Package RBGL. R topics documented: May 14, Version Title An interface to the BOOST graph library

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1 Version Title An interface to the BOOST raph library Packae RBGL May 14, 2018 Author Vince Carey Li Lon R. Gentleman Maintainer Bioconductor Packae Maintainer Depends raph, methods Imports methods Suests Rraphviz, XML, RUnit, BiocGenerics A fairly extensive and comprehensive interface to the raph alorithms contained in the BOOST library. License Artistic-2.0 URL LazyLoad yes biocviews GraphAndNetwork, Network R topics documented: astarsearch bandwidth bellman.ford.sp betweenness.centrality.clusterin bfs biconncomp boyermyrvoldplanaritytest brandes.betweenness.centrality chrobakpaynestraihtlinedrawin clusterincoef clusterincoefappr connectedcomp da.sp dijkstra.sp dominatortree edeconnectivity edmondsmaxcardinalitymatchin

2 2 R topics documented: edmondsoptimumbranchin extractpath FileDep floyd.warshall.all.pairs.sp profile raphgenerator hihlyconnsg incremental.components is.trianulated iskuratowskisubraph isomorphism isstraihtlinedrawin johnson.all.pairs.sp kcliques kcores lambdasets layout makebiconnectedplanar makeconnected makemaximalplanar max.flow maxclique maximumcycleratio mincut minimumcycleratio mstree.kruskal mstree.prim Orderin planarcanonicalorderin planarfacetraversal RBGL-defunct RBGL.overview removeselfloops separates sequential.vertex.colorin sloanstartendvertices sp.between stroncomp transitive.closure transitivity tsort wavefront Index 67

3 astarsearch 3 astarsearch Compute astarsearch for a raph Compute astarsearch for a raph Usae astarsearch() Aruments an instance of the raph class NOT IMPLEMENTED YET. TO BE FILLED IN a strin indicatin non-implementation Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) con <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) astarsearch(coex)

4 4 bandwidth bandwidth Compute bandwidth for an undirected raph Compute bandwidth for an undirected raph Usae bandwidth() Aruments an instance of the raph class with edemode undirected The bandwidth of an undirected raph G=(V, E) is the maximum distance between two adjacent vertices. See documentation on bandwidth in Boost Graph Library for more details. bandwidth the bandwidth of the iven raph Li Lon Boost Graph Library ( ) con <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) coex <- uraph(coex) bandwidth(coex)

5 bellman.ford.sp 5 bellman.ford.sp Bellman-Ford shortest paths usin boost C++ Alorithm for the sinle-source shortest paths problem for a raph with both positive and neative ede weihts. Usae bellman.ford.sp(,start=nodes()[1]) Aruments start instance of class raph character: node name for start of path This function interfaces to the Boost raph library C++ routines for Bellman-Ford shortest paths. Choose the appropriate alorithm to calculate the shortest path carefully based on the properties of the iven raph. See documentation on Bellman-Ford alorithm in Boost Graph Library for more details. A list with elements: all edes minimized true if all edes are minimized, false otherwise. distance penult The vector of distances from start to each node of ; includes Inf when there is no path from start. A vector of indices (in nodes()) of predecessors correspondin to each node on the path from that node back to start. For example, if the element one of this vector has value 10, that means that the predecessor of node 1 is node 10. The next predecessor is found by examinin penult[10]. start The start node that was supplied in the call to bellman.ford.sp. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( )

6 6 betweenness.centrality.clusterin See Also da.sp, dijkstra.sp, johnson.all.pairs.sp, sp.between con <- file(system.file("xml/conn2.xl",packae="rbgl"), open="r") dd <- fromgxl(con) close(con) bellman.ford.sp(dd) bellman.ford.sp(dd,nodes(dd)[2]) betweenness.centrality.clusterin Graph clusterin based on ede betweenness centrality Graph clusterin based on ede betweenness centrality Usae betweenness.centrality.clusterin(, threshold = -1, normalize = TRUE) Aruments threshold normalize an instance of the raph class with edemode undirected threshold to terminate clusterin process boolean, when TRUE, the ede betweenness centrality is scaled by 2/((n-1)(n-2)) where n is the number of vertices in ; when FALSE, the ede betweenness centrality is the absolute value To implement raph clusterin based on ede betweenness centrality. The alorithm is iterative, at each step it computes the ede betweenness centrality and removes the ede with maximum betweenness centrality when it is above the iven threshold. When the maximum betweenness centrality falls below the threshold, the alorithm terminates. See documentation on Clusterin alorithms in Boost Graph Library for details. A list of no.of.edes number of remainin edes after removal edes remainin edes ede.betweenness.centrality betweenness centrality of remainin edes Li Lon <li.lon@isb-sib.ch>

7 bfs 7 Boost Graph Library ( ) See Also brandes.betweenness.centrality con <- file(system.file("xml/conn.xl",packae="rbgl")) coex <- fromgxl(con) close(con) coex <- uraph(coex) betweenness.centrality.clusterin(coex, 0.5, TRUE) bfs Breadth and Depth-first search These functions return information on raph traversal by breadth and depth first search usin routines from the BOOST library. Usae bfs(object, node, checkconn=true) dfs(object, node, checkconn=true) Aruments object node checkconn instance of class raph from Bioconductor raph class node name where search starts; defaults to the node in first position in the node vector. loical for backwards compatibility; this parameter has no effect as of RBGL and will be removed in future versions. These two functions are interfaces to the BOOST raph library functions for breadth first and depth first search. Both methods handle unconnected raphs by applyin the alorithms over the connected components. Cormen et al note (p 542) that results of depth-first search may depend upon the order in which the vertices are examined... These different visitation orders tend not to cause problems in practice, as any DFS result can usually be used effectively, with essentially equivalent results.

