ITTC Science of Communication Networks The University of Kansas EECS 784 Graph Theory

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1 Science of Communication Networks The University of Kansas EECS 784 Graph Theory James P.G. Sterbenz Department of Electrical Engineering & Computer Science Information Technology & Telecommunications Research Center The University of Kansas 24 January 2017 rev James P.G. Sterbenz

2 Graphs Theory Outline GT.1 Graph types and representation GT.2 Graph properties and metrics GT.3 Distance and connectivity GT.4 Centrality GT.5 Special graphs: fuzzy, time-varying, hypergraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-2

3 Graph Theory Components Model network as a graph Vertex node: v (or n [L2009]) switch, Internet router, end system represented as Edge link: e i ={v j, v k } or e i =v j v k [GY2006] (or l i ={n j, n k } [L2009]) e i is joins and is incident to v j and v k v j v k denotes neighbours such that v j is adjacent to v k wired physical link, wireless link association, P2P relationship represented as or or or 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-3

4 Graph Theory Vertex Adjacency Adjacency: e i ={v j, v k } e i joins and is incident to v j and v k v j and v k are neighbours such that v j is adjacent to v k undirected adjacency v j v k or v j v k directed adjacency v j v k wired physical link, wireless link association, P2P relationship generally undirected bidirectional link may be directed unidirectional link later bidirectional link 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-4

5 Graph Theory Definition Graph G = (V; E ) E V V set of vertices V(G) = {,, } set of edges E(G) = {e 0, e 1, } mapping of e i {v j, v k } [ Bogdan Giuşcă] Example Königsberg graph with multiedges + (self-)loop V = {,,, } e 0 E = {e 0, e 1, e 2, e 3, e 4, e 5, e 6, e 7 } e 0,e 1 ={, }, e 2,e 3 ={, }, e 4 ={, }, e 5 ={, }, e 6 ={, }, e 7 ={, } 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-5 e 1 e 3 e 2 e 4 e 6 e 5 e 7

6 Graph Theory Terminology Edge and vertex functions edge set of a graph E(G) [D,W] or E G [G] vertex set of a graph V(G) [D,W] or V G [G] independent of names of edge or vertex set vertices & edges use distinct name space: E(G) V(G) = Ø may be indexed:,,, ; e 0, e 1, e 2, may use distinct symbols: v, w, x, ; e, f, g, Note: functions on vertices and edges omit subscript if graph to which they belong is obvious e.g. for G: E = E(G) = E G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-6 e 0 e 1 e 3 e 2 e 4 e 6 e 5 e 7

7 Graph Types Simple Simple graph contains no loops or multiedges in communication networks: loops generally not needed January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-7

8 Graph Types Multigraph Simple graph contains no loops or multiedges in communication networks: loops generally not needed Multigraph 1 contains multiedges 0 3 in communicatoin networks: represent trunk groups or multilink spans 2 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-8

9 Simple graph Graph Types Weighted contains no loops or multiedges Multigraph contains multiedges Weighted graph edges have a weight (scalar or vector) in communication networks: weight is a link attribute referred to as cost e.g. capacity, bandwidth utilised, delay, monetary cost (confusingly all refered to as link cost ) vector describes multiple attributes 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT

10 Graph Types Directed Simple graph contains no loops or multiedges Multigraph contains multiedges Weighted graph edges have a weight or cost Directed graph (or digraph) arcs are directed edges may be weighted digraph lecture NF in communication networks: needed for asymmetric links January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-10

11 Graph Types Mixed Simple graph 1 contains no loops or multiedges 0 3 Multigraph 1 2 contains multiedges 0 3 Weighted graph edges have a weight (scalar or vector) Directed graph (or digraph) edges are directed Mixed (partially directed) graph contains both undirected and directed edges January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-11

12 Forest graph with no cycles set of trees Tree T Graph Types Tree and Forest maximal component with no cycles vertices with d(t)=1 are leaves vertices with d(t)>1 are interior vertices any arbitrary vertex may be designated as the root T r in which case it is not a leaf even if d(t r )=1 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-12

13 Subgraph H G subset of vertices V H V G and edges E H E G Proper subgraph H G H G and H G Example? Set-Theoretic Operations Subgraph v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-13

