2.1 Undirected & unweighted network

Zachary Karate Club(Zachary 1977) is an example of undirected and unweighted network in this package. This dataset illustrates friendships between members of a university karate club. It is based on a faction membership after a social partion. The summary of important properties of this network is described below:

Edge Type: Friendship

Node Type: Person

Avg Edges: 77.50

Avg Nodes: 34.00

Graph properties: Unweighted, Undirected

This dataset can be easily accessed by using data() function:

data("zachary")
zachary

## IGRAPH 455c916 U--- 34 78 -- 
## + attr: id (v/n)
## + edges from 455c916:
##  [1]  1-- 2  1-- 3  2-- 3  1-- 4  2-- 4  3-- 4  1-- 5  1-- 6  1-- 7  5-- 7
## [11]  6-- 7  1-- 8  2-- 8  3-- 8  4-- 8  1-- 9  3-- 9  3--10  1--11  5--11
## [21]  6--11  1--12  1--13  4--13  1--14  2--14  3--14  4--14  6--17  7--17
## [31]  1--18  2--18  1--20  2--20  1--22  2--22 24--26 25--26  3--28 24--28
## [41] 25--28  3--29 24--30 27--30  2--31  9--31  1--32 25--32 26--32 29--32
## [51]  3--33  9--33 15--33 16--33 19--33 21--33 23--33 24--33 30--33 31--33
## [61] 32--33  9--34 10--34 14--34 15--34 16--34 19--34 20--34 21--34 23--34
## [71] 24--34 27--34 28--34 29--34 30--34 31--34 32--34 33--34

The result would have a class of “igraph” object.

2.2 Undirected & weighted network

Kangaroo(KONECT 2016a) is a sample of undirected and weighted network which indicates interactions among free-ranging grey kangaroos. The edge between two nodes shows a dominence interaction between two kanagaroos. The positive weight of each edge represents number of interaction between them. A brief explanation of it’s properties is clarified below:

Edge Type: Interaction

Node Type: Kangaroo

Avg Edges: 91

Nodes: 17

Graph properties: Weighted, Undirected

Edge weights: Positive weights

2.3 Directed & unweighted network

Macaque visual cortex(Felleman and Van Essen 1991) is a sample cortex network which is collected in 1991. In this dataset, vertices represents neocortical areas which involved in visual functions in Macaques. The direction displays the progress of synapsis from one to another. A summary of this can be as follows:

Edge Type: Pathway

Node Type: Cortical region

Avg Edges: 315.50

Nodes: 31.00

Graph properties: Directed, Unweighted

Edge weights: Positive weights

2.4 Directed & weighted network

Moreno rhesus(KONECT 2016b) is a directed and weighted network which describes grooming between free ranging rhesus macaques (Macaca mulatta) in Cayo Santiago during a two month period in 1963. In this dataset a vertice is identified as a monkey and the directed edge among them means grooming between them. The weights of the edges demonstrates how often this manner happened. The network summary is as follows:

Edge Type: Grooming

Node Type: Monkey

Avg Edges: 111

Nodes: 16

Graph properties: Directed, Weighted

Edge weights: Positive weights

2.5 Bipartite & directed network

Drugtarget(Barneh, Jafari, and Mirzaie 2015) is a bipartite, unconnected and directed network demonstrating interactions among Food and Drug Administration (FDA)-approved drugs and their corresponding protein targets. This network is a shrunken one in which metabolizing enzymes, carriers and transporters associated with drug metabolism are filtered and solely targets directly related to their pharmacological effects are included. A summary of this can be like:

Edge Type: inetraction

Node Type: drug, protein target

Avg Edges: 3766

Nodes: 1599

Graph properties: Bipartite, unconnected, directed

3.1 The segregation of “igraph” and “network” objects

“graph.extract.components” function is able to read “igraph” and “network” objects and returns their componets as a list of “igraph” objects. This function also has this ability to recognized bipartite graphs and user can decide that which project is suitable for his analysis. In order to use this function, we use zachary dataset and develop it in all of our functions.

