Week 1
Overview and basics of game theory
Graph theory
De nitions
• Graph - a way of specifying relationships amount a collection or items tissourcenode
◦ Directed graph/ diagraph - graph showing asymmetric relationships I a zistargetnode
◦ Weighted network - links have associated weights
◦ Bipartite network - there are two groups of nodes such that links only connect nodes from
di erent groups and not from the same group
◦ Multilayer network - there are di erent types of links, such as interlayer links
‣ Multiplex network - each layer is built upon the same set of nodes, couplings are
interlayer links coupling copies of the same node in di erent layers
• Temporal network - links are dynamic, node to node interactions occur at di erent
times
• All nodes and links existing during an interval are a snapshot of the network
• Nodes - set of objects
◦ Nodes are connected if there exists at least one path between them
• Edges/arcs - links between nodes
◦ Undirected arcs/ links
• Neighbours/ adjacent nodes - nodes connected by an edge
Network
G ME
Nodes
N is ofnetworkG
size JENifjisanodeinthenetwork
Edges E i.gl i jleEifthereisanedgebetweeniand
directed igleedoesn'timply j ilEE
undirected iDEE CjileE
• Path - a sequence of distinct edges that connects nodes
◦ Cycle - a path with at least three edges, in which the rst and last nodes are the same, but all
other nodes are distinct (ie. a ring)
‣ In communication and transportation networks these are present to allow for
redundency, so there are alternative routes
• Connectivity - a graph is connected if for every pair of nodes there is a path between them
• Giant component - a large complex component
Types of node
PIVOTAL GAINING
node Xispivotalforapairof Xis a iffora
gatekeeper B E
nodes YandZ
pairof
nodesYandZ
ifitliesonevery everypath
fromYtoZpassesthroughX
p
pathbetween
shortest Yandt F
c
E A Bis pivotalforAlandAD 9atekeeper therearetwo
if Aisagatekeeperandlocalgatekeeper
that
arenotc onnectedbyan Disalocalgatekeeperbut
1 isnotpivotalforanypair nota
neighbors
F B edge gatekeeper asallpairscanbeconnected
wopassingthroughD
C D
Overview and basics of game theory
Graph theory
De nitions
• Graph - a way of specifying relationships amount a collection or items tissourcenode
◦ Directed graph/ diagraph - graph showing asymmetric relationships I a zistargetnode
◦ Weighted network - links have associated weights
◦ Bipartite network - there are two groups of nodes such that links only connect nodes from
di erent groups and not from the same group
◦ Multilayer network - there are di erent types of links, such as interlayer links
‣ Multiplex network - each layer is built upon the same set of nodes, couplings are
interlayer links coupling copies of the same node in di erent layers
• Temporal network - links are dynamic, node to node interactions occur at di erent
times
• All nodes and links existing during an interval are a snapshot of the network
• Nodes - set of objects
◦ Nodes are connected if there exists at least one path between them
• Edges/arcs - links between nodes
◦ Undirected arcs/ links
• Neighbours/ adjacent nodes - nodes connected by an edge
Network
G ME
Nodes
N is ofnetworkG
size JENifjisanodeinthenetwork
Edges E i.gl i jleEifthereisanedgebetweeniand
directed igleedoesn'timply j ilEE
undirected iDEE CjileE
• Path - a sequence of distinct edges that connects nodes
◦ Cycle - a path with at least three edges, in which the rst and last nodes are the same, but all
other nodes are distinct (ie. a ring)
‣ In communication and transportation networks these are present to allow for
redundency, so there are alternative routes
• Connectivity - a graph is connected if for every pair of nodes there is a path between them
• Giant component - a large complex component
Types of node
PIVOTAL GAINING
node Xispivotalforapairof Xis a iffora
gatekeeper B E
nodes YandZ
pairof
nodesYandZ
ifitliesonevery everypath
fromYtoZpassesthroughX
p
pathbetween
shortest Yandt F
c
E A Bis pivotalforAlandAD 9atekeeper therearetwo
if Aisagatekeeperandlocalgatekeeper
that
arenotc onnectedbyan Disalocalgatekeeperbut
1 isnotpivotalforanypair nota
neighbors
F B edge gatekeeper asallpairscanbeconnected
wopassingthroughD
C D
