Applications to networksSI on a graph At each time slot, each link outgoing from an infected node spreads the disease with probability p g Susceptible Infected Can we apply Mean Field theory? ❒ Formally not, because in a graph the different nodes are not equivalent… ❒ …but we are stubbornDerive a Mean Field model ❒ Consider all the nodes equivalent ❒ e.g. assume that at each slot the graph changes, while keeping the average degree <d> Starting from an empty network we add a link with probability <d>/(N-1) k=1 Derive a Mean Field model ❒ Consider all the nodes equivalent ❒ e.g. assume that at each slot the graph changes, while keeping the average degree <d> Starting from an empty network we add a link with probability <d>/(N-1) k=2 Derive a Mean Field model ❒ i.e. at every slot we consider a sample of an ER graph with N nodes and probability <d>/(N-1) Starting from an empty network we add a link with probability <d>/(N-1) k=2 Derive a Mean Field model ❒ If I(k)=I, the prob. that a given susceptible node is infected is q I =1-(1-<d>/(N-1) p g) I ❒ and (I(k+1)-I(k)|I(k)=I) = d Bin(N-I, q I) k=2 Derive a Mean Field model ❒ If I(k)=I, the prob. that a given susceptible node is infected is q I =1-(1-<d>/(N-1) p g) I ❒ and (I(k+1)-I(k)|I(k)=I) = d Bin(N-I, q I) Equivalent to first SI model where p=<d>/(N-1) p g • We know that we need p (N) =p 0 /N 2 ❒ i (N) (k) ≈ µ 2 (kε(N))=1/((1/i 0-1) exp(-k
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