14 Set-Theoretic Operations Subgraph Subgraph H G subset of vertices V H V G and edges E H E G Proper subgraph H G H G and H G Example: H 1 G v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 1 v 4 e 3 e 6 e 8 e 2 e 4 e 7 e 5 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-14

15 Induced subgraph H G, H =: G[V H ] all edges E H E G are included that incident to V H Example: H 1 =: G[V H2 ]? Set-Theoretic Operations Induced Subgraph v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 1 v 4 e 3 e 6 e 8 e 2 e 4 e 7 e 5 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-15

16 Set-Theoretic Operations Induced Subgraph Induced subgraph H G, H =: G[V H ] all edges E H E G are included that incident to V H Example: H 1 not induced subgraph because e 4 missing H 2 =: G[V H2 ] v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 2 v 4 e 3 e 5 e 4 e 6 e 8 e 2 e 4 e 7 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-16

17 Set-Theoretic Operations Spanning Subgraph Spanning subgraph H G subgraph contains all vertices V H = V G but only a subset of edges E H E G Example: H 2 spanning subgraph? v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 2 v 4 e 3 e 5 e4 e 6 e 8 e 2 e 4 e 7 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-17

18 Set-Theoretic Operations Spanning Subgraph Spanning subgraph H G subgraph contains all vertices V T = V G but only a subset of edges E T E G Example: H 2 not spanning subgraph because v 4 missing H 3 G; V H3 = V G v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 3 v 4 e 0 e 3 e 2 e 4 e 7 e 5 e 8 e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-18

19 Set-Theoretic Operations Spanning Tree Spanning tree T G subgraph contains all vertices V T = V G and no cycles C T all connected graphs have at least one spanning tree Example: H 3 spanning tree? v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 3 v 4 e 0 e 3 e 2 e 4 e 7 e 5 e 8 e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-19

20 Set-Theoretic Operations Spanning Tree Spanning tree T G subgraph contains all vertices V T = V G and no cycles C T all connected graphs have at least one spanning tree Example: H 3 not spanning tree, e.g.,, v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G H 3 v 4 e 0 e 3 e 2 e 4 e 7 e 5 e 8 e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-20

21 Set-Theoretic Operations Spanning Tree Spanning tree T G subgraph contains all vertices V T = V G and no cycles C T all connected graphs have at least one spanning tree Example: H 3 not spanning tree, e.g.,, T 4 G; C T 4 after e 3, e 4 removed from H 3 v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G T 4 v 4 e 0 e 3 e 2 e 4 e 7 e 5 e 8 e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-21

22 Set-Theoretic Operations Spanning Tree Spanning tree T G subgraph contains all vertices V T = V G and no cycles C T subgraph edges in G T are chords of the tree T Example: H 3 not spanning tree, e.g.,, T 4 G; C T 4 after e 3, e 4 removed; e 1, e 3, e 4, e 5 chords v 5 v 5 v 4 e 0 e 1 e 3 e 5 e 6 e 8 G T 4 v 4 e 1 e 0 e 3 e 2 e 4 e 7 e 5 e 8 e 2 e 4 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-22

23 Complement G Set-Theoretic Operations Complement identical vertices V G = V G only possible edges not in original graph: E G = V G V G E G Example: v 5 v 5 G G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-23

24 Set-Theoretic Operations Line Graph Line Graph L(G) graph formed by swapping vertices and edges not to be confused with linear graph or network lecture RN Example: e 0 e 2 e 0 v 4 e 3 e 1 e 4 e 3 e 1 e 2 e 3 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-24

25 Set-Theoretic Operations Edge Operations Set-theoretic operations with edges and vertices edge addition edge deletion e 1 1 G e 4 e 1 1 G+e 6 e 4 1 e 1 e 3 G e 2 e 4 0 e e e 6 3 e 2 2 e 5 e 2 2 e 5 2 e 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-25

26 Set-Theoretic Operations Graph Operations Set-theoretic operations on multiple graphs union intersection difference 0 H G 3 1 G H G H G H January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-26

27 Simple Graph Types Complete Complete graph full mesh k -regular where k = G 1 all vertex pairs adjacent denoted K n [W2001, GY2006] or K n [D2010] with n vertices 1 K 4 2 K n (n 1)/2 edges (undirected) n (n 1) edges (directed) 0 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-27

28 r-partite graph r classes of vertices Simple Graph Types r -Partite every edge s incident vertices are in different classes [D] K 2, January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-28

