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

BACKGROUNDBACKGROUND

Organizational structures

Radner, Econometrica 61, 1109 (1993)

Garicano, J Political Economy 108, 874 (2000)

BACKGROUNDBACKGROUND

Computer networks

Ohira & Sawatari, Phys. Rev. E 58, 193 (1998)

Solé and Valverde, Physica 289A, 595 (2001)

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

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)

OOUTLINEUTLINE

Model of communication

Regular lattices

Optimization in complex networks

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

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

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

Routing algorithm: how the next node is selected?

r: information radius

dest shortest pathd r

dest randomd r

r=1

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

REGULAR LATTICES

Cayley trees

1 & 2 dimensional lattices

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)

Depending on the amount of generated packets, we observe a free phase or a collapsed phase.

Order parameter

To measure the transition between different regimes, we explore an order parameter

problems generatedproblems unsolved

η

t

N

pSt

0

1lim

The less congested structure is the flattest one.

Szz

pc

11

2/3

largest pc

Arenas, Díaz-Guilera and Guimerà, PRL 86, 3196 (2001)

Extension to other ordered lattices

1 2 /Dcp L1D:

2D:2 0.6D

cp S

Guimerà, Arenas and Díaz-Guilera, PRE submitted

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

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

More general queue model

0nfor

0nfor

a

a

n

k 1

1

2 01.0p 5 001.0p

OPTIMIZATION IN COMPLEX NETWORKSOPTIMIZATION IN COMPLEX NETWORKS

Building up complex networks:

links rewiring (random vs preferential)

General framework

We consider complex networks made-up via multiple mechanisms

Guimerà and Amaral, unpublished

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4

32

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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.

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

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

General framework: looking for optimal structures

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

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

How do the packets accumulate at single nodes?

Queue M/M/1 type

( )1in t

2

( )1in t

Queue model

Queue M/M/1:

probability distribution functions of:

time between arrivals

service time

are exponentials

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

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

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)

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

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!

At a given ratio of packet generation p which is the network structure that minimizes N(t)?

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