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Search and Congestion in Complex Search and Congestion in Complex Communication NetworksCommunication Networks
Albert Díaz-GuileraDepartament de Física Fonamental, Universitat de Barcelona
Alex Arenas, Dept. Eng. Informàtica i Matemàtiques. Rovira i Virgili
Antonio Cabrales, Dept. Economia, Univ. Pompeu Fabra
Francesc Giralt, Dept. Enginyeria Química, Univ. Rovira i Virgili
Roger Guimerà, Dept. Enginyeria Química, Univ. Rovira i Virgili
Fernando Vega-Redondo, Dept. Economia, Univ. Alacant
more information at http://www.ffn.ub.es/albert/
COSIN
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BACKGROUNDBACKGROUND
Organizational structures
Radner, Econometrica 61, 1109 (1993)
Garicano, J Political Economy 108, 874 (2000)
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BACKGROUNDBACKGROUND
Computer networks
Ohira & Sawatari, Phys. Rev. E 58, 193 (1998)
Solé and Valverde, Physica 289A, 595 (2001)
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BACKGROUNDBACKGROUND
Kleinberg, Nature 406, 845 (2000)
Tadic, Eur Phys J B 23, 221 (2001)
Adamic, Lukose, Puniyani, & Huberman, Phys Rev E 64, (2001)
Kim, Yoon, Han, & Jeong, cond-mat/0111232
Watts, Dodds, & Newman, Science
1 2
3
4
5Search in complex networks
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BACKGROUNDBACKGROUND
Goh, Kahng, & Kim, Phys Rev Lett 27, 278701 (2001)
Szabo, Alava, & Kertesz, cond –mat/0203278
Goh, Oh, Jeong, Kahng, & Kim, cond –mat/0205232
Load in complex networks (congestion)
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OOUTLINEUTLINE
Model of communication
Regular lattices
Optimization in complex networks
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MODEL OF COMMUNICATIONMODEL OF COMMUNICATION
Communicating agents: computers, employees
Communication channels: cables, email, phone
Information packets: packets, problems
Limited capability of the agents to deliver packets;
unlimited capability to store them in a queue
Routing algorithm
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Packets (problems) and destinations (solutions) are created at random. Packets flow towards their destination.
Origin (1)
(4) Destination
(3)
(2)
Packets are generated with a probability p per node and time step
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Limited capability to deliver packets
na number of packets at node a
ka capability to deliver packets of node a
qab quality of the channel between nodes
a and b
a
b
na
nb
ka
kb
qab
baab kkq
For each channel, we define its “quality”. It depends on the state of the two corresponding nodes.
aa n
k1
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Routing algorithm: how the next node is selected?
r: information radius
dest shortest pathd r
dest randomd r
r=1
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Dynamics
t=0
At each node, create a new packet with probability p.
For each packet in the net, calculate the quality qab of the channel through which the packet must flow. The packet jumps with probability qab.
Eliminate the packets that have reached their destination.
tt+1
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REGULAR LATTICES
Cayley trees
1 & 2 dimensional lattices
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Cayley trees
Notation: branching z (in the
example z=3)
Hierarchical organization of
knowledge
S size of the system
( )r
Origin (1)
Solution (4)
(3)
(2)
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Depending on the amount of generated packets, we observe a free phase or a collapsed phase.
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Order parameter
To measure the transition between different regimes, we explore an order parameter
problems generatedproblems unsolved
η
t
N
pSt
0
1lim
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The less congested structure is the flattest one.
Szz
pc
11
2/3
largest pc
Arenas, Díaz-Guilera and Guimerà, PRL 86, 3196 (2001)
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Extension to other ordered lattices
1 2 /Dcp L1D:
2D:2 0.6D
cp S
Guimerà, Arenas and Díaz-Guilera, PRE submitted
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Divergence of the average time to deliver a packet
• Cayley tree: 2
• 1D: 0.9
• 2D: 2.5
( )cp p
• = 1 by classical queue theory
Comparison of exponents
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Critical N with linking costs: ka is a decreasing function of the number of links
A hint for the optimal “group size”
Observe that the critical number of problems does not depend onthe number of levels
Guimerà, Arenas and Díaz-Guilera, Phys A 299, 247 (2001)
branching z
p c S
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More general queue model
0nfor
0nfor
a
a
n
k 1
1
2 01.0p 5 001.0p
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OPTIMIZATION IN COMPLEX NETWORKSOPTIMIZATION IN COMPLEX NETWORKS
Building up complex networks:
links rewiring (random vs preferential)
General framework
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We consider complex networks made-up via multiple mechanisms
Guimerà and Amaral, unpublished
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1
4
32
1 2
3
4
5
Nodes have local knowledge of the network (known first neighbors
i.e. r=1)
Global information (euclidean distance) about the lattice
From hierarchical lattices to complex networks.
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Influence of the different mechanisms in a communication network
Mechanism + -Ordered Informational Long average
content path length
Random Decrease in the Lost of informationaverage path lengthwithout causingcongestion
Preferential Decrease in the Congestionaverage path lengthwithout lost ofinformation
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Optimal communication structures depending on p
Guimerà, Arenas, Díaz-Guilera and Vega-Redondo, Proceedings WEHIA (2001)
Fraction of long range links
Tot
al lo
ad
Tot
al lo
ad
p small
p large
1 2 3
1
12 2
3
3
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General framework: looking for optimal structures
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What do we want to optimize?
For a given p, which is the structure that minimizes the number of packets?
Can we relate the number of packets to the topological properties of the network?
( ) ( )ii
N t n t
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Simplification of the model
The quality of the communication from node a to node bdepends only on the node that is going to send the information packet (not the receiver)
1ab a ba
a
q k qn
a
b
na
nb
ka
kb
qab
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How do the packets accumulate at single nodes?
Queue M/M/1 type
( )1in t
2
( )1in t
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Queue model
Queue M/M/1:
probability distribution functions of:
time between arrivals
service time
are exponentials
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The role of betweenness in congestion
1( )1
1
i
ii
pB
Nn tpB
N
Bi: “algorithmic betweenness”, average number of times that packets between any two pair of nodes go through i
= pBi/(N-1) = # packetsthat arrive to i on average
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Magnitude to optimize
*
/( 1)( ) ( )
1 /( 1)
<d> small p( 1)
(node with maximum betweenness) large p
ii
i i i
ii
pB NN t n t
pB N
pB Np
N
B
p small: search problemp large: congestion problem
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Relation between algorithmic properties and topology
Consider a packet that is at i whose destination is k; we define pij
k as the probability for the packet to go from i to j the next time step
Relationships between this probability and the algorithmic properties:
•distance: <d> = f (pijk)
•betweenness: Bn = g (pijk)
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pijk expressed in terms of the adjacency matrix
For the simple model:
0 : (1 )
1: (1 )
ijkij ik
ijj
ijkij ik jk ik ik
ijj
ar p
a
ar p a a
a
does not depend on the number of packetsif the packet is delivered, the prob to do it to node j
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Here we are
For the simple model
The goal is to minimize N(t) We have expressed N(t) in terms of the adjacency matrix Therefore now it is possible to minimize N(t) by exploring the space of possible adjacency matrices!
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At a given ratio of packet generation p which is the network structure that minimizes N(t)?
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CCONCLUSIONSONCLUSIONS
We have proposed a simple model for communication processes.
We characterize the phase transition from a free to a congested regime in regular lattices.
We find the optimum structures for small and large packet generation when: building-up networks with prescribed rules
looking directly at adjacency matrices of networks
We have found a relation between the dynamics, the algorithmic properties and the topological characteristics of the network