Introduction to the Modeling and Analysis of Complex Systems

17.1. NETWORK SIZE, DENSITY, AND PERCOLATION 375We can calculate the network percolation threshold for random graphs as follows. Letq be the probability for a randomly selected node to not belong to the largest connectedcomponent (LCC) of a network. If q < 1, then there is a giant component in the network. Inorder for a randomly selected node to be disconnected from the LCC, all of its neighborsmust be disconnected from the LCC too. Therefore, somewhat tautologicallyq = q k , (17.4)where k is the degree of the node in question. But in general, degree k is not uniquely determinedin a random network, so the right hand side should be rewritten as an expectedvalue, as∞∑q = P (k)q k , (17.5)k=0where P (k) is the probability for the node to have degree k (called the degree distribution;this will be discussed in more detail later). In an Erdős-Rényi random graph, thisprobability can be calculated by the following binomial distribution⎧⎨P (k) =⎩( n − 1k)p k (1 − p) n−1−k for 0 ≤ k ≤ n − 1,0 for k ≥ n,(17.6)because each node has n − 1 potential neighbors and k out of n − 1 neighbors need tobe connected to the node (and the rest need to be disconnected from it) in order for it tohave degree k. By plugging Eq. (17.6) into Eq. (17.5), we obtain∑n−1( ) n − 1q =p k (1 − p) n−1−k q k (17.7)kk=0∑n−1( ) n − 1=(pq) k (1 − p) n−1−k (17.8)kk=0= (pq + 1 − p) n−1 (17.9)= (1 + p(q − 1)) n−1 . (17.10)We can rewrite this as() n−1〈k〉(q − 1)q = 1 + , (17.11)nbecause the average degree 〈k〉 of an Erdős-Rényi graph for a large n is given by〈k〉 = np. (17.12)