Class 2: Graph theory and basic terminology Learning the language
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Transcript of Class 2: Graph theory and basic terminology Learning the language
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Class 2: Graph theory and basic terminology
Learning the language
Network Science: Graph Theory 2012
Prof. Albert-László BarabásiDr. Baruch Barzel, Dr. Mauro Martino
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Can one walk across the seven bridges
and never cross the same bridge twice?
THE BRIDGES OF KONIGSBERG
Network Science: Graph Theory 2012
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Can one walk across the seven bridges
and never cross the same bridge twice?
THE BRIDGES OF KONIGSBERG
1735: Leonhard Euler’s theorem:
(a) If a graph has nodes of odd degree, there is no path.
(b) If a graph is connected and has no odd degree nodes, it has at least one path.
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COMPONENTS OF A COMPLEX SYSTEM
components: nodes, vertices N
interactions: links, edges L
system: network, graph (N,L)
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network often refers to real systems•www, •social network•metabolic network.
Language: (Network, node, link)
graph: mathematical representation of a network•web graph, •social graph (a Facebook term)
Language: (Graph, vertex, edge)
We will try to make this distinction whenever it is appropriate, but in most cases we will use the two terms interchangeably.
NETWORKS OR GRAPHS?
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A COMMON LANGUAGE
Peter Mary
Albert
Albert
co-worker
friendbrothers
friend
Protein 1 Protein 2
Protein 5
Protein 9
Movie 1
Movie 3Movie 2
Actor 3
Actor 1 Actor 2
Actor 4
N=4L=4
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The choice of the proper network representation determines our ability to use network theory successfully. In some cases there is a unique, unambiguous representation. In other cases, the representation is by no means unique. For example,, the way we assign the links between a group of individuals will determine the nature of the question we can study.
CHOOSING A PROPER REPRESENTATION
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If you connect individuals that work with each other, you will explore the professional network.
CHOOSING A PROPER REPRESENTATION
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If you connect those that have a romantic and sexual relationship, you will be exploring the sexual networks.
CHOOSING A PROPER REPRESENTATION
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If you connect individuals based on their first name (all Peters connected to each other), you will be exploring what?
It is a network, nevertheless.
CHOOSING A PROPER REPRESENTATION
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Links: undirected (symmetrical)
Graph:
Directed links :URLs on the wwwphone calls metabolic reactions
UNDIRECTED VS. DIRECTED NETWORKS
Undirected Directed
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Links: directed (arcs).
Digraph = directed graph:
Undirected links :coauthorship linksActor networkprotein interactions
An undirected link is the superposition of two opposite directed links.
AG
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BC
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Node degree: the number of links connected to the node.
NODE DEGREESU
nd
irec
ted
In directed networks we can define an in-degree and out-degree.
The (total) degree is the sum of in- and out-degree.
Source: a node with kin= 0; Sink: a node with kout= 0.
2k inC 1k out
C 3Ck
Dir
ecte
d
AG
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BC
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We have a sample of values x1, ..., xN
Average (a.k.a. mean): typical value
<x> = (x1 + x1 + ... + xN)/N = Σi xi /N
A BIT OF STATISTICS
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N – the number of nodes in the graph
L – the number of links in the graph
N
iik
Nk
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outinN
1i
outi
outN
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ini
in kk ,kN
1k ,k
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AVERAGE DEGREEU
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The maximum number of links a network
of N nodes can have is:
A graph with degree L=Lmax is called a complete graph,
and its average degree is <k>=N-1
COMPLETE GRAPH
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Most networks observed in real systems are sparse:
L << Lmax or
<k> <<N-1.
WWW (ND Sample): N=325,729; L=1.4 106 Lmax=1012
<k>=4.51Protein (S. Cerevisiae): N= 1,870; L=4,470 Lmax=107
<k>=2.39 Coauthorship (Math): N= 70,975; L=2 105 Lmax=3 1010
<k>=3.9Movie Actors: N=212,250; L=6 106
Lmax=1.8 1013 <k>=28.78
(Source: Albert, Barabasi, RMP2002)
REAL NETWORKS ARE SPARSE
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The maximum number of links a network
of N nodes can have is:
METCALFE’S LAW
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Aij=1 if there is a link between node i and j
Aij=0 if nodes i and j are not connected to each other.
ADJACENCY MATRIX
Note that for a directed graph (right) the matrix is not symmetric.
