1. Search and Congestion in Complex Communication Networks Albert Díaz-Guilera Departament 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

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. BACKGROUND 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 5  Search in complex networks

5. BACKGROUND 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)

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; unlimited capability to store them in a queue  Routing algorithm

8. 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

9. Limited capability to deliver packets n a number of packets at node a k a capability to deliver packets of node a q ab quality of the channel between nodes a and b a b nana nbnb kaka kbkb q ab For each channel, we define its “quality”. It depends on the state of the two corresponding nodes.

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 q ab of the channel through which the packet must flow. The packet jumps with probability q ab. Eliminate the packets that have reached their destination. t  t+1

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

13. Cayley trees Notation: branching z (in the example z=3) Hierarchical organization of knowledge S size of the system Origin (1) Solution (4) (3) (2)

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

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

16. The less congested structure is the flattest one. largest p c 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 Cayley tree:   2 1D:   0.9 2D:   2.5  = 1 by classical queue theory Comparison of exponents

19. Critical N with linking costs: k a 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 Guimerà, Arenas and Díaz-Guilera, Phys A 299, 247 (2001) branching z p c S

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. 1 4 3 2 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.

24. Influence of the different mechanisms in a communication network Mechanism +- OrderedInformationalLong average contentpath length RandomDecrease in theLost of information average path length without causing congestion PreferentialDecrease in theCongestion average path length without lost of information

25. Optimal communication structures depending on p Guimerà, Arenas, Díaz-Guilera and Vega-Redondo, Proceedings WEHIA (2001) Fraction of long range links Total load p small p large 123 1 1 2 2 3 3

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 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) a b nana nbnb kaka kbkb q ab

29. Relation between dynamics and topology

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

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

32. The role of betweenness in congestion B i : “algorithmic betweenness”, average number of times that packets between any two pair of nodes go through i  = pB i /(N-1) = # packets that arrive to i on average

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

34. Relation between algorithmic properties and topology Consider a packet that is at i whose destination is k; we define p ij 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: = f (p ij k ) betweenness: B n = g (p ij k )

35. p ij k 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

36. 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!

37. Testing the assumptions

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

39. 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