DYNAMIC GRAPHSDivya Sindhu LekhaResear h Fellow, Department of Futures Studies, University ofKerala, Trivandrum-695034.September 22, 2011

Overview1 Introdu tion

Overview1 Introdu tion2 De�nitions

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs4 Appli ation Areas of Dynami Graphs

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs4 Appli ation Areas of Dynami Graphs5 Approa hes to Studying Dynami Graphs

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs4 Appli ation Areas of Dynami Graphs5 Approa hes to Studying Dynami Graphs6 Data Stru tures for Implementation

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs4 Appli ation Areas of Dynami Graphs5 Approa hes to Studying Dynami Graphs6 Data Stru tures for Implementation7 General Te hniques

Overview1 Introdu tion2 De�nitions3 A Representation of Dynami Graphs4 Appli ation Areas of Dynami Graphs5 Approa hes to Studying Dynami Graphs6 Data Stru tures for Implementation7 General Te hniques8 Con lusion

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.2 Graphs undergo dis rete hanges.

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.2 Graphs undergo dis rete hanges.3 Erdos and Renyi introdu e hanges in graph by adding weightto verti es/ edges by their existen e probability.

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.2 Graphs undergo dis rete hanges.3 Erdos and Renyi introdu e hanges in graph by adding weightto verti es/ edges by their existen e probability.4 Problem : Determine if the graph remains onne ted despiterandom dropout of edges/verti es.

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.2 Graphs undergo dis rete hanges.3 Erdos and Renyi introdu e hanges in graph by adding weightto verti es/ edges by their existen e probability.4 Problem : Determine if the graph remains onne ted despiterandom dropout of edges/verti es.5 Siljak Introdu ed dynami graphs in stability studies of omplexdynami systems.

Introdu tion1 Appli ations of graph algorithms - Communi ation Networks,VLSI Design, Graphi s, Assembly Planning.2 Graphs undergo dis rete hanges.3 Erdos and Renyi introdu e hanges in graph by adding weightto verti es/ edges by their existen e probability.4 Problem : Determine if the graph remains onne ted despiterandom dropout of edges/verti es.5 Siljak Introdu ed dynami graphs in stability studies of omplexdynami systems.6 To study the e�e t of un ertain inter onne tions between thesubsystems on stability of the overall system.

De�nitionsDemetres u, Fino hi, Italiano[3℄ de�nes dynami graphs in termsof updates on a graph.

De�nitionsDemetres u, Fino hi, Italiano[3℄ de�nes dynami graphs in termsof updates on a graph.De�nition 2.1An update on a graph is an operation that inserts/deletesedges/verti es of the graph OR hanges the attributes asso iatedwith edges/verti es su h as ost, olor.

De�nitionsDemetres u, Fino hi, Italiano[3℄ de�nes dynami graphs in termsof updates on a graph.De�nition 2.1An update on a graph is an operation that inserts/deletesedges/verti es of the graph OR hanges the attributes asso iatedwith edges/verti es su h as ost, olor.De�nition 2.2A dynami graph is a graph that is undergoing a sequen e ofupdates.

Kinell[4℄ de�nes dynami graphs in terms of timestep.

Kinell[4℄ de�nes dynami graphs in terms of timestep.De�nition 2.3A timestep t is an interval of time, su h that in any interval, notmore than one a tion is performed.

Kinell[4℄ de�nes dynami graphs in terms of timestep.De�nition 2.3A timestep t is an interval of time, su h that in any interval, notmore than one a tion is performed.De�nition 2.4A dynami graph is a graph G ∈ G at a given timestep t, denotedby G (t) or G (V ,E , t) where V ∈ G (t) and V ′ ∈ G (t + 1) mayonly di�er in one element ∀t ≥ 0.

Siljak[1℄ de�nes dynami graphs in a linear spa e as one-parametergroup of transformations of the graph spa e into itself.

Siljak[1℄ de�nes dynami graphs in a linear spa e as one-parametergroup of transformations of the graph spa e into itself.De�nition 2.5A dynami graph D is a one-parameter mapping Φ : R×D → D ofthe spa e D into itself satisfying the following axioms.