29 Complete bipartite graph Graph Theory Simple Graph Types all vertex pairs in different (disjoint) partite sets adjacent K r,s [W2001, GY2006, D2010] with r+s vertices 3 K 2,3 0 rs edges (undirected) 2rs edges (directed) January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-29

30 Graph Theory Simple Graph Types Complete bipartie graph K r,s all vertex pairs in different (disjoint) partite sets adjacent with n vertices with n (n 1)/2 edges 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-30

31 Graph Theory Isomorphism Isomorphic graphs G H structurally identical regardless of how drawn generally we consider them equal: G H G = H note that H appears to be non-planar (edges cross) G H v 4 v 4 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-31

32 Graph Theory Isomorphism Isomorphic graphs G H structurally identical regardless of how drawn generally we consider them equal: G H G = H but (isomorphic) H is planar (edges do not cross) G H v 4 v 4 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-32

33 Incidence matrix Q Graph Representation Data Structures rectangular matrix with a row/vertex and column/edge Adjacency matrix A square matrix with row, column entry for each vertex v i Path matrix P square matrix giving shortest path lengths between vertices 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-33

34 Graph Representation Incidence Matrix Incidence matrix Q (note: I refers to identity matrix) rectangular matrix rows r i are vertices v i columns c i are edges e i requires separate edge labelling Q(G) e 0 e 1 e 2 e 3 e 4 e 0 e 1 e 2 e 4 e 3 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-34

35 Graph Representation Incidence Matrix Incidence matrix Q (note: I refers to identity matrix) rectangular matrix rows r i are vertices v i columns c i are edges e i requires separate edge labelling Q(G) e 0 e 1 e 2 e 3 e e 0 e 1 e 2 e 4 e 3 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-35

36 Adjacency matrix A Graph Representation Adjacency Matrix square matrix with row, column entry for each vertex v i A(G) e 0 e 1 e 2 e 4 e 3 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-36

37 Adjacency matrix A Graph Representation Adjacency Matrix square matrix with row, column entry for each vertex v i defines edges e ij = v i v j between all vertex pairs for simple graph 0 = no edge, 1 = edge (simple graph) all diagonals 0 since no self-loops A(G) e 0 e 1 e 2 e 4 e 3 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-37

38 Adjacency matrix A Graph Representation Adjacency Matrix square matrix with row, column entry for each vertex v i defines edges e ij = v i v j between all vertex pairs for simple graph 0 = no edge, 1 = edge (simple graph) A(G) all diagonals 0 symmetric since undirected January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-38 e 0 e 1 e 2 e 4 e 3 G

39 Path matrix P Graph Representation Path Matrix square matrix with row, column entry for each vertex v i defines shortest path lengths d(v i v j ) between all vertex pairs all diagonals 0 P(G) e 0 e 1 e 2 e 4 e 3 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-39

40 Path matrix P Graph Representation Path Matrix square matrix with row, column entry for each vertex v i defines shortest path lengths d(v i v j ) between all vertex pairs all diagonals 0 for connected graphs all other entries nonzero p ij = min(pathlen(v i, v j )) P(G) symmetric since undirected _ 1 1 _ January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-40 e 0 e 1 e 2 e 4 e 3 G

41 Path matrix P Graph Representation Path Matrix square matrix with row, column entry for each vertex v i defines shortest path lengths d(v i v j ) between all vertex pairs all diagonals 0 for connected graphs all other entries nonzero p ij = min(pathlen(v i, v j )) P(G) symmetric since undirected January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-41 e 0 e 1 e 2 e 4 e 3 G

42 Graph Representation Laplacian Matrix Laplacian matrix L = D A square matrix with row, column entry for each vertex v i L e 0 e 3 e 1 e 2 e 4 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-42

43 Graph Representation Laplacian Matrix Laplacian matrix L = D A square matrix with row, column entry for each vertex v i degree matrix D ij = v i if i=j; 0 otherwise added to negative adjacency matrix A D A = L e 0 e 3 e 1 e 2 e 4 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-43

44 Graph Representation Laplacian Matrix Laplacian matrix L = D A square matrix with row, column entry for each vertex v i degree matrix D ij = v i if i=j; 0 otherwise added to negative adjacency matrix A D A _ _ _ _ = L e 0 e 3 e 1 e 2 e 4 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-44

