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Social Network Analysis Study Chap 4.3~ Jan. 11 2013 Jaram Park
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4.3 Directed GraphMany relations are directional. A relational is directional ifthe ties are oriented from one to another. - directional graph : Twitter graph - unidirectional graph: Facebook graph• 4.3.1 Subgraphs – Dyads – One of the most important subgraphs in a digraph (4가지 경우) null asymmetric asymmetric mutual
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4.3 Directed Graph• 4.3.2 Nodal Indegree and Outdegree Outdegree Indegree :measures of :measures of expensiveness receptivity, or popularity – Often summarize using the mean indegree or the mean outdegree – Types of Nodes in a Directed Graph • Isolate if dI(ni) = do(ni) = 0 • Transmitter if dI(ni) = 0 and do(ni) > 0 • Receiver if dI(ni) > 0 and do(ni) = 0 • Carrier or ordinary if dI(ni) >0 and do(ni) > 0
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4.3 Directed Graph• 4.3.3 Density of a Directed Graph – Density is equal to 1, then all dyads are mutual• 4.3.5 Directed Walks, Paths, Semipaths – One must consider the direction of the arcs – Directed walk: all arcs are “pointing” in the same direction. – Directed trail: no arc is included more than once – Directed path: no node and no arc is included more than once arcs all “pointing” in the same direction - In a semipath the direction of the arcs is irrelevant
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4.3 Directed Graph• 4.3.6 Reachability and Connectivity in Digraphs – Pairs of Nodes • (i) Weakly connected if they are joined by a semipath • (ii) Unilaterally connected if they are joined by a path from ni to nj, or path from nj to ni • (iii) Strongly connected if there is a path from ni to nj, and a path from nj to ni • (iv) Reculsively connected if they are strongly connected, and the path from ni to nj uses the same nodes and arcs as the path from nj to ni , in reverse order – Digraph Connectiedness • ( ) connected if all pairs of nodes are ( ) • In a weakly connected digraph, all pairs of nodes are connected by a semipath• 4.3.7 Geodesics, Distance and Diameter• 4.3.8 Special Kind of Directed Graphs
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4.3 Directed Graph• 4.3.7 Geodesics, Distance and Diameter – Path : Direction 고려! – Since the paths from ni to nj are likely to be different from nj to ni ,the geodesics from ni to nj may be different from the paths from nj to ni. – Diameter: the length of the longest geodesic between any pair of nodes – The diameter of a weakly or unilaterally connected directed graph is undefined.• 4.3.8 Special Kind of Directed Graphs – Complement and Converse of a Digraph – Tournaments
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4.4 Signed Graphs and Signed Directed Graphs• 4.4.1 Signed Graphs A signed graph is a graph whose lines carry the additional information of a valence: a positive or negative sign. (+,-) – Dyads : two possible states, {+, -} – Triads : four possible states, {zero, one, two, or three positive positive } – Cylcles • (+)(+) = + • (+)(-) = - • (-)(-) = +• 4.4.2 Signed Directed Graphs – Semicycles
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4.5 Valued Graphs and Valued Directed GraphsValued graphs are the appropriate graph theoreticrepresentation for valued relations : Gv(N,L,V)• 4.5.1 Nodes and Dyads – Nodes in Valued Graphs – Dyads in Valued Graphs• 4.5.2 Density in a Valued Graph – is defined as the ratio of the number of lines present to the maximum possible that could arise – is as the average of the values assigned to the lines/arcs
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4.5 Valued Graphs and Valued Directed Graphs• 4.5.3 Paths in Valued Graphs Interpretation of the lines (arcs) and values in the graph “strengths” or “values” of reachability – Value of a Path • The value of a path is thus the “weakest link” in the path. • A path at level c as a path between a pair of nodes such that each and every line in the path has a value greater than or equal to c – Reachability • The “stronger” the lines included in the path. In a valued graph, two nodes are reachable at level c if there is a path at level c between them. – Path Length • Sum of the values of the lines in it.
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4.8 Relations Mathematical relations • 4.8.1 Definition – The Cartesian product of two sets M,N – Algebraic notation. < ni , nj > Є R , iRj. • 4.8.2 Properties of Relations – Reflexive: if all possible < ni , ni > ties are present in R – Symmetric: iRj if and only if jRi, for all i and j – Antisymmetric: the presence of the < ni , nj > tie implies the absence of the < nj , ni > tie (e.x., tournament) – Transitive: if whenever iRj and jRk, then iRk, for all i,j, and k.
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4.9 Matrices • 4.9.1 Matrices for Graphs – The Sociomatrix : adjacency matrix, or sociomatrix • Size g x g (g rows and g columns) • Complete graph contains 1’s in all off-diagonal cells. • The sociomatrix records for each pair of nodes whether the nodes are adjacent or not – The Incidence Matrix (Table 4.3) • Present the information in a graph is called the incidence matrix,I or I(g) • 4.9.2 Matrices for Digraphs (Table 4.4) • 4.9.3 Matrices for Valued Graphs – The entry in cell (i,j) of X records the strength of the tie from actor i to actor j. • 4.9.4 Matrices for Two-Modes Networks • 4.9.5 Matrices for Hypergraphs
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4.9 Matrices • 4.9.6 Basic Matrix Operation – Vocabulary • Size • Cell • Symmetric – Matrix Permutation • Sometimes it is useful to rearrange the rows and columns in the sociomatrix to ( ) • Permutation is anay reodering of the objects simply by relabeling the rows and columns • Matrix permutations are also useful if the graph is ( ) – Transpose – Addition and Subtraction – Matrix Multiplication – Boolean Matrix Multiplication
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4.9 Matrices • 4.9.7 Computing Simple Network Properties – Walks and Reachability • The entries of Xp = {xij[p]} give exactly the number of walks of length p between ni and nj . • The longest possible path in a graph is equal to g-1 • Reachability matrix, XR = {xij[R]} – Geodesics and Distance • The first power p for which the (i,j) element is non-zero gives the length of the shortest path and is equal to d(i,j) • The diameter of a graph or digraph is the length of the largest geodesic in the graph or digraph – Computing Nodal Degrees – Computing Density
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4.10 Properties of Graphs, Relations,and Matrices
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Quiz – Types of Nodes in a Directed Graph • Show the relation using >, <, = • Isolate if dI(ni) do(ni) 0 • Transmitter if dI(ni) 0 and do(ni) 0 • Receiver if dI(ni) 0 and do(ni) 0 • Carrier or ordinary if dI(ni) 0 and do(ni) 0 – Pairs of Nodes 그림으로 나타내기 • (i) Weakly connected if they are joined by a semipath • (ii) Unilaterally connected if they are joined by a path from ni to nj, or path from nj to ni • (iii) Strongly connected if there is a path from ni to nj, and a path from nj to ni • (iv) Reculsively connected if they are strongly connected, and the path from ni to nj uses the same nodes and arcs as the path from nj to ni , in reverse order – Vocabulary • Size • Cell • Symmetric – Matrix Permutation • Sometimes it is useful to rearrange the rows and columns in the sociomatrix to ( ) • Permutation is anay reodering of the objects simply by relabeling the rows and columns • Matrix permutations are also useful if the graph is ( )
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