Epidemics on networks: Reducing disease transmission using health emergency declarations and peer communication

Azizi, Asma; Montalvo, Cesar; Espinoza, Baltazar; Kang, Yun; Castillo-Chavez, Carlos

Definition 4.2. Small-world network Suppose we have n nodes over a ring, where each node in the ring is connected to its k nearest neighbors in the ring. Then with probability p and independent from any other pair of edges we select two edges i 1 j 1 and i 2 j 2 such that j k is from the k nearest neighbor of i k for k ¼ 1; 2, and rewire them. Then the constructed network is called Watts-Strogatz Smal-world network (Watts & Strogatz, 1998) Gðn; k; pÞ. Definition 4.3. Preferential attachment Scale-free network Suppose we want to generate a network of n nodes. We start with one node and follow the following procedure n times: the probability of attaching a new node to the existing ones is proportional to their current degree:Pðconnect a new node i to existing node jÞ ¼ degðjÞ P k ðdegðkÞÞ , where the degðkÞ is the number of neighbors for node k, then with this probability we make new edge ij. The constructed network via this model is called BarabasieAlbert Scale-free network (Barab asi & Albert, 1999). The degree distribution of this network follows a power-law distribution.