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Search and Congestion in Complex Communication Networks

Search and Congestion in Complex Communication Networks.

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Search and Congestion in Complex Communication Networks

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  1. 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

  2. BACKGROUND • Organizational structures Radner, Econometrica 61, 1109 (1993) Garicano, J Political Economy 108, 874 (2000)

  3. BACKGROUND • Computer networks Ohira & Sawatari, Phys. Rev. E 58, 193 (1998) Solé and Valverde, Physica 289A, 595 (2001)

  4. 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

  5. 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

  6. OUTLINE • Model of communication • Regular lattices • Optimization in complex networks

  7. 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

  8. 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

  9. 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

  10. Routing algorithm: how the next node is selected? • r: information radius r=1

  11. 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

  12. REGULAR LATTICES • Cayley trees • 1 & 2 dimensional lattices

  13. (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

  14. Depending on the amount of generated packets, we observe a freephase or a collapsedphase.

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

  16. The less congested structure is the flattest one. largestpc Arenas, Díaz-Guilera and Guimerà, PRL 86, 3196 (2001)

  17. Extension to other ordered lattices 1D: 2D: Guimerà, Arenas and Díaz-Guilera, PRE submitted

  18. 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

  19. 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)

  20. More general queue model

  21. OPTIMIZATION IN COMPLEX NETWORKS • Building up complex networks: links rewiring (random vs preferential) • General framework

  22. We consider complex networks made-up via multiple mechanisms Guimerà and Amaral, unpublished

  23. 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

  24. 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

  25. 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)

  26. General framework: looking for optimal structures

  27. 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?

  28. 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)

  29. How do the packets accumulate at single nodes? Queue M/M/1 type

  30. Queue model • Queue M/M/1: probability distribution functions of: • time between arrivals • service time • are exponentials

  31. 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

  32. Magnitude to optimize p small: search problem p large: congestion problem

  33. 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)

  34. 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

  35. 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!

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

  37. 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

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