NetCoMi: Network Construction and Comparison for Microbiome Data

Documented in .calcProps

#' @title Calculate network properties
#'
#' @description Calculates network properties for a given adjacency matrix
#'
#' @param adjaMat adjacency matrix
#' @param dissMat dissimilarity matrix
#' @param assoMat association matrix
#' @param centrLCC logical indicating whether to compute centralities only 
#'   for the largest connected component (LCC). If \code{TRUE} 
#'   (default), centrality values of disconnected components are zero. 
#' @param avDissIgnoreInf logical indicating whether to ignore infinities when 
#'   calculating the average dissimilarity. If \code{FALSE} (default), infinity 
#'   values are set to 1.
#' @param sPathAlgo character indicating the algorithm used for computing
#'   the shortest paths between all node pairs. \code{\link[igraph]{distances}} 
#'   (igraph) is used for shortest path calculation. 
#'   Possible values are: "unweighted", "dijkstra" (default), "bellman-ford", 
#'   "johnson", or "automatic" (the fastest suitable algorithm is used). The 
#'   shortest paths are needed for the average (shortest) path length and 
#'   closeness centrality.
#' @param sPathNorm logical. If \code{TRUE} (default), shortest paths are 
#'   normalized by average dissimilarity (only connected nodes are considered),  
#'   i.e., a path is interpreted as steps with average dissimilarity. 
#'   If \code{FALSE}, the shortest path is the minimum sum of dissimilarities 
#'   between two nodes.
#' @param normNatConnect logical. If \code{TRUE} (default), the normalized 
#'   natural connectivity is returned.
#' @param weighted logical indicating whether the network is weighted.
#' @param isempty logical indicating whether the network is empty.
#' @param clustMethod character indicating the clustering algorithm. Possible
#'   values are \code{"hierarchical"} for a hierarchical algorithm based on
#'   dissimilarity values, or the clustering methods provided by the igraph
#'   package (see \code{\link[igraph]{communities}} for possible methods).
#'   Defaults to \code{"cluster_fast_greedy"} for association-based networks and
#'   to \code{"hierarchical"} for sample similarity networks.
#' @param clustPar list with parameters passed to the clustering functions.
#'   If hierarchical clustering is used, the parameters are passed to
#'   \code{\link[stats]{hclust}} as well as \code{\link[stats]{cutree}}.
#' @param weightClustCoef logical indicating whether (global) clustering 
#'   coefficient should be weighted (\code{TRUE}, default) or unweighted 
#'   (\code{FALSE}).
#' @param hubPar character vector with one or more elements (centrality 
#'   measures) used for identifying hub nodes. Possible values are \code{degree},
#'   \code{betweenness}, \code{closeness}, and \code{eigenvector}. If multiple
#'   measures are given, hubs are nodes with highest centrality for all selected
#'   measures. See details.
#' @param hubQuant quantile used for determining hub nodes. Defaults to 0.95.
#' @param lnormFit hubs are nodes with a centrality value above the 95\%
#'   quantile of the fitted log-normal distribution (if \code{lnormFit = TRUE})
#'   or of the empirical distribution of centrality values 
#'   (\code{lnormFit = FALSE}; default).
#' @param orbits numeric vector with integers from 0 to 14 defining the graphlet 
#'   orbits.
#' @param weightDeg logical. If \code{TRUE}, the weighted degree is used (see
#'   \code{\link[igraph]{strength}}). Default is \code{FALSE}. 
#'   Is automatically set to \code{TRUE} for a fully connected (dense) network.
#' @param normDeg,normBetw,normClose,normEigen logical. If \code{TRUE} 
#'   (default for all measures), a normalized version of the respective 
#'   centrality values is returned.
#' @param connectivity logical. If \code{TRUE} (default), edge and vertex 
#'   connectivity are calculated. Might be disabled to reduce execution time.
#' @param graphlet logical. If \code{TRUE} (default), graphlet-based network 
#'   properties are computed: orbit counts of graphlets with 2-4 nodes 
#'   (\code{ocount}) and Graphlet Correlation Matrix (\code{gcm}).
#' @param jaccard shall the Jaccard index be calculated?
#' @param jaccQuant quantile for the Jaccard index
#' @param verbose integer indicating the level of verbosity. Possible values:
#'   \code{"0"}: no messages, \code{"1"}: only important messages,
#'   \code{"2"}(default): all progress messages are shown. Can also be logical.
#'
#' @importFrom igraph graph_from_adjacency_matrix decompose.graph
#' @importFrom stats hclust as.dist cutree qlnorm quantile
#' @importFrom pulsar natural.connectivity