Siljak[1℄ de�nes dynami graphs in a linear spa e as one-parametergroup of transformations of the graph spa e into itself.De�nition 2.5A dynami graph D is a one-parameter mapping Φ : R×D → D ofthe spa e D into itself satisfying the following axioms.1 Φ(t0,D0) = D0, ∀t0 ∈ R, ∀D0 ∈ D

Siljak[1℄ de�nes dynami graphs in a linear spa e as one-parametergroup of transformations of the graph spa e into itself.De�nition 2.5A dynami graph D is a one-parameter mapping Φ : R×D → D ofthe spa e D into itself satisfying the following axioms.1 Φ(t0,D0) = D0, ∀t0 ∈ R, ∀D0 ∈ D2 Φ(t,D) is ontinuous, ∀t ∈ R, ∀D ∈ D

Siljak[1℄ de�nes dynami graphs in a linear spa e as one-parametergroup of transformations of the graph spa e into itself.De�nition 2.5A dynami graph D is a one-parameter mapping Φ : R×D → D ofthe spa e D into itself satisfying the following axioms.1 Φ(t0,D0) = D0, ∀t0 ∈ R, ∀D0 ∈ D2 Φ(t,D) is ontinuous, ∀t ∈ R, ∀D ∈ D3 Φ(t2,Φ(t1,D)) = Φ(t1 + t2,D), ∀t1, t2 ∈ R, ∀D ∈ D

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.1 Node-dynami graphs (V varies with time) e.g. Computers rashing/ re overing.

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.1 Node-dynami graphs (V varies with time) e.g. Computers rashing/ re overing.2 Edge-dynami graphs (E varies with time) e.g. Changingtopology of omputer network.

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.1 Node-dynami graphs (V varies with time) e.g. Computers rashing/ re overing.2 Edge-dynami graphs (E varies with time) e.g. Changingtopology of omputer network.3 Node-weighted dynami graphs (f : V → R varies with time)e.g. Changing omputing power.

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.1 Node-dynami graphs (V varies with time) e.g. Computers rashing/ re overing.2 Edge-dynami graphs (E varies with time) e.g. Changingtopology of omputer network.3 Node-weighted dynami graphs (f : V → R varies with time)e.g. Changing omputing power.4 Edge-weighted dynami graphs (g : E → R varies withtime)e.g. Computer network with hanging bandwidth.

Harary and Gupta[2℄ lassi�es dynami graphs into 5 basi types.1 Node-dynami graphs (V varies with time) e.g. Computers rashing/ re overing.2 Edge-dynami graphs (E varies with time) e.g. Changingtopology of omputer network.3 Node-weighted dynami graphs (f : V → R varies with time)e.g. Changing omputing power.4 Edge-weighted dynami graphs (g : E → R varies withtime)e.g. Computer network with hanging bandwidth.5 Fully-weighted dynami graphs (Both f and g varies withtime).

Demetres u, Fino hi, Italiano[3℄ lassi�es dynami graphs into 4basi types.

Demetres u, Fino hi, Italiano[3℄ lassi�es dynami graphs into 4basi types.1 Fully dynami graphs (unrestri ted insertions and deletions ofedges or verti es)

Demetres u, Fino hi, Italiano[3℄ lassi�es dynami graphs into 4basi types.1 Fully dynami graphs (unrestri ted insertions and deletions ofedges or verti es)2 Partially dynami graphs (only one type of update is allowed)

Demetres u, Fino hi, Italiano[3℄ lassi�es dynami graphs into 4basi types.1 Fully dynami graphs (unrestri ted insertions and deletions ofedges or verti es)2 Partially dynami graphs (only one type of update is allowed)3 In remental dynami graphs (only insertions allowed)

Demetres u, Fino hi, Italiano[3℄ lassi�es dynami graphs into 4basi types.1 Fully dynami graphs (unrestri ted insertions and deletions ofedges or verti es)2 Partially dynami graphs (only one type of update is allowed)3 In remental dynami graphs (only insertions allowed)4 De remental dynami graphs (only deletions allowed)

RepresentationThe dynami graph an be represented as the sequen e of itsadja en y matri es underneath. [8℄

Figure 1: The dynami graph RE ⊆{d1, d2, d3, d4} × {a1, a2, a3, a4}× {t1, t2, t3, t4}.

