Computer and Automation Research Institute Hungarian Academy of Sciences Generation of Robust Networks with Optimization under Budget Constraints (plus

Computer and Automation Research InstituteComputer and Automation Research Institute

Hungarian Academy of SciencesHungarian Academy of Sciences

Generation of Robust Networks Generation of Robust Networks with Optimization under Budget with Optimization under Budget

ConstraintsConstraints

(plus ongoing work)(plus ongoing work)

László GulyásMTA SZTAKI

[email protected]

Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences

EXYSTENCE Thematic Institute

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AgendaAgenda

• 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



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BackgroundBackground

• Software Engineering• Multi-Agent Systems• Agent-Based Modeling and Simulation• Complex Social Systems• Social Networks

– Bottom-Up Approach– Generative Social Models



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‘‘Engineering’ Robust NetworksEngineering’ 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.



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The Robustness of Internet 1/2The 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



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The Robustness of Internet 1/2The Robustness of Internet 1/2

• Random failures are extremely likely to effect only weakly connected nodes.

– Drawback: susceptibility to planned attacks.

#nod

es

#links



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Generation of Robust NetworksGeneration 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.



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Top-Down vs. Bottom-Up ApproachTop-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.



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The Model: OverviewThe 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).



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Details: Information AccessDetails: 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.



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Details: Budget ConstraintsDetails: Budget Constraints

• Homogenous case: – b = B for all agents.

• Heterogeneous case: – b’s are uniformly distributed in [1, B].



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Details: Pricing SchemesDetails: 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



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Results: Key FindingsResults: Key Findings

• Various combinations of pricing schemes and budget constraints yield robust networks.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Size-Independent PS.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Growing Costs PS.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– ‘Economies of Scale’ PS.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Size-Independent PS.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Growing Costs PS.



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• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– ‘Economies of Scale’ PS.



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Results: OverviewResults: 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.)



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Comparison to Standard NetworksComparison to Standard Networks

• Erdős-Rényi (‘random density‘) Network:



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Comparison to Standard NetworksComparison to Standard Networks

• Watts-Strogatz (‘Small-World’) Network :



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Special Network TopologiesSpecial 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.



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Special Network TopologiesSpecial 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.



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SummarySummary

• 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.



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Ongoing Work…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.



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Discrete Choices on NetworksDiscrete Choices on Networks

• Econometrics approach: discrete choice theory.

• Principles:– Social Influence– Social Dynamics– Coupled Dynamics– Unknown Social Network/Dynamics

Universality Classes.



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FrameworkFramework

• Dynamic Social Discrete Choice Model: (A, C, G, R D)

– A={a1, …, aN} – agents

– C={c1, …, cM} – alternatives

– GAA – interaction network

– R=A G {rij} – decision rules (prob. dist.)

– D:G AG – network dynamics



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Framework: ConstraintsFramework: 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.



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Previous WorkPrevious 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.



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Focus: Social DynamicsFocus: 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.



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Endogenous DynamicsEndogenous Dynamics



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The Endogenous Network Model – The Endogenous Network Model – Binary CaseBinary 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=

0 1 zi

+L

-L

T



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The Endogenous Network Model – The Endogenous Network Model – Binary Case (cont.)Binary Case (cont.)

di 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.

di may only partially be fulfilled.



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Preliminary ResultsPreliminary 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.



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Summary of Threshold SystemsSummary of Threshold Systems



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Preliminary Experiments with Preliminary Experiments with Biased Initial NetworksBiased Initial Networks

• Positive feedback only (in network dynamics) is not enough to to tip the steady balance.



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Preliminary Experiments with Preliminary Experiments with Biased Initial Networks (cont.)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?



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Closing WordsClosing 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 InstituteComputer and Automation Research Institute

Hungarian Academy of SciencesHungarian Academy of Sciences

Thank you!Thank you!

Documents

Computer and Automation Research Institute Hungarian Academy of Sciences Generation of Robust Networks with Optimization under Budget Constraints (plus