aitia international inc. and lorand eotvos university, budapest robust networks from local...
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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]
AITIA International Inc. and Lorand Eotvos University, BudapestAITIA International Inc. and Lorand Eotvos University, Budapest
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.
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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
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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
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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
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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
0 2 4 6 8 10
<d>
<E(b
C)>
/<E
(bC
-of-
SF
)>
ER
SF
(Averaged over 10 samples. Relative to SF.)
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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.
– Inverse of epidemics / terror networks
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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.• Aldridge et al.’s work on ‘local preferential attachment’.• 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:
• PS0: PS(i) = C
• Growing Costs:
• PS1: PS(i) = C*B / i
• Decreasing Costs (‘economies of scale’):
• PS2: 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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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.)
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Results: Key Findings 2/3Results: Key Findings 2/3
(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
<d>
<E
(bC
)>/<
E(b
C-o
f-S
F)>
HoPS=0
HoPS=1
HoPS=2
HePS=0
HePS=1
HePS=2
SF
ER
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Results: Key Findings 3/3Results: Key Findings 3/3
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
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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)
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Nature of Generated Networks (#1)Nature of Generated Networks (#1)
• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Growing Costs PS. (PS1)
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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)
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Nature of Generated Networks (#2)Nature of Generated Networks (#2)
• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Size-Independent PS. (PS0)
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Nature of Generated Networks (#4)Nature of Generated Networks (#4)
• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Growing Costs PS. (PS1)
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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)
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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.
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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.
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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.
• 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.