Graph drawing and graph analysis utilities

Yifan Hu, Xiangdong Wen

Wolfram Research Inc

Abstract

Graph drawing algorithms automatically layout connectivity information in a visually appealing way. Applications of graph drawing include software engineering (flow charts), bioinformatics (protein network), genealogy, sociology (social network), VLSI design, and visualization of hypertext links. We will present new system functions GraphPlot, TreePlot and LayeredGraphPlot in the coming version of Mathematica, the algorithms, design and usage. In addition, we will discuss new functions for the analysis of graphs, including PageRank and CommunityStructureAssignment, as well as an interactive tool GraphEdit for inputting and edit graphs.

Outline of the talk

Why Graph Drawing and Analysis?

Graph Drawing functions

GraphUtilities package

Why Graph Drawing and Graph Analysis?

Graph: g = {V, E}, V: vertices, E: edges.

Graph encapsulates complex relations and connections.

Increased interest from areas such as bioinformatics, social sciences.

Of interest: visualizing complex relations; discoverying hidden pattern and gaining knowledge from such relations.

Tools: graph drawing, graph analysis,...

Good graph drawing usually more helpful than a texture description in displaying complex relations.

Graph analysis helps to gain quantitative knowledge.

Graph Drawing: the problem

Graph drawing: the process from {1 -> 5, 1 -> 6, 2 -> 4, 2 -> 6, 3 -> 4, 3 -> 5, 3 -> 6, 4 -> 6, 5 -> 6} to

A difficult problem: hard to quantify aesthetics/beauty; hard to find the best drawing manually other than for very small graphs.

An automatic procedure is thus useful.

Graph Drawing: pre 6.0 status

Very efficient and high quality graph drawing algorithms for straight edge graph drawing are available in DiscreteMath`GraphPlot (V 5.1).

Works well for undirected graph (directed graph are treated as undirected).

Graph Drawing: Key ingradient

forced directed algorithms

SocialNetwork={"李"->"胡","王"->"胡","李"->"王","Carol"->"Andre","Carol"->"Fernando","Carol"->"Diane","Andre"->"Diane","李"->"黄","Andre"->"Fernando","黄"->"徐","Andre"->"Beverly","Jane"->"李","Fernando"->"Diane","Fernando"->"Garth","Fernando"->"Heather","Diane"->"Beverly","Diane"->"Garth","Diane"->"Ed","Beverly"->"Garth","Beverly"->"Ed","Garth"->"Ed","Garth"->"Heather","Jane"->"胡","Heather"->"Jane","徐"->"王"};
GraphPlot[SocialNetwork,VertexLabeling->True]

The spring electrical embedding model

Graph Drawing: Key Ingradients

multilevel graph coarsening to give a "global view"

Graph Drawing: Key Ingradients

Octree data structure to overcome O(|V((|)^2)) all to all force calculation

Has overall complexity of O(|V| Log[|V|] + |E|).

Graph Drawing: What's New in 6.0

GraphPlot, GraphPlot3D, and TreePlot are now system functions.

Tooltips for vertices are automatically on

GraphPlot[{4->3,5->2,5->3,5->4,6->1,6->2,6->4,6->5,7->1,7->2,7->3,7->5}]

Labeling is much easier

GraphPlot[{4->3,5->2,5->3,5->4,6->1,6->2,6->4,6->5,7->1,7->2,7->3,7->5},VertexLabeling->True]

Using arrows for edges are also much easier

GraphPlot[{4->3,5->2,5->3,5->4,6->1,6->2,6->4,6->5,7->1,7->2,7->3,7->5},DirectedEdges->True,VertexLabeling->True]

Support of multiedges and self-loops

GraphPlot[{1->2,2->1,3->1,3->2,4->1,4->2,4->4},VertexLabeling->True]

Here we make the self - loop smaller and multiedges more prominent

GraphPlot[{1->2,2->1,3->1,3->2,4->1,4->2,4->4},VertexLabeling->True,SelfLoopStyle->0.3,MultiedgeStyle->0.3]

Edge labels are displayed if supplied

GraphPlot[{1->2,2->1,3->1,3->2,4->1,4->2,{4->4,"loop"}},VertexLabeling->True]

You can display edge labels using tooltips

GraphPlot[{1->5,1->6,2->4,2->6,3->4,3->5,{3->6,"edge 3->6"},4->6,5->6},EdgeLabeling->Automatic,VertexLabeling->True]

Or doing it this way

GraphPlot[{Tooltip[1,"vertex 1"]->5,1->6,2->4,2->6,3->4,3->5,Tooltip[3->6,"edge 3->6"],4->6,5->6},VertexLabeling->True]

VertexRenderingFunction and EdgeRenderingFunction: fine controls

VertexRenderingFunction is in the form vf[coord, vtxname, vtxind]

Plot edges with gray arrows with ends set back from vertices by 0.1

GraphPlot[{1->2,2->3,3->4,4->5,5->6,6->7,7->1,11->12,12->13,13->14,14->15,15->16,16->17,17->11,1->11,2->12,3->13,4->14,5->15,6->16,7->17},EdgeRenderingFunction->({GrayLevel[0.5],Arrow[#1,0.1]}&)]

VertexRenderingFunction and EdgeRenderingFunction: fine controls

EdgeRenderingFunction is in the form ef[coords, vtxnames, eglbl, vtxinds]

This plots a 3D graph using spheres for vertices and cylinders for edges.

GraphPlot3D[{2->1,5->1,6->1,3->2,11->2,4->3,16->3,5->4,21->4,26->5,7->6,10->6,8->7,30->7,9->8,42->8,10->9,38->9,12->10,12->11,15->11,13->12,14->13,37->13,15->14,33->14,17->15,17->16,20->16,18->17,19->18,32->18,20->19,53->19,22->20,22->21,25->21,23->22,24->23,52->23,25->24,48->24,27->25,27->26,30->26,28->27,29->28,47->28,30->29,43->29,32->31,35->31,54->31,33->32,34->33,35->34,36->34,56->35,37->36,40->36,38->37,39->38,40->39,41->39,57->40,42->41,45->41,43->42,44->43,45->44,46->44,58->45,47->46,50->46,48->47,49->48,50->49,51->49,59->50,52->51,55->51,53->52,54->53,55->54,60->55,57->56,60->56,58->57,59->58,60->59},EdgeRenderingFunction->({Cylinder[#1,.1]}&),VertexRenderingFunction->({Sphere[#1,.25]}&),Background->Black]

VertexRenderingFunction and EdgeRenderingFunction: fine controls

This plot each vertex using a football, using Inset

gr = ;
gr2=GraphPlot[g={1->2,2->3,1->3},VertexRenderingFunction->(Inset[gr,#1]&),ImageSize->100]

gr3=GraphPlot[g,VertexRenderingFunction->(Inset[gr2,#1]&),ImageSize->200]

gr4=GraphPlot[g,VertexRenderingFunction->(Inset[gr3,#1]&),ImageSize->400]

GraphPlot[g,VertexRenderingFunction->(Inset[gr4,#1]&),ImageSize->800]

LayeredGraphPlot is another new functions

hierarchical styled drawing of directed graphs

LayeredGraphPlot[{4->3,5->3,5->4,6->1,6->2,6->4,6->5,6->3},VertexLabeling->True]

LayeredGraphPlot

A food chain examples

LayeredGraphPlot[{"John"->"plants","lion"->"John","tiger"->"John","tiger"->"deer","lion"->"deer","deer"->"plants","mosquito"->"lion","frog"->"mosquito","mosquito"->"tiger","John"->"cow","cow"->"plants","mosquito"->"deer","mosquito"->"John","snake"->"frog","vulture"->"snake"},Left,VertexLabeling->True,ImageSize->800]

LayeredGraphPlot is another new functions

A flow chart

LayeredGraphPlot[{"ratiotest"->"ratiolimit","getdata"->"dir_inf _bad","getdata"->"evalderiv","NSPpreliminaries"->"getdata","findsp"->"termlimit","findsp"->"ratiotest","findsp"->"nSumW","findsp"->"nSumEM","findsp"->"nProductEM","findsp"->"nProductW","nSumW"->"nSumFinite","nSumW"->"nSumWtail","nSumEM"->"evalterm","nSumEM"->"nSumFinite","nSumEM"->"evalderiv","nProductEM"->"nPFinite","nProductW"->"nPFinite","nProductW"->"nPWtail","oNSum"->"CleanOptionList","oNSum"->"multidim","oNSum"->"NSPpreliminaries","oNSum"->"findsp","nSumFinite"->"evalterm","nSumWtail"->"evalterm","nSumWtail"->"epsalg","oNProduct"->"CleanOptionList","oNProduct"->"multidim","oNProduct"->"NSPpreliminaries","oNProduct"->"findsp","nPFinite"->"evalterm","nPWtail"->"evalterm","nPWtail"->"epsalg","oSequenceLimit"->"epsalg"},Left,VertexLabeling->True,AspectRatio->1,ImageSize->600]

Dealing with disconnected graphs

For disconnected graphs, a new option PackingStyle is available.