graph_extract_components(zachary)

## [[1]]
## IGRAPH 1f8476d U--- 34 78 -- 
## + attr: id (v/n)
## + edges from 1f8476d:
##  [1]  1-- 2  1-- 3  2-- 3  1-- 4  2-- 4  3-- 4  1-- 5  1-- 6  1-- 7  5-- 7
## [11]  6-- 7  1-- 8  2-- 8  3-- 8  4-- 8  1-- 9  3-- 9  3--10  1--11  5--11
## [21]  6--11  1--12  1--13  4--13  1--14  2--14  3--14  4--14  6--17  7--17
## [31]  1--18  2--18  1--20  2--20  1--22  2--22 24--26 25--26  3--28 24--28
## [41] 25--28  3--29 24--30 27--30  2--31  9--31  1--32 25--32 26--32 29--32
## [51]  3--33  9--33 15--33 16--33 19--33 21--33 23--33 24--33 30--33 31--33
## [61] 32--33  9--34 10--34 14--34 15--34 16--34 19--34 20--34 21--34 23--34
## [71] 24--34 27--34 28--34 29--34 30--34 31--34 32--34 33--34

This results the only component of the zachary graph. This function is laso applicable for bipartite networks. Using the num.proj argument, user can decide on which projection is inetrested to work on. As an example of bipartite graphs, we uses drugtarget network as follows:

data(drugtarget)

drug_comp <- graph_extract_components( drugtarget, directed = TRUE , bipartite_proj=TRUE, num_proj=2)

## Warning in bipartite_projection(x): vertex types converted to logical

head(drug_comp)

## [[1]]
## IGRAPH 1fced2f UNW- 131 560 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edges from 1fced2f (vertex names):
##  [1] O00555--O15399 O15399--O60391 O43497--P00915 P00915--P00918
##  [5] O15554--P03886 P03372--P04278 P00915--P05023 P00918--P05023
##  [9] P03372--P06401 P07550--P08172 P08172--P08173 P07550--P08588
## [13] P08172--P08588 P08588--P08908 P08172--P08912 P08173--P08912
## [17] P07550--P08913 P08588--P09172 P03372--P10275 P04278--P10275
## [21] P07550--P11229 P08172--P11229 P08173--P11229 P08588--P11229
## [25] P08912--P11229 O00555--P14416 O43497--P14416 P08172--P14416
## [29] P08173--P14416 P08588--P14416 P08908--P14416 P08912--P14416
## + ... omitted several edges
## 
## [[2]]
## IGRAPH 1fced2f UNW- 6 15 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edges from 1fced2f (vertex names):
##  [1] O00764--P34896 O00764--P35520 P34896--P35520 O00764--Q96GD0
##  [5] P34896--Q96GD0 P35520--Q96GD0 O00764--Q9NVS9 P34896--Q9NVS9
##  [9] P35520--Q9NVS9 Q96GD0--Q9NVS9 O00764--Q9Y617 P34896--Q9Y617
## [13] P35520--Q9Y617 Q96GD0--Q9Y617 Q9NVS9--Q9Y617
## 
## [[3]]
## IGRAPH 1fcefa1 UNW- 2 1 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edge from 1fcefa1 (vertex names):
## [1] O14646--P11388
## 
## [[4]]
## IGRAPH 1fcefa1 UNW- 7 10 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edges from 1fcefa1 (vertex names):
##  [1] O14659--P00374 O14659--P04818 P00374--P04818 P04818--P23921
##  [5] P09884--P23921 P04818--P27707 P23921--P27707 P04818--P30085
##  [9] P23921--P30085 P27707--P30085
## 
## [[5]]
## IGRAPH 1fcf212 UNW- 2 1 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edge from 1fcf212 (vertex names):
## [1] O14987--P21731
## 
## [[6]]
## IGRAPH 1fcf212 UNW- 4 5 -- 
## + attr: id (v/n), name (v/c), weight (e/n)
## + edges from 1fcf212 (vertex names):
## [1] O15528--P11473 P11473--Q02318 O15528--Q6VVX0 P11473--Q6VVX0
## [5] Q02318--Q6VVX0

It will return all components of the second projection of the network.

