library(ideanet)

head(florentine_nodes)head(florentine_edges)

nw_flor<-netwrite(nodelist = florentine_nodes,node_id ="id",i_elements = florentine_edges$source,j_elements = florentine_edges$target,type = florentine_edges$type,directed =FALSE,net_name ="florentine")

flor_cluster<-role_analysis(method ="cluster",graph = nw_flor$igraph_list,nodes = nw_flor$node_measures,directed =FALSE,min_partitions =2,max_partitions =7,viz =TRUE,cluster_summaries =TRUE,fast_triad =TRUE)

Cluster Memberships

Depending on the amount of partitioning applied during clustering,individual nodes may vary in terms of cluster membership. Users caninspect cluster membership of individual nodes at each level ofpartitioning using thecluster_assignments object:

head(flor_cluster$cluster_assignments)

Hereid contains each node’s simplified identifier as itappears in thenode_measures dataframe produced bynetwrite. Columns beginning with thecut_prefix indicate a specific level of partitioning. In most cases, we areinterested in finding a single solution that best categorizes nodes intodifferent types (“roles”) according to their relational characteristics.role_analysis determines the optimal level of partitioningby taking the distance matrix used in the clustering process andconverting it into a similarity matrix. This similarity matrix is thentreated as a dense network whose modularity varies according to themembership of nodes within derived clusters. Finally,role_analysis designates the level of partitioning whosecluster assignments produce the highest modularity score as the bestfit. In effect, this converts a multirelational role problem into asingle-relation community detection problem in a dense network.

Cluster assignments at this identified optimal level are stored inthemax_mod column, and values in this column are generallythose that users will want to use. However, if users require clusters tohave a minimum size as specified by themin_partition_sizeargument, they will want smaller clusters identified inmax_mod to be subsumed into a parent cluster. When this isthe case, thebest_fit column will contain the closestcompromise betweenmax_mod and the user’sspecifications.

Cluster Dendrogram

To determine the number of clusters produced at the optimal level ofpartitioning, you can simply identify the maximum value contained inmax_mod. However,role_analysis generates twodiagnostic visualizations that provide a faster way of interpretingclustering output. Thecluster_dendrogram visualizationillustrates the cluster membership of nodes at each level ofpartitioning while also indicating membership of nodes at the optimalpartitioning level:

flor_cluster$cluster_dendrogram

Modularity Plot

Whilecluster_dendrogram shows where nodes fall at eachlevel of partitioning,cluster_modularity shows how themodularity score of the similarity matrix changes at each level ofpartitioning:

flor_cluster$cluster_modularity

Note: this plot may not appear in R Markdown documents, but willappear in a plot window if called in the R console.

Looking at this plot and the dendrogram together, we see that nodesin the network have been assigned to one of seven different clusters(including one isolate node; isolates are assigned their own cluster inour approach), and that this partitioning produces the best fit asdetermined by modularity score. We also see that while most clusterscontain about 2-4 nodes, node 8 appears to be unique enough in itsrelational position to constitute its own cluster.

Cluster Summaries

We now know that nodes in this network fall into one of sevenpositions or “roles.” A proper understanding of these results requiresmore, however. If clusters are supposed to represent different kinds ofroles that nodes occupy in the network, we’ll want to knowwhycertain nodes are placed in one cluster over another and how theseclusters differ from one another. Thecluster_summariesdataframe provides a numerical overview of differences between inferredclusters, allowing us to make progress to this end.

flor_cluster$cluster_summaries

cluster_summaries provides both crude and standardizedaverages of the relational measures used to determine clustermembership. These include various measures of network centrality, aswell as the frequency with which nodes occupy specific positions indifferent kinds of triads that appear in the network (motifs). Rightaway, we see that the single node in cluster 6 differs from itscounterparts in other clusters. This node has a considerably higherdegree, betweenness, and closeness centrality measures, among others. Wealso see that our cluster of isolates (cluster 7) appears at the end ofthis data frame, with all of its values set toNA givenisolates’ lack of connection to other nodes in the network.