Appli ation AreasWide variety of appli ation areas in lude

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming Languages

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen e

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer Networks

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear h

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru tures

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilers

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilersDatabase

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilersDatabaseFault Toleran e

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilersDatabaseFault Toleran eConne tivity in Graphs

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilersDatabaseFault Toleran eConne tivity in GraphsSigned Graphs

Appli ation AreasWide variety of appli ation areas in ludeComputer Programming LanguagesArti� ial Intelligen eComputer NetworksOperation Resear hData Stru turesCompilersDatabaseFault Toleran eConne tivity in GraphsSigned GraphsGraph Games

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph Theory

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.If more than one omponent of a dynami graph hange withtime, then the invariants an be studied mu h in the fashion ofpartial derivatives.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.If more than one omponent of a dynami graph hange withtime, then the invariants an be studied mu h in the fashion ofpartial derivatives.Logi Programming (LP)

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.If more than one omponent of a dynami graph hange withtime, then the invariants an be studied mu h in the fashion ofpartial derivatives.Logi Programming (LP)Dynami graphs an be modeled using a logi program.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.If more than one omponent of a dynami graph hange withtime, then the invariants an be studied mu h in the fashion ofpartial derivatives.Logi Programming (LP)Dynami graphs an be modeled using a logi program.The logi program an be exe uted to dis over interestingproperties of the dynami graph being modeled.

Approa hes to studyWay ba k in 1997, Harary and Gupta[2℄ suggested two possibleapproa hes that an be used in studying dynami graphs.Knowledge from Stati Graph TheoryA dynami graph an be viewed as a dis rete sequen e ofstati graphs.We an study dynami graphs by spe ifying the properties thatremain invariant with time.If more than one omponent of a dynami graph hange withtime, then the invariants an be studied mu h in the fashion ofpartial derivatives.Logi Programming (LP)Dynami graphs an be modeled using a logi program.The logi program an be exe uted to dis over interestingproperties of the dynami graph being modeled.LP - Relational programming paradigm, Nondeterminism (Aptfor ombinatorial sear hes)

Data Stru tures Implementing Dynami Graphs

Data Stru tures Implementing Dynami GraphsTopology Trees

Data Stru tures Implementing Dynami GraphsTopology TreesET Trees

Data Stru tures Implementing Dynami GraphsTopology TreesET TreesTop Trees

Data Stru tures: Topology Trees

Data Stru tures: Topology TreesIntrodu ed by Frederi kson(1985) to maintain dynami treesupon insertions and deletions of edges.

Data Stru tures: Topology TreesIntrodu ed by Frederi kson(1985) to maintain dynami treesupon insertions and deletions of edges.Given a tree T of a forest, a luster is a onne ted subgraphof T .

Data Stru tures: Topology TreesIntrodu ed by Frederi kson(1985) to maintain dynami treesupon insertions and deletions of edges.Given a tree T of a forest, a luster is a onne ted subgraphof T .The ardinality of a luster is the number of its verti es.

Data Stru tures: Topology TreesIntrodu ed by Frederi kson(1985) to maintain dynami treesupon insertions and deletions of edges.Given a tree T of a forest, a luster is a onne ted subgraphof T .The ardinality of a luster is the number of its verti es.The external degree of a luster is the number of tree edgesin ident to it.

Data Stru tures: Topology TreesIntrodu ed by Frederi kson(1985) to maintain dynami treesupon insertions and deletions of edges.Given a tree T of a forest, a luster is a onne ted subgraphof T .The ardinality of a luster is the number of its verti es.The external degree of a luster is the number of tree edgesin ident to it.De�nition 6.1A topology tree is a hierar hi al representation of T : ea h levelpartitions the verti es of T into lusters. Cluster at level 0 ontainsone vertex ea h. Clusters at level l ≥ 1 form a partition of theverti es of the tree T ′ obtained by shrinking ea h luster at levell -1 into a single vertex.

