AITIA International Inc. and Lorand Eotvos University, Budapest ROBUST NETWORKS FROM LOCAL OPTIMIZATION A Bottom-Up Model to Generate Networks with Skewed.

AITIA International Inc. and Lorand Eotvos University, BudapestAITIA International Inc. and Lorand Eotvos University, Budapest

ROBUST NETWORKS FROM LOCAL OPTIMIZATIONROBUST NETWORKS FROM LOCAL OPTIMIZATIONA Bottom-Up Model to Generate Networks with A Bottom-Up Model to Generate Networks with

Skewed Degree DistributionsSkewed Degree Distributions

László GulyásAITIA International Inc.

Lorand Eotvos University, [email protected]


August 5, 2006. Collegium Budapest 2

This work is an extended version ofThis 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.



OverviewOverview

• 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



The Robustness of Internet 1/3The 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



The Robustness of Internet The Robustness of Internet 22/3/3

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

– Drawback: susceptibility to planned attacks.

– Opposite goal than in• Epidemics stopping• Destroying terrorist

networks#n

odes

#links



The Robustness of Internet The Robustness of Internet 33/3/3

• 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)

N=2000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

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<d>

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C)>

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-of-

SF

)>

ER

SF

(Averaged over 10 samples. Relative to SF.)



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.

– Inverse of epidemics / terror networks



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.• Aldridge et al.’s work on ‘local preferential attachment’.• Agent-based model capable of producing robust

networks.• Scale-free networks as a special case.



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).



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.



Details: Budget ConstraintsDetails: Budget Constraints

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

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



Details: Pricing SchemesDetails: 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



Results: Key FindingsResults: Key Findings

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



Results: Key Findings 1/3Results: Key Findings 1/3

N=2000

0

0.001

0.002

0.003

0.004

0.005

0.006

1 2 4 6 8

<d>

<E

(bC

)>

HoPS=0

HoPS=1

HoPS=2

HePS=0

HePS=1

HePS=2

ER

SF

(Averaged over 10 samples.)




(Averaged over 10 samples. Relative to SF.)

N=2000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6 7 8 9

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E(b

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HePS=0

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SF

ER




1 2 4 6 8HoPS=0

HoPS=1

HoPS=2

HePS=0

HePS=1

HePS=2

<E(bC)>/<E(bC-of-SF)>

<d>

Pricing & Budget

N=2000

0-0.5 0.5-1 1-1.5 1.5-2



Nature of Generated Networks (#3)Nature of Generated Networks (#3)

• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Size-Independent PS. (PS0)




• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Growing Costs PS. (PS1)



Nature of Generated Networks (##)Nature of Generated Networks (##)

• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– ‘Economies of Scale’ PS. (PS2)




• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Size-Independent PS. (PS0)




• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Growing Costs PS. (PS1)



Nature of Generated Networks (##)Nature of Generated Networks (##)

• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– ‘Economies of Scale’ PS. (PS2)



Nature of Generated NetworksNature 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.



Special Network TopologiesSpecial 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.



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.



SummarySummary

• 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.

AITIA International Inc. and Lorand Eotvos University, Budapest ROBUST NETWORKS FROM LOCAL OPTIMIZATION A Bottom-Up Model to Generate Networks with Skewed.

Documents

budapest robust networks

lorand eotvos university

networks of internet

free networks

links slide

international workshop

robustness of internet

nodes connectivity