
5. Assessment of conditional probability of deviation from means This step assesses the conditional probability of deviation of two centrality measures (or any other two continuous variables) from their corresponding means in opposite directions.
   • The independent centrality measure (variable) is considered as the condition variable and the other as the desired one.
   • As you will see in the results, the whole data is also randomly splitted into half in order to further test the validity of conditional probability assessments.
   • The higher the conditional probability the more two centrality measures behave in contrary manners.

MyData <- centrality.measures        # Preparing the data

My.metrics.assessment <- double.cent.assess(data = MyData,       # Association assessment
                                            nodes.colname = "name",
                                            dependent.colname = "BetweennessCentrality",
                                            independent.colname = "NeighborhoodConnectivity")

print(My.metrics.assessment)
#> $Summary_statistics
#>         BetweennessCentrality NeighborhoodConnectivity
#> Min.              0.000000000                   1.2000
#> 1st Qu.           0.000000000                  66.0000
#> Median            0.000000000                 156.0000
#> Mean              0.005813357                 132.3443
#> 3rd Qu.           0.000340000                 179.3214
#> Max.              0.529464720                 192.0000
#> 
#> $Normality_results
#>                               p.value
#> BetweennessCentrality    1.415450e-50
#> NeighborhoodConnectivity 9.411737e-30
#> 
#> $Dependent_Normality
#> [1] "Non-normally distributed"
#> 
#> $Independent_Normality
#> [1] "Non-normally distributed"
#> 
#> $GAM_nonlinear.nonmonotonic.results
#>      edf  p-value 
#> 8.992406 0.000000 
#> 
#> $Association_type
#> [1] "nonlinear-nonmonotonic"
#> 
#> $HoeffdingD_Statistic
#>         D_statistic P_value
#> Results  0.01770279   1e-08
#> 
#> $Dependence_Significance
#>                       Hoeffding
#> Results Significantly dependent
#> 
#> $NNS_dep_results
#>         Correlation Dependence
#> Results  -0.7948106  0.8647164
#> 
#> $ConditionalProbability
#> [1] 55.35386
#> 
#> $ConditionalProbability_split.half.sample
#> [1] 55.90331

Note: It should also be noted that as a single regression line does not fit all models with a certain degree of freedom, based on the size and correlation mode of the variables provided, this function might return an error due to incapability of running step 2. In this case, you may follow each step manually or as an alternative run the other function named double.cent.assess.noRegression which does not perform any regression test and consequently it is not required to determine the dependent and independent variables.

• Nature of association (without considering dependence direction)

The function double.cent.assess.noRegression could be used to automatically assess both the distribution mode of centrality measures (two continuous variables) and the nature of their association. The analyses done through this formula are as follows:

1. Normality assessment:
   • Variables with lower than 5000 observations: Shapiro-Wilk test
   • Variables with over 5000 observations: Anderson–Darling test
     
     
2. Dependence assessment:
   • Hoeffding’s independence test: Hoeffding’s test of independence is a test based on the population measure of deviation from independence which computes a D Statistics ranging from -0.5 to 1: Greater D values indicate a higher dependence between variables.
   • Descriptive non-linear non-parametric dependence test: This assessment is based on non-linear non-parametric statistics (NNS) which outputs a dependence value ranging from 0 to 1. For further details please refer to NNS: Nonlinear Nonparametric Statistics: Greater values indicate a higher dependence between variables.
     
     
3. Correlation assessment: As the correlation between most of the centrality measures follows a non-monotonic form, this part of the assessment is done based on the non-linear non-parametric statistics (NNS) which itself calculates the correlation based on partial moments and outputs a correlation value ranging from -1 to 1. For further details please refer to NNS: Nonlinear Nonparametric Statistics.
   
   
4. Assessment of conditional probability of deviation from means This step assesses the conditional probability of deviation of two centrality measures (or any other two continuous variables) from their corresponding means in opposite directions.
   • The centrality2 variable is considered as the condition variable and the other (centrality1) as the desired one.
   • As you will see in the results, the whole data is also randomly splitted into half in order to further test the validity of conditional probability assessments.
   • The higher the conditional probability the more two centrality measures behave in contrary manners.

MyData <- centrality.measures        # Preparing the data

My.metrics.assessment <- double.cent.assess.noRegression(data = MyData,       # Association assessment
                                                         nodes.colname = "name",
                                                         centrality1.colname = "BetweennessCentrality",
                                                         centrality2.colname = "NeighborhoodConnectivity")

print(My.metrics.assessment)
#> $Summary_statistics
#>         BetweennessCentrality NeighborhoodConnectivity
#> Min.              0.000000000                   1.2000
#> 1st Qu.           0.000000000                  66.0000
#> Median            0.000000000                 156.0000
#> Mean              0.005813357                 132.3443
#> 3rd Qu.           0.000340000                 179.3214
#> Max.              0.529464720                 192.0000
#> 
#> $Normality_results
#>                               p.value
#> BetweennessCentrality    1.415450e-50
#> NeighborhoodConnectivity 9.411737e-30
#> 
#> $Centrality1_Normality
#> [1] "Non-normally distributed"
#> 
#> $Centrality2_Normality
#> [1] "Non-normally distributed"
#> 
#> $HoeffdingD_Statistic
#>         D_statistic P_value
#> Results  0.01770279   1e-08
#> 
#> $Dependence_Significance
#>                       Hoeffding
#> Results Significantly dependent
#> 
#> $NNS_dep_results
#>         Correlation Dependence
#> Results  -0.7948106  0.8647164
#> 
#> $ConditionalProbability
#> [1] 55.35386
#> 
#> $ConditionalProbability_split.half.sample
#> [1] 55.68163

Identification of the most influential network nodes (hubs)

IHS : IHS is the first integrative method for the identification of network hubs. The IHS formula integrates three network centrality measure including degree centrality, betweenness centrality, and neighborhood connectivity in such a way that both synergize their effects and remove their biases.

MyData <- centrality.measures        # Preparing the data

My.vertices.IHS <- ihs(DC = centrality.measures$Degree,       # Calculation of IHS
                       BC = centrality.measures$BetweennessCentrality,
                       NC = centrality.measures$NeighborhoodConnectivity)

print(head(My.vertices.IHS))
#> [1] 196.2039215   2.9822273   0.1572078   6.1757221   0.3199993   0.5095222