Exploring Game-Theoretic Formation of Realistic Networks · Web: ... //hpi.de/friedrich/home.html...
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Tobias Friedrich, Pascal Lenzner, Christopher Weyand Hasso Plattner Institute E-mail: fi[email protected] Web: https://hpi.de/friedrich/home.html Prof.-Dr.-Helmert-Straße 2-3 14482 Potsdam, Germany Exploring Game-Theoretic Formation of Realistic Networks Properties of Real-world Networks: small-world property: diameter in O ( log n ) , small average path length high clustering: many triangles and small cliques, average clustering coefficient in [0.2, 0.8] power-law degree distribution: probability that a node has degree k is proportional to k - β , for some constant 2 ≤ β ≤ 5 average clustering coefficient: 0.351 diameter: 9 avg path length: 3.801 fitted power-law exponent: 2.3 Snapshot of AS-level graph of the Internet (01 January 2000). Data from Stanford Large Network Database [1], plotted using Gephi with Yifan-Hu and Force Atlas layout. The graph has 3570 nodes colored by their modularity class and 7750 edges. Node sizes are proportional to their degree. Networks via Game-Theory • nodes are selfish and rational agents • strategy of an agent = subset of other agents • agents strive for good position in the network • costly links are formed according to strategies • strategies of all agents determines the edge-set • each agent tries to minimize some cost function • consider pure Nash equilibrium of the game Strategic Network Augmentation 1. start with sparse connected initial network 2. activate agents in round-robin/random fashion 3. active agent buys local edge to improve centrality 4. edge-cost is proportional to node degree 5. iterate until all agents are happy Our Model: [1] http://snap.stanford.edu/data/as.html Final equilibrium graph generated via global strategic network augmentation, plotted using Gephi with Force Atlas layout. The graph has 1000 nodes colored by their modularity class. Node sizes are proportional to their degree. 0.0001 0.001 0.01 0.1 1 1 10 100 Probability Degree k Cumulative degree distribution (log-log plot) Probability that a random node has degree ≥ k ƒ (n)= n -1.3 Average local clustering coefficient Power-law exponent diameter: 4 avg path length: 2.71 fitted power-law exponent: 3.2 maximum degree: 205 minimum degree: 7 nodes: 1000 edges: 7423 average degree: 14.84 Average local clustering coefficient Power-law exponent Results for the global model ( n = 10000) average clustering coefficient: 0.321 Results for the local model ( n = 20000) 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 alpha 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 exponent mean min/max 0.00 0.25 0.50 0.75 1.00 1.25 stdev 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 alpha 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 local_clustering mean min/max 0.00 0.05 0.10 0.15 0.20 stdev 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 alpha 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 exponent mean min/max 0.00 0.05 0.10 0.15 0.20 0.25 stdev 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 alpha 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 local_clustering mean min/max 0.00 0.02 0.05 0.08 0.10 0.12 stdev
Transcript of Exploring Game-Theoretic Formation of Realistic Networks · Web: ... //hpi.de/friedrich/home.html...
Tobias Friedrich, Pascal Lenzner, Christopher Weyand
Hasso Plattner Institute
E-mail: [email protected]
Web: https://hpi.de/friedrich/home.html
Prof.-Dr.-Helmert-Straße 2-3 14482 Potsdam, Germany
Exploring Game-Theoretic Formation of Realistic Networks
Properties of Real-world Networks:
small-world property:
diameter in O(logn),small average path length
high clustering:
many triangles and small cliques,average clustering coefficient in[0.2, 0.8]
power-law degree distribution:
probability that a node has degreek is proportional to k−β, for someconstant 2 ≤ β ≤ 5
average clusteringcoefficient: 0.351
diameter: 9avg path length: 3.801
fitted power-lawexponent: 2.3
Snapshot of AS-level graph of the Internet (01 January 2000). Data from StanfordLarge Network Database [1], plotted using Gephi with Yifan-Hu and Force Atlaslayout. The graph has 3570 nodes colored by their modularity class and 7750edges. Node sizes are proportional to their degree.
Networks via Game-Theory• nodes are selfish and rational agents
• strategy of an agent = subset of other agents
• agents strive for good position in the network
• costly links are formed according to strategies
• strategies of all agents determines the edge-set
• each agent tries to minimize some cost function
• consider pure Nash equilibrium of the game
Strategic Network Augmentation1. start with sparse connected initial network
2. activate agents in round-robin/random fashion
3. active agent buys local edge to improve centrality
4. edge-cost is proportional to node degree
5. iterate until all agents are happy
Our Model:
[1] http://snap.stanford.edu/data/as.html
Final equilibrium graph generated via global strategicnetwork augmentation, plotted using Gephi with Force Atlaslayout. The graph has 1000 nodes colored by theirmodularity class. Node sizes are proportional to their degree.
0.0001
0.001
0.01
0.1
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1 10 100
Pro
bab
ility
Degree k
Cumulative degree distribution (log-log plot)
Probability that a random node has degree ≥ kƒ (n) = n−1.3
Average local clustering coefficientPower-law exponent
diameter: 4avg path length: 2.71
fitted power-lawexponent: 3.2
maximum degree: 205minimum degree: 7
nodes: 1000edges: 7423
average degree: 14.84 Average local clustering coefficientPower-law exponent
Results for the global model (n = 10000)
average clusteringcoefficient: 0.321
Results for the local model (n = 20000)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0alpha
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