The default value is "NestestGrid", the other value is "ClosestPacking", which implement a new polyomino packing algorithm.

This is the default nested-grid packing

g=(n=100;SeedRandom[321];
Flatten[Table[Table[x=RandomReal[1,{n+1}];
j=Random[Integer,{2,6}];
Table[x[[i+1]]->x[[Mod[i^j,n]+1]],{i,0,n}]],{10}]]);
GraphPlot[g]

Dealing with disconnected graphs

This is the result of PackingStyle -> "ClosestPacking"

GraphPlot[g,PackingStyle->"ClosestPacking"]

Polyomino packing algorithm: the polyominos

Polyomino packing algorithm: adjusting padding

GraphPlot[g,PackingStyle->{"ClosestPacking","PolyominoPadding"->1},Plot->True]

Challenges: taking care of vertex sizes

Sometimes you need to layout vertices so that some vertices occupies more space

Example: a vertex with long name want more space

Undocumented support, which works to some extent for GraphPlot

It works by adjusting the repulsive/attractive forces

GraphPlot[{1->2,2->3,3->4,4->1},"VertexSizes"->{1,2,3,1},VertexLabeling->True]

<<GraphUtilities`;

sz = RandomInteger[{1,4},{16}];GraphPlot[GridGraph[4,4],"VertexSizes"->sz,VertexRenderingFunction->(Disk[#1,sz[[#3]]*0.1]&),VertexLabeling->True]

If a few vertices have huge vertex weight, things can be a little execpected

sz = Table[1,{16}];sz[[6]]=4;GraphPlot[GridGraph[4,4],"VertexSizes"->sz,VertexRenderingFunction->(Disk[#1,sz[[#3]]*0.2]&),VertexLabeling->True]

Challenges: taking care of vertex sizes

For LayeredGraphPlot the support is better

LayeredGraphPlot[g={4->3,555555->3,22222222222222->3,555555->4,6->1,6->22222222222222,6->4,6->555555,6->3},VertexLabeling->True]

LayeredGraphPlot[g,VertexLabeling->True,"VertexSizes"->Map[{Sqrt[StringLength[ToString[#]]],1.}&,VertexList[g]]]

Challenges: taking care of edge weights

Users may want to draw some edges longer than others: take edge weights into account

GraphPlot[{{1,1,0,2},{1,1,1,0},{0,1,1,1},{2,0,1,1}},VertexLabeling->True]

Undocumented support based on tuning the repulsive/attractive force

GraphPlot[{{1,1,0,2},{1,1,1,0},{0,1,1,1},{2,0,1,1}},Method->{"SpringElectricalEmbedding","EdgeWeighted"->True},VertexLabeling->True]

Challenges: constraints

User want certain node to be at certain position

GraphPlot[{1->2,2->3,3->4,4->1},VertexLabeling->True,VertexCoordinateRules->{1->{1,1},2->{2,2},3->{4,2}},Frame->True,FrameTicks->True]

A larger example

GraphPlot[GridGraph[8,8],VertexLabeling->True,VertexCoordinateRules->{1->{1,1},2->{2,2},3->{4,2}},Frame->True,FrameTicks->True]

But there are difficulties

GraphPlot[GridGraph[9,9],VertexLabeling->True,VertexCoordinateRules->{1->{1,1},2->{2,2},3->{4,2}},Frame->True,FrameTicks->True]

Examples

Interatice thesarus

SynonymPlot[word_String,Grouped_:False] := DynamicModule[
{x, g, g1, g2, ng, hash, res,hashfun,i},
x = WordData[word, "Synonyms"];
If [Length[x] == 0 || ! ListQ[x], Return[Graphics[{}]]];
ng = Length[x];
hash= .;
hash[_]:=0;
hashfun[y_List,val_?NumericQ]:=(Map[(hash[#]=val)&,y]);Thread[hashfun[x,Range[ng]/ng]];
hash[i_Integer]:= i/ng;
g1 = Thread[Rule[word, Flatten[x]]];
g2 = Map[Outer[Rule, #, #] &, x];
g = Flatten[{g1, g2}];If[Grouped,i=0;g=Flatten[Map[{"run"->++i,Thread[i->#](*,Outer[Rule, #, #]*)}&,x]];];
res = GraphPlot[g, EdgeRenderingFunction -> ({GrayLevel[0.8], If [#2[[1]] =!= word &&!IntegerQ[#2[[1]]]&& #2[[2]] =!= word && !IntegerQ[#2[[2]]], {}, Line[#1]]} &), VertexRenderingFunction ->(If[#2 === word, {Black,Text[Framed[#2, Background -> Yellow], #1]}, {Blend["Memphis",hash[#2]], If[IntegerQ[#2],Disk[#,0.1],Text[Framed[ Button[#2, Dynamic[res = SynonymPlot[#2,Grouped]], Appearance -> None],Background -> GrayLevel[1, 0.6]], #1]]}] &), SelfLoopStyle -> False,MultiedgeStyle -> False, VertexLabeling -> True,Method -> {"SpringElectricalEmbedding","RepulsiveForcePower"->-1.2},ImageSize -> 600];
Dynamic[res]
];
SynonymPlot["good"]

A version that animates

SynonymPlot[word_String] := DynamicModule[
{x, g, g1, g2, ng, hash, res,coord,z,now,sta,plotrg,coord2},
x = WordNetData[word, "Synonyms"];If [Length[x] == 0 || ! ListQ[x], Return[Graphics[{}]]];
ng = Length[x];
hash[_] := 0;
auroraColor[v_] := Blend[{, , , , , }, v];
g1 = Thread[Rule[word, Flatten[x]]];
g2 = Map[Outer[Rule, #, #] &, x];
g = Flatten[{g1, g2}];
coord=GraphPlot[g,
Method -> {"SpringElectricalModel", "RepulsiveForcePower" -> -1.2}][[1,2,2]];
sta = RandomReal[1,{Length[coord]}];
plotrg =Map[{Min[#],Max[#]}&, Transpose[coord]];
now = FromDate[Date[]];
res=Dynamic[
z=1/(1+(FromDate[Refresh[Date[],UpdateInterval->0.1]]-now));
z=z^2;
If[z>0.001,coord2 = z*sta+(1-z)*coord,coord2=coord];
 GraphPlot[g,EdgeRenderingFunction -> ({GrayLevel[0.8],  If [#2[[1]] =!= word && #2[[2]] =!= word, {}, Line[#1]]} &), VertexRenderingFunction ->(If[#2 === word, {Black, Text[Framed[#2, Background -> Yellow], #1]}, {auroraColor[#3/ng], Text[Framed[ Button[#2, Dynamic[res = SynonymPlot[#2]], Appearance -> None],Background -> GrayLevel[1, 0.6]], #1]}] &), SelfLoopStyle -> False,MultiedgeStyle -> False, VertexLabeling -> True, VertexCoordinateRules->coord2,ImageSize -> 600,PlotRange->plotrg]
];
Dynamic[res]

];

SynonymPlot["run"]

Examples

E. coli transcription networks

This is based on

R Milo, S Shen-Orr, S Itzkovitz, N Kashtan, D Chklovskii & U Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science, 298:824-827 (2002)

"...Transcriptional regulation networks in cells orchestrate gene expression. In this network the 'nodes' are operons, and each 'edge' is directed from an operon that encodes a transcription factor to an operon that it directly regulates (an operon is one or more genes transcribed on the same mRNA)...."