45 Graph Representation Laplacian Matrix Laplacian matrix L = D A square matrix with row, column entry for each vertex v i degree matrix D ij = v i if i=j; 0 otherwise added to negative adjacency matrix A Property? D A L = e 0 e 3 e 1 e 2 e 4 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-45

46 Graph Representation Laplacian Matrix Laplacian matrix L = D A square matrix with row, column entry for each vertex v i degree matrix D ij = v i if i=j; 0 otherwise added to negative adjacency matrix A rows and columns sum to 0 D A L = e 0 e 3 e 1 e 2 e 4 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-46

47 Graphs Theory Graph Properties and Metrics GT.1 Graph types and representation GT.2 Graph properties and metrics GT.3 Distance and connectivity GT.4 Centrality GT.5 Special graphs: fuzzy, time-varying, hypergraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-47

48 Graph Properties Scale Scale order of a graph: number of vertices G [D] or n(g) [W] empty graph Ø = (Ø, Ø) has G = 0 trivial graph has G = 1 infinite graph has G = size of a graph: number of edges G [D] or e(g) [W] Examples G =? G =? common in networking to use size for number of nodes = order 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-48 e 0 e 1 e 3 e 2 e 4 e 6 e 5 G e 7

49 Graph Properties Scale Scale order of a graph: number of vertices G [D] or n(g) [W] empty graph Ø = (Ø, Ø) has G = 0 trivial graph has G = 1 infinite graph has G = size of a graph: number of edges G [D] or e(g) [W] Examples G = 4 G = 8 common in networking to use size for number of nodes = order 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-49 e 0 e 1 e 3 e 2 e 4 e 6 e 5 G e 7

50 Graph Properties Neighbourhood Neighbourhood of a vertex (open) neighbourhood N G (v i ) or N(v i ) [W,GY,D] set of all vertices adjacent to v i closed neighbourhood N G [v i ] or N [v i ] [GY] N [v i ] = N G (v i ) {v i } Examples N( ) =? N [ ] =? 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-50 v 4 e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

51 Graph Properties Neighbourhood Neighbourhood of a vertex (open) neighbourhood N G (v i ) or N(v i ) [W,GY,D] set of all vertices adjacent to v i closed neighbourhood N G [v i ] or N [v i ] [GY] N [v i ] = N G (v i ) {v i } Examples N( ) = {,,, v 4 } N( ) = 4 N [ ] = {,,,, v 4 } N [ ] = 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-51 v 4 e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

52 Graph Properties Degree Degree of a vertex number of incident edges d(v i ) [D,W], deg(v i ) [GY] d( v ) a i i j V v 4 e 0 e 5 v 5 e 1 e 7 e 2 e 4 e 6 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-52 e 3

53 Graph Properties Degree Degree of a vertex number of incident edges d(v i ) [D,W], deg(v i ) [GY] min degree of graph: (G)=min(d G (v i )) [D,W], min (G) [GY] max degree of graph: (G)=max(d G (v i )) [D,W], max (G) [GY] avg degree of a graph: 1 d( ) d( v ) i V (G) d (G) (G) k-regular graph: d(v)=k v G Examples =? =? d =? 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-53 v 4 v V e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

54 Graph Properties Degree Degree of a vertex number of incident edges d(v i ) [D,W], deg(v i ) [GY] min degree of graph: (G)=min(d G (v i )) [D,W], min (G) [GY] max degree of graph: (G)=max(d G (v i )) [D,W], max (G) [GY] avg degree of a graph: 1 d( ) d( v ) i V (G) d (G) (G) k-regular graph: d(v)=k v G Examples = d(v 5 ) = 1 = d( ) = d( ) = 4 d = (1+3(3)+2(4))/6 = 3 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-54 v 4 v V e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

55 Graph Properties Degree Distribution Degree distribution of a graph distribution of degrees d(v i ) Example = 0 = 4 d = (1+3(3)+2(4))/6 = d v 6 v 4 e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-55

56 Graph Properties k -Regular Graphs k-regular graph uniform degree distribution: d(v)=k v G Regular graph: all vertices have the same degree k d(v)=k v G denoted K n [W, GY] or K n [D] with n vertices 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-56

