Social and Economic Networks: Lecture 2, Representing Networks

22 Pages·2013·0.24 MB·English

Social and Economic Networks: Lecture 2,
Representing Networks
Alper Duman
Izmir University Economics,
March 5, 2013
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
Measuring Networks
(cid:73) A graph (N,g) consists of a set of nodes
N = {1,2,..,n} and a real-valued matrix g, where g
ij
represents the relation between i and j
(cid:73) This is called adjacency matrix.
(cid:73) If all g entries are either 0 or 1, the graph is called
ij
unweighted; otherwise it is a weighted graph/network.
(cid:73) If all g = g then the graph is undirected; otherwise it is
ij ji
a directed graph/network.
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
For N = {1,2,3} the adjacency matrix is
 
0 1 0
g = 1 0 1
0 1 0
Draw the network given the above adjacency matrix g
The same network can be represented by an edgelist;
g = {{1,2},{2,3}}
How many nodes and edges does g has?
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
Paths and Cycles
(cid:73) In order to capture indirect interactions in a network it is
essential to model paths through the network.
(cid:73) A path in a network g ∈ G(N) between nodes i and j is a
sequence of links i i ,i i ,....,i i such that i i ∈ g
1 2 2 3 K−1 K k k+1
for each k ∈ {1,2,....,K −1}, with i = 1 and i = j and
i K
such that each node in the sequence i ,...i is distinct.
1 K
(cid:73) A walk in a network g ∈ G(N) between nodes i and j is a
sequence of links i i ,i i ,....,i i such that i i ∈ g
1 2 2 3 K−1 K k k+1
for each k ∈ {1,2,....,K −1}, with i = 1 and i = j
i K
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
(cid:73) A cycle is a walk that starts and ends at the same node,
so that the only node that appears twice is the
starting/ending node.
(cid:73) A cycle can be constructed from any path by adding a
link from the end to the starting node
(cid:73) A geodesic between nodes i and j is a shortest path
between these nodes.
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
Let g = 0, then gL gives us how many walks there are of
ii
length L between any nodes.
For example, find how many walks of length 2 there are
between node 1 and 3 given the below adjacency matrix g
 
0 1 1 0
1 0 0 1
g =  
1 0 0 1
0 1 1 0
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
Components and Connected Subgraphs
(cid:73) A network (N,g) is connected if every two nodes in the
network are connected by some path in the network.
(cid:73) A component of a network (N,g) is a non-empty
subnetwork (N(cid:48),g) such that ∅ =(cid:54) N(cid:48) ⊂ N , g(cid:48) ⊂ g, and
(N(cid:48),g(cid:48)) is connected and if i ⊂ N(cid:48) and ij ⊂ g then j ⊂ N(cid:48)
and ij ⊂ g(cid:48).
(cid:73) Thus the components of a network are the distinct
maximal connected subgraphs of a network.
(cid:73) The set of components of a network (N,g) is denoted
C(N,g).
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
(cid:73) A network is connected if and only if it consists of a
single component; Π(N,g) = N.
(cid:73) Components of a network partition the nodes into groups
within which nodes are path-connected.
(cid:73) A link ij is a bridge in the network of g if g −ij has more
components than g.
(cid:73) For a directed network, directions are important. If all
nodes of a subgraph is connected through directed links,
then that subgraph is a strongly connected component.
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
Trees, Stars, Circles and Complete Networks
(cid:73) A tree is a network that has no cycles. (Families are trees)
(cid:73) A forest is a network such that each component is a tree.
(cid:73) A star is a network in which there exists some node i
such that every link in the network involves node i.
(cid:73) There is only one center node i in a star.
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks
(cid:73) A complete network is one in which all possible links are
present so that g = 1 for all i (cid:54)= j.
ij
(cid:73) A circle is a network that has a single cycle and is such
that each node in the network has exactly two neighbours.
(cid:73) The neighbourhood of a node i is the set of nodes that i
is linked to; N (g) = j : g = 1.
i ij
(cid:73) The degree of a node is the number of links that involves
that node, which is the cardinal measure of the node’s
neighbourhood; d (g) = NumberN (g).
i i
AlperDuman IzmirUniversityEconomics, SocialandEconomicNetworks:Lecture2,RepresentingNetworks