The data is found at http://www.weizmann.ac.il/mcb/UriAlon/groupNetworksData.html

The network

TranscriptionNetwork={{"acrR"->"acrAB",2},{"ada_alkB"->"aidB",1},{"ada_alkB"->"alkA",1},{"adiA_adiY"->"adiA",1},{"alpA"->"slp",1},{"appY"->"appCBA",1},{"araC"->"araBAD",3},{"araC"->"araE",3},{"araC"->"araFG_araH_1H_2",3},{"araC"->"araJ",1},{"arcA"->"aceBAK",2},{"arcA"->"appCBA",1},{"arcA"->"appY",1},{"arcA"->"betIBA",2},{"arcA"->"cydAB",1},{"arcA"->"cyoABCDE",2},{"arcA"->"dctA",2},{"arcA"->"fadBA",2},{"arcA"->"focA_pflB",1},{"arcA"->"fumA",2},{"arcA"->"fumC",2},{"arcA"->"glcDEFGB",2},{"arcA"->"glpACB",2},{"arcA"->"gltA",2},{"arcA"->"icdA",2},{"arcA"->"lctPRD",2},{"arcA"->"mdh",2},{"arcA"->"nuoABCEFGHIJKLMN",2},{"arcA"->"sdhCDAB_b0725_sucABCD",2},{"arcA"->"sodA",2},{"argR"->"argCBH",2},{"argR"->"argD",2},{"argR"->"argE",2},{"argR"->"argF",2},{"argR"->"argI",2},{"argR"->"carAB",2},{"arsR"->"arsEFG",2},{"asnC"->"asnA",1},{"atoC"->"atoDAB",1},{"atoC"->"atoDAE",1},{"betIBA"->"betT",2},{"birA_murA"->"bioA",2},{"birA_murA"->"bioBFCD",2},{"cadC"->"cadBA",1},{"caiF"->"caiTABCDE",1},{"caiF"->"fixABCX",1},{"cbl"->"ssuEADCB",1},{"cbl"->"tauABCD",1},{"cpxAR"->"cpxP",1},{"cpxAR"->"dsbA",1},{"cpxAR"->"ecfI",1},{"cpxAR"->"htrA",1},{"cpxAR"->"motABcheAW",2},{"cpxAR"->"rotA",1},{"cpxAR"->"skp_lpxDA_fabZ",1},{"cpxAR"->"tsr",2},{"cpxAR"->"xprB_dsbC_recJ",1},{"crp"->"acs",1},{"crp"->"aldB",1},{"crp"->"ansB",1},{"crp"->"araBAD",1},{"crp"->"araC",1},{"crp"->"araE",1},{"crp"->"araFG_araH_1H_2",1},{"crp"->"araJ",1},{"crp"->"caiF",1},{"crp"->"caiTABCDE",1},{"crp"->"cirA",1},{"crp"->"cpdB",1},{"crp"->"cyaA",2},{"crp"->"dadAX",1},{"crp"->"dctA",2},{"crp"->"dcuB_fumB",2},{"crp"->"deoCABD",3},{"crp"->"dsdXA",1},{"crp"->"ebgAC",1},{"crp"->"epd_pgk",1},{"crp"->"fadL",2},{"crp"->"fixABCX",1},{"crp"->"flhDC",1},{"crp"->"focA_pflB",1},{"crp"->"fucAO",1},{"crp"->"fucPIKUR",1},{"crp"->"fur",1},{"crp"->"galETKM",3},{"crp"->"galS",1},{"crp"->"glgCAP",1},{"crp"->"glgS",1},{"crp"->"glnALG",1},{"crp"->"glpACB",1},{"crp"->"glpD",1},{"crp"->"glpFK",1},{"crp"->"glpTQ",1},{"crp"->"gltA",1},{"crp"->"gntKU",1},{"crp"->"gntT",1},{"crp"->"ivbL_ilvBN",1},{"crp"->"lacZYA",1},{"crp"->"malEFG",1},{"crp"->"malI",2},{"crp"->"malK_lamB_malM",1},{"crp"->"malS",1},{"crp"->"malT",1},{"crp"->"malXY",3},{"crp"->"manXYZ",1},{"crp"->"melAB",1},{"crp"->"melR",1},{"crp"->"mglBAC",1},{"crp"->"nagBACD",1},{"crp"->"nagE",1},{"crp"->"nupG",3},{"crp"->"ompA",2},{"crp"->"ppiA",3},{"crp"->"proP",3},{"crp"->"ptsHI_crr",3},{"crp"->"rhaBAD",3},{"crp"->"rhaT",1},{"crp"->"rpoH",3},{"crp"->"sdhCDAB_b0725_sucABCD",1},{"crp"->"speC",2},{"crp"->"srlAEBD_gutM_srlR_gutQ",1},{"crp"->"tdcABCDEFG",1},{"crp"->"tnaLAB",1},{"crp"->"tsx",3},{"crp"->"ubiG",1},{"crp"->"udp",1},{"crp"->"uhpT",1},{"crp"->"yhfA",3},{"crp"->"yiaKLMNOPQRS",1},{"csgDEFG"->"csgBA",1},{"cspA"->"gyrA",1},{"cspA"->"hns",1},{"cynR"->"cynTSX",1},{"cysB"->"cysDNC",1},{"cysB"->"cysJIH",1},{"cysB"->"cysK",1},{"cysB"->"cysPUWAM",1},{"cysB"->"tauABCD",1},{"cytR"->"deoCABD",2},{"cytR"->"nupC",2},{"cytR"->"nupG",2},{"cytR"->"ppiA",2},{"cytR"->"rpoH",2},{"cytR"->"tsx",3},{"cytR"->"udp",2},{"deoR"->"deoCABD",2},{"deoR"->"nupG",2},{"deoR"->"tsx",2},{"dnaA"->"nrdAB",1},{"dnaA"->"rpoH",2},{"dsdC"->"dsdXA",1},{"ebgR"->"ebgAC",2},{"envY_ompT"->"ompC",1},{"envY_ompT"->"ompF",1},{"evgA"->"ompC",1},{"exuR"->"exuT",2},{"exuR"->"uxaCA",2},{"exuR"->"uxuABR",2},{"fadR"->"fabA",1},{"fadR"->"fadBA",2},{"fadR"->"fadL",2},{"fadR"->"iclMR",2},{"fadR"->"uspA",2},{"fecIR"->"fecABCDE",1},{"fhlA"->"fdhF",1},{"fhlA"->"hycABCDEFGH",1},{"fhlA"->"hypABCDE",1},{"flhDC"->"flgAMN",1},{"flhDC"->"flgBCDEFGHIJK",1},{"flhDC"->"flhBAE",1},{"flhDC"->"fliAZY",1},{"flhDC"->"fliE",1},{"flhDC"->"fliFGHIJK",1},{"flhDC"->"fliLMNOPQR",1},{"fliAZY"->"flgBCDEFGHIJK",1},{"fliAZY"->"flgKL",1},{"fliAZY"->"flgMN",1},{"fliAZY"->"