3.2 The segregation of other graph formats

If you had an edgelist, an adjacency matrix or a graphNEL format of a network, the “misc.extract.components” can be useful. This function extracts the componets of other formats of graph. For illustration, we convert “zachary” graph to an edgelist to be able to use it for this function.

library(igraph)
zachary_edgelist <- as_edgelist(zachary)

misc_extract_components(zachary_edgelist)

## [[1]]
## IGRAPH 1fe4cf9 D--- 34 78 -- 
## + edges from 1fe4cf9:
##  [1]  1-> 2  1-> 3  2-> 3  1-> 4  2-> 4  3-> 4  1-> 5  1-> 6  1-> 7  5-> 7
## [11]  6-> 7  1-> 8  2-> 8  3-> 8  4-> 8  1-> 9  3-> 9  3->10  1->11  5->11
## [21]  6->11  1->12  1->13  4->13  1->14  2->14  3->14  4->14  6->17  7->17
## [31]  1->18  2->18  1->20  2->20  1->22  2->22 24->26 25->26  3->28 24->28
## [41] 25->28  3->29 24->30 27->30  2->31  9->31  1->32 25->32 26->32 29->32
## [51]  3->33  9->33 15->33 16->33 19->33 21->33 23->33 24->33 30->33 31->33
## [61] 32->33  9->34 10->34 14->34 15->34 16->34 19->34 20->34 21->34 23->34
## [71] 24->34 27->34 28->34 29->34 30->34 31->34 32->34 33->34

3.3 Giant component extraction

In the most of research topics of network analysis, network features are related to the largest connected component of a graph(Newman 2010). In order to get that for an “igraph” or a “network” object, “giant.component.extract” function is specified. For using this function we can do:

giant_component_extract(zachary)

## [[1]]
## IGRAPH 1fe7dd8 U--- 34 78 -- 
## + attr: id (v/n)
## + edges from 1fe7dd8:
##  [1]  1-- 2  1-- 3  2-- 3  1-- 4  2-- 4  3-- 4  1-- 5  1-- 6  1-- 7  5-- 7
## [11]  6-- 7  1-- 8  2-- 8  3-- 8  4-- 8  1-- 9  3-- 9  3--10  1--11  5--11
## [21]  6--11  1--12  1--13  4--13  1--14  2--14  3--14  4--14  6--17  7--17
## [31]  1--18  2--18  1--20  2--20  1--22  2--22 24--26 25--26  3--28 24--28
## [41] 25--28  3--29 24--30 27--30  2--31  9--31  1--32 25--32 26--32 29--32
## [51]  3--33  9--33 15--33 16--33 19--33 21--33 23--33 24--33 30--33 31--33
## [61] 32--33  9--34 10--34 14--34 15--34 16--34 19--34 20--34 21--34 23--34
## [71] 24--34 27--34 28--34 29--34 30--34 31--34 32--34 33--34
## 
## [[2]]
##       [,1] [,2]
##  [1,]    1    2
##  [2,]    1    3
##  [3,]    2    3
##  [4,]    1    4
##  [5,]    2    4
##  [6,]    3    4
##  [7,]    1    5
##  [8,]    1    6
##  [9,]    1    7
## [10,]    5    7
## [11,]    6    7
## [12,]    1    8
## [13,]    2    8
## [14,]    3    8
## [15,]    4    8
## [16,]    1    9
## [17,]    3    9
## [18,]    3   10
## [19,]    1   11
## [20,]    5   11
## [21,]    6   11
## [22,]    1   12
## [23,]    1   13
## [24,]    4   13
## [25,]    1   14
## [26,]    2   14
## [27,]    3   14
## [28,]    4   14
## [29,]    6   17
## [30,]    7   17
## [31,]    1   18
## [32,]    2   18
## [33,]    1   20
## [34,]    2   20
## [35,]    1   22
## [36,]    2   22
## [37,]   24   26
## [38,]   25   26
## [39,]    3   28
## [40,]   24   28
## [41,]   25   28
## [42,]    3   29
## [43,]   24   30
## [44,]   27   30
## [45,]    2   31
## [46,]    9   31
## [47,]    1   32
## [48,]   25   32
## [49,]   26   32
## [50,]   29   32
## [51,]    3   33
## [52,]    9   33
## [53,]   15   33
## [54,]   16   33
## [55,]   19   33
## [56,]   21   33
## [57,]   23   33
## [58,]   24   33
## [59,]   30   33
## [60,]   31   33
## [61,]   32   33
## [62,]    9   34
## [63,]   10   34
## [64,]   14   34
## [65,]   15   34
## [66,]   16   34
## [67,]   19   34
## [68,]   20   34
## [69,]   21   34
## [70,]   23   34
## [71,]   24   34
## [72,]   27   34
## [73,]   28   34
## [74,]   29   34
## [75,]   30   34
## [76,]   31   34
## [77,]   32   34
## [78,]   33   34

