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Graphs Definitions, Measures, Graphology, Pathology, Degree Distributions

Graphs Definitions, Measures, Graphology, Pathology, Degree Distributions. Network Science: Graph Theory 2012. COMPONENTS OF A COMPLEX SYSTEM. components : nodes , vertices N. interactions : links, edges L. system : network, graph (N,L).

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Graphs Definitions, Measures, Graphology, Pathology, Degree Distributions

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  1. Graphs Definitions, Measures, Graphology, Pathology, Degree Distributions Network Science: Graph Theory 2012

  2. COMPONENTS OF A COMPLEX SYSTEM • components: nodes, vertices N • interactions: links, edges L • system: network, graph (N,L) Network Science: Graph Theory 2012

  3. NETWORKS OR GRAPHS? • network often refers to real systems • www, • social network • metabolic network. • Language: (Network, node, link) • graph: mathematical representation of a network • web graph, • social graph (a Facebook term) • Language: (Graph, vertex, edge) • We will try to make this distinction whenever it is appropriate, but in most cases we will use the two terms interchangeably. Network Science: Graph Theory 2012

  4. A COMMON LANGUAGE friend Movie 1 co-worker Mary Actor 2 Peter Actor 1 Albert Actor 4 Movie 3 friend brothers Movie 2 Albert Actor 3 Protein 2 Protein 1 Protein 5 N=4 L=4 Protein 9 Network Science: Graph Theory 2012

  5. CHOOSING A PROPER REPRESENTATION The choice of the proper network representation determines our ability to use network theory successfully. In some cases there is a unique, unambiguous representation. In other cases, the representation is by no means unique. For example,, the way we assign the links between a group of individuals will determine the nature of the question we can study. Network Science: Graph Theory 2012

  6. UNDIRECTED VS. DIRECTED NETWORKS Undirected Directed Links: undirected (symmetrical) Graph: Links: directed (arcs). Digraph = directed graph: L A D M B An undirected link is the superposition of two opposite directed links. F C I D G B E G A H C F Directed links : URLs on the www phone calls metabolic reactions Undirected links : coauthorship links Actor network protein interactions Network Science: Graph Theory 2012

  7. NODE DEGREES Node degree: the number of links connected to the node. A Undirected B D B C • In directed networks we can define an in-degree and out-degree. The (total) degree is the sum of in- and out-degree. • Source: a node with kin= 0; Sink: a node withkout= 0. Directed E G A F

  8. Graph density • No. of the connections that may exist between n nodes • directed graph emax= n*(n-1) • undirected graph emax= n*(n-1)/2 • What fraction are present? • density = e/ emax • For example, out of 12 possible connections, this graph has 7, giving it a density of 7/12 = 0.583

  9. Graph density • Would this measure be useful for comparing networks of different sizes (different numbers of nodes)? • As n → ∞, a graph whose density reaches • 0 is a sparse graph • a constant is a dense graph

  10. AVERAGE DEGREE Recall: <x> = (x1 + x1 + ... + xN)/N = Σi xi /N Undirected j • N – the number of nodes in the graph • L – the number of links in the graph i D B C Directed E A F Network Science: Graph Theory 2012

  11. COMPLETE GRAPH The maximum number of links a network of N nodes can have is: • A graph with degree L=Lmaxis called a complete graph, and its average degree is <k>=N-1 Network Science: Graph Theory 2012

  12. REAL NETWORKS ARE SPARSE Most networks observed in real systems are sparse: L << Lmax or <k> <<N-1. WWW (ND Sample): N=325,729; L=1.4 106Lmax=1012 <k>=4.51 Protein (S. Cerevisiae): N= 1,870; L=4,470 Lmax=107 <k>=2.39 Coauthorship (Math): N= 70,975; L=2 105 Lmax=3 1010 <k>=3.9 Movie Actors: N=212,250; L=6 106 Lmax=1.8 1013 <k>=28.78 (Source: Albert, Barabasi, RMP2002) Network Science: Graph Theory 2012

  13. ADJACENCY MATRIX Aij=1 if there is a link between node i and j Aij=0 if nodes i and j are not connected to each other. 4 4 3 3 2 2 1 1 Undirected is not symmetric. Directed is symmetric. Network Science: Graph Theory 2012

  14. ADJACENCY MATRIX a bcdefgh a 0 1 0 0 1 0 1 0 b 1 0 1 0 0 0 0 1 c 0 1 0 1 0 1 1 0 d 0 0 1 0 1 0 0 0 e 1 0 0 1 0 0 0 0 f0 0 1 0 0 0 1 0 g1 0 1 0 0 0 0 0 h0 1 0 0 0 0 0 0 a e h b d f g c Network Science: Graph Theory 2012