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12 3
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a b c d e f g ha 0 1 0 0 1 0 1 0b 1 0 1 0 0 0 0 1c 0 1 0 1 0 1 1 0d 0 0 1 0 1 0 0 0e 1 0 0 1 0 0 0 0f 0 0 1 0 0 0 1 0g 1 0 1 0 0 0 0 0h 0 1 0 0 0 0 0 0
ADJACENCY MATRIX
b
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ADJACENCY MATRIX AND NODE DEGREESU
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Dir
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3
GRAPHOLOGY 1
Undirected Directed
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Actor network, protein-protein interactions WWW, citation networksNetwork Science: Graph Theory 2012
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GRAPHOLOGY 2
Unweighted(undirected)
Weighted(undirected)
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1
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23
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protein-protein interactions, www Call Graph, metabolic networksNetwork Science: Graph Theory 2012
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GRAPHOLOGY 3
Self-interactions Multigraph(undirected)
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14
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Protein interaction network, www Social networks, collaboration networksNetwork Science: Graph Theory 2012
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GRAPHOLOGY 4
Complete Graph(undirected)
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Actor network, protein-protein interactionsNetwork Science: Graph Theory 2012
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GRAPHOLOGY: Real networks can have multiple characteristics
WWW > directed multigraph with self-interactions
Protein Interactions > undirected unweighted with self-interactions
Collaboration network > undirected multigraph or weighted.
Mobile phone calls > directed, weighted.
Facebook Friendship links > undirected, unweighted.
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bipartite graph (or bigraph) is a graph whose nodes can be divided into two disjoint sets U and V such that every link connects a node in U to one in V; that is, U and V are independent sets.
Examples:
Hollywood actor networkCollaboration networksDisease network (diseasome)
BIPARTITE GRAPHS
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Gene network
GENOME
PHENOMEDISEASOME
Disease network
Goh, Cusick, Valle, Childs, Vidal & Barabási, PNAS (2007)
GENE NETWORK – DISEASE NETWORK
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HUMAN DISEASE NETWORK
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A path is a sequence of nodes in which each node is adjacent to the next one
Pi0,in of length n between nodes i0 and in is an ordered collection of n+1 nodes and n links
•A path can intersect itself and pass through the same link repeatedly. Each time a link is crossed, it is counted separately
•A legitimate path on the graph on the right: ABCBCADEEBA
• In a directed network, the path can follow only the direction of an arrow.
PATHS
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The distance (shortest path, geodesic path) between two
nodes is defined as the number of edges along the shortest
path connecting them.
*If the two nodes are disconnected, the distance is infinity.
In directed graphs each path needs to follow the direction of
the arrows.
Thus in a digraph the distance from node A to B (on an AB
path) is generally different from the distance from node B to A
(on a BCA path).
DISTANCE IN A GRAPH Shortest Path, Geodesic Path
DC
A
B
DC
A
B
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Nij,number of paths between any two nodes i and j: Length n=1: If there is a link between i and j, then Aij=1 and Aij=0 otherwise.
Length n=2: If there is a path of length two between i and j, then AikAkj=1, and AikAkj=0 otherwise.The number of paths of length 2:
Length n: In general, if there is a path of length n between i and j, then Aik…Alj=1 and Aik…Alj=0 otherwise.The number of paths of length n between i and j is*
*holds for both directed and undirected networks.
NUMBER OF PATHS BETWEEN TWO NODES Adjacency Matrix
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Distance between node 1 and node 4:
1.Start at 1.
FINDING DISTANCES: BREADTH FIRST SEATCH
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Distance between node 1 and node 4:1.Start at 1.2.Find the nodes adjacent to 1. Mark them as at distance 1. Put them in a queue.
FINDING DISTANCES: BREADTH FIRST SEATCH
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Distance between node 1 and node 4:1.Start at 1.2.Find the nodes adjacent to 1. Mark them as at distance 1. Put them in a queue.3.Take the first node out of the queue. Find the unmarked nodes adjacent to it in the graph. Mark them with the label of 2. Put them in the queue.
FINDING DISTANCES: BREADTH FIRST SEATCH
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Distance between node 1 and node 4:
1.Repeat until you find node 4 or there are no more nodes in the queue.2.The distance between 1 and 4 is the label of 4 or, if 4 does not have a label, infinity.
FINDING DISTANCES: BREADTH FIRST SEATCH
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Diameter: dmax the maximum distance between any pair of nodes in the graph.