This function extracts the strongests components of the input network as “igraph” objects.

4.1 Proper centrality suggestions

All of the introduced centrality measures are not appropriate for all types of networks. So, to figure out which of them is suitable, “proper.centralities” is specifid. This funtion distinguishes proper centrality types based on network topology. To use this, we can do:

proper_centralities(zachary)

##  [1] "subgraph centrality scores"                      
##  [2] "Topological Coefficient"                         
##  [3] "Average Distance"                                
##  [4] "Barycenter Centrality"                           
##  [5] "BottleNeck Centrality"                           
##  [6] "Centroid value"                                  
##  [7] "Closeness Centrality (Freeman)"                  
##  [8] "ClusterRank"                                     
##  [9] "Decay Centrality"                                
## [10] "Degree Centrality"                               
## [11] "Diffusion Degree"                                
## [12] "DMNC - Density of Maximum Neighborhood Component"
## [13] "Eccentricity Centrality"                         
## [14] "eigenvector centralities"                        
## [15] "K-core Decomposition"                            
## [16] "Geodesic K-Path Centrality"                      
## [17] "Katz Centrality (Katz Status Index)"             
## [18] "Kleinberg's authority centrality scores"         
## [19] "Kleinberg's hub centrality scores"               
## [20] "clustering coefficient"                          
## [21] "Lin Centrality"                                  
## [22] "Lobby Index (Centrality)"                        
## [23] "Markov Centrality"                               
## [24] "Radiality Centrality"                            
## [25] "Shortest-Paths Betweenness Centrality"           
## [26] "Current-Flow Closeness Centrality"               
## [27] "Closeness centrality (Latora)"                   
## [28] "Communicability Betweenness Centrality"          
## [29] "Community Centrality"                            
## [30] "Cross-Clique Connectivity"                       
## [31] "Entropy Centrality"                              
## [32] "EPC - Edge Percolated Component"                 
## [33] "Laplacian Centrality"                            
## [34] "Leverage Centrality"                             
## [35] "MNC - Maximum Neighborhood Component"            
## [36] "Hubbell Index"                                   
## [37] "Semi Local Centrality"                           
## [38] "Closeness Vitality"                              
## [39] "Residual Closeness Centrality"                   
## [40] "Stress Centrality"                               
## [41] "Load Centrality"                                 
## [42] "Flow Betweenness Centrality"                     
## [43] "Information Centrality"

It returns the full names of suitable centrality types for the input graph. The input must have a class of “igraph” object.