  15. ADJACENCY MATRIX AND NODE DEGREES 4 Undirected 3 2 1 Directed 4 3 2 1 Network Science: Graph Theory 2012

  16. GRAPHOLOGY • 1 Undirected Directed 4 4 1 1 2 2 3 3 Actor network, protein-protein interactions WWW, citation networks Network Science: Graph Theory 2012

  17. GRAPHOLOGY • 2 Unweighted (undirected) Weighted (undirected) 4 4 1 1 2 2 3 3 protein-protein interactions, www Call Graph, metabolic networks Network Science: Graph Theory 2012

  18. GRAPHOLOGY • 3 Self-interactions • Multigraph • (undirected) 4 4 1 1 2 2 3 3 Protein interaction network, www Social networks, collaboration networks Network Science: Graph Theory 2012

  19. GRAPHOLOGY • 4 Complete Graph (undirected) 4 1 2 3 Actor network, protein-protein interactions Network Science: Graph Theory 2012

  20. GRAPHOLOGY: Real networks can have multiple characteristics WWW > directed multigraph with self-interactions Protein Interactions > undirected unweighted with self-interactions Collaboration network > undirected multigraph or weighted. Mobile phone calls > directed, weighted. Facebook Friendship links > undirected, unweighted. Network Science: Graph Theory 2012

  21. BIPARTITE GRAPHS bipartite graph(or bigraph) is a graph whose nodes can be divided into two disjoint setsU and V such that every link connects a node in U to one in V; that is, U and V are independent sets. Examples: Hollywood actor network Collaboration networks Disease network (diseasome) Network Science: Graph Theory 2012

  22. PATHS A path is a sequence of nodes in which each node is adjacent to the next one Pi0,in of length nbetween nodes i0 and in is an ordered collection of n+1 nodes and n links B • A path can intersect itself and pass through the same link repeatedly. Each time a link is crossed, it is counted separately • A legitimate path on the graph on the right: ABCBCADEEBA • In a directed network, the path can follow only the direction of an arrow. A E C D Network Science: Graph Theory 2012

  23. DISTANCE IN A GRAPH Shortest Path, Geodesic Path The distance (shortest path, geodesic path) between two nodes is defined as the number of edges along the shortest path connecting them. *If the two nodes are disconnected, the distance is infinity. In directed graphs each path needs to follow the direction of the arrows. Thus in a digraph the distance from node A to B (on an AB path) is generally different from the distance from node B to A (on a BCA path). B A C D B A C D Network Science: Graph Theory 2012

  24. NUMBER OF PATHS BETWEEN TWO NODES Adjacency Matrix • Nij, number of paths between any two nodes i and j: • Length n=1:If there is a link between i and j, then Aij=1 and Aij=0 otherwise. • Length n=2:If there is a path of length two between i and j, then AikAkj=1, and AikAkj=0 otherwise. • The number of paths of length 2: • Length n: In general, if there is a path of length n between i and j, then Aik…Alj=1 and Aik…Alj=0 otherwise. • The number of paths of length n between i and j is Network Science: Graph Theory 2012

  25. FINDING DISTANCES: BREADTH FIRST SEARCH • Distance between node1and node 4: • Start at1. 1 1 1 3 4 3 1 2 4 3 2 1 1 1 3 1 4 2 3 3 4 4 1 3 4 4 4 2 2 3 Network Science: Graph Theory 2012

  26. FINDING DISTANCES: BREADTH FIRST SEATCH • Distance between node1and node 4: • Start at1. • Find the nodes adjacent to1. Mark them as at distance 1. Put them in a queue. 3 4 3 2 4 3 2 1 1 1 1 1 1 3 4 2 3 3 4 4 1 1 3 4 4 4 2 2 3 Network Science: Graph Theory 2012

  27. FINDING DISTANCES: BREADTH FIRST SEATCH • Distance between node1and node 4: • Start at1. • Find the nodes adjacent to1. Mark them as at distance 1. Put them in a queue. • Take the first node out of the queue. Find the unmarked nodes adjacent to it in the graph. Mark them with the label of 2. Put them in the queue. 3 4 3 2 2 4 3 1 1 2 2 1 1 1 1 1 1 3 4 2 2 3 3 4 4 1 1 1 3 4 4 4 2 2 2 2 3 Network Science: Graph Theory 2012