.calcProps <- function(adjaMat,
                       dissMat,
                       assoMat,
                       avDissIgnoreInf,
                       sPathNorm,
                       sPathAlgo,
                       normNatConnect,
                       weighted,
                       isempty,
                       clustMethod,
                       clustPar,
                       weightClustCoef,
                       hubPar,
                       hubQuant,
                       lnormFit,
                       connectivity,
                       graphlet,
                       orbits,
                       weightDeg,
                       normDeg,
                       normBetw,
                       normClose,
                       normEigen,
                       centrLCC,
                       jaccard = FALSE,
                       jaccQuant = NULL,
                       verbose = 0) {
  
  
  
  #== Create igraph objects and decompose graph ================================
  # Create graph from adjacency matrix
  
  if (verbose == 2) {
    message("Create igraph objects ... ", appendLF = FALSE)
  }
  
  net <- igraph::graph_from_adjacency_matrix(adjaMat, weighted=T,
                                             mode="undirected", diag=F)
  
  nNodes <- ncol(adjaMat)
  
  # decomposed graph
  dg_net <- igraph::decompose.graph(net)
  
  # size and number of the connected components
  dgcount <- unlist(lapply(dg_net, igraph::vcount))
  nComp <- length(dgcount)
  compSize <- t(as.matrix(table(dgcount)))
  compSize <- rbind(as.numeric(colnames(compSize)), compSize[1, ])
  dimnames(compSize) <- list(c("size:", "   #:"), rep("", ncol(compSize)))
  compSize <- compSize[, order(compSize[1,], decreasing = TRUE), drop = FALSE]
  
  if (isempty) {
    emptyvec <- numeric(nNodes)
    names(emptyvec) <- colnames(adjaMat)
    
    output <- list()
    
    output$nComp <- nNodes
    output$compSize<- compSize 
    output$lccNames <- names(emptyvec)[1]
    output$clust <- emptyvec
    output$tree <- NULL
    output$clust_lcc <- emptyvec[1]
    output$tree_lcc <- NULL
    output$deg <- output$deg_unnorm <- emptyvec
    output$betw <- output$betw_unnorm <- emptyvec
    output$close <- output$close_unnorm <- emptyvec
    output$eigen <- output$eigen_unnorm <- emptyvec
    output$lccSize <- 1
    output$lccSizeRel <- 1/nNodes
    output$avDiss <- output$avDiss_lcc <- 1
    output$avPath <- output$avPath_lcc <- 1
    output$vertconnect <- output$vertconnect_lcc <- 0
    output$edgeconnect <- output$edgeconnect_lcc <- 0
    output$natConnect <- output$natConnect_lcc <- 0
    output$clustCoef <- output$clustCoef_lcc <- 0
    output$density <- output$density_lcc <- 0
    output$modul <- output$modul_lcc <- 0
    output$pep <- output$pep_lcc <- 0
    output$hubs <- output$topdeg <- output$topbetw <- output$topclose <- 
      output$topeigen <- character(0)
    
    return(output)
  }
  
  # Largest connected component (LCC)
  indLCC <- which.max(unlist(lapply(dg_net, function(x) length(igraph::V(x)))))
  
  net_lcc <- dg_net[[indLCC]]
  lccSize <- length(igraph::V(net_lcc))
  lccSizeRel <- lccSize / nNodes
  
  # Adjacency of the LCC
  adjaMat_lcc <- as.matrix(igraph::as_adjacency_matrix(net_lcc, attr="weight"))
  diag(adjaMat_lcc) <- 1
  
  # Names of nodes in the LCC
  lccNames <- colnames(adjaMat_lcc)
  
  if (!weighted) {
    dissMat[!is.infinite(dissMat)] <- 1
    diag(dissMat) <- 0
  }
  
  # Dissimilarity of the LCC
  dissMat_lcc <- dissMat[lccNames, lccNames]
  
  #== Graph objects of dissimilarity matrices ==================================
  
  dissMatnoInf <- dissMat
  dissMatnoInf[is.infinite(dissMatnoInf)] <- 0
  
  dissMatnoInf_lcc <- dissMat_lcc
  dissMatnoInf_lcc[is.infinite(dissMatnoInf_lcc)] <- 0
  