Figure 2: Clustering of a tree T with restri tions (1)external degree ≤ 3and (2) ardinality ≤ 2.

Topology Tree

Figure 3: Topology Tree of T

Topology Trees: Fa ts

Topology Trees: Fa ts1 The number of nodes in ea h level is a onstraint fra tion ofnodes in previous level.

Topology Trees: Fa ts1 The number of nodes in ea h level is a onstraint fra tion ofnodes in previous level.2 Number of levels is O(logn).

Topology Trees: Fa ts1 The number of nodes in ea h level is a onstraint fra tion ofnodes in previous level.2 Number of levels is O(logn).3 (Frederi kson's Theorem) The update of a topology tree anbe supported in O(logn) time.

Data Stru tures: ET Trees

Data Stru tures: ET TreesIntrodu ed by Henzinger and King(1999) to maintain dynami forest with vertex weight.

Data Stru tures: ET TreesIntrodu ed by Henzinger and King(1999) to maintain dynami forest with vertex weight.Updates allowed: (1) Cut arbitrary edges (2) Insert edgesbetween trees (3) Add/Remove weight on verti es.

Data Stru tures: ET TreesIntrodu ed by Henzinger and King(1999) to maintain dynami forest with vertex weight.Updates allowed: (1) Cut arbitrary edges (2) Insert edgesbetween trees (3) Add/Remove weight on verti es.Supported queries: (1) Conne ted(u, v) (2) Size(v) (3)Minkey(v).

Data Stru tures: ET TreesIntrodu ed by Henzinger and King(1999) to maintain dynami forest with vertex weight.Updates allowed: (1) Cut arbitrary edges (2) Insert edgesbetween trees (3) Add/Remove weight on verti es.Supported queries: (1) Conne ted(u, v) (2) Size(v) (3)Minkey(v).An Euler tour of T is a maximal losed walk over the graphobtained by repla ing ea h edge of T by 2 dire ted edges inopposite dire tions (length = 2n-2).

Data Stru tures: ET TreesIntrodu ed by Henzinger and King(1999) to maintain dynami forest with vertex weight.Updates allowed: (1) Cut arbitrary edges (2) Insert edgesbetween trees (3) Add/Remove weight on verti es.Supported queries: (1) Conne ted(u, v) (2) Size(v) (3)Minkey(v).An Euler tour of T is a maximal losed walk over the graphobtained by repla ing ea h edge of T by 2 dire ted edges inopposite dire tions (length = 2n-2).De�nition 6.2An ET tree is a dynami balan ed binary tree over some Euler touraround T .

Figure 4: Euler tour and ET Tree of a tree T .

ET Trees: Fa ts

ET Trees: Fa ts1 Despite the repetition of verti es in an Euler tour an ET Treehas O(n) nodes.

ET Trees: Fa ts1 Despite the repetition of verti es in an Euler tour an ET Treehas O(n) nodes.2 Both updates and queries an be supported in O(logn) time.

Data Stru tures: Top Trees

Data Stru tures: Top TreesIntrodu ed by Alstrup et. al.(1997) to maintain e� ientlyinformation about paths in trees (from topology tree -partition edges).

Data Stru tures: Top TreesIntrodu ed by Alstrup et. al.(1997) to maintain e� ientlyinformation about paths in trees (from topology tree -partition edges).Given a tree T of a forest, a luster is a onne ted subtree ofT with atmost two boundary verti es (have edges out of thesubtree).

Data Stru tures: Top TreesIntrodu ed by Alstrup et. al.(1997) to maintain e� ientlyinformation about paths in trees (from topology tree -partition edges).Given a tree T of a forest, a luster is a onne ted subtree ofT with atmost two boundary verti es (have edges out of thesubtree).Two lusters are neighbors if their interse tion ontainsexa tly one vertex.

Data Stru tures: Top TreesIntrodu ed by Alstrup et. al.(1997) to maintain e� ientlyinformation about paths in trees (from topology tree -partition edges).Given a tree T of a forest, a luster is a onne ted subtree ofT with atmost two boundary verti es (have edges out of thesubtree).Two lusters are neighbors if their interse tion ontainsexa tly one vertex.Operations: (1) Merge (2) Split.

