Complex Networks

NETWORK SCIENCE The science of the 21st century

Times
cited

Years Network Science: Introduction January 10, 2011


Times
cited

Years

Network Science: Introduction January 10, 2011


Why now?


THE EMERGENCE OF NETWORK SCIENCE

Data Availability: Movie Actor Network, 1998;
World Wide Web, 1999.
C elegans neural wiring diagram 1990
Citation Network, 1998
Metabolic Network, 2000;
PPI network, 2001

Universality: The architecture of networks emerging in various
domains of science, nature, and technology are
more similar to each other than one would have
expected.

The (urgent) need to Despite the challenges complex systems offer us, we
cannot afford to not address their behavior, a view
understand complexity: increasingly shared both by scientists and policy
makers. Networks are not only essential for this
journey, but during the past decade some of the most
important advances towards understanding complexity
were provided in context of network theory.


EPIDEMIC FORECAST Predicting the H1N1 pandemic

Real Projected


DOCUMENTARY

Thex


Graph theory and basic terminology
Learning the language

Network Science: Graph Theory January 24, 2011

COMPONENTS OF A COMPLEX SYSTEM

components: nodes, vertices N

interactions: links, edges L

system: network, graph (N,L)


NETWORKS OR GRAPHS?

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.


A COMMON LANGUAGE

friend Movie 1
co-worker
Peter Mary Actor 1 Actor 2

Albert Movie 3 Actor 4
brothers friend Movie 2

Albert
Actor 3

Protein 1 Protein 2
Protein 5

Protein 9
N=4
L=4

CHOOSING A PROPER REPRESENTATION

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, for a group of individuals, the way you assign
the links will determine the nature of the question you can
study.



If you connect individuals
that work with each other,
you will explore
the professional network.



If you connect those that
have a sexual relationship,
you will be exploring the
sexual networks.



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.


UNDIRECTED VS. DIRECTED NETWORKS

Undirected Directed

Links: undirected (symmetrical) Links: directed (arcs).

Graph: Digraph = directed graph:

L
A
D
M B An undirected
F
C link is the
I superposition of
D two opposite
directed links.
B G
E
G
H A
C
F

Undirected links : Directed links :
coauthorship links URLs on the www
Actor network phone calls
protein interactions metabolic reactions


ADJACENCY MATRIX

4
4

3
3 2
2
1
1

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.

Note that for a directed graph (right) the matrix is not symmetric.


ADJACENCY MATRIX

a e

a bcdefgh
a 0 1 0 0 1 0 1 0
b 1 0 1 0 0 0 0 1
c 0 1 0 1 0 1 1 0
d 0 0 1 0 1 0 0 0
h b d e 1 0 0 1 0 0 0 0
f0 0 1 0 0 0 1 0
g1 0 1 0 0 0 0 0
h0 1 0 0 0 0 0 0
f

g c


NODE DEGREES

Node degree: the number of links connected to the node.
Undirected

j

i

4

3
2
1


NODE DEGREES

In directed networks we can define an in-degree and out-degree.
D The (total) degree is the sum of in- and out-degree.
B
C
Directed

kC  2
in
kC  1
out
kC  3
E
G
A

F Source: a node with kin= 0; Sink: a node withkout= 0.

4

3
2
1

A BIT OF STATISTICS

We have a sample of values x1, ..., xN

Average(a.k.a. mean): typical value
<x> = (x1 + x1 + ... + xN)/N = Σi xi /N

Standard deviation:fluctuations around typical value
σx= √Σi (xi - <x>)2/N


AVERAGE DEGREE

N
1
k
Undirected

j k  i
N i 1

i

N – the number of nodes in the graph

N N
1 1
k , k 
D
  k iout , k in  k out
B in in out
C
k i
N i 1 N i 1
Directed

E
A

F


COMPLETE GRAPH

The maximum number of links a network
of N nodes can have is:

A graph with degree L=Lmaxis called a complete graph,
and its average degree is <k>=N-1


SPARSE GRAPH

Most networks observed in real systems are sparse:

L <<Lmax (or <k><<N-1).