flhBAE",1},{"fliAZY"->"fliC",1},{"fliAZY"->"fliDST",1},{"fliAZY"->"fliE",1},{"fliAZY"->"fliFGHIJK",1},{"fliAZY"->"fliLMNOPQR",1},{"fliAZY"->"motABcheAW",1},{"fliAZY"->"tarTapcheRBYZ",1},{"fliAZY"->"tsr",1},{"fnr"->"acs",1},{"fnr"->"ansB",1},{"fnr"->"arcA",1},{"fnr"->"aspA",1},{"fnr"->"caiF",1},{"fnr"->"cydAB",2},{"fnr"->"cyoABCDE",2},{"fnr"->"dcuB_fumB",1},{"fnr"->"dmsABC",1},{"fnr"->"fdnGHI",1},{"fnr"->"focA_pflB",1},{"fnr"->"frdABCD",1},{"fnr"->"glpACB",1},{"fnr"->"hypABCDE",1},{"fnr"->"icdA",2},{"fnr"->"narGHJI",1},{"fnr"->"narK",1},{"fnr"->"ndh",2},{"fnr"->"nirBDC_cysG",1},{"fnr"->"nuoABCEFGHIJKLMN",2},{"fnr"->"sdhCDAB_b0725_sucABCD",2},{"fnr"->"tdcABCDEFG",1},{"FruR"->"aceBAK",2},{"FruR"->"adhE",2},{"FruR"->"fruBKA",2},{"FruR"->"icdA",1},{"FruR"->"ppsA",1},{"FruR"->"ptsHI_crr",3},{"FruR"->"pykF",2},{"fucPIKUR"->"fucAO",1},{"fur"->"cirA",2},{"fur"->"entCEBA",2},{"fur"->"fecIR",2},{"fur"->"fepA_entD",2},{"fur"->"fepB",2},{"fur"->"fepDGC",2},{"fur"->"fhuACDB",2},{"fur"->"sodA",2},{"fur"->"tonB",2},{"GalR"->"galETKM",2},{"GalR"->"galS",2},{"galS"->"mglBAC",2},{"gatR_1"->"gatYZABCDR_2",2},{"gcvA"->"gcvTHP",3},{"gcvR"->"gcvTHP",2},{"glcC"->"glcDEFGB",1},{"glnALG"->"glnHPQ",3},{"glnALG"->"nac",1},{"glpR"->"glpACB",2},{"glpR"->"glpD",2},{"glpR"->"glpFK",2},{"glpR"->"glpTQ",2},{"gntR"->"edd_eda",2},{"gntR"->"gntKU",2},{"gntR"->"gntT",2},{"hcaR"->"hcaA1A2CBD_yphA",1},{"himA"->"aceBAK",1},{"himA"->"caiTABCDE",2},{"himA"->"carAB",3},{"himA"->"dps",1},{"himA"->"ecpD_htrE",1},{"himA"->"focA_pflB",1},{"himA"->"glcDEFGB",1},{"himA"->"glnHPQ",3},{"himA"->"himD",2},{"himA"->"hycABCDEFGH",1},{"himA"->"hypABCDE",1},{"himA"->"narGHJI",1},{"himA"->"narK",1},{"himA"->"nuoABCEFGHIJKLMN",2},{"himA"->"ompC",2},{"himA"->"ompF",3},{"himA"->"ompR_envZ",2},{"himA"->"osmE",2},{"himA"->"pspABCDE",1},{"himA"->"sodA",2},{"himA"->"tdcABCDEFG",1},{"hns"->"caiF",2},{"hns"->"flhDC",1},{"hns"->"fliAZY",1},{"hns"->"nhaA",1},{"hns"->"osmC",1},{"hns"->"rcsAB",1},{"hns"->"stpA",2},{"hydHG"->"zraP",1},{"iclMR"->"aceBAK",2},{"iclMR"->"acs",2},{"ilvY"->"ilvC",1},{"kdpDE"->"kdpABC",1},{"lacI"->"lacZYA",2},{"leuO"->"leuLABCD",1},{"lexA_dinF"->"polB",2},{"lexA_dinF"->"recA",2},{"lexA_dinF"->"recN",2},{"lexA_dinF"->"rpsU_dnaG_rpoD",2},{"lexA_dinF"->"ssb",2},{"lexA_dinF"->"sulA",2},{"lexA_dinF"->"umuDC",2},{"lexA_dinF"->"uvrA",2},{"lexA_dinF"->"uvrB",2},{"lexA_dinF"->"uvrC",2},{"lexA_dinF"->"uvrD",2},{"lrp"->"gcvTHP",1},{"lrp"->"gltBDF",1},{"lrp"->"ilvIH",1},{"lrp"->"kbl_tdh",3},{"lrp"->"livJ",2},{"lrp"->"livKHMGF",2},{"lrp"->"lysU",2},{"lrp"->"ompC",2},{"lrp"->"ompF",1},{"lrp"->"oppABCDF",2},{"lrp"->"osmC",1},{"lrp"->"sdaA",2},{"lrp"->"serA",1},{"lrp"->"stpA",1},{"lysR"->"lysA",1},{"lysR"->"tdcABCDEFG",1},{"malI"->"malXY",2},{"malT"->"malEFG",1},{"malT"->"malK_lamB_malM",1},{"malT"->"malPQ",1},{"malT"->"malS",1},{"malT"->"malZ",1},{"marRAB"->"fpr",1},{"marRAB"->"fumC",1},{"marRAB"->"nfo",1},{"marRAB"->"sodA",1},{"marRAB"->"zwf",1},{"melR"->"melAB",1},{"metJ"->"metA",2},{"metJ"->"metC",2},{"metJ"->"metF",2},{"metJ"->"metR",2},{"metR"->"glyA",1},{"metR"->"metA",1},{"metR"->"metH",1},{"mhpR"->"mhpABCDFE",1},{"mlc"->"malT",2},{"mlc"->"manXYZ",2},{"mlc"->"ptsG",2},{"mlc"->"ptsHI_crr",2},{"modE"->"modABC",2},{"nac"->"gdhA",2},{"nac"->"putAP",1},{"nadR"->"nadB",2},{"nadR"->"pncB",2},{"nagBACD"->"glmUS",3},{"nagBACD"->"manXYZ",2},{"nagBACD"->"nagE",2},{"narL"->"adhE",2},{"narL"->"caiF",2},{"narL"->"dcuB_fumB",2},{"narL"->"dmsABC",2},{"narL"->"fdnGHI",1},{"narL"->"focA_pflB",2},{"narL"->"frdABCD",2},{"narL"->"narGHJI",1},{"narL"->"narK",1},{"narL"->"nirBDC_cysG",1},{"narL"->"nrfABCDEFG",1},{"narL"->"nuoABCEFGHIJKLMN",1},{"narL"->"torCAD",2},{"nhaR"->"nhaA",1},{"nlpD_rpoS"->"acs",1},{"nlpD_rpoS"->"adhE",1},{"nlpD_rpoS"->"aldB",1},{"nlpD_rpoS"->"alkA",1},{"nlpD_rpoS"->"appY",1},{"nlpD_rpoS"->"cpxAR",1},{