  28. FINDING DISTANCES: BREADTH FIRST SEATCH • Distance between node1and node 4: • Repeat until you find node 4 or there are no more nodes in the queue. • The distance between1and4is the label of4or, if4does not have a label, infinity. 3 4 3 2 4 3 2 1 1 1 3 4 2 3 3 4 4 1 3 4 4 4 2 2 3 Network Science: Graph Theory 2012

  29. NETWORK DIAMETER AND AVERAGE DISTANCE • Diameter:dmaxthe maximum distance between any pair of nodes in the graph; along the shortest path. • Average path length/distance, <d>, for a connected graph: • where dijis the distance from node i to node j Network Science: Graph Theory 2012

  30. THE BRIDGES OF KONIGSBERG 1736: Konigsberg (Prussia) 7 bridges over river Pregelconnected 4 land masses. Can one walk across the seven bridges and never cross the same bridge twice? Euler PATH or CIRCUIT: return to the starting point by traveling each link of the graph once and only once. Network Science: Graph Theory 2012 http://www.numericana.com/answer/graphs.htm

  31. EULERIAN GRAPH: it has an Eulerian path Every vertex of this graph has an even degree, therefore this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle. http://en.wikipedia.org/wiki/Euler_circuit Network Science: Graph Theory 2012

  32. EULER CIRCUITS IN DIRECTED GRAPHS B D A C E If a digraph is strongly connected and the in-degree of each node is equal to its out-degree, then there is an Euler circuit G F Otherwise there is no Euler circuit. in a circuit we need to enter each node as many times as we leave it. Network Science: Graph Theory 2012

  33. 1 1 • PATHOLOGY: summary 2 2 5 5 Path Shortest Path 3 3 4 4 A sequence of nodes such that each node is connected to the next node along the path by a link. The path with the shortest length between two nodes (distance). Network Science: Graph Theory 2012

  34. 1 1 • PATHOLOGY: summary 2 2 5 5 Diameter Average Path Length 3 3 4 4 The longest shortest path in a graph The average of the shortest paths for all pairs of nodes. Network Science: Graph Theory 2012

  35. 1 1 • PATHOLOGY: summary 2 2 5 5 Cycle Self-avoiding Path 3 3 4 4 A path with the same start and end node. A path that does not intersect itself. Network Science: Graph Theory 2012

  36. 1 1 • PATHOLOGY: summary 2 2 5 5 Eulerian Path Hamiltonian Path 3 3 4 4 A path that traverses each link exactly once. A path that visits each node exactly once. Network Science: Graph Theory 2012

  37. CONNECTIVITY OF UNDIRECTED GRAPHS Connected (undirected) graph: any two vertices can be joined by a path. A disconnected graph is made up by two or more connected components. B B A Largest Component: Giant Component A C F D C F D F The rest: Isolates F G G Bridge: if we erase it, the graph becomes disconnected. Network Science: Graph Theory 2012

  38. CONNECTIVITY OF UNDIRECTED GRAPHS Adjacency Matrix The adjacency matrix of a network with several components can be written in a block-diagonal form, so that nonzero elements are confined to squares, with all other elements being zero: Figure after Newman, 2010 Network Science: Graph Theory 2012

  39. THREE CENTRAL QUANTITIES IN NETWORK SCIENCE Degree distribution pk Average path length <d> Clustering coefficient C Network Science: Graph Theory 2012

  40. STATISTICS REMINDER We have a sample of values x1, ..., xN Distribution of x (a.k.a. PDF): probability that a randomly chosen value is xP(x) = (# values x) / NΣiP(xi) = 1 always! Histograms >>> Network Science: Graph Theory 2012

  41. pdf -- the continuous case Will work on the board Network Science: Graph Theory 2012

  42. P(k) k • DEGREE DISTRIBUTION Degree distributionP(k): probability that a randomly chosen vertex has degree k Nk = # nodes with degree k P(k) = Nk / N ➔ plot 0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 4 Network Science: Graph Theory 2012

  43. DEGREE DISTRIBUTION discrete representation: pkis the probability that a node has degree k. continuum description: p(k) is the pdf of the degrees, where represents the probability that a node’s degree is between k1 and k2. Normalization condition: where Kmin is the minimal degree in the network. Network Science: Graph Theory 2012

  44. CLUSTERING COEFFICIENT • Clustering coefficient: What portion of your neighbors are connected? Captures the degree to which the neighbors of a given node link to each other. • Node i with degree Ki • ei is the number of links between the Kineighbors of node i • Ciin [0,1] Network Science: Graph Theory 2012

  45. THREE CENTRAL QUANTITIES IN NETWORK SCIENCE A. Degree distribution: pk B. Path length: <d> C. Clustering coefficient: Network Science: Graph Theory 2012

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