2. 4.3 Directed Graph
Many relations are directional. A relational is directional if
the 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 4.5 Valued Graphs and
Valued Directed Graphs
Valued graphs are the appropriate graph theoretic
representation 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
9. 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.
12. 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.
13. 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
14. 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
15. 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
17. 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 ( )
