Network Basics. CMSC 498J: Social Media Computing. Department of Computer Science University of Maryland Spring Hadi Amiri

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1 Network Basics CMSC 498J: Social Media Computing Department of Computer Science University of Maryland Spring 2016 Hadi Amiri

2 Lecture Topics Graphs as Models of Networks Graph Theory Nodes, links, node degree, etc Graph density Complete Graph Distance and Diameter Adjacency matrix Graph Connectivity Reachability Sub-graphs Graph Types 2

3 Graph Theory A graph consists of N: a set of nodes (items, entities, people, etc), and E: a set of links or edges between nodes Graph is a way to specify relationships / links amongst a set of nodes. We define N= N size of N E= E size of E 3

4 Graph Theory. Cnt. Nodes i and j are adjacent or neighbors if: There is an edge btw them! i є N j є N (i, j) є E i j 4

5 Sample Graphs 1. Lives Near Graph nodes Links or edges Graph Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

6 Sample Graphs 2. Brand Proximity Graph Source: Inferring brand proximities from user-generated content. Paul Dwyer. J of Brand Management 19,

7 Graphs as Models of Networks ARPANET: Early Internet precursor December 1970 with13 nodes 7

8 Graphs as Models of Networks- Cnt. Only the connectivity matters Could capture distance as weights if needed 8

9 Graphs as Models of Networks- Cnt. Graph terminology often derived from transportation metaphors E.g. shortest path, flow, diameter 9

10 Graphs as Models of Networks- Cnt. Abstract graph theory is interesting in itself But in network science, items typically represent real-world entities Several examples (from Lecture 1.) Communication networks Companies, telephone wires Social networks People, friendship/contacts Information networks Web sites, hyperlinks 10

11 Basic Graph Concepts 11

12 Node Degree d(i) Given Node i, its degree d(i) is: the number nodes adjacent to it. Degree Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

13 Graph Density How many edges are possible? a j b i c d 13

14 Graph Density- Cnt. 5 a j b i c d 14

15 Graph Density- Cnt a j b i c d 15

16 Graph Density- Cnt a j b i c d 16

17 Graph Density- Cnt a j b i c d 17

18 Graph Density- Cnt a j b i c d 18

19 Graph Density- Cnt. (N-1) + (N-2) + (N-3) =? a j b i c d 19

20 Graph Density- Cnt. (N-1) + (N-2) + (N-3) = N * (N-1) / 2 a j b i c d 20

21 Graph Density- Cnt. Graph Density of a given graph G is determined by: the proportion of all possible edges that are present in the graph, i.e. With N nodes and E edges, graph density is: Number of edges in G / Number of all possible edges in G E N (N 1) 2 21

22 Graph Density- Cnt. Graph Density Degree E N (N 1) 2 6 / [6*(6-1)/2] = 6/15 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

23 Graph Density- Cnt. What is the density of this graph? N= 16 E= 20 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

24 Complete Graph If all edges are present, then all nodes are adjacent (neighbors), and the graph is a Complete Graph. a j b i c d What is the density of a complete graph? 24

25 Distance and Diameter Distance btw node i and j: d(i,j) Length of the shortest path between i and j Diameter of a graph Diameter of a graph is the maximum value of d(i,j) for all i and j Next session! for now: The path with min number of edges. 25

26 Distance and Diameter- Cnt. distance What is the distance and diameter of a complete graph? Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

27 Adjacency Matrix A = n 1 n 2 n 3 n 4 n 5 n n n n n Each row or column represents a node! A = A T Properties of adjacency matrix next session 27

28 Graph Connectivity Indirect connections between nodes We discuss about: Walks Trails Paths 28

29 Graph Connectivity- Cnt. Walk A sequence of nodes and edges that starts and ends with nodes where each node is incident to the edges following and preceding it. Trail A walk is a walk with distinct edges Path A walk with distinct nodes & edges. The length of a walk, trail, or path is the number of edges in it. 29

30 Graph Connectivity- Cnt. Walk A sequence of nodes and edges that starts and ends with nodes where each node is incident to the edges following and preceding it. Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

31 Graph Connectivity- Cnt. Walk A sequence of nodes and edges that starts and ends with nodes where each node is incident to the edges following and preceding it. Sample Walk: W=n 1 l 2 n 4 l 3 n 2 l 3 n 4 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

32 Graph Connectivity- Cnt. Trail A trail is a walk in which all edges are distinct, although some node(s) may be included more than once. Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

33 Graph Connectivity- Cnt. Trail A trail is a walk in which all edges are distinct, although some node(s) may be included more than once. Sample Trail: T=n 4 l 3 n 2 l 4 n 3 l 5 n 4 l 2 n 1 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

34 Graph Connectivity- Cnt. Path A path is a walk in which all nodes and all edges are distinct. Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