8 8 biconncomp For bfs a vector of node indices in order of BFS visit. For dfs a list of two vectors of nodes, with elements discover (order of DFS discovery), and finish (order of DFS completion). VJ Carey <stvjc@channin.harvard.edu> Boost Graph Library ( ) con1 <- file(system.file("xml/bfsex.xl",packae="rbgl"), open="r") dd <- fromgxl(con1) close(con1) bfs(dd, "r") bfs(dd, "s") con2 <- file(system.file("xml/dfsex.xl",packae="rbgl"), open="r") dd2 <- fromgxl(con2) close(con2) dfs(dd2, "u") biconncomp Compute biconnected components for a raph Compute biconnected components for a raph Usae biconncomp() articulationpoints() Aruments an instance of the raph class

9 boyermyrvoldplanaritytest 9 A biconnected raph is a connected raph that remains connected when any one of its vertices, and all the edes incident on this vertex, is removed and the raph remains connected. A biconnected component of a raph is a subraph which is biconnected. An inteer label is assined to each ede to indicate which biconnected component it s in. A vertex in a raph is called an articulation point if removin it increases the number of connected components. See the documentation for the Boost Graph Library for more details. For biconncomp: a vector whose lenth is no. of biconnected components, each entry is a list of nodes that are on the same biconnected components. For articulationpoints: a vector of articulation points in the raph. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) con <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) biconncomp(coex) articulationpoints(coex) boyermyrvoldplanaritytest boyermyrvoldplanaritytest boyermyrvoldplanaritytest description Usae boyermyrvoldplanaritytest() Aruments instance of class raphnel from Bioconductor raph class

10 10 brandes.betweenness.centrality loical, TRUE if test succeeds Li Lon Boost Graph Library ( ) con <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") coex <- fromgxl(con) edemode(coex) = "undirected" boyermyrvoldplanaritytest(coex) # only shows runnability, need better case brandes.betweenness.centrality Compute betweenness centrality for an undirected raph Compute betweenness centrality for an undirected raph Usae brandes.betweenness.centrality() Aruments an instance of the raph class with edemode undirected Brandes.betweenness.centrality computes the betweenness centrality of each vertex or each ede in the raph, usin an alorithm by U. Brandes. Betweenness centrality of a vertex v is calculated as follows: N_st(v) = no. of shortest paths from s to t that pass throuh v, N_st = no. of shortest paths from s to t, betweenness centrality of v = sum(n_st(v)/n_st) for all vertices s!= v!= t. Betweenness centrality of an ede is calculated similarly. The relative betweenness centrality for a vertex is to scale the betweenness centrality of the iven vertex by 2/(n**2-3n + 2) where n is the no. of vertices in the raph. Central point dominance measures the maximum betweenness of any vertex in the raph. See documentation on brandes betweenness centrality in Boost Graph Library for more details.

11 chrobakpaynestraihtlinedrawin 11 A list of betweenness.centrality.vertices betweenness centrality of each vertex betweenness.centrality.edes betweenness centrality of each ede relative.betweenness.centrality.vertices relative betweenness centrality of each vertex dominance maximum betweenness of any point in the raph Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) See Also betweenness.centrality.clusterin con <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) coex <- uraph(coex) brandes.betweenness.centrality(coex) chrobakpaynestraihtlinedrawin chrobakpaynestraihtlinedrawin chrobakpaynestraihtlinedrawin description Usae chrobakpaynestraihtlinedrawin() Aruments instance of class raphnel from Bioconductor raph class

12 12 chrobakpaynestraihtlinedrawin M. Chrobak, T. Payne, A Linear-time Alorithm for Drawin a Planar Graph on the Grid, Information Processin Letters 54: , A matrix with rows x and y, and columns correspondin to raph nodes. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) V <- LETTERS[1:7] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+1], V[2+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+3], V[0+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+4], V[5+1], ) <- addede(v[1+5], V[6+1], ) <- addede(v[1+6], V[3+1], ) <- addede(v[1+0], V[4+1], ) <- addede(v[1+1], V[3+1], ) <- addede(v[1+3], V[5+1], ) <- addede(v[1+2], V[6+1], ) <- addede(v[1+1], V[4+1], ) <- addede(v[1+1], V[5+1], ) <- addede(v[1+1], V[6+1], ) x3 <- chrobakpaynestraihtlinedrawin() x3 plot(t(x3)) el = edel() for (i in seq_len(lenth(nodes()))) sements( rep(x3["x",i], lenth(el[[i]]$edes)), rep(x3["y",i], lenth(el[[i]]$edes)), x3["x", nodes()[el[[i]]$edes]], x3["y", nodes()[el[[i]]$edes]] )

13 clusterincoef 13 clusterincoef Calculate clusterin coefficient for an undirected raph Calculate clusterin coefficient for an undirected raph Usae clusterincoef(, Weihted=FALSE, vw=deree()) Aruments Weihted vw an instance of the raph class calculate weihted clusterin coefficient or not vertex weihts to use when calculatin weihted clusterin coefficient For an undirected raph G, let delta(v) be the number of trianles with v as a node, let tau(v) be the number of triples, i.e., paths of lenth 2 with v as the center node. Let V be the set of nodes with deree at least 2. Define clusterin coefficient for v, c(v) = (delta(v) / tau(v)). Define clusterin coefficient for G, C(G) = sum(c(v)) / V, for all v in V. Define weihted clusterin coefficient for, Cw(G) = sum(w(v) * c(v)) / sum(w(v)), for all v in V. Clusterin coefficient for raph G. Li Lon <li.lon@isb-sib.ch> Approximatin Clusterin Coefficient and Transitivity, T. Schank, D. Waner, Journal of Graph Alorithms and Applications, Vol. 9, No. 2 (2005). See Also clusterincoefappr, transitivity, raphgenerator con <- file(system.file("xml/conn.xl",packae="rbgl")) <- fromgxl(con) close(con) cc <- clusterincoef() ccw1 <- clusterincoef(, Weihted=TRUE) vw <- c(1, 1, 1, 1, 1,1, 1, 1) ccw2 <- clusterincoef(, Weihted=TRUE, vw)

