Computer and Automation Research Institute Hungarian Academy of Sciences Generation of Robust Networks with Optimization under Budget Constraints (plus ongoing work) László Gulyás MTA SZTAKI

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 3 Background Software Engineering Multi-Agent Systems Agent-Based Modeling and Simulation Complex Social Systems Social Networks –Bottom-Up Approach –Generative Social Models

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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 follows a power law. #nodes[#links = x] = x -a

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 7 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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).

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 11 Details: Budget Constraints Homogenous case: –b = B for all agents. Heterogeneous case: –b’s are uniformly distributed in [1, B].

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 13 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 14 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –Size-Independent PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 15 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –Growing Costs PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 16 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Homogenous Budget Constraints. –‘Economies of Scale’ PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 17 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –Size-Independent PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 18 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –Growing Costs PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 19 Results: Key Findings Various combinations of pricing schemes and budget constraints yield robust networks. –Heterogeneous Budget Constraints. –‘Economies of Scale’ PS.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.)

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 21 Comparison to Standard Networks Erdős-Rényi (‘random density‘) Network:

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 22 Comparison to Standard Networks Watts-Strogatz (‘Small-World’) Network :

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 24 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 27 Discrete Choices on Networks Econometrics approach: discrete choice theory. Principles: –Social Influence –Social Dynamics –Coupled Dynamics –Unknown Social Network/Dynamics  Universality Classes.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 28 Framework Dynamic Social Discrete Choice Model: (A, C, G, R D) –A={a 1, …, a N } – agents –C={c 1, …, c M } – alternatives –G  A  A – interaction network –R=A  G  {r ij } – decision rules (prob. dist.) –D:G  A  G – network dynamics

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 30 Previous Work 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=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={Erdős-Rényi network} D={ Dynamic exogenous rewiring with prob. q } Gulyás & Dugundji (Unpublished) Do not alter the qualitative outcome. Even for q=1!  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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 32 Endogenous Dynamics

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 33 The Endogenous Network Model – Binary Case u  [0,1]: prob. of change per agent, per step. z i  [0,1]: ratio of same-decision neighbors. d i  [0,N-1]: number of same-dec. neighbors.  d i = 0 1 zizi +L -L T

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 34 The Endogenous Network Model – Binary Case (cont.)  d i defines 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.   d i may only partially be fulfilled.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 37 Summary of Threshold Systems

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 39 Preliminary Experiments with Biased Initial Networks Positive feedback only (in network dynamics) is not enough to to tip the steady balance.

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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?

Computer and Automation Research Institute, Hungarian Academy of Sciences EXYSTENCE Thematic Institute 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.

Computer and Automation Research Institute Hungarian Academy of Sciences Thank you!