  # Whole network
  dissnet <- igraph::graph_from_adjacency_matrix(dissMatnoInf, 
                                                 weighted=T,
                                                 mode="undirected", 
                                                 diag=F)
  
  # LCC
  dissnet_lcc <- igraph::graph_from_adjacency_matrix(dissMatnoInf_lcc, 
                                                     weighted=T,
                                                     mode="undirected", 
                                                     diag=F)
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #== Clustering ============================================================
  
  clust <- clust_lcc <- NULL
  tree <- tree_lcc <- NULL
  
  if (clustMethod != "none") {
    
    if (verbose == 2) {
      message("Compute clustering (", clustMethod, ") ... ", appendLF = FALSE)
    }
    
    if (clustMethod == "hierarchical") {
      
      if (is.null(clustPar$method)) {
        clustPar$method <- "average"
      }
      
      dissMat.tmp <- dissMat
      dissMat.tmp[is.infinite(dissMat.tmp)] <- 1
      
      dissMat_lcc.tmp <- dissMat_lcc
      dissMat_lcc.tmp[is.infinite(dissMat_lcc.tmp)] <- 1
      
      tree <- stats::hclust(stats::as.dist(dissMat.tmp), 
                            method = clustPar$method)
      
      tree_lcc <- stats::hclust(as.dist(dissMat_lcc.tmp), 
                                method = clustPar$method)
      rm(dissMat.tmp, dissMat_lcc.tmp)
      
      if (is.null(clustPar$k) & is.null(clustPar$h)) {
        clustPar$k <- 3
      }
      
      clust <- do.call(stats::cutree, list(tree = tree, 
                                           k = clustPar$k, 
                                           h = clustPar$h))
      
      clust_lcc <- do.call(stats::cutree, list(tree = tree_lcc, 
                                               k = clustPar$k, 
                                               h = clustPar$h))
      
      names(clust) <- rownames(adjaMat)
      names(clust_lcc) <- rownames(adjaMat_lcc)
      
    } else {
      
      if (clustMethod == "cluster_edge_betweenness") {
        clustres <- do.call(getExportedValue("igraph", clustMethod),  
                            c(list(graph = net, weights = E(dissnet)$weight),
                              clustPar))
        
        clustres_lcc <- do.call(getExportedValue("igraph", clustMethod),  
                                c(list(graph = net_lcc, 
                                       weights = E(dissnet_lcc)$weight),
                                  clustPar))
      } else {
        clustres <- do.call(getExportedValue("igraph", clustMethod), 
                            c(list(net), clustPar))
        clustres_lcc <- do.call(getExportedValue("igraph", clustMethod), 
                                c(list(net_lcc), clustPar))
      }
      
      clust <- clustres$membership
      names(clust) <- clustres$names
      
      # LCC
      clust_lcc <- clustres_lcc$membership
      names(clust_lcc) <- clustres_lcc$names
    }
    
    cltab <- table(clust)
    cltab_lcc <- table(clust_lcc)
    
    # cluster 0 assigned to elements in a cluster with only one element
    clust[clust %in% which(cltab == 1)] <- 0
    clust_lcc[clust_lcc %in% which(cltab_lcc == 1)] <- 0
    
    if (verbose == 2) {
      message("Done.")
    }
    
  }
  
  
  #== Shortest paths ===========================================================
  
  if (verbose == 2) {
    message("Compute shortest paths ... ", appendLF = FALSE)
  }
  
  # Whole network
  sPath <- .suppress_warnings(igraph::distances(dissnet, algorithm = sPathAlgo), 
                              startsWith, "Unweighted algorithm chosen")
  
  # LCC
  sPath_lcc <- .suppress_warnings(igraph::distances(dissnet_lcc, 
                                                    algorithm = sPathAlgo), 
                                  startsWith, "Unweighted algorithm chosen")

  if (verbose == 2) {
    message("Done.")
  }
  
  #== Average dissimilarity/distance ===========================================

  if (weighted) {
    dissVec <- dissMat[lower.tri(dissMat)]
    dissVec_lcc <- dissMat_lcc[lower.tri(dissMat_lcc)]
    
    if (avDissIgnoreInf) {
      dissVec[is.infinite(dissVec)] <- NA
      dissVec_lcc[is.infinite(dissVec_lcc)] <- NA
    } else {
      dissVec[is.infinite(dissVec)] <- 1
      dissVec_lcc[is.infinite(dissVec_lcc)] <- 1
    }
    