14 14 clusterincoefappr clusterincoefappr Approximate clusterin coefficient for an undirected raph Approximate clusterin coefficient for an undirected raph Usae clusterincoefappr(, k=lenth(nodes()), Weihted=FALSE, vw=deree()) Aruments Weihted vw k an instance of the raph class calculate weihted clusterin coefficient or not vertex weihts to use when calculatin weihted clusterin coefficient parameter controls total expected runtime It is quite expensive to compute cluster coefficient and transitivity exactly for a lare raph by computin the number of trianles in the raph. Instead, clusterincoefappr samples triples with appropriate probability, returns the ratio between the number of existin edes and the number of samples. MORE ABOUT CHOICE OF K. See reference for more details. Approximated clusterin coefficient for raph. Li Lon <li.lon@isb-sib.ch> Approximatin Clusterin Coefficient and Transitivity, T. Schank, D. Waner, Journal of Graph Alorithms and Applications, Vol. 9, No. 2 (2005). See Also clusterincoef, transitivity, raphgenerator

15 connectedcomp 15 con <- file(system.file("xml/conn.xl",packae="rbgl")) <- fromgxl(con) close(con) k = lenth(nodes()) cc <- clusterincoefappr(, k) ccw1 <- clusterincoefappr(, k, Weihted=TRUE) vw <- c(1, 1, 1, 1, 1,1, 1, 1) ccw2 <- clusterincoefappr(, k, Weihted=TRUE, vw) connectedcomp Identify Connected Components in an Undirected Graph Usae The connected components in an undirected raph are identified. If the raph is directed then the weakly connected components are identified. connectedcomp() Aruments raph with edemode undirected Uses a depth first search approach to identifyin all the connected components of an undirected raph. If the input,, is a directed raph it is first transformed to an undirected raph (usin uraph). A list of lenth equal to the number of connected components in. Each element of the list contains a vector of the node labels for the nodes that are connected. Vince Carey <stvjc@channin.harvard.edu> Boost Graph Library ( ) See Also conncomp,stroncomp, uraph, same.component

16 16 da.sp con <- file(system.file("gxl/kmstex.xl",packae="raph"), open="r") km <- fromgxl(con) close(con) km <- raph::addnode(c("f","g","h"), km) km <- addede("g", "H", km, 1) km <- addede("h", "G", km, 1) ukm <- uraph(km) ukm edes(ukm) connectedcomp(ukm) da.sp DAG shortest paths usin boost C++ Alorithm for the sinle-source shortest-paths problem on a weihted, directed acyclic raph (DAG) Usae da.sp(,start=nodes()[1]) Aruments start instance of class raph source node for start of paths These functions are interfaces to the Boost raph library C++ routines for sinle-source shortestpaths on a weihted directed acyclic raph. Choose appropriate shortest-path alorithms carefully based on the properties of the input raph. See documentation in Boost Graph Library for more details. A list with elements: distance penult start The vector of distances from start to each node of ; includes Inf when there is no path from start. A vector of indices (in nodes()) of predecessors correspondin to each node on the path from that node back to start. For example, if the element one of this vector has value 10, that means that the predecessor of node 1 is node 10. The next predecessor is found by examinin penult[10]. The start node that was supplied in the call to da.sp. Li Lon <li.lon@isb-sib.ch>

17 dijkstra.sp 17 Boost Graph Library ( ) See Also bellman.ford.sp, dijkstra.sp, johnson.all.pairs.sp, sp.between con <- file(system.file("xml/conn2.xl",packae="rbgl"), open="r") dd <- fromgxl(con) close(con) da.sp(dd) da.sp(dd,nodes(dd)[2]) dijkstra.sp Dijkstra s shortest paths usin boost C++ dijkstra s shortest paths Usae dijkstra.sp(,start=nodes()[1], ew=unlist(edeweihts())) Aruments start ew instance of class raph character: node name for start of path numeric: ede weihts. These functions are interfaces to the Boost raph library C++ routines for Dijkstra s shortest paths. For some raph subclasses, computin the ede weihts can be expensive. If you are callin dijkstra.sp in a loop, you can pass the ede weihts explicitly to avoid the ede weiht creation cost. A list with elements: distance penult The vector of distances from start to each node of ; includes Inf when there is no path from start. A vector of indices (in nodes()) of predecessors correspondin to each node on the path from that node back to start

18 18 dominatortree. For example, if the element one of this vector has value 10, that means that the predecessor of node 1 is node 10. The next predecessor is found by examinin penult[10]. start The start node that was supplied in the call to dijkstra.sp. VJ Carey <stvjc@channin.harvard.edu> Boost Graph Library ( ) See Also bellman.ford.sp, da.sp, johnson.all.pairs.sp, sp.between con1 <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") dd <- fromgxl(con1) close(con1) dijkstra.sp(dd) dijkstra.sp(dd,nodes(dd)[2]) con2 <- file(system.file("xml/ospf.xl",packae="rbgl"), open="r") ospf <- fromgxl(con2) close(con2) dijkstra.sp(ospf,nodes(ospf)[6]) dominatortree Compute dominator tree from a vertex in a directed raph Compute dominator tree from a vertex in a directed raph Usae dominatortree(, start=nodes()[1]) lenauertarjandominatortree(, start=nodes()[1]) Aruments start a directed raph, one instance of the raph class a vertex in raph