57 Degree of a digraph D vertex v i indegree d + (v i ) [D,W]: # of arcs whose head is incident to v i outdegree d (v i ) [D,W]: # of arcs whose tail is incident to v i {min, max} {in, out}degree relations: + (D) d + (D) + (D) Examples (D) d (D) (D) =? =? d =? + =? + =? d + =? Graph Properties Degree of a Digraph 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-57 v 4 e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

58 Degree of a digraph D vertex v i indegree d + (v i ) [D,W]: # of arcs whose head is incident to v i outdegree d (v i ) [D,W]: # of arcs whose tail is incident to v i {min, max} {in, out}degree relations: + (D) d + (D) + (D) Examples (D) d (D) (D) + = 0 + = 4 d + = (0+2(1)+2+3+4)/6 = 1.83 = 1 = 3 Graph Properties Degree of a Digraph d = (2(1)+3(2)+3)/6 = January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-58 v 4 e 0 e 3 e 2 e 4 e 6 e 5 v 5 e 1 e 7

59 Assortativity Graph Theory Assortativity probability of attachment to similar vertices typically in degree distribution 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-59

60 Graph Theory Assortativity Assortativity probability of attachment to similar vertex degrees Example assortative? January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-60

61 Graph Theory Assortativity Assortativity probability of attachment to similar vertex degrees Example assortative 8 vertices d=4 8 vertices d= January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-61

62 Graph Theory Assortativity Assortativity probability of attachment to similar vertex degrees Example assortative: yes 8 vertices d=4 incident to 7 d=4 vertices 1 d=9 vertex 8 vertices d=8 incident to 7 d=8 vertices 1 d=4 vertex January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-62

63 Graphs Theory Connectivity GT.1 Graph types and representation GT.2 Graph properties and metrics GT.3 Distance and connectivity GT.3 Centrality GT.4 Special graphs: fuzzy, time-varying, hypergraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-63

64 Graph Properties Walks, Paths, Cycles Walk through a graph nonempty alternating sequence e 0 e 1 e k 1 v k such that e i = {v k, v i+1 } length of walk is k [W2001, D2001] walk is closed iff = v k e 0 e 5 e 1 e 3 e 2 e 4 e 6 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-64

65 Graph Properties Walks, Paths, Cycles Walk through a graph nonempty alternating sequence e 0 e 1 e k 1 v k such that e i = {v k, v i+1 } length of walk is k [W2001, D2001] walk is closed iff = v k Paths and walks path P is a walk whose vertices are distinct cycle is a closed path trail is a walk whose edges are distinct Use in networks? only relevant for multigraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-65 e 0 e 1 e 3 e 2 e 4 e 6 e 5 e 7

66 Graph Properties Walks, Paths, Cycles Walk through a graph nonempty alternating sequence e 0 e 1 e k 1 v k such that e i = {v k, v i+1 } length of walk is k [W2001, D2001] walk is closed iff = v k Paths and walks path P is a walk whose vertices are distinct cycle is a closed path trail is a walk whose edges are distinct only relevant for multigraphs communication along paths in networks no reason to traverse vertex more than once! 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-66 e 0 e 1 e 3 e 2 e 4 e 6 e 5 e 7

67 Graph Properties Shortest Paths Path P is a walk whose vertices are distinct also distinct edges for simple graphs Shortest path P with fewest vertices path P st between vertices v s and v t : v s v t path P sit between vertices v s and v t including v i : v s v i v t path P i between any vertices including v i : v x v i v y shortest path need not be unique generally want shortest path in comm. net Unique shortest path P* i th shortest path P i 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-67 e 0 e 1 e 3 e 2 e 4 e 6 e 5 e 7

68 Length hopcount Graph Properties Length and Distance number of edges links on a walk, path, or trail cycle of length k is a k-cycle C k Distance shortest path hopcount d G (v i,v j ) length of the shortest path from v i to v j if no such path exists d G (v i,v j ) = Average path length average hopcount mean distance over all vertex pairs i, j d( v, v ) i nn ( 1) e 4 most real-world complex networks have small APL e small world effect 2 e 6 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-68 j e 0 e 1 e 3 e 5 e 7

69 Distance Metrics Diameter and radius (very important in networking) eccentricity (v i ) is max distance to any other vertex max(d(v i, v j )) v j V G = E / V = ½ d (G) [W2001] radius rad(g) [D2010] shortest eccentricity in G, min( (v i ) v j V G ) length of shortest shortest path diameter diam(g) [D2010] greatest distance in G, max( (v i ) v j V G ) length of longest shortest path important network property rad(g) diam(g) 2 rad(g) 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-69 e 0 e 1 e 3 e 2 e 4 e 6 e 5 e 7

