Robust Networks from Local Optimization: A Bottom-Up Model for Skewed Degree Distributions

1. ROBUST NETWORKS FROM LOCAL OPTIMIZATIONA Bottom-Up Model to Generate Networks with Skewed Degree Distributions László GulyásAITIA International Inc.Lorand Eotvos University, Budapestlgulyas@aitia.ai

2. 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. Collegium Budapest

3. 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 Collegium Budapest

4. 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 followsa power law. • #nodes[#links = x] = x-a Collegium Budapest

5. 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 Collegium Budapest

6. 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.) Collegium Budapest

7. Generation of Robust Networks • Purpose: • Explanation: • Internet evolved to be robust spontaneouslyin 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 Collegium Budapest

8. 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. Collegium Budapest

9. 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). Collegium Budapest

10. 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. Collegium Budapest

11. Details: Budget Constraints • Homogenous case: • b = B for all agents. • Heterogeneous case: • b’s are uniformly distributed in [1, B]. Collegium Budapest

12. 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 Collegium Budapest

13. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. Collegium Budapest

14. Results: Key Findings 1/3 • (Averaged over 10 samples.) Collegium Budapest

15. Results: Key Findings 2/3 • (Averaged over 10 samples. Relative to SF.) Collegium Budapest

16. Results: Key Findings 3/3 Collegium Budapest

17. Nature of Generated Networks (#3) • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • Size-Independent PS. (PS0) Collegium Budapest

18. Nature of Generated Networks (#1) • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • Growing Costs PS. (PS1) Collegium Budapest

19. Nature of Generated Networks (##) • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • ‘Economies of Scale’ PS. (PS2) Collegium Budapest

20. Nature of Generated Networks (#2) • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • Size-Independent PS. (PS0) Collegium Budapest

21. Nature of Generated Networks (#4) • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • Growing Costs PS. (PS1) Collegium Budapest

22. Nature of Generated Networks (##) • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • ‘Economies of Scale’ PS. (PS2) Collegium Budapest

23. 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. Collegium Budapest

24. 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. Collegium Budapest

25. Special Network Topologies • ‘Scale-Free’ (power law) Networks: • The ‘economies of scale’ PS and heterogeneousbudget constraints also yield to a power law distribution of in-edges. Collegium Budapest

26. 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. Collegium Budapest