19 edeconnectivity 19 As stated in documentation on Lenauer Tarjan dominator tree in Boost Graph Library: A vertex u dominates a vertex v, if every path of directed raph from the entry to v must o throuh u. This function builds the dominator tree for a directed raph. Output is a vector, ivin each node its immediate dominator. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) con1 <- file(system.file("xml/dominator.xl",packae="rbgl"), open="r") 1 <- fromgxl(con1) close(con1) dominatortree(1) lenauertarjandominatortree(1) edeconnectivity computed ede connectivity and min disconnectin set for an undirected raph computed ede connectivity and min disconnectin set for an undirected raph Usae edeconnectivity() Aruments an instance of the raph class with edemode undirected Consider a raph G consistin of a sinle connected component. The ede connectivity of G is the minimum number of edes in G that can be cut to produce a raph with two (disconnected) components. The set of edes in this cut is called the minimum disconnectin set.

20 20 edmondsmaxcardinalitymatchin A list: connectivity mindisconset the inteer describin the number of edes that must be severed to obtain two components a list (of lenth connectivity) of pairs of node names describin the edes that need to be cut to obtain two components Vince Carey <stvjc@channin.harvard.edu> Boost Graph Library ( ) See Also mincut, edmonds.karp.max.flow, push.relabel.max.flow con <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) edeconnectivity(coex) edmondsmaxcardinalitymatchin edmondsmaxcardinalitymatchin edmondsmaxcardinalitymatchin description Usae edmondsmaxcardinalitymatchin() Aruments instance of class raphnel from Bioconductor raph class a list with two components: a loical named Is max matchin and a character matrix named Matchin with two rows vertex and matched vertex, entries are node labels.

21 edmondsoptimumbranchin 21 Li Lon Boost Graph Library ( ) V <- LETTERS[1:18] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[4+1], ); <- addede(v[1+1], V[5+1], ); <- addede(v[1+2], V[6+1], ); <- addede(v[1+3], V[7+1], ); <- addede(v[1+4], V[5+1], ); <- addede(v[1+6], V[7+1], ); <- addede(v[1+4], V[8+1], ); <- addede(v[1+5], V[9+1], ); <- addede(v[1+6], V[10+1], ); <- addede(v[1+7], V[11+1], ); <- addede(v[1+8], V[9+1], ); <- addede(v[1+10], V[11+1], ); <- addede(v[1+8], V[13+1], ); <- addede(v[1+9], V[14+1], ); <- addede(v[1+10], V[15+1], ); <- addede(v[1+11], V[16+1], ); <- addede(v[1+14], V[15+1], ); x9 <- edmondsmaxcardinalitymatchin() x9 <- addede(v[1+12], V[13+1], ); <- addede(v[1+16], V[17+1], ); x10 <- edmondsmaxcardinalitymatchin() x10 edmondsoptimumbranchin edmondsoptimumbranchin edmondsoptimumbranchin description Usae edmondsoptimumbranchin()

22 22 extractpath Aruments instance of class raphnel from Bioconductor raph class This is an implementation of Edmonds alorithm to find optimum branchin in a directed raph. See references for details. A list with three elements: edelist, weihts, and nodes for the optimum branchin traversal Li Lon <li.lon@isb-sib.ch> See Edmonds Alorithm on V <- LETTERS[1:4] <- new("raphnel", nodes=v, edemode="directed") <- addede(v[1+0],v[1+1],, 3) <- addede(v[1+0],v[2+1],, 1.5) <- addede(v[1+0],v[3+1],, 1.8) <- addede(v[1+1],v[2+1],, 4.3) <- addede(v[1+2],v[3+1],, 2.2) x11 <- edmondsoptimumbranchin() x11 extractpath convert a dijkstra.sp predecessor structure into the path joinin two nodes determine a path between two nodes in a raph, usin output of dijkstra.sp. Usae extractpath(s, f, pens) Aruments s f pens index of startin node in nodes vector of the raph from which pens was derived index of endin node in nodes vector predecessor index vector as returned in the preds component of dijkstra.sp output

23 FileDep 23 numeric vector of indices of nodes alon shortest path Vince Carey Boost Graph Library ( ) See Also allshortestpaths data(filedep) dd <- dijkstra.sp(filedep) extractpath(1,9,dd$pen) FileDep FileDep: a raphnel object representin a file dependency dataset example in boost raph library FileDep: a raphnel object representin a file dependency dataset example in boost raph library Usae data(filedep) an instance of raphnel Boost Graph Library ( )

24 24 floyd.warshall.all.pairs.sp # this is how the raph of data(filedep) was obtained library(raph) fd <- file(system.file("xml/filedep.xl",packae="rbgl"), open="r") show(fromgxl(fd)) if (require(rraphviz)) { data(filedep) plot(filedep) } close(fd) floyd.warshall.all.pairs.sp compute shortest paths for all pairs of nodes compute shortest paths for all pairs of nodes Usae floyd.warshall.all.pairs.sp() Aruments raph object with ede weihts iven Compute shortest paths between every pair of vertices for a dense raph. It works on both undirected and directed raph. The result is iven as a distance matrix. The matrix is symmetric for an undirected raph, and asymmetric (very likely) for a directed raph. For a sparse raph, the johnson.all.pairs.sp functions should be used instead. See documentation on these alorithms in Boost Graph Library for more details. A matrix of shortest path lenths between all pairs of nodes in the raph. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( )