70 Distance Circumference and Girth Girth and circumference (limited use in networking) girth is length of shortest cycle g(g) is min(length(c k C G))) if C G then g(g) = circumference is length of longest cycle circim(g) is max(length(c k C G))) if C G then circum(g) = 0 e 0 e 5 v 5 v 4 e 1 e 3 e 2 e 4 e 6 e 7 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-70

71 Graph G is connected Connectivity Introduction if there exists a path between every vertex pair graph that is not connected is disconnected communication networks should generally be connected DTNs (disruption tolerant networks) are frequently disconn. Connected component of CC(G) connected subgraph of G Giant component of GC(G) maximal connected subgraph of G Strongly connected component of a digraph SCC(D) all v D reachable from one another 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-71

72 Graph G is k-connected Connectivity k-connected remains connected with removal of any k 1 edges or vertices 2-connected is normally called biconnected communication nets should generally be at least biconnected why? Connectivity of a graph (G) = max(k) such that G is k-connected 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-72

73 Graph G is k-connected Connectivity k-connected remains connected with removal of any k 1 edges or vertices 2-connected is normally called biconnected communication nets should generally be at least biconnected resilience: tolerance to removal of link or node Connectivity of a graph (G) = max(k) such that G is k-connected 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-73

74 Connectivity Articulation Points and Bridges Cut vertex or articulation point vertex whose removal disconnects a component Bridge edge whose removal disconnects a component 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-74

75 Connectivity Articulation Point Cut vertex or articulation point vertex whose removal disconnects a component Example articulation points? v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-75

76 Connectivity Articulation Point Cut vertex or articulation point vertex whose removal disconnects a component Example articulation points:, v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-76

77 Connectivity Articulation Point Cut vertex or articulation point vertex whose removal disconnects a component Example articulation points:, components? v 4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-77

78 Connectivity Articulation Point Cut vertex or articulation point vertex whose removal disconnects a component Example articulation points:, components: v 4, v 5, {v 4,v 5 },, {, } v 6 v 4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-78

79 Bridge Connectivity Bridge edge whose removal disconnects a component Example bridges? v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-79

80 Bridge Connectivity Bridge edge whose removal disconnects a component Example bridge: {, v 6 } v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-80

81 Bridge Connectivity Bridge edge whose removal disconnects a component Example bridge: {, v 6 } v 4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-81

82 Biconnected k-connected for k=2 Connectivity Biconnectivity Example: not biconnected articulation points:, bridge: {, v 6 } v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-82

83 Connectivity Biconnectivity Biconnected k-connected for k=2 no single vertex or edge removal partitions the graph no single node or link failure disconnects the network Example: biconnected {v 5, } spans articulation point: {, v 6 } spans articulation point: {, v 6 } alternative to bridge: {, v 6 } v 4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-83

84 Connectivity Clique Clique C G fully connected set of vertices in a graph we normally ignore: single vertices (clique of order 1) pairs of vertices connected by an edge (clique of order 2) maximal clique cannot be extended by the addition of any adjacent vertex that forms a larger clique maximum clique is the largest clique in a given graph G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-84

85 Connectivity Clique Clique C G fully connected set of vertices in a graph Example: maximal cliques? Hint: there are 3 maximum clique? v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-85

86 Connectivity Clique Clique C G fully connected set of vertices in a graph Example: maximal cliques: V C1 = {, v 4, v 5 }; V C2 = {,,, } ; V C3 = {, v 6 } maximum clique? v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-86

87 Connectivity Clique Clique C G fully connected set of vertices in a graph Example: maximal cliques: V C1 = {, v 4, v 5 }; V C2 = {,,, } ; V C3 = {, v 6 } maximum clique: V C2 = 4 v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-87

88 Connectivity Clustering Coëfficient Clustering coëfficient cc(g) measures degree to which nodes cluster with one another local: global: 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-88

89 Connectivity Clustering Coëfficient : Local Local clustering coëfficient cc(v i ) fraction of edges among neighbours to form a clique number of edges in clique: C(N(v i )) = k(k 1)/2; d(v i ) = k cc(v i ) = 2 N(v i ) /k(k 1)) 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-89

