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search and congestion in complex communication networks
Search and Congestion in Complex Communication Networks

Albert Díaz-GuileraDepartament de Física Fonamental, Universitat de BarcelonaAlex Arenas, Dept. Eng. Informàtica i Matemàtiques. Rovira i Virgili Antonio Cabrales, Dept. Economia, Univ. Pompeu FabraFrancesc 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

background
BACKGROUND
  • Organizational structures

Radner, Econometrica 61, 1109 (1993)

Garicano, J Political Economy 108, 874 (2000)

background1
BACKGROUND
  • Computer networks

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

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

background2

4

3

1

2

BACKGROUND
  • Search in complex networks

5

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

background3
BACKGROUND
  • Load in complex networks (congestion)

Goh, Kahng, & Kim, Phys Rev Lett 27, 278701 (2001)

Szabo, Alava, & Kertesz, cond –mat/0203278

Goh, Oh, Jeong, Kahng, & Kim, cond –mat/0205232

o utline
OUTLINE
  • Model of communication
  • Regular lattices
  • Optimization in complex networks
model of communication
MODEL OF COMMUNICATION
  • Communicating agents: computers, employees
  • Communication channels: cables, email, phone
  • Information packets:packets, problems
  • Limited capability of the agents to deliver packets;unlimitedcapability to store them in a queue
  • Routing algorithm
slide8
Packets (problems) and destinations (solutions) are created at random. Packets flow towards their destination.

(2)

Origin (1)

(3)

(4) Destination

  • Packets are generated with a probability p per node and time step
limited capability to deliver packets

na

a

ka

qab

nb

kb

b

Limited capability to deliver packets

For each channel, we define its“quality”.It depends

on the state of the two

corresponding nodes.

  • 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
dynamics
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.
  • tt+1
regular lattices
REGULAR LATTICES
  • Cayley trees
  • 1 & 2 dimensional lattices
cayley trees

(2)

Origin (1)

(3)

Solution (4)

Cayley trees
  • Notation: branching z (in the example z=3)
  • Hierarchical organization of knowledge
  • S size of the system
order parameter
Order parameter

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

the less congested structure is the flattest one
The less congested structure is the flattest one.

largestpc

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

extension to other ordered lattices
Extension to other ordered lattices

1D:

2D:

Guimerà, Arenas and Díaz-Guilera, PRE submitted

divergence of the average time to deliver a packet
Divergence of the average time  to deliver a packet

Comparison of exponents

  • Cayley tree:   2
  • 1D:   0.9
  • 2D:   2.5
  •  = 1 by classical queue theory
critical n with linking costs k a is a decreasing function of the number of links
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 on

the number of levels

pc S

branching z

Guimerà, Arenas and Díaz-Guilera, Phys A 299, 247 (2001)

optimization in complex networks
OPTIMIZATION IN COMPLEX NETWORKS
  • Building up complex networks:

links rewiring (random vs preferential)

  • General framework
from hierarchical lattices to complex networks

5

4

3

1

2

From hierarchical lattices to complex networks.

2

3

1

4

  • Nodes have local knowledge of the network (known first neighbors i.e. r=1)
  • Global information (euclidean distance) about the lattice
influence of the different mechanisms in a communication network
Influence of the different mechanisms in a communication network

Mechanism+-

Ordered Informational Long average

content path length

Random Decrease in the Lost of information

average path length

without causing

congestion

Preferential Decrease in the Congestion

average path length

without lost of

information

optimal communication structures depending on p
Optimal communication structures depending on p

p small

3

1

Total load

Total load

p large

2

2

3

1

Fraction of long range links

1

2

3

Guimerà, Arenas, Díaz-Guilera and Vega-Redondo, Proceedings WEHIA (2001)

what do we want to optimize
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?

simplification of the model
Simplification of the model

na

a

ka

qab

nb

kb

b

The quality of the communication from node a to node b

depends only on the node that is going to send the information packet (not the receiver)

queue model
Queue model
  • Queue M/M/1:

probability distribution functions of:

        • time between arrivals
        • service time
      • are exponentials
the role of betweenness in congestion
The role of betweenness in congestion

 = pBi/(N-1)= # packets

that arrive toi on average

Bi: “algorithmicbetweenness”, average number of

times that packets between any two pair of nodes

go through i

magnitude to optimize
Magnitude to optimize

p small: search problem

p large: congestion problem

relation between algorithmic properties and topology
Relation between algorithmic properties and topology

Consider a packet that is at i whose destination is k; we definepijk 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)
p ij k expressed in terms of the adjacency matrix
pijk expressed in terms of the adjacency matrix

For the simple model:

does not depend on the number of packets

if the packet is delivered, the prob to do it to node j

here we are
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
At a given ratio of packet generation p: which is the network structure that minimizes N(t)?
c onclusions
CONCLUSIONS
  • 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