Data Stru tures: Top Trees

Data Stru tures: Top TreesDe�nition 6.3A Top tree is a binary tree su h that:

Data Stru tures: Top TreesDe�nition 6.3A Top tree is a binary tree su h that:1 Leaves = Edges; Internal nodes = Clusters.

Data Stru tures: Top TreesDe�nition 6.3A Top tree is a binary tree su h that:1 Leaves = Edges; Internal nodes = Clusters.2 Internal node = Union of its 2 hildren whi h are neighbors.

Data Stru tures: Top TreesDe�nition 6.3A Top tree is a binary tree su h that:1 Leaves = Edges; Internal nodes = Clusters.2 Internal node = Union of its 2 hildren whi h are neighbors.3 Root represents the entire tree.

Data Stru tures: Top TreesDe�nition 6.3A Top tree is a binary tree su h that:1 Leaves = Edges; Internal nodes = Clusters.2 Internal node = Union of its 2 hildren whi h are neighbors.3 Root represents the entire tree.4 Height = O(logn).

Figure 5: Clusters of a tree T .

Figure 6: Top Tree of T .

General Te hniques

General Te hniquesClustering: (Frederi kson) Partitioning graph into onne tedsubgraphs su h that ea h update involves only a small numberof su h lusters.

General Te hniquesClustering: (Frederi kson) Partitioning graph into onne tedsubgraphs su h that ea h update involves only a small numberof su h lusters.Sparsi� ation: (Eppstein) A divide-and- onquer te hnique toredu e the dependen e of time bounds on the number ofedges.

General Te hniquesClustering: (Frederi kson) Partitioning graph into onne tedsubgraphs su h that ea h update involves only a small numberof su h lusters.Sparsi� ation: (Eppstein) A divide-and- onquer te hnique toredu e the dependen e of time bounds on the number ofedges.Randomization: (Henzinger and King) Allow to design e� ientrandomized algorithms.

Con lusion.

Con lusion.Dynami graphs arise in virtually all areas of s ien e.

Con lusion.Dynami graphs arise in virtually all areas of s ien e.Used as a �exible modeling on ept for studying stru tural hanges in a broad range of omplex systems.

Con lusion.Dynami graphs arise in virtually all areas of s ien e.Used as a �exible modeling on ept for studying stru tural hanges in a broad range of omplex systems.In omputer s ien e, all data stru tures an be regarded asimplementations of dynami graphs.

Con lusion.Dynami graphs arise in virtually all areas of s ien e.Used as a �exible modeling on ept for studying stru tural hanges in a broad range of omplex systems.In omputer s ien e, all data stru tures an be regarded asimplementations of dynami graphs.Dynami graphs are also found in hemistry, anthropology,psy hology, and all other areas to whi h graph theory an befruitfully applied.

Referen es I[1℄ D.D.Siljak, Dynami Graphs:Plenary Paper. The InternationalConferen e on Hybrid Systems and Appli ations, University ofLouisiana, Lafayette, LA, May 22-26, (2006)[2℄ F. Harary and G. Gupta, Dynami Graph Models, Mathl. Com-put. Modelling Vol. 25, No. 7, pp. 79-87 (1997).[3℄ C. Demetres u, I. Fino hi and G. F. Italiano, Dynami GraphAlgorithms, University of Rome.[4℄ A. Kinell, Dynami Graph Labelling (2004)[5℄ R. F. Werne k, Design and Analysis of Data Stru tures for Dy-nami Trees, Dissertation, Department of Computer S ien e,Prin eton University, June, (2006)

Referen es II[6℄ D. D. Sleator and R. E. Tarjan, A Data Stru ture for Dynami Trees , Journal of Computer and System S ien es, Vol. 26, No.3, June (1983) 362-391.[7℄ P. N. Klein: Algorithms for Planar Graphs Le ture 7: Dynami -tree data stru ture, (2009).[8℄ K. T. Nguyen, L. Cerf1, M. Plantevit, and J. Bouli aut: Dis- overing Inter-Dimensional Rules in Dynami Graphs, (2010)

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