"nlpD_rpoS"->"dps",1},{"nlpD_rpoS"->"ftsQAZ",1},{"nlpD_rpoS"->"katG",1},{"nlpD_rpoS"->"narZYWV",1},{"nlpD_rpoS"->"nhaA",1},{"nlpD_rpoS"->"osmC",1},{"nlpD_rpoS"->"osmY",1},{"nlpD_rpoS"->"proP",1},{"ompR_envZ"->"csgBA",1},{"ompR_envZ"->"csgDEFG",1},{"ompR_envZ"->"fadL",2},{"ompR_envZ"->"flhDC",2},{"ompR_envZ"->"ompC",1},{"ompR_envZ"->"ompF",3},{"oxyR"->"ahpCF",1},{"oxyR"->"dps",1},{"oxyR"->"gorA",1},{"oxyR"->"katG",1},{"phoBR"->"phnCDE_f73_phnFGHIJKLMNOP",1},{"phoBR"->"phoA",1},{"phoBR"->"phoE",1},{"phoBR"->"pstSCAB_phoU",1},{"pspF"->"pspABCDE",1},{"purR"->"codBA",2},{"purR"->"cvpA_purF_ubiX",2},{"purR"->"gcvTHP",2},{"purR"->"glnB",2},{"purR"->"glyA",2},{"purR"->"guaBA",2},{"purR"->"prsA",2},{"purR"->"purC",2},{"purR"->"purEK",2},{"purR"->"purHD",2},{"purR"->"purL",2},{"purR"->"purMN",2},{"purR"->"pyrC",2},{"purR"->"pyrD",2},{"purR"->"speA",2},{"purR"->"ycfC_purB",2},{"rbsR"->"rbsDACBK",2},{"rcsA"->"ftsQAZ",1},{"rcsA"->"wza_wzb_b2060_wcaA_wcaB",1},{"rhaSR"->"rhaBAD",1},{"rhaSR"->"rhaT",1},{"rob"->"aslB",1},{"rob"->"fumC",1},{"rob"->"galETKM",2},{"rob"->"inaA",1},{"rob"->"marRAB",1},{"rob"->"mdlA",1},{"rob"->"nfo",1},{"rob"->"sodA",1},{"rob"->"ybaO",1},{"rob"->"ybiS",1},{"rob"->"yfhD",1},{"rob"->"zwf",1},{"rpiR_alsBACEK"->"rpiB",2},{"rpoE_rseABC"->"cutC",1},{"rpoE_rseABC"->"dapA_nlpB_purA",1},{"rpoE_rseABC"->"ecfABC",1},{"rpoE_rseABC"->"ecfD",1},{"rpoE_rseABC"->"ecfF",1},{"rpoE_rseABC"->"ecfG",1},{"rpoE_rseABC"->"ecfH",1},{"rpoE_rseABC"->"ecfI",1},{"rpoE_rseABC"->"ecfJ",1},{"rpoE_rseABC"->"ecfK",1},{"rpoE_rseABC"->"ecfLM",1},{"rpoE_rseABC"->"fkpA",1},{"rpoE_rseABC"->"htrA",1},{"rpoE_rseABC"->"ksgA_epaG_epaH",1},{"rpoE_rseABC"->"lpxDA_fabZ",1},{"rpoE_rseABC"->"mdoGH",1},{"rpoE_rseABC"->"nlpB_purA",1},{"rpoE_rseABC"->"ostA_surA_pdxA",1},{"rpoE_rseABC"->"rfaDFCL",1},{"rpoE_rseABC"->"rpoD",1},{"rpoE_rseABC"->"rpoH",1},{"rpoE_rseABC"->"skp_lpxDA_fabZ",1},{"rpoE_rseABC"->"uppS_cdsA_ecfE",1},{"rpoE_rseABC"->"xprB_dsbC_recJ",1},{"rpoH"->"clpP",1},{"rpoH"->"dnaKJ",1},{"rpoH"->"grpE",1},{"rpoH"->"hflB",1},{"rpoH"->"htpG",1},{"rpoH"->"htpY",1},{"rpoH"->"ibpAB",1},{"rpoH"->"lon",1},{"rpoH"->"mopA",1},{"rpoH"->"mopB",1},{"rpoN"->"atoC",1},{"rpoN"->"dctA",1},{"rpoN"->"fdhF",1},{"rpoN"->"fhlA",1},{"rpoN"->"glnALG",1},{"rpoN"->"glnHPQ",1},{"rpoN"->"hycABCDEFGH",1},{"rpoN"->"hypA",1},{"rpoN"->"nac",1},{"rpoN"->"nycA",1},{"rpoN"->"pspABCDE",1},{"rpoN"->"rtcR",1},{"rpoN"->"zraP",1},{"rtcR"->"rtcAB",2},{"soxR"->"soxS",1},{"soxS"->"acnA",1},{"soxS"->"fpr",1},{"soxS"->"fumC",1},{"soxS"->"nfo",1},{"soxS"->"sodA",1},{"soxS"->"zwf",1},{"tdcAR"->"tdcABCDEFG",1},{"torR"->"torCAD",1},{"treR"->"treBC",2},{"trpR"->"aroH",2},{"trpR"->"aroL_yaiA_aroM",2},{"trpR"->"mtr",2},{"trpR"->"trpLEDCBA",2},{"tyrR"->"aroF_tyrA",2},{"tyrR"->"aroG",2},{"tyrR"->"aroL_yaiA_aroM",2},{"tyrR"->"aroP",2},{"tyrR"->"mtr",1},{"tyrR"->"tyrB",2},{"tyrR"->"tyrP",3},{"uhpA"->"uhpT",1},{"uidR"->"uidRABC",2},{"uxuABR"->"uidRABC",2},{"xapR"->"xapAB",1},{"xylFGHR"->"xylAB",1},{"yhdG_fis"->"adhE",1},{"yhdG_fis"->"alaWX",1},{"yhdG_fis"->"aldB",3},{"yhdG_fis"->"argU",1},{"yhdG_fis"->"argW",1},{"yhdG_fis"->"argX_hisR_leuT_proM",1},{"yhdG_fis"->"aspV",1},{"yhdG_fis"->"leuQPV",1},{"yhdG_fis"->"leuX",1},{"yhdG_fis"->"lysT_valT_lysW",1},{"yhdG_fis"->"metT_leuW_glnUW_metU_glnVX",1},{"yhdG_fis"->"metY_yhbC_nusA_infB",1},{"yhdG_fis"->"nrdAB",1},{"yhdG_fis"->"pdhR_aceEF_lpdA",1},{"yhdG_fis"->"pheU",1},{"yhdG_fis"->"pheV",1},{"yhdG_fis"->"proK",1},{"yhdG_fis"->"proL",1},{"yhdG_fis"->"proP",1},{"yhdG_fis"->"sdhCDAB_b0725_sucABCD",1},{"yhdG_fis"->"serT",1},{"yhdG_fis"->"serX",1},{"yhdG_fis"->"thrU_tyrU_glyT_thrT",1},{"yhdG_fis"->"thrW",1},{"yhdG_fis"->"tyrTV",1},{"yhdG_fis"->"valUXY_lysV",1},{"yiaJ"->"yiaKLMNOPQRS",2},{"yjbK"->"znuABC",2},{"yjdHG"->"dctA",1},{"yjdHG"->"dcuB_fumB",1},{"yjdHG"->"frdABCD",1},{"zntR"->"zntA",1}};