Average path length (or distance), <d>, for a connected graph:
where dij is the
distance from node i to node j
In an undirected graph dij =dji , so we only need to count them once:
NETWORK DIAMETER AND AVERAGE DISTANCE
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Can one walk across the seven bridges
and never cross the same bridge twice?
THE BRIDGES OF KONIGSBERG
Euler PATH or CIRCUIT: return to the starting point by traveling each link of the graph once and only once.
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Every vertex of this graph has an even degree, therefore this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle.
http://en.wikipedia.org/wiki/Euler_circuit
EULERIAN GRAPH: it has an Eulerian path
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If a digraph is strongly connected and the in-degree of each node is equal to its out-degree, then there is an Euler circuit
Otherwise there is no Euler circuit.in a circuit we need to enter each node as
many times as we leave it.
A
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EULER CIRCUITS IN DIRECTED GRAPHS
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PATHOLOGY: summary
2 5
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Path
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Shortest Path
A sequence of nodes such that each node is connected to the next node
along the path by a link.
The path with the shortest length between two nodes
(distance). Network Science: Graph Theory 2012
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PATHOLOGY: summary
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Diameter
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Average Path Length
The longest shortest path in a graph
The average of the shortest paths for all pairs of nodes.
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PATHOLOGY: summary
2 5
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Cycle
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1Self-avoiding Path
A path with the same start and end node.
A path that does not intersect itself.
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PATHOLOGY: summary
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Eulerian Path Hamiltonian Path
A path that visits each node exactly once.
A path that traverses each link exactly once.
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Connected (undirected) graph: any two vertices can be joined by a path.A disconnected graph is made up by two or more connected components.
Bridge: if we erase it, the graph becomes disconnected.
Largest Component: Giant Component
The rest: Isolates
CONNECTIVITY OF UNDIRECTED GRAPHS
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The adjacency matrix of a network with several components can be written in a block-diagonal form, so that nonzero elements are confined to squares, with all other elements being zero:
Figure after Newman, 2010
CONNECTIVITY OF UNDIRECTED GRAPHS Adjacency Matrix
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Strongly connected directed graph: has a path from each node to
every other node and vice versa (e.g. AB path and BA path).
Weakly connected directed graph: it is connected if we disregard theedge directions.
Strongly connected components can be identified, but not every node is partof a nontrivial strongly connected component.
In-component: nodes that can reach the scc,
Out-component: nodes that can be reached from the scc.
CONNECTIVITY OF DIRECTED GRAPHS
D C
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Degree distribution pk
THREE CENTRAL QUANTITIES IN NETWORK SCIENCE
Average path length <d>
Clustering coefficient C
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We have a sample of values x1, ..., xN
Distribution of x (a.k.a. PDF): probability that a randomly chosen value is x
P(x) = (# values x) / N
Σi P(xi) = 1 always!
STATISTICS REMINDER
Histograms >>>
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Degree distribution P(k): probability that a randomly chosen vertex has degree k
Nk = # nodes with degree k
P(k) = Nk / N plot➔
k
P(k)
1 2 3 4
0.10.20.30.40.50.6
DEGREE DISTRIBUTION
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discrete representation: pk is the probability that a node has degree k.
continuum description: p(k) is the pdf of the degrees, where
represents the probability that a node’s degree is between k1 and k2.
Normalization condition:
where Kmin is the minimal degree in the network.
DEGREE DISTRIBUTION
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Clustering coefficient:
what portion of your neighbors are connected?
Node i with degree ki
Ci in [0,1]
CLUSTERING COEFFICIENT
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Degree distribution: P(k)
Path length: <d>
Clustering coefficient:
THREE CENTRAL QUANTITIES IN NETWORK SCIENCE
Network Science: Graph Theory 2012
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A. Degree distribution: pk
B. Path length: <d>
C. Clustering coefficient:
THREE CENTRAL QUANTITIES IN NETWORK SCIENCE
Network Science: Graph Theory 2012
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The average path-length varies as
Constant degree, constant clustering coefficient.
ONE DIMENSIONAL LATTICE: nodes on a line
Network Science: Graph Theory 2012
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2/1NLl
nodes insidefor 15
6C
In general, the average distance varies as
where D is the dimensionality of the lattice. Constant degree (coordination number),
constant clustering coefficient.
D/1Nl
5.0max
l
1l
Nl Nl61max
TWO DIMENSIONAL LATTICE
Network Science: Graph Theory 2012
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Network Science: Graph Theory 2012