    ### average dissimilarity
    avDiss <- mean(dissVec, na.rm = TRUE)
    if (is.na(avDiss)) avDiss <- 1
    
    avDiss_lcc <- mean(dissVec_lcc, na.rm = TRUE)
    if (is.na(avDiss_lcc)) avDiss_lcc <- 1
    
    # "normalized" shortest paths (units with average dissimilarity)
    if (sPathNorm) {
      sPath <- sPath / avDiss
      sPath_lcc <- sPath_lcc / avDiss_lcc
    }
  } else {
    avDiss <- avDiss_lcc <- 1
  }

  #== global network properties ================================================
  
  if (verbose == 2) {
    message("\nCompute global properties:")
    
    message("Average dissimilarity ... ", appendLF = FALSE)
  }
  
  ### Average shortest path length 
  
  sPathVec <- sPath[lower.tri(sPath)]
  sPathVec[is.infinite(sPathVec)] <- NA
  avPath <- mean(sPathVec, na.rm = TRUE)
  if (is.na(avPath)) avPath <- 1
  
  sPathVec_lcc <- sPath_lcc[lower.tri(sPath_lcc)]
  avPath_lcc <- mean(sPathVec_lcc, na.rm = TRUE)
  if (is.na(avPath_lcc)) avPath_lcc <- 1
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Connectivity 
  
  if (verbose == 2) {
    message("Edge / vertex connectivity ... ", appendLF = FALSE)
  }
  
  if (connectivity) {
    # vertex connectivity
    vertconnect <- igraph::vertex_connectivity(net)
    vertconnect_lcc <- igraph::vertex_connectivity(net_lcc)
    
    # edge connectivity
    edgeconnect <- igraph::edge_connectivity(net)
    edgeconnect_lcc <- igraph::edge_connectivity(net_lcc)
  } else {
    vertconnect <- vertconnect_lcc <- NA
    edgeconnect <- edgeconnect_lcc <- NA
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Natural connectivity
  
  if (verbose == 2) {
    message("Natural connectivity ... ", appendLF = FALSE)
  }
  
  natConnect <- pulsar::natural.connectivity(adjaMat, norm = normNatConnect)
  natConnect_lcc <- pulsar::natural.connectivity(adjaMat_lcc,
                                                 norm = normNatConnect)
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Clustering coefficient
  
  if (verbose == 2) {
    message("Clustering coefficient ... ", appendLF = FALSE)
  }
  
  if (weightClustCoef) {
    # Complete network
    clustCoef.tmp <- igraph::transitivity(net, type = "barrat")
    clustCoef <- mean(clustCoef.tmp, na.rm = TRUE)
    if (is.nan(clustCoef)) clustCoef <- 0
    
    # LCC
    clustCoef.tmp <- igraph::transitivity(net_lcc, type = "barrat")
    clustCoef_lcc <- mean(clustCoef.tmp, na.rm = TRUE)
    if (is.nan(clustCoef_lcc)) clustCoef_lcc <- 0
    
    rm(clustCoef.tmp)
  } else {
    # Complete network
    clustCoef <- igraph::transitivity(net, type = "global")
    if (is.na(clustCoef)) clustCoef <- 0
    
    # LCC
    clustCoef_lcc <- igraph::transitivity(net_lcc, type = "global")
    if (is.na(clustCoef_lcc)) clustCoef_lcc <- 0
  }
  
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Modularity
  
  if (verbose == 2) {
    message("Modularity ... ", appendLF = FALSE)
  }
  
  # Complete network
  if (clustMethod != "none") {
    modul <- igraph::modularity(net, (clust+1))
  } else {
    modul <- NA
  }
  