The largest connected component in this network

TranscriptionNetworkLgComp={{"ada_alkB"->"aidB",1},{"ada_alkB"->"alkA",1},{"appY"->"appCBA",1},{"araC"->"araBAD",3},{"araC"->"araE",3},{"araC"->"araFG_araH_1H_2",3},{"araC"->"araJ",1},{"arcA"->"aceBAK",2},{"arcA"->"appCBA",1},{"arcA"->"appY",1},{"arcA"->"betIBA",2},{"arcA"->"cydAB",1},{"arcA"->"cyoABCDE",2},{"arcA"->"dctA",2},{"arcA"->"fadBA",2},{"arcA"->"focA_pflB",1},{"arcA"->"fumA",2},{"arcA"->"fumC",2},{"arcA"->"glcDEFGB",2},{"arcA"->"glpACB",2},{"arcA"->"gltA",2},{"arcA"->"icdA",2},{"arcA"->"lctPRD",2},{"arcA"->"mdh",2},{"arcA"->"nuoABCEFGHIJKLMN",2},{"arcA"->"sdhCDAB_b0725_sucABCD",2},{"arcA"->"sodA",2},{"argR"->"argCBH",2},{"argR"->"argD",2},{"argR"->"argE",2},{"argR"->"argF",2},{"argR"->"argI",2},{"argR"->"carAB",2},{"atoC"->"atoDAB",1},{"atoC"->"atoDAE",1},{"betIBA"->"betT",2},{"caiF"->"caiTABCDE",1},{"caiF"->"fixABCX",1},{"cpxAR"->"cpxP",1},{"cpxAR"->"dsbA",1},{"cpxAR"->"ecfI",1},{"cpxAR"->"htrA",1},{"cpxAR"->"motABcheAW",2},{"cpxAR"->"rotA",1},{"cpxAR"->"skp_lpxDA_fabZ",1},{"cpxAR"->"tsr",2},{"cpxAR"->"xprB_dsbC_recJ",1},{"crp"->"acs",1},{"crp"->"aldB",1},{"crp"->"ansB",1},{"crp"->"araBAD",1},{"crp"->"araC",1},{"crp"->"araE",1},{"crp"->"araFG_araH_1H_2",1},{"crp"->"araJ",1},{"crp"->"caiF",1},{"crp"->"caiTABCDE",1},{"crp"->"cirA",1},{"crp"->"cpdB",1},{"crp"->"cyaA",2},{"crp"->"dadAX",1},{"crp"->"dctA",2},{"crp"->"dcuB_fumB",2},{"crp"->"deoCABD",3},{"crp"->"dsdXA",1},{"crp"->"ebgAC",1},{"crp"->"epd_pgk",1},{"crp"->"fadL",2},{"crp"->"fixABCX",1},{"crp"->"flhDC",1},{"crp"->"focA_pflB",1},{"crp"->"fucAO",1},{"crp"->"fucPIKUR",1},{"crp"->"fur",1},{"crp"->"galETKM",3},{"crp"->"galS",1},{"crp"->"glgCAP",1},{"crp"->"glgS",1},{"crp"->"glnALG",1},{"crp"->"glpACB",1},{"crp"->"glpD",1},{"crp"->"glpFK",1},{"crp"->"glpTQ",1},{"crp"->"gltA",1},{"crp"->"gntKU",1},{"crp"->"gntT",1},{"crp"->"ivbL_ilvBN",1},{"crp"->"lacZYA",1},{"crp"->"malEFG",1},{"crp"->"malI",2},{"crp"->"malK_lamB_malM",1},{"crp"->"malS",1},{"crp"->"malT",1},{"crp"->"malXY",3},{"crp"->"manXYZ",1},{"crp"->"melAB",1},{"crp"->"melR",1},{"crp"->"mglBAC",1},{"crp"->"nagBACD",1},{"crp"->"nagE",1},{"crp"->"nupG",3},{"crp"->"ompA",2},{"crp"->"ppiA",3},{"crp"->"proP",3},{"crp"->"ptsHI_crr",3},{"crp"->"rhaBAD",3},{"crp"->"rhaT",1},{"crp"->"rpoH",3},{"crp"->"sdhCDAB_b0725_sucABCD",1},{"crp"->"speC",2},{"crp"->"srlAEBD_gutM_srlR_gutQ",1},{"crp"->"tdcABCDEFG",1},{"crp"->"tnaLAB",1},{"crp"->"tsx",3},{"crp"->"ubiG",1},{"crp"->"udp",1},{"crp"->"uhpT",1},{"crp"->"yhfA",3},{"crp"->"yiaKLMNOPQRS",1},{"csgDEFG"->"csgBA",1},{"cspA"->"gyrA",1},{"cspA"->"hns",1},{"cytR"->"deoCABD",2},{"cytR"->"nupC",2},{"cytR"->"nupG",2},{"cytR"->"ppiA",2},{"cytR"->"rpoH",2},{"cytR"->"tsx",3},{"cytR"->"udp",2},{"deoR"->"deoCABD",2},{"deoR"->"nupG",2},{"deoR"->"tsx",2},{"dnaA"->"nrdAB",1},{"dnaA"->"rpoH",2},{"dsdC"->"dsdXA",1},{"ebgR"->"ebgAC",2},{"envY_ompT"->"ompC",1},{"envY_ompT"->"ompF",1},{"evgA"->"ompC",1},{"fadR"->"fabA",1},{"fadR"->"fadBA",2},{"fadR"->"fadL",2},{"fadR"->"iclMR",2},{"fadR"->"uspA",2},{"fecIR"->"fecABCDE",1},{"fhlA"->"fdhF",1},{"fhlA"->"hycABCDEFGH",1},{"fhlA"->"hypABCDE",1},{"flhDC"->"flgAMN",1},{"flhDC"->"flgBCDEFGHIJK",1},{"flhDC"->"flhBAE",1},{"flhDC"->"fliAZY",1},{"flhDC"->"fliE",1},{"flhDC"->"fliFGHIJK",1},{"flhDC"->"fliLMNOPQR",1},{"fliAZY"->"flgBCDEFGHIJK",1},{"fliAZY"->"flgKL",1},{"fliAZY"->"flgMN",1},{"fliAZY"->"flhBAE",1},{"fliAZY"->"fliC",1},{"fliAZY"->"fliDST",1},{"fliAZY"->"fliE",1},{"fliAZY"->"fliFGHIJK",1},{"fliAZY"->"fliLMNOPQR",1},{"fliAZY"->"motABcheAW",1},{"fliAZY"->"tarTapcheRBYZ",1},{"fliAZY"->"tsr",1},{"fnr"->"acs",1},{"fnr"->"ansB",1},{"fnr"->"arcA",1},{"fnr"->"aspA",1},{"fnr"->"caiF",1},{"fnr"->"cydAB",2},{"fnr"->"cyoABCDE",2},{"fnr"->"dcuB_fumB",1},{"fnr"->"dmsABC",1},{"fnr"->"fdnGHI",1},{"fnr"->"focA_pflB",1},{"fnr"->"frdABCD",1},{"fnr"->"glpACB",1},{"fnr"->"hypABCDE",1},{"fnr"->"icdA",2},{"fnr"->"narGHJI",1},{"fnr"->"narK",1},{"fnr"->"ndh",2},{"fnr"->"nirBDC_cysG",1},{"fnr"->"nuoABCEFGHIJKLMN",2},{"fnr"->"sdhCDAB_b0725_sucABCD",2},{"fnr"->"tdcABCDEFG",1},{"FruR"->"aceBAK",2},{"FruR"->"adhE",2},{"FruR"->"fruBKA",2},{"FruR"->"icdA",1},{"FruR"->"ppsA",1},{"FruR"->"ptsHI_crr",3},{"FruR"->"pykF",2},{"fucPIKUR"->"fucAO",1},{"fur"->"cirA",2},{"fur"->"entCEBA",2},{"fur"->"fecIR",2},{"fur"->"fepA_entD",2},{"fur"->"fepB",2},{"fur"->"fepDGC",2},{"fur"->"fhuACDB",2},{"fur"->"sodA",2},{"fur"->"tonB",2},{"GalR"->"galETKM",2},{"GalR"->"galS",2},{"galS"->"mglBAC",2},{"gcvA"->"gcvTHP",3},{"gcvR"->"gcvTHP",2},{"glcC"->"glcDEFGB",1},{"glnALG"->"glnHPQ",3},{"glnALG"->"nac",1},{"glpR"->"glpACB",2},{"glpR"->"glpD",2},{"glpR"->"glpFK",2},{"glpR"->"glpTQ",2},{"gntR"->"edd_eda",2},{"gntR"->"gntKU",2},{"gntR"->"gntT",2},{"himA"->"aceBAK",1},{"himA"->"caiTABCDE",2},{"himA"->"carAB",3},{"himA"->"dps",1},{"himA"->"ecpD_htrE",1},{"himA"->"focA_pflB",1},{"himA"->"glcDEFGB",1},{"himA"->"glnHPQ",3},{"himA"->"himD",2},{"himA"->"hycABCDEFGH",1},{"himA"->"hypABCDE",1},{"himA"->"narGHJI",1},{"himA"->"narK",1},{"himA"->