# Temporal betweenness centrality `tbc` returns the temporal betweenness centrality for each node in a dynamic network (sequence of graph snapshots). ```r tbc(x, type = NULL, startsnapshot = 1, endsnapshot = length(x), vertexindices = NULL, directed = FALSE, normalize = TRUE, centrality_evolution = FALSE) ``` ## Arguments - `x`: A list of adjacency matrices or a list of adjacency lists. - `type`: Data format of `x`. Possible formats are `"M"` for a list of adjacency matrices (containing only 1s and 0s) and `"L"` for a list of adjacency lists (adjacency lists of the igraph package are supported). Default is `NULL`. - `startsnapshot`: Numeric. Entry of `x` to start the calculation of `tbc`. Default is 1. - `endsnapshot`: Numeric. Entry of `x` to end the calculation of `tbc`. Default is the last element of `x`. - `vertexindices`: Numeric. A vector of nodes. Only shortest temporal paths ending at nodes in `vertexindices` are considered for calculating `tbc`. Can be used to parallel the calculation of `tbc` (see section Examples). Default is `NULL`. - `directed`: Logical. Set `TRUE` if the dynamic network is a directed network. Default is `FALSE`. - `normalize`: Logical. Set `TRUE` if centrality values should be normalized with $1/((|V|-1)*(|V|-2)*m)$ where $|V|$ is the number of nodes and $m =$ `endsnapshot` $-$ `startsnapshot`. Default is `TRUE`. - `centrality_evolution`: Logical. Set `TRUE` if an additional matrix should be returned containing the centrality values at each snapshot. Rows correspondent to nodes, columns correspondent to snapshots. Default is `FALSE`. ## Returns The (normalized) temporal betweenness centrality (`TBC`) values of all nodes. If `centrality_evolution` is `TRUE`, an additional matrix will be returned (`CentEvo`), containing the temporal c("$(|V|\n$", "$ x T)$") matrix is returned (`CentEvo`), containing the temporal centrality value at each snapshot between `startsnapshot` and `endsnapshot`. ## Details `tbc` calculates the temporal betweenness centrality (Kim and Anderson, 2012). To keep the computational effort linear in the number of snapshots the Reversed Evolution Network algorithm (REN; Hanke and Foraita, 2017) is used to find all shortest temporal paths. ## Warning Using adjacency matrices as input exponentially increases the required memory. Use adjacency lists to save memory. ## Examples ```r # Create a list of adjacency matrices, plot the corresponding graphs # (using the igraph package) and calculate tbc A1 <- matrix(c(0,1,0,0,0,0, 1,0,1,0,0,0, 0,1,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0), ncol=6) A2 <- matrix(c(0,0,0,0,0,0, 0,0,1,0,0,0, 0,1,0,1,1,0, 0,0,1,0,0,0, 0,0,1,0,0,0, 0,0,0,0,0,0), ncol=6) A3 <- matrix(c(0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0), ncol=6) A4 <- matrix(c(0,1,0,0,0,0, 1,0,0,1,0,0, 0,0,0,0,0,0, 0,1,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0), ncol=6) library(igraph) par(mfrow=c(2,2)) Layout <- layout_in_circle(graph_from_adjacency_matrix(A1, mode = "undirected")) plot(graph_from_adjacency_matrix(A1, "undirected"), layout=Layout) plot(graph_from_adjacency_matrix(A2, "undirected"), layout=Layout) plot(graph_from_adjacency_matrix(A3, "undirected"), layout=Layout) plot(graph_from_adjacency_matrix(A4, "undirected"), layout=Layout) As <- list(A1,A2,A3,A4) tbc(As, "M", centrality_evolution=TRUE) ### Create list of adjacency lists Ls <- lapply(seq_along(As), function(i){ sapply(1:6, function(j){which(As[[i]][j,]==1)}) }) tbc(Ls, "L", centrality_evolution=TRUE) ### Run tbc in parallel ### library(parallel) # Calculate the number of cores cores_avail <- detectCores()-1 # Initiate cluster cl <- makeCluster(2) clusterExport(cl, c("As", "tbc")) TBC <- parLapply(cl, 1:6, function(x){ tbc(As, "M", vertexindices = x) } ) stopCluster(cl) Reduce("+", TBC) ``` ## References Kim, Hyoungshick and Anderson, Ross (2012). **Temporal node centrality in complex networks**. Physical Review E, 85 (2). Hanke, Moritz and Foraita, Ronja (2017). **Clone temporal centrality measures for incomplete sequences of graph snapshots**. BMC Bioinformatics, 18 (1). ## See Also `tcc,tdc`

tbc function