  # LCC
  if (clustMethod != "none") {
    modul_lcc <- igraph::modularity(net_lcc, (clust_lcc+1))
  } else {
    modul_lcc <- NA
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Density (relative number of edges)
  
  if (verbose == 2) {
    message("Density ... ", appendLF = FALSE)
  }
  
  density <- igraph::edge_density(net)
  density_lcc <- igraph::edge_density(net_lcc)
  
  if (verbose == 2) {
    message("Done.")
  }
  #-------------------------------
  ### Ratio of positive to negative associations
  
  if (verbose == 2) {
    message("P-N-Ratio ... ", appendLF = FALSE)
  }
  
  # Complete network
  if (is.null(assoMat)) { # no negative dissimilarities possible
    pep <- 100
    pep_lcc <- 100
    
  } else {
    edge_all <- sum(assoMat[lower.tri(assoMat)] != 0)
    edge_pos <- sum(assoMat[lower.tri(assoMat)] > 0)
    pep <- edge_pos / edge_all * 100
    
    # LCC
    assoMat_lcc <- assoMat[rownames(dissMat_lcc), colnames(dissMat_lcc)]
    
    edge_all <- sum(assoMat_lcc[lower.tri(assoMat_lcc)] != 0)
    edge_pos <- sum(assoMat_lcc[lower.tri(assoMat_lcc)] > 0)
    pep_lcc <- edge_pos / edge_all * 100
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #== Centrality measures ======================================================
  
  if (verbose == 2) {
    message("\nCompute centralities:")
    
    message("Degree ... ", appendLF = FALSE)
  }
  
  ### degree
  if (weightDeg) {
    deg <- deg_unnorm <- igraph::strength(net)
    deg_lcc <- strength(net_lcc)
    
  } else {
    deg <- igraph::degree(net, normalized = normDeg)
    deg_unnorm <- igraph::degree(net)
  }
  
  if (centrLCC) {
    deg[!names(deg) %in% lccNames] <- 0
    deg_unnorm[!names(deg_unnorm) %in% lccNames] <- 0
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### betweenness centrality (based on distances)
  
  if (verbose == 2) {
    message("Betweenness centrality ... ", appendLF = FALSE)
  }
  
  if (centrLCC) {
    betw <- betw_unnorm <- rep(0, ncol(adjaMat))
    names(betw) <- names(betw_unnorm) <- colnames(adjaMat)
    
    betw.tmp <- igraph::betweenness(dissnet_lcc, normalized = normBetw)
    betw_unnorm.tmp <- igraph::betweenness(dissnet_lcc)
    
    betw[names(betw.tmp)] <- betw.tmp
    betw_unnorm[names(betw_unnorm.tmp)] <- betw_unnorm.tmp
    
  } else {
    betw <- igraph::betweenness(dissnet, normalized = normBetw)
    betw_unnorm <- igraph::betweenness(dissnet)
  }
  
  betw[is.nan(betw)] <- 0
  betw_unnorm[is.nan(betw_unnorm)] <- 0
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### closeness centrality (based on distances)
  
  if (verbose == 2) {
    message("Closeness centrality ... ", appendLF = FALSE)
  }
  
  if (centrLCC) {
    close_unnorm <- rep(0, ncol(adjaMat))
    names(close_unnorm) <- colnames(adjaMat)
    
    sPath_rev <- 1/sPath_lcc
    diag(sPath_rev) <- NA
    
    close_unnorm.tmp <- sapply(1:nrow(sPath_rev), function(i) {
      sum(sPath_rev[i,], na.rm = TRUE)
    })
    names(close_unnorm.tmp) <- lccNames
    close_unnorm[lccNames] <- close_unnorm.tmp
    
    if (normClose) {
      # normalize closeness by n-1 
      close <- close_unnorm / (lccSize - 1)
    } else {
      close <- close_unnorm
    }
    
  } else {
    #if (nComp > 1 && sPathNorm) {
    #warning("Normalized shortest paths depend on average dissimilarity, which
    #may be biased because infinite paths between unconnected nodes are ignored. 
    #Hence, closeness centrality may also be biased and should either be based 
    #on non-normalized shortest paths or be computed only for the LCC.")
    #}
    
    sPath_rev <- 1/sPath
    diag(sPath_rev) <- NA
    
    close_unnorm <- sapply(1:nrow(sPath_rev), function(i) {
      sum(sPath_rev[i,], na.rm = TRUE)
    })
    names(close_unnorm) <- colnames(adjaMat)
    
    if (normClose) {
      # normalize closeness by n-1 
      close <- close_unnorm / (nNodes - 1)
    } else {
      close <- close_unnorm
    }
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #-------------------------------
  ### Eigenvector centrality
  
  if (verbose == 2) {
    message("Eigenvector centrality ... ", appendLF = FALSE)
  }
  
  if (!centrLCC && nComp > 1) {
    
    dgcount[dgcount == 1] <- 0
    
    vnumb <- sum(unlist(dgcount))
    
    ev <- numeric(0)
    
    for (i in seq_along(dg_net)) {
      ev <- c(ev,
              igraph::eigen_centrality(
                dg_net[[i]], scale = FALSE)$vector * (dgcount[i] / vnumb))
    }
    
    eigen_unnorm <- ev[colnames(adjaMat)]
    
    if (normEigen) {
      eigen <- eigen_unnorm / max(eigen_unnorm)
    } else {
      eigen <- eigen_unnorm
    }
    
