Closing the Random Graph Gap in Tuza's Conjecture through the Online Triangle Packing Process

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

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On the Structure of a Random Graph with Nonuniform Distribution

Proceedings of the Bakuriani Colloquium in Honour of Yu.V. Prohorov, Bakuriani, Georgia, USSR, 24 February–4 March, 1990 ◽

10.1515/9783112313626-034 ◽

1991 ◽

pp. 445-456

Keyword(s):

Random Graph ◽

Nonuniform Distribution

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A Note on the Majority Dynamics in Inhomogeneous Random Graphs

Results in Mathematics ◽

10.1007/s00025-021-01436-z ◽

2021 ◽

Vol 76 (3) ◽

Author(s):

Yilun Shang

Keyword(s):

Random Graph ◽

Initial State ◽

Time Step ◽

Dynamic Monopoly ◽

Inhomogeneous Random Graphs ◽

Assignment Probability ◽

Edge Probability ◽

Consensus Reaching Process ◽

Consensus Reaching ◽

Inhomogeneous Random Graph

AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

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