90 Connectivity Clustering Coëfficient : Local Local clustering coëfficient cc(v i ) fraction of edges among neighbours to form a clique number of edges in clique: C(N(v i )) = k(k 1)/2; d(v i ) = k cc(v i ) = 2 N(v i ) /k(k 1)) Examples cc(v i ) =? v 7 v 7 G 1 G 2 G 3 G 4 v 7 v 7 v 8 v 8 v 8 v 4 v 8 v 9 v 9 v 9 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-90

91 Connectivity Clustering Coëfficient : Local Local clustering coëfficient cc(v i ) fraction of edges among neighbours to form a clique number of edges in clique: C(N(v i )) = k(k 1)/2; d(v i ) = k cc(v i ) = 2 N(v i ) /k(k 1)) Examples cc( ) = 0; cc( ) = 1 / 3 ; cc( ) = 2 / 3 ; cc(v 4 ) = 1 v 7 v 7 G 1 G 2 G 3 G 4 v 7 v 7 v 8 v 8 v 8 v 4 v 8 v 9 v 9 v 9 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-91

92 Connectivity Clustering Coëfficient: Average Average clustering coëfficient cc(g) average clustering coëfficient across all vertices n 1 cc( v ) i i 0 cc( ) ; vi n if cc(g) cc(g rand ) of same order, G is small world network lecture SW 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-92

93 Connectivity Clustering Coëfficient: Global Global clustering coëfficient cc(g) degree to which graph contains triangular subgraphs (cliques) 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-93

94 Graphs Theory Centrality GT.1 Graph types and representation GT.2 Graph properties and metrics GT.3 Connectivity GT.3 Centrality GT.4 Special graphs: fuzzy, time-varying, hypergraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-94

95 Centrality Introduction Centrality is a measure of vertex or edge importance how central is in the graph by some importance measure Multiple centrality measures degree centrality betweeness centrality closeness centrality 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-95

96 Centrality Degree Centrality Degree centrality : importance by degree degree of the vertex degree centrality of a graph G = i max[d(v i )] Degree centrality of G? v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-96

97 Centrality Degree Centrality Degree centrality : importance by degree degree of the vertex degree centrality of a graph: C D (G) = i max[d(v i )] Degree centrality of C D (G) = d( ) = 5 v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-97

98 Centrality Degree Centrality Degree centrality : importance by degree degree of the vertex degree centrality of a graph: C D (G) = i max[d(v i )] Degree centrality of C D (G) = d( ) = 5 Communication networks provide significant connectivity to other nodes {,,,v 4,v 5 } v4 v 6 v 5 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-98

99 Coreness of a graph Centrality Coreness notion of core vs. periphery more global form of centrality Non-iterative coreness k-core of a graph: removing all v i such that d(v i ) < k core(g) = 0 all vertices remain v4 v 7 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-99

100 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k core(g) = 1 all connected vertices remain v4 v 7 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-100

101 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k core(g) = 2 stub vertices (d =1) removed v4 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-101

102 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k core(g) = 3 moderate values represent backbone of network v4 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-102

103 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k core(g) = 4 higher values may be less useful for communication nets v4 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-103

104 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k subgraph representing core(g) = 5 v4 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-104

105 Centrality Coreness Coreness of a graph (non-iterative) k-core of a graph: removing all v i such that d(v i ) < k core(g) = 6 when core(g) = (G ) = Ø no vertices remain v4 v 6 v 5 G 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-105

106 Coreness of a graph Centrality Coreness iterative coreness: recompute on subgraph after each step core(h) = [HRI+2008] 3 H 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-106

107 Coreness of a graph Centrality Coreness iterative coreness: recompute on subgraph after each step core(h) = 2 stub vertices (d =1) removed [HRI+2008] 3 H 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-107

108 Coreness of a graph Centrality Coreness iterative coreness: recompute on subgraph after each step core(h) = 3 vertices (d 2) removed: backbone remains [HRI+2008] 3 H 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-108

109 Centrality Betweeness Betweeness centrality of a vertex or edge number of shortest paths passing through Communication networks these nodes and links will carry many end-to-end flows 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-109