  } else {
    eigen.tmp <- igraph::eigen_centrality(net_lcc, scale = normEigen)$vector
    eigen_unnorm.tmp <- igraph::eigen_centrality(net_lcc, scale = FALSE)$vector
    
    eigen <- eigen_unnorm <- numeric(nNodes)
    names(eigen) <- names(eigen_unnorm) <- colnames(adjaMat)
    eigen[names(eigen.tmp)] <- eigen.tmp
    eigen_unnorm[names(eigen_unnorm.tmp)] <- eigen_unnorm.tmp
  }
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #== hubs and Jaccard index ================================================
  
  if (verbose == 2) {
    if (lnormFit) {
      message("\nCompute hubs (based on log-normal quantiles... ", 
              appendLF = FALSE)
    } else {
      message("\nCompute hubs (based on empirical quantiles) ... ", 
              appendLF = FALSE)
    }
    
  }
  
  if (lnormFit) {
    # identify nodes with highest centrality value
    pdeg <- try(MASS::fitdistr(deg_unnorm[deg_unnorm>0], "lognormal")$estimate, 
                silent = TRUE)
    
    if (inherits(pdeg, "try-error")) {
      topdeg <- hubdeg <- NULL
      
    } else {
      hubdeg <- names(deg_unnorm[deg_unnorm > stats::qlnorm(hubQuant, 
                                                            pdeg[1], 
                                                            pdeg[2])])
      
      if (jaccard) {
        topdeg <- names(deg_unnorm[deg_unnorm > stats::qlnorm(jaccQuant, 
                                                              pdeg[1], 
                                                              pdeg[2])])
      } else {
        topdeg <- NULL
      }
    }
    
    pbetw <- try(MASS::fitdistr(betw_unnorm[betw_unnorm>0], 
                                "lognormal")$estimate, 
                 silent = TRUE)
    
    if (inherits(pbetw, "try-error")) {
      topbetw <- hubbetw <- NULL
    } else {
      hubbetw <- names(betw_unnorm[betw_unnorm > stats::qlnorm(hubQuant, 
                                                               pbetw[1], 
                                                               pbetw[2])])
      if (jaccard) {
        topbetw <- names(betw_unnorm[betw_unnorm > stats::qlnorm(jaccQuant, 
                                                                 pbetw[1], 
                                                                 pbetw[2])])
      } else {
        topbetw <- NULL
      }
    }
    
    pclose <- try(MASS::fitdistr(close_unnorm[close_unnorm>0], 
                                 "lognormal")$estimate, 
                  silent = TRUE)
    
    if (inherits(pclose, "try-error")) {
      topclose <- hubclose <- NULL
    } else {
      hubclose <- names(close_unnorm[close_unnorm > stats::qlnorm(hubQuant, 
                                                                  pclose[1], 
                                                                  pclose[2])])
      if (jaccard) {
        topclose <- names(close_unnorm[close_unnorm > stats::qlnorm(jaccQuant, 
                                                                    pclose[1], 
                                                                    pclose[2])])
      } else {
        topclose <- NULL
      }
    }
    
    peigen <- try(MASS::fitdistr(eigen_unnorm[eigen_unnorm>0], 
                                 "lognormal")$estimate, 
                  silent = TRUE)
    
    if (inherits(peigen, "try-error")) {
      topeigen <- hubeigen <- NULL
    } else {
      hubeigen <- names(eigen_unnorm[eigen_unnorm > stats::qlnorm(hubQuant, 
                                                                  peigen[1], 
                                                                  peigen[2])])
      if (jaccard) {
        topeigen <- names(eigen_unnorm[eigen_unnorm > stats::qlnorm(jaccQuant, 
                                                                    peigen[1], 
                                                                    peigen[2])])
      } else {
        topeigen <- NULL
      }
    }
    