"nuoABCEFGHIJKLMN",2},{"himA"->"ompC",2},{"himA"->"ompF",3},{"himA"->"ompR_envZ",2},{"himA"->"osmE",2},{"himA"->"pspABCDE",1},{"himA"->"sodA",2},{"himA"->"tdcABCDEFG",1},{"hns"->"caiF",2},{"hns"->"flhDC",1},{"hns"->"fliAZY",1},{"hns"->"nhaA",1},{"hns"->"osmC",1},{"hns"->"rcsAB",1},{"hns"->"stpA",2},{"hydHG"->"zraP",1},{"iclMR"->"aceBAK",2},{"iclMR"->"acs",2},{"lacI"->"lacZYA",2},{"lrp"->"gcvTHP",1},{"lrp"->"gltBDF",1},{"lrp"->"ilvIH",1},{"lrp"->"kbl_tdh",3},{"lrp"->"livJ",2},{"lrp"->"livKHMGF",2},{"lrp"->"lysU",2},{"lrp"->"ompC",2},{"lrp"->"ompF",1},{"lrp"->"oppABCDF",2},{"lrp"->"osmC",1},{"lrp"->"sdaA",2},{"lrp"->"serA",1},{"lrp"->"stpA",1},{"lysR"->"lysA",1},{"lysR"->"tdcABCDEFG",1},{"malI"->"malXY",2},{"malT"->"malEFG",1},{"malT"->"malK_lamB_malM",1},{"malT"->"malPQ",1},{"malT"->"malS",1},{"malT"->"malZ",1},{"marRAB"->"fpr",1},{"marRAB"->"fumC",1},{"marRAB"->"nfo",1},{"marRAB"->"sodA",1},{"marRAB"->"zwf",1},{"melR"->"melAB",1},{"metJ"->"metA",2},{"metJ"->"metC",2},{"metJ"->"metF",2},{"metJ"->"metR",2},{"metR"->"glyA",1},{"metR"->"metA",1},{"metR"->"metH",1},{"mlc"->"malT",2},{"mlc"->"manXYZ",2},{"mlc"->"ptsG",2},{"mlc"->"ptsHI_crr",2},{"nac"->"gdhA",2},{"nac"->"putAP",1},{"nagBACD"->"glmUS",3},{"nagBACD"->"manXYZ",2},{"nagBACD"->"nagE",2},{"narL"->"adhE",2},{"narL"->"caiF",2},{"narL"->"dcuB_fumB",2},{"narL"->"dmsABC",2},{"narL"->"fdnGHI",1},{"narL"->"focA_pflB",2},{"narL"->"frdABCD",2},{"narL"->"narGHJI",1},{"narL"->"narK",1},{"narL"->"nirBDC_cysG",1},{"narL"->"nrfABCDEFG",1},{"narL"->"nuoABCEFGHIJKLMN",1},{"narL"->"torCAD",2},{"nhaR"->"nhaA",1},{"nlpD_rpoS"->"acs",1},{"nlpD_rpoS"->"adhE",1},{"nlpD_rpoS"->"aldB",1},{"nlpD_rpoS"->"alkA",1},{"nlpD_rpoS"->"appY",1},{"nlpD_rpoS"->"cpxAR",1},{"nlpD_rpoS"->"dps",1},{"nlpD_rpoS"->"ftsQAZ",1},{"nlpD_rpoS"->"katG",1},{"nlpD_rpoS"->"narZYWV",1},{"nlpD_rpoS"->"nhaA",1},{"nlpD_rpoS"->"osmC",1},{"nlpD_rpoS"->"osmY",1},{"nlpD_rpoS"->"proP",1},{"ompR_envZ"->"csgBA",1},{"ompR_envZ"->"csgDEFG",1},{"ompR_envZ"->"fadL",2},{"ompR_envZ"->"flhDC",2},{"ompR_envZ"->"ompC",1},{"ompR_envZ"->"ompF",3},{"oxyR"->"ahpCF",1},{"oxyR"->"dps",1},{"oxyR"->"gorA",1},{"oxyR"->"katG",1},{"pspF"->"pspABCDE",1},{"purR"->"codBA",2},{"purR"->"cvpA_purF_ubiX",2},{"purR"->"gcvTHP",2},{"purR"->"glnB",2},{"purR"->"glyA",2},{"purR"->"guaBA",2},{"purR"->"prsA",2},{"purR"->"purC",2},{"purR"->"purEK",2},{"purR"->"purHD",2},{"purR"->"purL",2},{"purR"->"purMN",2},{"purR"->"pyrC",2},{"purR"->"pyrD",2},{"purR"->"speA",2},{"purR"->"ycfC_purB",2},{"rcsA"->"ftsQAZ",1},{"rcsA"->"wza_wzb_b2060_wcaA_wcaB",1},{"rhaSR"->"rhaBAD",1},{"rhaSR"->"rhaT",1},{"rob"->"aslB",1},{"rob"->"fumC",1},{"rob"->"galETKM",2},{"rob"->"inaA",1},{"rob"->"marRAB",1},{"rob"->"mdlA",1},{"rob"->"nfo",1},{"rob"->"sodA",1},{"rob"->"ybaO",1},{"rob"->"ybiS",1},{"rob"->"yfhD",1},{"rob"->"zwf",1},{"rpoE_rseABC"->"cutC",1},{"rpoE_rseABC"->"dapA_nlpB_purA",1},{"rpoE_rseABC"->"ecfABC",1},{"rpoE_rseABC"->"ecfD",1},{"rpoE_rseABC"->"ecfF",1},{"rpoE_rseABC"->"ecfG",1},{"rpoE_rseABC"->"ecfH",1},{"rpoE_rseABC"->"ecfI",1},{"rpoE_rseABC"->"ecfJ",1},{"rpoE_rseABC"->"ecfK",1},{"rpoE_rseABC"->"ecfLM",1},{"rpoE_rseABC"->"fkpA",1},{"rpoE_rseABC"->"htrA",1},{"rpoE_rseABC"->"ksgA_epaG_epaH",1},{"rpoE_rseABC"->"lpxDA_fabZ",1},{"rpoE_rseABC"->"mdoGH",1},{"rpoE_rseABC"->"nlpB_purA",1},{"rpoE_rseABC"->"ostA_surA_pdxA",1},{"rpoE_rseABC"->"rfaDFCL",1},{"rpoE_rseABC"->"rpoD",1},{"rpoE_rseABC"->"rpoH",1},{"rpoE_rseABC"->"skp_lpxDA_fabZ",1},{"rpoE_rseABC"->"uppS_cdsA_ecfE",1},{"rpoE_rseABC"->"xprB_dsbC_recJ",1},{"rpoH"->"clpP",1},{"rpoH"->"dnaKJ",1},{"rpoH"->"grpE",1},{"rpoH"->"hflB",1},{"rpoH"->"htpG",1},{"rpoH"->"htpY",1},{"rpoH"->"ibpAB",1},{"rpoH"->"lon",1},{"rpoH"->"mopA",1},{"rpoH"->"mopB",1},{"rpoN"->"atoC",1},{"rpoN"->"dctA",1},{"rpoN"->"fdhF",1},{"rpoN"->"fhlA",1},{"rpoN"->"glnALG",1},{"rpoN"->"glnHPQ",1},{"rpoN"->"hycABCDEFGH",1},{"rpoN"->"hypA",1},{"rpoN"->"nac",1},{"rpoN"->"nycA",1},{"rpoN"->"pspABCDE",1},{"rpoN"->"rtcR",1},{"rpoN"->"zraP",1},{"rtcR"->"rtcAB",2},{"soxR"->"soxS",1},{"soxS"->"acnA",1},{"soxS"->"fpr",1},{"soxS"->"fumC",1},{"soxS"->"nfo",1},{"soxS"->"sodA",1},{"soxS"->"zwf",1},{"tdcAR"->"tdcABCDEFG",1},{"torR"->"torCAD",1},{"uhpA"->"uhpT",1},{"yhdG_fis"->"adhE",1},{"yhdG_fis"->"alaWX",1},{"yhdG_fis"->"aldB",3},{"yhdG_fis"->"argU",1},{"yhdG_fis"->"argW",1},{"yhdG_fis"->"argX_hisR_leuT_proM",1},{"yhdG_fis"->"aspV",1},{"yhdG_fis"->"leuQPV",1},{"yhdG_fis"->"leuX",1},{"yhdG_fis"->"lysT_valT_lysW",1},{"yhdG_fis"->"metT_leuW_glnUW_metU_glnVX",1},{"yhdG_fis"->"metY_yhbC_nusA_infB",1},{"yhdG_fis"->"nrdAB",1},{"yhdG_fis"->"pdhR_aceEF_lpdA",1},{"yhdG_fis"->"pheU",1},{"yhdG_fis"->"pheV",1},{"yhdG_fis"->"proK",1},{"yhdG_fis"->"proL",1},{"yhdG_fis"->"proP",1},{"yhdG_fis"->"sdhCDAB_b0725_sucABCD",1},{"yhdG_fis"->"serT",1},{"yhdG_fis"->"serX",1},{"yhdG_fis"->"thrU_tyrU_glyT_thrT",1},{"yhdG_fis"->"thrW",1},{"yhdG_fis"->"tyrTV",1},{"yhdG_fis"->"valUXY_lysV",1},{"yiaJ"->"yiaKLMNOPQRS",2},{"yjdHG"->"dctA",1},{"yjdHG"->"dcuB_fumB",1},{"yjdHG"->"frdABCD",1}};