WWW (ND Sample): N=325,729; <k>=4.51
Protein (S. Cerevisiae): N=1870; <k>=2.39
Coauthorship (Math): N=70 975; <k>=3.9
Movie Actors: N=212 250; <k>=28.78
(Source: Albert, Barabasi, RMP2002)

Consequence: Their adjacency matrix is filled with zeros!


ACTOR NETWORK
Austin Powers: Let’s make
The spy who it legal
shagged me

Robert Wagner

Wild Things
What Price Glory

Barry Norton

A Few Monsieur
Good Men Verdoux

ACTOR NETWORK

Nodes: actors
Links: cast jointly

Days of Thunder (1990) Far
and Away (1992) Eyes
Wide Shut (1999)

N = 212,250 actors k=28.78


IMBD SCALE FREE


GRAPHOLOGY 1

Undirected Directed
4
4
1 1

2 2
3 3

Actor network, protein-protein interactions WWW, citation networks

GRAPHOLOGY 2

Unweighted Weighted
(undirected) 4 (undirected) 4
1 1

2 2
3 3

protein-protein interactions, www Call Graph, metabolic networks

GRAPHOLOGY 3

Self-interactions Multigraph
(undirected)
4 4
1 1

2 2
3 3

Protein interaction network, www Social networks, collaboration networks

GRAPHOLOGY 4

Complete Graph
(undirected)
4
1

2
3

Actor network, protein-protein interactions

GRAPHOLOGY X

WWW >>directed multigraph with self-interactions

Protein Interactions >>undirected unweighted with self-interactions

Protein Complex >>unweighted complete graph with self-interactions

Collaboration network >>undirected multigraph or weighted.

Mobile phone calls >>directed, weighted.

Facebook Friendship links >>undirected, unweighted.


BIPARTITE GRAPHS

bipartite graph(or bigraph) is a graph
whose nodes can be divided into two
disjoint setsU 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 network
Collaboration networks
Disease network (diseasome)


GENE NETWORK – DISEASE NETWORK

DISEASOME PHENOME

GENOME

Gene network
Disease network

Goh, Cusick, Valle, Childs, Vidal &Barabási, PNAS (2007)

STATISTICS REMINDER

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

ΣiP(xi) = 1 always!

Histograms >>>


DEGREE DISTRIBUTION

Degree distributionP(k): probability that
a randomly chosen vertex has degree k

Nk = # nodes with degree k
P(k) = Nk / N  plot
P(k)
0.6
0.5
0.4
0.3
0.2
0.1

1 2 3 4 k


DEGREE DISTRIBUTION

discrete representation: pkis 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.


PATHS

A path is a sequence of nodes in which each node is adjacent to the next one

Pi0,in of length nbetween nodes i0 and in is an ordered collection of n+1 nodes and n links

B

A
•A path can intersect itself and pass through the same
link repeatedly. Each time a link is crossed, it is counted
E
separately
C
•A legitimate path on the graph on the right:
D
ABCBCADEEBA

•In a directed network, the path can follow only the
direction of an arrow.


NUMBER OF PATHS BETWEEN TWO NODES Adjacency Matrix

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.


DISTANCE IN A GRAPH Shortest Path, Geodesic Path

B The distance (shortest path, geodesic path) between two
nodes is defined as the number of edges along the shortest
A
path connecting them.

*If the two nodes are disconnected, the distance is infinity.
C
D

In directed graphs each path needs to follow the direction of
the arrows.
B
Thus in a digraph the distance from node A to B (on an AB
A
path) is generally different from the distance from node B to A
(on a BCA path).

C
D


FINDING DISTANCES: BREADTH FIRST SEATCH

Distance between node1and node 4:

1.Start at1.

1




1.Start at1.
2.Find the nodes adjacent to1. Mark them as at distance 1. Put them in a queue.

1 1 1

1




1.Start at1.
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.