25 profile 25 See Also johnson.all.pairs.sp con <- file(system.file("xml/conn.xl", packae="rbgl"), open="r") coex <- fromgxl(con) close(con) floyd.warshall.all.pairs.sp(coex) profile Compute profile for a raph Compute profile for a raph Usae profile() Aruments an instance of the raph class The profile of a iven raph is the sum of bandwidths for all the vertices in the raph. See documentation on this function in Boost Graph Library for more details. profile the profile of the raph Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) con <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) profile(coex)

26 26 raphgenerator raphgenerator Generate an undirected raph with adjustable clusterin coefficient Generate an undirected raph with adjustable clusterin coefficient Usae raphgenerator(n, d, o) Aruments n d o no. of nodes in the enerated raph parameter for preferential attachment parameter for triple eneration The raph enerator works accordin to the prefential attachment rule. It also enerates raphs with adjustable clusterin coefficient. Parameter d specifies how many preferred edes a new node has. Parameter o limits how many triples to add to a new node. See reference for details. no. of nodes no. of edes edes No. of nodes in the enerated raph No. of edes in the enerated raph Edes in the enerated raph Li Lon <li.lon@isb-sib.ch> Approximatin Clusterin Coefficient and Transitivity, T. Schank, D. Waner, Journal of Graph Alorithms and Applications, Vol. 9, No. 2 (2005). See Also clusterincoef, transitivity, clusterincoefappr n <- 20 d <- 6 o <- 3 <- raphgenerator(n, d, o)

27 hihlyconnsg 27 hihlyconnsg Compute hihly connected subraphs for an undirected raph Compute hihly connected subraphs for an undirected raph Usae hihlyconnsg(, sat=3, ldv=c(3,2,1)) Aruments sat ldv an instance of the raph class with edemode undirected sinleton adoption threshold, positive inteer heuristics to remove lower deree vertice, a decreasin sequence of positive inteer A raph G with n vertices is hihly connected if its connectivity k(g) > n/2. The HCS alorithm partitions a iven raph into a set of hihly connected subraphs, by usin minimum-cut alorithm recursively. To improve performance, the approach is refined by adoptin sinletons, removin low deree vertices and merin clusters. On sinleton adoption: after each round of partition, some hihly connected subraphs could be sinletons (i.e., a subraph contains only one node). To reduce the number of sinletons, therefore reduce number of clusters, we try to et "normal" subraphs to "adopt" them. If a sinleton, s, has n neihbours in a hihly connected subraph c, and n > sat, we add s to c. To adapt to the modified subraphs, this adoption process is repeated until no further such adoption. On lower deree vertices: when the raph has low deree vertices, minimum-cut alorithm will just repeatedly separate these vertices from the rest. To reduce such expensive and non-informative computation, we "remove" these low deree vertices first before applyin minimum-cut alorithm. Given ldv = (d1, d2,..., dp), (d[i] > d[i+1] > 0), we repeat the followin (i from 1 to p): remove all the hihly-connected-subraph found so far; remove vertices with derees < di; find hihlyconnected-subraphs; perform sinleton adoptions. The Boost implementation does not support self-loops, therefore we sinal an error and suest that users remove self-loops usin the function removeselfloops function. This chane does affect deree, but the oriinal article makes no specific reference to self-loops. A list of clusters, each is iven as vertices in the raph. Li Lon <li.lon@isb-sib.ch> A Clusterin Alorithm based on Graph Connectivity by E. Hartuv, R. Shamir, 1999.

28 28 incremental.components See Also edeconnectivity, mincut, removeselfloops con <- file(system.file("xml/hcs.xl",packae="rbgl")) coex <- fromgxl(con) close(con) hihlyconnsg(coex) incremental.components Compute connected components for an undirected raph Usae Compute connected components for an undirected raph init.incremental.components() incremental.components() same.component(, node1, node2) Aruments node1 node2 an instance of the raph class one vertex of the iven raph another vertex of the iven raph This family of functions work toether to calculate the connected components of an undirected raph. The alorithm is based on the disjoint-sets. It works where the raph is rowin by addin new edes. Call "init.incremental.components" to initialize the calculation on a new raph. Call "incremental.components" to re-calculate connected components after rowin the raph. Call "same.component" to learn if two iven vertices are in the same connected components. Currently, the codes can only handle ONE incremental raph at a time. When you start workin on another raph by callin "init.incremental.components", the disjoint-sets info on the previous raph is lost. See documentation on Incremental Connected Components in Boost Graph Library for more details. Output from init.incremental.components is a list of component numbers for each vertex in the raph. Output from incremental.components is a list of component numbers for each vertex in the raph. Output from same.component is true if both nodes are in the same connected component, otherwise it s false.

29 is.trianulated 29 Li Lon Boost Graph Library ( ) See Also conncomp, connectedcomp, stroncomp con <- file(system.file("xml/conn2.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) init.incremental.components(coex) incremental.components(coex) v1 <- 1 v2 <- 5 same.component(coex, v1, v2) is.trianulated Decide if a raph is trianulated Decide if a raph is trianulated Usae is.trianulated() Aruments an instance of the raph class An undirected raph G = (V, E) is trianulated (i.e. chordal) if all cycles [v1, v2,..., vk] of lenth 4 or more have a chord, i.e., an ede [vi, vj] with j!= i +/- 1 (mod k) An equivalent definition of chordal raphs is: G is chordal iff either G is an empty raph, or there is an v in V such that 1. the neihborhood of v (i.e., v and its adjacent nodes) forms a clique, and 2. recursively, G-v is chordal

