Generation of Robust Networks with Optimization under Budget Constraints (plus ongoing work)

1. Generation of Robust Networks with Optimization under Budget Constraints(plus ongoing work) László GulyásMTA SZTAKIgulyas@sztaki.hu

2. Agenda • Background • Engineering • Agent-Based Simulation • Modeling Complex Social Systems/Networks • ‘Engineering’ Robust Networks • Past project • A localized, agent-based approach • Ongoing work (‘Teaser’) • Discrete Choices on (Endogenous) Networks

3. Background • Software Engineering • Multi-Agent Systems • Agent-Based Modeling and Simulation • Complex Social Systems • Social Networks • Bottom-Up Approach • Generative Social Models

4. ‘Engineering’ Robust Networks • Past project (under publication) • Presented at IWES’04 • ‘Networked’ version of previous work (at Lyon TI). • Generative approach: • Agent-based model. • Maximizing agents. • Limited information access • Limited cognitive abilities. • A bottom-up, localized version of the Preferential Attachment model.

5. The Robustness of Internet 1/2 • Random failures of nodes have little effect on the overall connectivity. • 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

6. The Robustness of Internet 1/2 • Random failures are extremely likely to effect only weakly connected nodes. • Drawback: susceptibility to planned attacks. #nodes #links

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.

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. • I propose an agent-based model capable of producing robust networks. • Scale-free networks as a special case.

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).

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.

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

12. Details: Pricing Schemes • Size-Independent: • PS1: PS(i) = C • Growing Costs: • PS2: PS(i) = C*B / i • Decreasing Costs (‘economies of scale’): • PS3: PS(i) = i / C

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

14. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • Size-Independent PS.

15. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • Growing Costs PS.

16. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Homogenous Budget Constraints. • ‘Economies of Scale’ PS.

17. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • Size-Independent PS.

18. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • Growing Costs PS.

19. Results: Key Findings • Various combinations of pricing schemes and budget constraints yield robust networks. • Heterogeneous Budget Constraints. • ‘Economies of Scale’ PS.

20. Results: Overview • All three pricing schemes lead to the over-representation of low-degree nodes. • This bias is stronger with the size-independent and growing costs PS. • Homogeneous and heterogeneous budget constraints yield qualitatively similar networks. • Except for the decreasing pricing scheme: ‘star topology’. (Very robust against random failures, but often undesirable.)

21. Comparison to Standard Networks • Erdős-Rényi (‘random density‘) Network:

22. Comparison to Standard Networks • Watts-Strogatz(‘Small-World’) Network :

23. Special Network Topologies • ‘Scale-Free’ (power law) Networks: • The particular ‘growing costs’ PS is a hyperbolic function of the number of nodes. • Scale-free networks with both homogenous and heterogeneous budget constraints.

24. 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.

25. Summary • A bottom-up approach to generate robust networks was presented. • Also capable of producing special network topologies, including scale-free networks. • Driving force: control over information access.

26. Ongoing Work… • Discrete Choices on Dynamic, Endogenous Networks • Background & Motivation: • Rush-hour traffic jams in the Netherlands. • Modeling Residential/Transportation Mode Choices with Social Influences. • Binary/Multinomial/Nested Choices • Generative, agent-based approach. • Empirical extensions.

27. Discrete Choices on Networks • Econometrics approach: discrete choice theory. • Principles: • Social Influence • Social Dynamics • Coupled Dynamics • Unknown Social Network/Dynamics  Universality Classes.

28. Framework • Dynamic Social Discrete Choice Model: (A, C, G, R D) • A={a1, …, aN} – agents • C={c1, …, cM} – alternatives • GAA – interaction network • R=A G {rij} – decision rules (prob. dist.) • D:G AG – network dynamics

29. Framework: Constraints • Social Influence: the agents’ utilities of the alternatives is a linear function of the average choice of their neighbors. • Rules from Probabilistic Logit Model • An ‘Ising-type’ model, BUT: • From the point of view of the agents. • We are interested in system behavior as a function of the network, not as a function of the ‘uncertainty’ (temperature) parameter.

30. Previous Work • M=2, G={Erdős-Rényi networks} or G={Watts-Strogatz networks} • Dugundji & Gulyás(2003) • The latter (2) of the previous two regimes splits: • 100% outcome, (2), only if • The network is fully connected, and • Has the small-world property. • M=2, G={full network} (“mean-field” case) • Aoki (1995), Brock & Durlauf (2001): • Two regimes depending on ‘sensitivity’/’certainty’: • The population is equally split (randomized). (1) • 100% outcome. (2) • M=3, G={full network} (“mean-field” case) • Brock & Durlauf (2002) • Two regimes: • Equal split. • Three 100% outcomes. • M=3, G={Erdős-Rényi networks} or G={Watts-Strogatz networks} • Gulyás & Dugundji (Forthcoming) • Just like the M=2 case: • 100% outcomes only if • The network is fully connected, and • Has the small-world property. • M=2, G={Erdős-Rényi network}D={Dynamic exogenousrewiring with prob. q} • Gulyás & Dugundji (Unpublished) • Do not alter the qualitative outcome. • Even for q=1!

31. Focus: Social Dynamics • Social Dynamics, Dynamic Networks. • Exogenous changes don’t make much difference. • Equal split or 100% dominance. • In contrast, the real world produces cycles. • Intuition: Endogenous network dynamics.

32. Endogenous Dynamics

33. The Endogenous Network Model – Binary Case • u[0,1]: prob. of change per agent, per step. • zi[0,1]: ratio of same-decision neighbors. • di[0,N-1]: number of same-dec. neighbors. • di= T +L zi 0 1 -L

34. The Endogenous Network Model – Binary Case (cont.) • didefines a class of ‘future networks’. • Probabilistic [uniform] choice. • Subject to keeping network density constant: • Each new neighbor ‘costs’ one link to the opposite group. • Technical constraints: • Non-multiplex network. • Sufficient number of opposite-decision links.  dimay only partially be fulfilled.

35. Preliminary Results • Initial network: • Erdős-Rényi (random) networks. • Uniform initial choice distribution: • Only positive feedback in D. (T=1.0) • The effect of the speed of the dynamics (u). • Threshold systems (negative feedback). (T<1.0) • Biased initial choice distribution: • The “identification problem”. • The role of negative feedback.

36. Preliminary Results • Initial network: • Erdős-Rényi (random) networks. • Uniform initial choice distribution: • Only positive feedback in D. (T=1.0) • The effect of the speed of the dynamics (u). • Threshold systems (negative feedback). (T<1.0) • Biased initial choice distribution: • The “identification problem”. • The role of negative feedback.

37. Summary of Threshold Systems

38. Preliminary Results • Initial network: • Erdős-Rényi (random) networks. • Uniform initial choice distribution: • Only positive feedback in D. (T=1.0) • The effect of the speed of the dynamics (u). • Threshold systems (negative feedback). (T<1.0) • Biased initial choice distribution: • The “identification problem”. • The role of negative feedback.

39. Preliminary Experiments with Biased Initial Networks • Positive feedback only (in network dynamics) is not enough to to tip the steady balance.

40. Preliminary Experiments with Biased Initial Networks (cont.) • Negative feedback (T<1) and maybe uneven initial choice distribution seem to be capable of inducing dynamics. • However, 100% outcomes seem to be extremely hard to achieve. • Cycles, just like in the real world?

41. Closing Words • Past and ongoing work on generative, agent-based models of social networks. • A bottom-up model of network formation. • Understanding the effect of various networks topologies on the global performance of a ‘well-understood’ model. • Understanding the effect of dynamic, endogenous networks.

42. Thank you!