  } else {
    hubdeg <- names(deg[deg > quantile(deg, hubQuant)])
    if (jaccard) {
      topdeg <- names(deg[deg > quantile(deg, jaccQuant)])
    } else {
      topdeg <- NULL
    }
    
    hubbetw <- names(betw[betw > quantile(betw, hubQuant)])
    if (jaccard) {
      topbetw <- names(betw[betw > quantile(betw, jaccQuant)])
    } else {
      topbetw <- NULL
    }
    
    hubclose <- names(close[close > quantile(close, hubQuant)])
    if (jaccard) {
      topclose <- names(close[close > quantile(close, jaccQuant)])
    } else {
      topclose <- NULL
    }
    
    hubeigen <- names(eigen[eigen > quantile(eigen, hubQuant)])
    if (jaccard) {
      topeigen <- names(eigen[eigen > quantile(eigen, jaccQuant)])
    } else {
      topeigen <- NULL
    }
  }
  
  ###############
  # identify hub nodes
  hubparams <-  list()
  if ("degree" %in% hubPar) {
    hubparams <- c(hubparams, list(hubdeg))
  }
  if ("betweenness" %in% hubPar) {
    hubparams <- c(hubparams, list(hubbetw))
  }
  if ("closeness" %in% hubPar) {
    hubparams <- c(hubparams, list(hubclose))
  }
  if ("eigenvector" %in% hubPar) {
    hubparams <- c(hubparams, list(hubeigen))
  }
  
  hubs <- Reduce(intersect, hubparams)
  
  if (verbose == 2) {
    message("Done.")
  }
  
  #== Graphlet-based measures ==================================================
  
  if (graphlet) {
    if (verbose == 2) {
      message("Compute GCM ... ", appendLF = FALSE)
    }
    
    graphlet <- calcGCM(adjaMat, orbits = orbits)
    gcmtest <- testGCM(graphlet, verbose = FALSE)
    graphlet$pvals <- gcmtest$pvals1
    graphlet$pAdjust <- gcmtest$pAdjust1
    graphlet <- graphlet[c(2, 1, 3, 4)]
    class(graphlet) <- "GCM"
    
    graphlet_lcc <- calcGCM(adjaMat_lcc, orbits = orbits)
    gcmtest <- testGCM(graphlet_lcc, verbose = FALSE)
    graphlet_lcc$pvals <- gcmtest$pvals1
    graphlet_lcc$pAdjust <- gcmtest$pAdjust1
    graphlet_lcc <- graphlet_lcc[c(2, 1, 3, 4)]
    class(graphlet_lcc) <- "GCM"
    
    if (verbose == 2) {
      message("Done.")
    }
    
  } else {
    graphlet <- graphlet_lcc <- NA
  }
  
  #========================================================================
  
  output <-list(nComp = nComp,
                compSize = compSize,
                lccNames = lccNames,
                clust = clust,
                tree = tree,
                clust_lcc = clust_lcc,
                tree_lcc = tree_lcc,
                deg = deg,
                deg_unnorm = deg_unnorm,
                betw = betw,
                betw_unnorm = betw_unnorm,
                close = close,
                close_unnorm = close_unnorm,
                eigen = eigen,
                eigen_unnorm = eigen_unnorm,
                lccSize = lccSize,
                lccSizeRel = lccSizeRel,
                avDiss = avDiss,
                avDiss_lcc = avDiss_lcc,
                avPath = avPath,
                avPath_lcc = avPath_lcc,
                vertconnect = vertconnect,
                vertconnect_lcc = vertconnect_lcc,
                edgeconnect = edgeconnect,
                edgeconnect_lcc = edgeconnect_lcc,
                natConnect = natConnect,
                natConnect_lcc = natConnect_lcc,
                clustCoef = clustCoef,
                clustCoef_lcc = clustCoef_lcc,
                density = density,
                density_lcc = density_lcc,
                modul = modul,
                modul_lcc = modul_lcc,
                pep = pep,
                pep_lcc = pep_lcc,
                hubs = hubs,
                topdeg = topdeg,
                topbetw = topbetw,
                topclose = topclose,
                topeigen = topeigen,
                graphlet = graphlet,
                graphlet_lcc = graphlet_lcc,
                adjaMat_lcc = adjaMat_lcc)
  
  return(output)
}