30 30 iskuratowskisubraph The return value is TRUE if is trianulated and FALSE otherwise. An error is thrown if the raph is not undirected; you miht use uraph to compute the underlyin raph. Li Lon Combinatorial Optimization: alorithms and complexity (p. 403) by C. H. Papadimitriou, K. Steilitz con1 <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con1) close(con1) is.trianulated(coex) con2 <- file(system.file("xml/hcs.xl",packae="rbgl"), open="r") coex <- fromgxl(con2) close(con2) is.trianulated(coex) iskuratowskisubraph iskuratowskisubraph iskuratowskisubraph description Usae iskuratowskisubraph() Aruments instance of class raphnel from Bioconductor raph class a list with three elements: Is planar (loical), Is there a Kuratowski subraph (loical), and a two-row character matrix Edes of Kuratowski subraph with rows from and two and node names as entries. Li Lon <li.lon@isb-sib.ch>

31 isomorphism 31 Boost Graph Library ( ) V <- LETTERS[1:6] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+0], V[2+1], ) <- addede(v[1+0], V[3+1], ) <- addede(v[1+0], V[4+1], ) <- addede(v[1+0], V[5+1], ) <- addede(v[1+1], V[2+1], ) <- addede(v[1+1], V[3+1], ) <- addede(v[1+1], V[4+1], ) <- addede(v[1+1], V[5+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+2], V[4+1], ) <- addede(v[1+2], V[5+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+3], V[5+1], ) <- addede(v[1+4], V[5+1], ) x4 <- iskuratowskisubraph() x4 isomorphism Compute isomorphism from vertices in one raph to those in another raph Compute isomorphism from vertices in one raph to those in another raph Usae isomorphism(1, 2) Aruments 1 2 one instance of the raph class one instance of the raph class

32 32 isstraihtlinedrawin As stated in documentation on isomorphism in Boost Graph Library: An isomorphism is a 1-to- 1 mappin of the vertices in one raph to the vertices of another raph such that adjacency is preserved. Another words, iven raphs G1 = (V1,E1) and G2 = (V2,E2) an isomorphism is a function f such that for all pairs of vertices a,b in V1, ede (a,b) is in E1 if and only if ede (f(a),f(b)) is in E2. Output is true if there exists an isomorphism between 1 and 2, otherwise it s false. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) con1 <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") 1 <- fromgxl(con1) close(con1) con2 <- file(system.file("xml/conn2.xl",packae="rbgl"), open="r") 2 <- fromgxl(con2) close(con2) isomorphism(1, 2) isstraihtlinedrawin isstraihtlinedrawin isstraihtlinedrawin description Usae isstraihtlinedrawin(, drawin) Aruments drawin instance of class raphnel from Bioconductor raph class coordinates of node positions

33 johnson.all.pairs.sp 33 loical, TRUE if drawin is a straiht line layout for Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) V <- LETTERS[1:7] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+1], V[2+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+3], V[0+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+4], V[5+1], ) <- addede(v[1+5], V[6+1], ) <- addede(v[1+6], V[3+1], ) <- addede(v[1+0], V[4+1], ) <- addede(v[1+1], V[3+1], ) <- addede(v[1+3], V[5+1], ) <- addede(v[1+2], V[6+1], ) <- addede(v[1+1], V[4+1], ) <- addede(v[1+1], V[5+1], ) <- addede(v[1+1], V[6+1], ) x3 <- chrobakpaynestraihtlinedrawin() x8 <- isstraihtlinedrawin(, x3) x8 johnson.all.pairs.sp compute shortest path distance matrix for all pairs of nodes compute shortest path distance matrix for all pairs of nodes Usae johnson.all.pairs.sp()

34 34 kcliques Aruments raph object for which edematrix and edeweihts are defined Uses BGL alorithm. matrix of shortest path lenths, read from row node to col node Vince Carey Boost Graph Library ( ) See Also bellman.ford.sp, da.sp, dijkstra.sp, sp.between con <- file(system.file("dot/joh.xl", packae="rbgl"), open="r") z <- fromgxl(con) close(con) johnson.all.pairs.sp(z) kcliques Find all the k-cliques in an undirected raph Find all the k-cliques in an undirected raph Usae kcliques() Aruments an instance of the raph class

35 kcores 35 Notice that there are different definitions of k-clique in different context. In computer science, a k-clique of a raph is a clique, i.e., a complete subraph, of k nodes. In Social Network Analysis, a k-clique in a raph is a subraph where the distance between any two nodes is no reater than k. Here we take the definition in Social Network Analysis. Let D be a matrix, D[i][j] is the shortest path from node i to node j. Alorithm is outlined as followin: (1) use Johnson s alorithm to fill D; let N = max(d[i][j]) for all i, j; (2) each ede is a 1-clique by itself; (3) for k = 2,..., N, try to expand each (k-1)-clique to k-clique: (3.1) consider a (k-1)-clique the current k-clique KC; (3.2) repeat the followin: if for all nodes j in KC, D[v][j] <= k, add node v to KC; (3.3) eliminate duplicates; (4) the whole raph is N-clique. A list of lenth N; k-th entry (k = 1,..., N) is a list of all the k-cliques in raph. Li Lon <li.lon@isb-sib.ch> Social Network Analysis: Methods and Applications. By S. Wasserman and K. Faust, pp con <- file(system.file("xml/snacliqueex.xl",packae="rbgl")) coex <- fromgxl(con) close(con) kcliques(coex) kcores Find all the k-cores in a raph Find all the k-cores in a raph Usae kcores(, EdeType=c("in", "out")) Aruments EdeType an instance of the raph class what types of edes to be considered when is directed

