Tobias Friedrich, Pascal Lenzner, Christopher Weyand

Hasso Plattner Institute

E-mail: fistname.lastname@hpi.de

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.

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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)

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