35 Graph Connectivity- Cnt. Path A path is a walk in which all nodes and all edges are distinct. Sample Path: P=n 1 l 2 n 4 l 3 n 2 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

36 Graph Connectivity- Cnt. Is this a Walk? Trail? Path? Yes, Yes, No We call a closed walk with distinct nodes & edges Cycle! n 2 l 4 n 3 l 5 n 4 l 3 n 2 Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

37 Reachability If there is a path between nodes i and j, then i and j are reachable from each other. 37

38 Connected Graph A graph is connected if every pair of its nodes are reachable from each other i.e. there is a path between them. Disconnected Graph How can we make this graph connected? Connected Graph and this graph disconnected? Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

39 Sub-graphs Graph G s is a sub-graph of G if its nodes and edges are a subset of G s nodes and edges respectively. 39

40 Sub-graphs- Cnt. Graph G s is a sub-graph of G if its nodes and edges are a subset nodes and edges of G respectively. a a b b G s1 i c i c d d a G G s2 i c d 40

41 Graph Types We study a few types of graphs: Bipartite graphs Digraphs Multigraphs Hypergraphs 41

42 Graph Types- Bipartite Graphs A bipartite graph is an undirected graph in which nodes can be partitioned into two (disjoint) sets N 1 and N 2 such that: (u, v) E implies either u N 1 and v N 2 or vice versa. In other words, all edges go between the two sets N 1 and N 2 but are not allowed within N 1 and N 2. N 1 =movies a b c d N 2 =actors x y z N 1 ={a,b,c,d} N 2 ={x,y,z} 42

43 Graph Types- Digraphs Digraphs or Directed Graphs Edges are directed Adjacency: There is a direct edge btw nodes! i є N j є N (i, j) є E i j 43

44 Graph Types- Digraphs- Cnt. Node Indegree and Outdegree Indegree The indegree of a node, d I (i), is the number of nodes that links i, Outdegree The outdegree of a node, d O (i), is the number of nodes that are linked by i, Indegree: number of edges terminating at i. Outdegree: number of edges originating at i. 44

45 Graph Types- Digraphs- Cnt. d I ( n j A = ) n i 1 A ij d O ( n ) i n j A ij A!= A T 45

46 Graph Types- Digraphs- Cnt. Density of Digraph: Number of all possible edges in Digraph? N * (N-1) E N (N 1) 46

47 Graph Types- Digraphs- Cnt. Connectivity Walks Trails Paths The same as before just links are directed! 47

48 Graph Types- Multigraphs A Multigraph (or multivariate graph) G consists of: a set of nodes, and two or more sets of edges, E + = {E 1, E 2,, E r }, r is the number of sets of edges 48

49 Multigraph 1. Source: the geography of transport systems 49

50 Multigraph 2. 50

51 Graph Types- Multigraphs- Cnt. Each E i indicated one type of relationship, e.g.: E 1 : lives near relationship E 2 : friends at the beginning of the year E 3 : friends at the end of the year 51

52 Graph Types- Multigraphs- Cnt. Number of edges btw any two nodes in a multigraph? E + = {E 1, E 2,, E r }, r is the number of sets of edges Undirected multigraph [0, r] Directed multigraph [0, 2*r] 52

53 Graph Types- Hypergraphs A hypergraph is a graph in which an edge can connect any number of nodes. In a hypergraph, E is a set of non-empty subsets of N called hyperedges. 53

54 Graph Types- Hypergraphs- Cnt. A hypergraph is a graph in which an edge can connect any number of nodes. In a hypergraph, E is a set of non-empty subsets of N called hyperedges. N={v 1, v 2, v 3, v 4, v 5, v 6, v 7 } E={e 1, e 2, e 3, e 4 }= {{v 1, v 2, v 3 }, {v 2, v 3 }, {v 3, v 5, v 6 }, {v 4 }} 54

55 Weighted Graphs Edges may carry additional information Tie strength how good are two nodes as friends? Distance how long is the distance btw two cities? Delay how long does the transmission take btw two cities? Signs two nodes are friends or enemies? Such graphs are called weighted or signed graphs and we will study them later. 55

56 Isomorphic Graphs Isomorphic Two graphs are isomorphic if: there is a one-to-one mapping btw their nodes that preserves adjacency! Source: Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,

57 Questions? 57

58 Announcements Please register on Piazza for important announcements forum discussions Hadi's office hours (UPDATED): 12:15-1:30pm, or by appointment. HWs due time and date (UPDATED): 11:00am on Tuesday classes late within 2:30 hour (i.e. until the end of Hadi's office hours): 10% reduction in grade; after that: zero mark. 58

59 Reading Ch.02 Graphs [NCM] Ch. 04 Social network analysis: Methods and applications. Wasserman, Stanley. Cambridge university press,