36 36 lambdasets A k-core in a raph is a subraph where each node is adjacent to at least a minimum number, k, of the other nodes in the subraph. A k-core in a raph may not be connected. The core number for each node is the hihest k-core this node is in. A node in a k-core will be, by definition, in a (k-1)-core. The implementation is based on the alorithm by V. Bataelj and M. Zaversnik, The example snacoreex.xl is in the paper by V. Bataelj and M. Zaversnik, A vector of the core numbers for all the nodes in. Li Lon <li.lon@isb-sib.ch> Social Network Analysis: Methods and Applications. By S. Wasserman and K. Faust, pp An O(m) Alorithm for Cores decomposition of networks, by V. Bataelj and M. Zaversnik, con1 <- file(system.file("xml/snacoreex.xl",packae="rbgl")) kcoex <- fromgxl(con1) close(con1) kcores(kcoex) con2 <- file(system.file("xml/conn2.xl",packae="rbgl")) kcoex2 <- fromgxl(con2) close(con2) kcores(kcoex2) kcores(kcoex2, "in") kcores(kcoex2, "out") lambdasets Find all the lambda-sets in an undirected raph Find all the lambda-sets in an undirected raph Usae lambdasets()

37 layout 37 Aruments an instance of the raph class From reference (1), p. 270: A set of nodes is a lambda-set if any pair of nodes in the lambda set has larer ede connectivity than any pair of nodes consistin of one node from within the lamda set and a second node from outside the lamda set. As stated in reference (2), a lambda set is a maximal subset of nodes who have more ede-independent paths connectin them to each other than to outsiders. A lambda set could be characterized by the minimum ede connectivity k amon its members, and could be called lambda-k sets. Let N be maximum ede connectivity of raph, we output all the lambda-k set for all k = 1,..., N. Maximum ede connectivity, N, of the raph, and A list of lenth N; k-th entry (k = 1,..., N) is a list of all the lambda-k sets in raph. Li Lon <li.lon@isb-sib.ch> (1) Social Network Analysis: Methods and Applications. By S. Wasserman and K. Faust, pp (2) LS sets, lambda sets and other cohesive subsets. By S. P. Boratti, M. G. Everett, P. R. Shirey, Social Networks 12 (1990) p con <- file(system.file("xml/snalambdaex.xl",packae="rbgl")) coex <- fromgxl(con) close(con) lambdasets(coex) layout Layout an undirected raph in 2D suspended june Layout an undirected raph in 2D suspended june Usae circlelayout(, radius=1) # does not compile with boost 1.49 kamadakawaisprinlayout(, ede_or_side=1, es_lenth=1 ) fruchtermanreinoldforcedirectedlayout(, width=1, heiht=1) randomgraphlayout(, minx=0, maxx=1, miny=0, maxy=1)

38 38 layout Aruments radius ede_or_side es_lenth width heiht minx maxx miny maxy an instance of the raph class with edemode undirected radius of a reular n-polyon boolean indicatin the lenth is for an ede or for a side, default is for an ede the lenth of an ede or a side for layout the width of the dislay area, all x coordinates fall in [-width/2, width/2] the heiht of the display area, all y coordinates fall in [-heiht/2, heiht/2] minimum x coordinate maximum x coordinate minimum y coordinate maximum y coordinate If you want to simply draw a raph, you should consider usin packae Rraphviz. The layout options in packae Rraphviz: neato, circo and fdp, correspond to kamadakawaisprinlayout, circlelayout and fruchtermanreinoldforcedirectedlayout, respectively. Function circlelayout layouts the raph with the vertices at the points of a reular n-polyon. The distance from the center of the polyon to each point is determined by the radius parameter. Function kamadakawaisprinlayout provides Kamada-Kawai sprin layout for connected, undirected raphs. User provides either the unit lenth e of an ede in the layout or the lenth of a side s of the display area. Function randomgraphlayout places the points of the raph at random locations. Function fruchtermanreinoldforcedirectedlayout performs layout of unweihted, undirected raphs. It s a force-directed alorithm. The BGL implementation doesn t handle disconnected raphs very well, since it doesn t explicitly ive each connected component a reion proportional to its size. See documentation on this function in Boost Graph Library for more details. A (2 x n) matrix, where n is the number of nodes in the raph, each column ives the (x, y)- coordinates for the correspondin node. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) See Also layoutgraph

39 makebiconnectedplanar 39 ## Not run: con <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) coex <- uraph(coex) circlelayout(coex) kamadakawaisprinlayout(coex) randomgraphlayout(coex) fruchtermanreinoldforcedirectedlayout(coex, 10, 10) ## End(Not run) makebiconnectedplanar makebiconnectedplanar makebiconnectedplanar description Usae makebiconnectedplanar() Aruments instance of class raphnel from Bioconductor raph class From An undirected raph is connected if, for any two vertices u and v, there s a path from u to v. An undirected raph is biconnected if it is connected and it remains connected even if any sinle vertex is removed. Finally, a planar raph is maximal planar (also called trianulated) if no additional ede (with the exception of self-loops and parallel edes) can be added to it without creatin a nonplanar raph. Any maximal planar simple raph on n > 2 vertices has exactly 3n - 6 edes and 2n - 4 faces, a consequence of Euler s formula. If a planar raph isn t connected, isn t biconnected, or isn t maximal planar, there is some set of edes that can be added to the raph to make it satisfy any of those three properties while preservin planarity. Many planar raph drawin alorithms make at least one of these three assumptions about the input raph, so there are functions in the Boost Graph Library that can help: make_connected adds a minimal set of edes to an undirected raph to make it connected. make_biconnected_planar adds a set of edes to a connected, undirected planar raph to make it biconnected while preservin planarity.