2
2 1 1 1

2

1

2 2


FINDING DISTANCES: BREADTH FIRST SEARCH


1.Start at1.
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.
4.…
5.Take the first node, w, out of the queue. Find the unmarked nodes adjacent to it in the
graph. Mark them with the label of w+1. Put them in the queue.

2 w
w 2 1 1 1 w+1
w+1
2




1.Repeat until you find node 4 or there are no more nodes in the queue.
2.The distance between1and4is the label of4or, if4does not have a label, infinity.

3 4
3

2
4 3
3 2 1 1 1 4

2
3 3
4 1 4

4 3
4 4
2 2

3


FINDING DISTANCES: BREADTH FIRST SEATCH ALGORITHM

For a weighted network, we have Dijkstra’s algorithm.

http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

http://www.yaldex.com/games-programming/0672323699_ch12lev1sec7.html

RECORDING DISTANCES

B

A B C D A
A 0 lABlAClAD
B lBA 0 lBClBD
Fill out the matrix
C lCAlCB 0 lCD
D lDAlDBlDC 0 C
D

Q: How many entries will you need for an N- node graph? B

A

A: N(N-1) in a digraph, N(N-1)/2 in a symmetrical graph.
Let’s use the notation
C
D


NETWORK DIAMETER AND AVERAGE DISTANCE

Diameter: the maximum distance between any pair of nodes in the graph.

Average path length/distance for a connected graph (component) or a strongly
connected (component of a) digraph.

where lij is the distance from node i to node j

In an undirected (symmetrical) graph lij =lji, we only need to count them once


CONNECTIVITY OF UNDIRECTED GRAPHS

Connected (undirected) graph: any two vertices can be joined by a path.
A disconnected graph is made up by two or more connected components.

B
B
A
A Largest Component:
Giant Component

C
D F C
D F
F
F The rest: Isolates
G
G

Bridge: if we erase it, the graph becomes disconnected.


CONNECTIVITY OF UNDIRECTED GRAPHS Adjacency Matrix

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 DIRECTED GRAPHS

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 the
edge directions.

Strongly connected components can be identified, but not every node is part
of a nontrivial strongly connected component.

B
E
A F
B

A

D E
C
D C G
F

G

In-component: nodes that can reach the scc,
Out-component: nodes that can be reached from the scc.

HISTORICAL DETOUR: THE BRIDGES OF KONIGSBERG

Can one walk across the seven bridges and never cross the same bridge twice?

Euler circuit: return to the starting point by traveling each link of the graph
once and only once.
http://maps.google.com/maps?oe=utf-
8&client=firefox-
a&q=kaliningrad&ie=UTF8&hq=&hnear=Kalining
rad,+Kaliningrad+Oblast,+Russia&gl=us&ll=54.70

Euler’s theorem: 7293,20.510788&spn=0.009248,0.025878&t=h&
z=15

(a) If a graph has nodes of odd degree, it cannot have an Euler circuit.
(b) If a graph is connected and has no odd degree nodes, it has at
least one Euler circuit.

How would we need to modify the graph so it has an Euler circuit?

EULERIAN GRAPH

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

EULER CIRCUITS IN DIRECTED GRAPHS
B
If a digraph is strongly connected and the in-
degree of each node is equal to its out-degree,
D
then there is an Euler circuit

A C
Q: Give one possible Euler circuit
E

F G

Otherwise there is no Euler circuit.
This is because in a circuit we need to
enter each node as many times as we
leave it.

CLUSTERING COEFFICIENT

Clustering coefficient:
what portion of your neighbors are connected?

Node i with degree ki

Ciin [0,1]



Clustering coefficient: what portion of your
neighbors are connected?

3 6
1
8
5
4
2 7
10
9

i=8: k8=2, e8=1, TOT=2*1/2=1  C8=1/1=1



Clustering coefficient: what portion of your
neighbors are connected?

3 6
1
8
5
4
2 7
10
9

i=4: k4=4, e4=2, TOTAL=4*3/2=6  C4=2/6=1/3


KEY MEASURES

Degree distribution: P(k)

Path length: l

Clustering coefficient:


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