110 Closeness centrality Centrality Closeness assumes that important vertices are close to all others Closeness of a vertex inverse of farness sum of the hopcounts to all other vertices close( v ) 1 / d( v, d ) Closeness of a graph 1 far( v ) far( v ) d( v, d ) 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-110 n 1 i i j j 0 j i i i j i j 0 j i 1 close( ) close( v ) i n v i G n 1

111 Graphs and Network Topology Special Graphs GT.1 Graph types and representation GT.2 Graph properties and metrics GT.3 Connectivity GT.3 Centrality GT.4 Special graphs: fuzzy, time-varying, hypergraphs 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-111

112 Special Graphs Multilevel Graph Multilevel graph G = (G 0, G 1, G L 1 ) [CPS2012] ordered from lowest to highest of L levels graph G i = (V I ; E i ) at level i level i is a numerical layer or descriptive label e.g. G 2, G 2.5, G 3, G 3.5, G 4 e.g. G HBH, G L3, G AS, G E2E (i, j) + i j: vertices at layer above are subset of layer below: V i V j edges at each level are arbitrary connection of vertices partition at lower layer forces partition at the layer above: (u, v) V j : conn i (u, v)=false conn j (u, v)=false 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-112

113 Multilevel Network Analysis Multilevel Graph Model Connected network Disconnected network Partitioned network Multilevel model for unweighted & undirected graphs Two requirements for multilevel graph model: nodes at the above level are subset of lower level nodes that are disconnected below are disconnected above 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-113

114 Special Graphs Multiprovider Graph Multilprovider graph G* = (G L3i, G AS ) [ÇPS2012] intradomain provider graphs G L3i interdomain AS graph G AS 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-114

115 Abstraction of Internet Topology Multilevel and Multiprovider View E2E topology users and apps AS 1 AS level topology AS 2 AS 3 AS 4 ISP 1 router level topology IXP 1 ISP 2 IXP 2 ISP 3 IXP 3 ISP 4 physical level topology 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-115

116 Special Graphs Time-Varying Graph Time-varying graph G = (V, E,,,, ) [CSQS2011] with labelled edges E V V L and vertices V set of vertices V(G) = {,, } correspond to links set of edges E(G) = {e 0, e 1, } correspond to nodes mapping of e i endpoint pairs {v j, v k } gives net topology lifetime of the system T presence of an edge at given time : E {0,1} latency of an edge at a given time : E T presence of an edge at given time : E {0,1} presence of an vertex at given time : V {0,1} vertex labelling added in [ZGB+2012] 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-116

117 Network and Challenge Modelling Representation of Dynamic MANET Topologies MANET model example [ZGB+2012] strong link weights represent high link availability January 5 October KU EECS Modelling 784 Science Wireless of Nets Challenges Graph Theory 117 SCN-GT-117

118 Fuzzy graph Special Graphs Fuzzy Graph fuzzy logic used to determine set membership 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-118

119 Hypergraph Special Graphs Hypergraph edges connect an arbitrary number of vertices may be greater than two 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-119

120 Graphs, Flows, and Net Topology References and Further Reading [L2009] Ted G. Lewis, Network Science, Wiley, 2009 [N2010] M.E.J. Newman, Networks: An Introduction, Oxford, January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-120

121 Graph Theory References and Further Reading [D2010] Reinhard Dietsel, Graph Theory, 4th ed., Graduate Texts in Mathematics 173, Springer, 2010 [GY2006] Jonathan L. Gross and Jay Yellen, Graph Theory and its Applications, 2nd ed., Chapan and Hall, 2006 [W2001] Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, 2001 [MN2000] John N. Mordeson and Premchand S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Physica-Verlag Springer, 2000 [V2009] Vitaly I. Voloshin, Introduction to Graph and Hypergraph Theory, Nova Science Publishers, January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-121

122 Graph Theory References and Further Reading [CRTB2007] L. da F. Costa, F.A. Rodrigues, G. Travieso, and P.R. Villas Boas, Characterization of Complex Networks: A Survey of Measurements, Advances in Physics, Taylor & Francis, vol.56 no.1, Feb. 2007, pp [BLM+2006] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex Networks: Structure and Dynamics, Physics Reports, Elsevier, (424) 2006, pp January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-122

123 Graphs, Flows, and Net Topology References and Further Reading [AMO1993] Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice Hall, January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-123

124 End of Foils 24 January 2017 KU EECS 784 Science of Nets Graph Theory SCN-GT-124