1. AITIA International Inc. and Lorand Eotvos University, Budapest ROBUST NETWORKS FROM LOCAL OPTIMIZATION A Bottom-Up Model to Generate Networks with Skewed Degree Distributions László Gulyás AITIA International Inc. Lorand Eotvos University, Budapest lgulyas@aitia.ai

2. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest2 This work is an extended version of László Gulyás: “A Generative Model of Power Law Distributions with Optimizing Agents with Constrained Information Access”, European Conference on Complex Systems, Paris, November 2005. László Gulyás: „Generation of Robust Networks with Optimization under Budget Constraints”, In Proceedings of The 5th International Workshop on Emergent Synthesis (IWES'04), Budapest, 2004.

3. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest3 Overview Robust networks: –The robustness of Internet. Generation of Robust Networks: –Top-down approaches vs. a bottom-up model. An emergent approach: –Controlling the actor’s information access. –An agent-based model with market metaphors. Results of numerical experiments Summary

4. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest4 The Robustness of Internet 1/3 Random failures of nodes have little effect on the overall connectivity. (Barabási-Albert) –The networks of Internet have a characteristic (“scale-free”) structure. –The distribution of the #links per node follows a power law. #nodes[#links = x] = x -a

5. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest5 The Robustness of Internet 2/3 Random failures are likely to effect only weakly connected nodes. –Drawback: susceptibility to planned attacks. –Opposite goal than in Epidemics stopping Destroying terrorist networks #nodes #links

6. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest6 The Robustness of Internet 3/3 Replication of Barabási-Albert’s with a formal measure: –Expected betweenness centrality. –How many paths are likely to be cut by the failure of a single node. ER – Erdos-Renyi SF – Scale-Free (Albert-Barabási) (Averaged over 10 samples. Relative to SF.)

7. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest7 Generation of Robust Networks Purpose: –Explanation: Internet evolved to be robust spontaneously in a distributed manner. It is an intriguing question to explain how and why. –Engineering: It is of practical interest to be able to generate robust networks without total top-down control. –Inverse of epidemics / terror networks

8. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest8 Top-Down vs. Bottom-Up Approach The prevailing explanation: –Preferential Attachment Model (Albert&Barabási) (for the generation of scale-free networks): Incremental addition of nodes. Each node has a fixed number of links. Newcomers attach to existing nodes with probability proportional to the nodes’ connectivity. No bottom-up explanation so far. Aldridge et al.’s work on ‘local preferential attachment’. Agent-based model capable of producing robust networks. Scale-free networks as a special case.

9. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest9 The Model: Overview Incremental addition of nodes (agents). A fixed E number of links per agent. –Initially: E fully connected nodes. Agents maximize their connectivity by linking to the nodes with the highest degrees. –Subject to their information access: –They buy information from a Central Authority (CA), limited by their personal budget constraints b. The price of information: –Independent of the agents in question, but may depend on the size of the network, according to a pricing scheme (PS).

10. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest10 Details: Information Access Agents have no previous information concerning the network. –Therefore they cannot specify the node they are interested in. –However, they can list the nodes they already have knowledge about. –The CA returns random node not contained by the list, together with its degree.

11. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest11 Details: Budget Constraints Homogenous case: –b = B for all agents. Heterogeneous case: –b’s are uniformly distributed in [1, B].

12. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest12 Details: Pricing Schemes Size-Independent: PS0: PS(i) = C Growing Costs: PS1:PS(i) = C*B / i Decreasing Costs ( ‘ economies of scale ’ ): PS2:PS(i) = i / C

13. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest13 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks.

14. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest14 Results: Key Findings 1/3 (Averaged over 10 samples.)

15. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest15 Results: Key Findings 2/3 (Averaged over 10 samples. Relative to SF.)

16. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest16 Results: Key Findings 3/3

17. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest17 Nature of Generated Networks (#3) Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –Size-Independent PS. (PS0)

18. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest18 Nature of Generated Networks (#1) Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –Growing Costs PS. (PS1)

19. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest19 Nature of Generated Networks (##) Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –‘ Economies of Scale ’ PS. (PS2)

20. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest20 Nature of Generated Networks (#2) Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –Size-Independent PS. (PS0)

21. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest21 Nature of Generated Networks (#4) Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –Growing Costs PS. (PS1)

22. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest22 Nature of Generated Networks (##) Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –‘ Economies of Scale ’ PS. (PS2)

23. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest23 Nature of Generated Networks PS1 seems to be better than PS0. Homogenous budget seems to work better than heterogeneous. PS2 seems to be non-robust. –Albeit they sometimes produce actual scale-free networks.

24. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest24 Special Network Topologies ‘Scale-Free’ (power law) Networks: –The particular ‘ growing costs ’ P1 is a hyperbolic function of the number of nodes. Scale-free networks with both homogenous and heterogeneous budget constraints.

25. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest25 Special Network Topologies ‘Scale-Free’ (power law) Networks: –The ‘ economies of scale ’ PS and heterogeneous budget constraints also yield to a power law distribution of in- edges.

26. AITIA International Inc. and Lorand Eotvos University, Budapest August 5, 2006.Collegium Budapest26 Summary A bottom-up approach to generate robust networks was presented. –Also capable of producing special network topologies, including scale-free networks. Used economic metaphors, but mainly to ease thinking and communications. The key is: control over information access. –Perhaps old, but a generally useful concept for complex systems with autonomous entities.