4.2 Various types of centralities computation

In the next step, proper centralities and those which are looking for can be chosen. In order to compute proper centrality types resulted from the “proper.centralities”, you can use “calculate.centralities” function as below.

calculate_centralities(zachary, include = "Degree Centrality")

## $`Degree Centrality`
##  [1] 16  9 10  6  3  4  4  4  5  2  3  1  2  5  2  2  2  2  2  3  2  2  2
## [24]  5  3  3  2  4  3  4  4  6 12 17

In this function, you have the ability to specify some centrality types that is not your favor to calculate by the “except” argument:

pr_cent <- proper_centralities(zachary)

##  [1] "subgraph centrality scores"                      
##  [2] "Topological Coefficient"                         
##  [3] "Average Distance"                                
##  [4] "Barycenter Centrality"                           
##  [5] "BottleNeck Centrality"                           
##  [6] "Centroid value"                                  
##  [7] "Closeness Centrality (Freeman)"                  
##  [8] "ClusterRank"                                     
##  [9] "Decay Centrality"                                
## [10] "Degree Centrality"                               
## [11] "Diffusion Degree"                                
## [12] "DMNC - Density of Maximum Neighborhood Component"
## [13] "Eccentricity Centrality"                         
## [14] "eigenvector centralities"                        
## [15] "K-core Decomposition"                            
## [16] "Geodesic K-Path Centrality"                      
## [17] "Katz Centrality (Katz Status Index)"             
## [18] "Kleinberg's authority centrality scores"         
## [19] "Kleinberg's hub centrality scores"               
## [20] "clustering coefficient"                          
## [21] "Lin Centrality"                                  
## [22] "Lobby Index (Centrality)"                        
## [23] "Markov Centrality"                               
## [24] "Radiality Centrality"                            
## [25] "Shortest-Paths Betweenness Centrality"           
## [26] "Current-Flow Closeness Centrality"               
## [27] "Closeness centrality (Latora)"                   
## [28] "Communicability Betweenness Centrality"          
## [29] "Community Centrality"                            
## [30] "Cross-Clique Connectivity"                       
## [31] "Entropy Centrality"                              
## [32] "EPC - Edge Percolated Component"                 
## [33] "Laplacian Centrality"                            
## [34] "Leverage Centrality"                             
## [35] "MNC - Maximum Neighborhood Component"            
## [36] "Hubbell Index"                                   
## [37] "Semi Local Centrality"                           
## [38] "Closeness Vitality"                              
## [39] "Residual Closeness Centrality"                   
## [40] "Stress Centrality"                               
## [41] "Load Centrality"                                 
## [42] "Flow Betweenness Centrality"                     
## [43] "Information Centrality"

calc_cent <- calculate_centralities(zachary, except  = pr_cent[5:40])

The result would be a list of computed centralities.

4.3 Recognition of most informative measures

In order to figure out the order of most important centrality types based on your graph structure, “pca.centralities” function can be used. This applies principal component analysis on the computed centrality values(Husson, Lê, and Pages 2010). For this, the result of “calculate.centralities” method is needed:

pca_centralities( calc_cent )

A display of most informative centrality measures based on principal component analysis. The red line indicates the random threshold of contribution. This barplot represents contribution of variable values based on the number of dimensions.

For choosing the number of principal components, we considered cumulative percentage of variance values which are more than 80 as the cut off which can be edited using “cut.off” argument. It returns a plot for visualizing contribution values of the computed centrality measures due to the number of principal components. The “scale.unit” argument gives the ability to whether it should normalize the input or not.

pca_centralities( calc_cent , scale.unit = FALSE )

A representation of most informative centrality measures based on principal component analysis between unscaled(not normalized) centrality values.