Drawing the network

The network consists of many components.

GraphPlot[TranscriptionNetwork,PackingStyle->{"ClosestPacking","PolyominoPadding"->3},EdgeLabeling->Automatic,VertexRenderingFunction->({EdgeForm[Yellow],Blue,Tooltip[Disk[#,0.12],#2]}&)]

Spread out the vertices more.

GraphPlot[TranscriptionNetwork,Method->{"SpringElectricalEmbedding","RepulsiveForcePower"->-1.2},PackingStyle->{"ClosestPacking","PolyominoPadding"->3},EdgeLabeling->Automatic,VertexRenderingFunction->({EdgeForm[Yellow],Blue,Tooltip[Disk[#,0.13],#2]}&)]

This is an alternative way of spreading out the vertices.

GraphPlot[TranscriptionNetwork,Method->{"SpringElectricalEmbedding","InferentialDistance"->4},PackingStyle->{"ClosestPacking","PolyominoPadding"->3},EdgeLabeling->Automatic,VertexRenderingFunction->({EdgeForm[Yellow],Blue,Tooltip[Disk[#,0.13],#2]}&)]

Examples: Protein (an oxidoreductase)

This plots a oxidoreductase protein from "R Milo, S Itzkovitz, N Kashtan, R Levitt, S Shen-Orr, I Ayzenshtat, M Sheffer& U Alon, Superfamilies of designed and evolved networks, Science,303:1538-42 (2004).", The data was found at http://www.weizmann.ac.il/mcb/UriAlon/Papers/networkMotifs/1AORInter_st.txt

GraphPlot[{{1->2,1},{1->9,2},{1->17,2},{2->3,1},{2->9,2},{3->49,2},{4->5,2},{4->42,2},{5->6,2},{5->9,2},{5->17,2},{5->42,1},{6->7,1},{6->8,2},{6->17,2},{6->28,2},{6->42,2},{7->8,2},{7->16,2},{7->28,2},{7->44,2},{8->10,2},{8->11,2},{8->14,2},{8->16,2},{8->18,2},{8->19,2},{8->28,2},{9->17,2},{10->11,1},{10->12,2},{10->13,2},{10->14,2},{10->17,2},{10->18,2},{11->12,2},{11->18,2},{12->13,2},{12->14,2},{12->18,2},{13->14,2},{13->15,2},{13->16,2},{13->18,2},{13->32,2},{14->15,1},{14->16,2},{14->18,2},{14->32,1},{14->36,2},{15->16,2},{15->32,2},{15->36,2},{15->37,2},{16->17,2},{16->18,2},{16->32,2},{16->33,2},{16->37,2},{18->19,2},{19->20,2},{19->43,2},{20->21,2},{20->43,2},{21->22,2},{22->27,2},{22->28,2},{22->29,2},{23->24,2},{23->25,2},{23->26,2},{23->27,2},{23->30,2},{23->31,2},{23->39,2},{24->25,2},{24->31,2},{24->39,2},{25->26,2},{25->31,2},{26->27,2},{26->30,2},{26->31,2},{26->32,2},{27->30,2},{28->29,2},{29->85,2},{30->31,2},{30->32,2},{30->33,2},{32->36,2},{32->85,2},{33->36,2},{35->36,2},{35->79,2},{35->82,2},{36->37,2},{37->38,2},{39->40,2},{39->41,2},{39->47,2},{40->41,2},{40->46,2},{40->48,2},{43->44,2},{44->45,1},{45->46,1},{47->48,2},{50->51,1},{50->52,2},{50->59,1},{50->67,2},{51->52,2},{51->59,2},{51->67,2},{52->53,1},{53->93,2},{53->99,2},{54->55,2},{54->92,2},{55->56,2},{55->59,2},{55->67,2},{55->92,1},{56->57,1},{56->58,2},{56->67,2},{56->78,2},{56->92,2},{57->58,2},{57->66,2},{57->78,2},{57->94,2},{58->60,2},{58->61,2},{58->64,2},{58->66,2},{58->68,2},{58->69,2},{58->78,2},{59->67,2},{60->61,2},{60->62,2},{60->64,2},{60->67,2},{60->68,2},{61->62,2},{61->68,2},{62->63,2},{62->68,2},{63->64,2},{63->65,2},{63->66,2},{63->68,2},{63->82,2},{64->65,1},{64->66,2},{64->68,2},{64->82,1},{64->86,2},{65->66,2},{65->82,2},{65->86,2},{65->87,2},{66->67,2},{66->68,2},{66->82,2},{66->83,2},{66->87,2},{68->69,2},{69->70,2},{69->93,2},{70->71,2},{71->72,2},{72->77,2},{72->78,2},{72->79,2},{73->74,2},{73->75,2},{73->76,2},{73->77,2},{73->80,2},{73->81,2},{73->89,2},{74->75,2},{74->81,2},{74->89,2},{75->76,2},{75->81,2},{76->77,2},{76->80,2},{76->81,2},{76->82,2},{77->80,2},{78->79,2},{80->81,2},{80->82,2},{80->83,2},{82->86,2},{85->86,2},{86->87,2},{87->88,2},{89->90,2},{89->91,2},{89->97,2},{90->91,2},{90->96,2},{90->98,2},{93->94,2},{94->95,1},{95->96,1},{97->98,2}},EdgeLabeling->Automatic,MultiedgeStyle->False,VertexRenderingFunction->({EdgeForm[Yellow],Blue,Tooltip[Disk[#,0.13],#2]}&)]

A graph from electrical  engineering

This graph comes from a sparse matrix describing a “Square Dielectric Waveguide”.

g=Import["ftp://math.nist.gov/pub/MatrixMarket2/NEP/dwave/dwa512.mtx.gz"];

GraphPlot3D[g,VertexRenderingFunction-> None]

GraphUtilities package

This was called DiscreteMath`GraphPlot

This is a table of functions in the GraphUtilities package, 
Bold = new.

({{AdjacencyMatrix, AllPairsShortestPath}, {Bicomponents, ClosenessCentrality}, {CommunityModularity, CommunityStructureAssignment}, {CommunityStructurePartition, EdgeList}, {GraphCoordinates, GraphCoordinates1D}, {GraphCoordinates3D, GraphDistance}, {GraphEdit, InBetweenCoordinate}, {LinkRank, LinkRankRules}, {MaximalBipartiteMatching, MaximalIndependentEdgeSet}, {MaximalIndependentVertexSet, MinCut}, {MinimumBandwidthOrdering, PageRank}, {PageRankRules, PseudoDiameter}, {StrongComponents, ShortestPath}, {ToCombinatoricaGraph, VertexList}, { ,  }})

GraphUtilities package: PageRank function

The success of Google is due to no small part to the PageRank algorithm.

People who used earlier search engines know how you have to read pages to find your result.

PageRank ranks the importance of result.

One way of looking at it: the more links (votes) to a page, the more important a page is. In turn, a vote from an important page has more weight.

Alternative: a random surfer, and PageRank of a page is the probability of the surfer at that page.

CommunityStructureAssignment

This calculates the PageRank and PageRankRules of a small graph.

<<GraphUtilities`;

g={"home"->"contact","home"->"links","home"->"work","links"->"friend site"};
prr=PageRankRules[g]

This shows PageRank and the graph together.

gr=GraphPlot[g,VertexCoordinateRules->{{0,1},{.5,0},{1,1},{0,0},{1,0}},EdgeRenderingFunction->(Arrow[#1,.2]&),VertexRenderingFunction->({GrayLevel[.9],EdgeForm[Blue],Disk[#1,.2],Black,Text[#2,#1+{0,0.05}],Text[#2/.prr,#1-{0,0.05}]}&)]

Adding some back links boost the PageRank of the home page a lot.

g={"home"->"contact","home"->"links","home"->"work","links"->"friend site","friend site"->"home","work"->"home","contact"->"home"};
prr=PageRankRules[g];GraphPlot[g,VertexCoordinateRules->{{0,1},{.5,0},{1,1},{0,0},{1,0}},EdgeRenderingFunction->(Arrow[#1,.2]&),VertexRenderingFunction->({GrayLevel[.9],EdgeForm[Blue],Disk[#1,.2],Black,Text[#2,#1+{0,0.05}],Text[#2/.prr,#1-{0,0.05}]}&)]

PageRank of function web pages

Function website data

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f[{x_,y_},k_]:=Map[x->#[[1]]&,Take[y,Min[k,Length[y]]]];
functionWebGraph=Flatten[Map[f[#,Infinity]&,data]];functionWebGraph2=Flatten[Map[f[#,3]&,data]];

PageRank of function pages

TableForm[Map[(List@@#)&,Take[Sort[PageRankRules[functionWebGraph],(#1[[2]]>#2[[2]]&)],30]]]

GraphUtilities package: CommunityStructureAssignment function

Community structure: groups of vertices such that there is a higher density of edges within groups than between them.

Interest comes from Prof. Seth Chandler

Example

g={1->2,2->3,3->1,3->4,4->5,5->6,6->4,6->7,7->8,8->9,9->10,10->7,10->1};
assignment = CommunityStructureAssignment[g]

color = 0.7*assignment/Max[assignment];
GraphPlot[g,VertexRenderingFunction->({EdgeForm[Black],Hue[color[[#3]]],Disk[#,.1],Black,Text[#2,#]}&)]

Southwest network

g=(<<SouthwestNetwork);

comm=CommunityStructureAssignment[g]

GraphPlot[g,Method->None,VertexRenderingFunction->({Hue[comm[[#3]]/6],Tooltip[Disk[#1,.02],#2]}&)]

GraphUtilities package: GraphEdit function

It is convenient to have a way to input a graph interactively.

It will be part of GraphUtilities package.

GraphEdit[{1->2}]

GraphEdit[GridGraph[5,5]]

Conclusions

Very efficient and high quality undirected graph drawing algorithms are available now as system functions.

Version 6.0 also has an efficient and high quality directed graph drawing algorithm.

Better packing algorithms for disconnected components.

GraphUtilities package has many new functions, including PageRank, CommunityStructureAssignment, and interactive graph editing function GraphEdit.

Many challenges: planar graphs, coordinate constraints.