While recognized here, these differences are also visualized in thecluster_summaries_cent object. Because the network examinedhere is multirelational,cluster_summaries_cent plots thesedifferences for each unique relationship type in the network, as well asfor the overall network:

flor_cluster$cluster_summaries_cent$marriage

flor_cluster$cluster_summaries_cent$business

flor_cluster$cluster_summaries_cent$summary_graph

Those familiar with positions and motifs in networks know that asmany as 36 types of positions can exist in a network, which can beunwieldy to inspect alongside other measures. Consequently, differencesin triad positions are visualized separately incluster_summaries_triad:

flor_cluster$cluster_summaries_triad$marriage

flor_cluster$cluster_summaries_triad$business

flor_cluster$cluster_summaries_triad$summary_graph

Overall, the node in cluster 6 tends to have the highest values onmost measures used to identify roles in the network. Those familiar withthe substantive setting of this network will not be surprised to learnthat this node represents the Medici family, which was known for itspower and influence in Renaissance Florence. Additionally, nodes incluster 2 tend to appear in more clustered parts of this network due totheir business ties. If one is curious to see where the Medici andfamilies in other role positions appear relative to one another in thenetwork, one can quickly take the information contained incluster_assignments and assign it as a node-level attributein anigraph object for visualization:

igraph::V(nw_flor$florentine)$role<- flor_cluster$cluster_assignments$best_fitplot(nw_flor$florentine,vertex.color =as.factor(igraph::V(nw_flor$florentine)$role),vertex.label =NA)

Heatmaps

A final point of consideration in positional analysis involvesknowing whether nodes in a particular role tend to form ties amongthemselves or with nodes in other roles. When using hierarchicalclustering,role_analysis generates a series of heatmaps,contained in a list, to visualize the frequency of tie formation withinand between clusters. Each heatmap measures connections across clustersusing different measures, and the names of these measures are used toextract their corresponding plot from the list:

flor_cluster$cluster_relations_heatmaps$chisq# Chi-squared

flor_cluster$cluster_relations_heatmaps$density# Density

flor_cluster$cluster_relations_heatmaps$density_std# Density (Standardized)

flor_cluster$cluster_relations_heatmaps$density_centered# Density (Zero-floored)

Looking at the density-based heatmaps here, one finds a high level ofconnection between the Medici family and families belonging to cluster4. One can also see that families in cluster 2 have a high propensity tobe tied to families in cluster 5.

flor_concor<-role_analysis(method ="concor",graph = nw_flor$igraph_list,nodes = nw_flor$node_measures,directed =FALSE,min_partitions =1,max_partitions =4,viz =TRUE)

Block Memberships

flor_concor$concor_assignments%>%  dplyr::select(id, family, dplyr::starts_with("block"), best_fit)

Modularity Plot

As with the hierarchical clustering method, the optimal level ofpartitioning for CONCOR is determined according to the maximization ofmodularity in a similarity matrix. One can inspect how modularitychanges at different levels of partitioning using theconcor_modularity visualization:

flor_concor$concor_modularity

Visualizing CONCOR assignments in a conventional networkvisualization entails a similar process to that used for hierarchicalclustering.

igraph::V(nw_flor$florentine)$concor<- flor_concor$concor_assignments$best_fitplot(nw_flor$florentine,vertex.color =as.factor(igraph::V(nw_flor$florentine)$concor),vertex.label =NA)

Block Tree

In lieu of a dendrogram, users can see how smaller partitions branchoff of larger parents with theconcor_block_treevisualization. Likecluster_dendrogram, this visualizationallows users to quickly gauge the relative size of blocks inferred byCONCOR:

flor_concor$concor_block_tree

Heatmaps

Finally, users can also assess the level of connection across CONCORblocks using theconcor_relations_heatmaps object:

flor_concor$concor_relations_heatmaps$chisq

flor_concor$concor_relations_heatmaps$density

flor_concor$concor_relations_heatmaps$density_std

flor_concor$concor_relations_heatmaps$density_centered

On the whole, using CONCOR tells us that nodes in the Florentinenetwork fall into one of only two blocks (plus a third block for ourisolate), and that nodes within these roles tend to interact amongthemselves rather than with nodes in the other block. These simplerresults are less informative than those produced by the hierarchicalclustering method. But this is not to say that CONCOR is an inferiorapproach to positional analysis. Interpreting results from positionalanalysis often entails more subjectivity than other network analysismethods. Although two partitions may maximize modularity, users may findthat a higher level of partitioning produces blocks with importantsubstantive differences. Were we to accept four blocks as a moreappropriate fit than two, we see our inferred blocks start to resemblethe groups we inferred using hierarchical clustering. Moreover, thisresemblance also comes with only a small drop in modularity:

igraph::V(nw_flor$florentine)$concor2<- flor_concor$concor_assignments$block_2plot(nw_flor$florentine,vertex.color =as.factor(igraph::V(nw_flor$florentine)$concor2),vertex.label =NA)

With this in mind, we encourage users to thoroughly consider how theytreat their data when usingrole_analysis and to use theirbest judgment when interpreting its output.