Another method for distinguishing which centrality measure has more information or in another words has more costs is using (t-SNE) t-Distributed Stochastic Neighbor Embedding analysis(Van Der Maaten 2014). This is a non-linear dimensionality reduction algorithm used for high-dimensional data. “tsne.centralities”function applies t-sne on centrality measure values like below:

tsne_centralities( calc_cent, dims = 2, perplexity = 1, scale=TRUE)

A display of most informative centrality measures based on t-Distributed Stochastic Neighbor Embedding analysis among scaled(not normalized) centrality values.

This returns the bar plot of computed cost values of each centrality measure on a plot. In order to access only computed values of PCA and t-sne methods, “summary.pca_centralities” and “tsne.centralities” functions can be helpful.

5 Comparison visualization of centrality measures

To visualize the results of network centrality analysis some convenient functions have been developed as it described below.

5.1 Graph visualization based on the type of centralities

After evaluating centrality measures, demonstrating high values of centralities in some nodes gives an overall insight about the network to the researcher. By using “visualize.graph” function, you will be able to illustrate the input graph based on the specified centrality value. If the centrality measure values were computed, “computed.centrality.value” argument is recommended. Otherwise, using “centrality.type” argument, the function will compute centrality based on the input name of centrality type. For practice, we specifies “Degree Centrality”. Here,

visualize_graph( zachary , centrality.type="Degree Centrality")

Graph illustration based on centrality measure. The size of nodes represent the degree centrality values.

5.2 Heatmap of centrality measure values

On of the way of complex large network visualizations(more than 100 nodes and 200 edges) is using heatmap(Pryke, Mostaghim, and Nazemi 2007). “visualize.heatmap” function demonstrates a heatmap plot between the centrality values. The input is a list containing the computed values.

visualize_heatmap( calc_cent , scale = TRUE  )

Observed centrality measure heatmap. The colors from blue to red displays scaled centrality values.

5.3 Correlation between computed centrality measures

Comprehending pair correlation among centralities is a popular analysis for researchers(Dwyer et al. 2006). In order to that, “visualize.correlations” method is appropriate. In this you are able to specify the type of correlation which you are enthusiastic to obtain.

visualize_correlations(calc_cent,"pearson")

A display of correlation among computed centrality measures. The red to blue highlighted circles represent the top to bottom Pearson correlation coefficients(Benesty et al. 2009) which differ from -1 to 1. The higher the value becomes larger, circles’ sizes get larger too.

5.4 Node dendrogram based on a centrality type

In order to visualize a simple clustering across the nodes of a graph based on a specific centrality measure, we can use the “visualize.dendogram” function. This function draw a dendrogram plot in which colors indicate the clusters.

visualize_dendrogram(zachary, k=4)

Circular dendrogram plot of vertices based on specified centrality measure. Each color represents a cluster.

5.5 Regression across centrality measures

In this package additionally to correlation calculation, ability to apply linear regression for each pair of centralities has been prepared to realize the association between centralities. For visualization, “visualize.association” method is an approprite function to use:

subgraph_cent <- calc_cent[[1]]
Topological_coef <- calc_cent[[2]]

visualize_association(  subgraph_cent , Topological_coef)

## $linear.regression
## 
## Call:
## lm(formula = df[, 2] ~ df[, 1])
## 
## Coefficients:
## (Intercept)      df[, 1]  
##   3.898e-16   -7.059e-01  
## 
## 
## $visualization

## Warning: Computation failed in `stat_summary()`:
## Hmisc package required for this function

Association plot between two centrality variables. The red line is an indicator of linear regression line among them.

5.6 Pairwise correlation between centrality types

To access the distribution of centrality values and their corresponding pair correlation value, “visualize.pair.correlation” would be helpful. The Pearson correlation(Benesty et al. 2009) has been used for this method.

visualize_pair_correlation( subgraph_cent , Topological_coef)

Pairwise Pearson correlation between two centrality values.

The result is a scatterplot visualizing correlation values.