40 40 makeconnected make_maximal_planar adds a set of edes to a biconnected, undirected planar raph to make it maximal planar. The function documented here implements the second approach. A list with two elements: Is planar is a loical indicatin achievement of planarity, and new raph, a raphnel instance that is biconnected and planar. Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) V <- LETTERS[1:11] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+3], V[0+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+4], V[5+1], ) <- addede(v[1+5], V[3+1], ) <- addede(v[1+5], V[6+1], ) <- addede(v[1+6], V[7+1], ) <- addede(v[1+7], V[8+1], ) <- addede(v[1+8], V[5+1], ) <- addede(v[1+8], V[9+1], ) <- addede(v[1+0], V[10+1], ) x6 <- makebiconnectedplanar() x6 makeconnected makeconnected makeconnected description Usae makeconnected()

41 makemaximalplanar 41 Aruments instance of class raphnel from Bioconductor raph class a raphnel instance with a minimal set of edes added to achieve connectedness. an instance of raphnel Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) V <- LETTERS[1:11] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+5], V[6+1], ) <- addede(v[1+6], V[7+1], ) <- addede(v[1+8], V[9+1], ) <- addede(v[1+9], V[10+1], ) <- addede(v[1+10], V[8+1], ) x5 <- makeconnected() x5 makemaximalplanar makemaximalplanar makemaximalplanar description Usae makemaximalplanar()

42 42 max.flow Aruments instance of class raphnel from Bioconductor raph class see a list with two elements, Is planar:, a loical indicatin state of raph, and new raph, a raph- NEL instance Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) V <- LETTERS[1:10] <- new("raphnel", nodes=v, edemode="undirected") <- addede(v[1+0], V[1+1], ) <- addede(v[1+1], V[2+1], ) <- addede(v[1+2], V[3+1], ) <- addede(v[1+3], V[4+1], ) <- addede(v[1+4], V[5+1], ) <- addede(v[1+5], V[6+1], ) <- addede(v[1+6], V[7+1], ) <- addede(v[1+7], V[8+1], ) <- addede(v[1+8], V[9+1], ) x7 <- makemaximalplanar() x7 max.flow Compute max flow for a directed raph Compute max flow for a directed raph

43 max.flow 43 Usae edmonds.karp.max.flow(, source, sink) push.relabel.max.flow(, source, sink) kolmoorov.max.flow(, source, sink) Aruments source sink an instance of the raph class with edemode directed node name (character) or node number (int) for the source of the flow node name (character) or node number (int) for the sink of the flow Given a directed raph G=(V, E) of a sinle connected component with a vertex source and a vertex sink. Each arc has a positive real valued capacity, currently it s equivalent to the weiht of the arc. The flow of the network is the net flow enterin the vertex sink. The maximum flow problem is to determine the maximum possible value for the flow to the sink and the correspondin flow values for each arc. See documentation on these alorithms in Boost Graph Library for more details. A list of maxflow edes flows the max flow from source to sink the nodes of the arcs with non-zero capacities the flow values of the arcs with non-zero capacities Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) See Also mincut, edeconnectivity con <- file(system.file("xml/dijkex.xl",packae="rbgl"), open="r") <- fromgxl(con) close(con) ans1 <- edmonds.karp.max.flow(, "B", "D") ans2 <- edmonds.karp.max.flow(, 3, 2) ans3 <- push.relabel.max.flow(, 2, 4) # 3 and 2 equivalent to "C" and "B" # 2 and 4 equivalent to "B" and "D"

44 44 maxclique ans4 <- push.relabel.max.flow(, "C", "B") # error in the followin now, 14 june 2014 #ans5 <- kolmoorov.max.flow(, "B", "D") #ans6 <- kolmoorov.max.flow(, 3, 2) maxclique Find all the cliques in a raph Find all the cliques in a raph Usae maxclique(, nodes=null, edemat=null) Aruments nodes edemat an instance of the raph class vector of node names, to be supplied if is not 2 x p matrix with indices of edes in nodes, one-based, only to be supplied if code is not Notice the maximum clique problem is NP-complete, which means it cannot be solved by any known polynomial alorithm. We implemented the alorithm by C. Bron and J. Kerbosch, It is an error to supply both and either of the other aruments. If is not supplied, no checkin of the consistency of nodes and edemat is performed. maxclique list of all cliques in Li Lon <li.lon@isb-sib.ch> Findin all cliques of an undirected raph, by C. Bron and J. Kerbosch, Communication of ACM, Sept 1973, Vol 16, No. 9.

45 maximumcycleratio 45 con1 <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con1) close(con1) maxclique(coex) con2 <- file(system.file("xml/hcs.xl",packae="rbgl"), open="r") coex <- fromgxl(con2) close(con2) maxclique(coex) maximumcycleratio maximumcycleratio maximumcycleratio description Usae maximumcycleratio() Aruments instance of class raphnel from Bioconductor raph class NOT IMPLEMENTED A list with messae indicatin not implemented Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( )

46 46 mincut mincut Compute min-cut for an undirected raph Compute min-cut for an undirected raph Usae mincut() Aruments an instance of the raph class with edemode undirected Given an undirected raph G=(V, E) of a sinle connected component, a cut is a partition of the set of vertices into two non-empty subsets S and V-S, a cost is the number of edes that are incident on one vertex in S and one vertex in V-S. The min-cut problem is to find a cut (S, V-S) of minimum cost. For simplicity, the returned subset S is the smaller of the two subsets. A list of mincut S V-S the number of edes to be severed to obtain the minimum cut the smaller subset of vertices in the minimum cut the other subset of vertices in the minimum cut Li Lon <li.lon@isb-sib.ch> Boost Graph Library ( ) See Also edeconnectivity con <- file(system.file("xml/conn.xl",packae="rbgl"), open="r") coex <- fromgxl(con) close(con) mincut(coex)