Complex Network Properties and Models

Informally define the term „Small World Effect“! (1 sentence).

What is  the  disadvantage  of  the  expression                                                                                                    for  the  average  path  length  in  a  network?      (1   sentence).   Suggest  an  alternative  expression  avoiding  this  disadvantage!

State the expression for the fraction of nodes with degree k  (==the probability of a node having degree k) in case of a power law degree distribution!

If a power law distribution has an exponent of (-­α): What is the exponent of the corresponding cumulative distribution?

Sketch a power law distribution in a log-log coordinate system!

Give a precise expression for the degree distribution p_k in a Poisson Random Graph model G_n,p ! Qualitatively sketch p_k for n → ∞, keeping the mean degree z fixed!

Emergence of a giant component in a Poisson random graph model:

Explain the following expression for the fraction of nodes u that do not belong to the giant component

For p_k = derive a transcendent equation for S=1-u!

Sketch S qualitatively as a function of z! Explain in 1 sentence the relation to phase transitions!

Watts-Strogatz model: informally describe the model for D=1!

Watts-Strogatz model: Interpret / explain the following diagram in view of the rewiring parameter p! (1-2 sentences)

Watts Strogatz model, variant (2) (only additional shortcuts) : degree distribution:

Model of Price: Informally motivate the terms on the right side of the recursion equation for the net change of np_k per added vertex:

Network resilience: What is the difference in effect between randomly removing nodes with

(a) constant probability q or

(b) removing the highest degree nodes?

(Explain informally in 1-2 sentences!)

Models of Epidemiology: quantities: susceptibles s, infectives i, recovered r, infection rate β, recovery rate γ: