Introduction to the Modeling and Analysis of Complex Systems

338 CHAPTER 16. DYNAMICAL NETWORKS I: MODELING“Measure the difference between your neighbor and yourself, and then sum upall those differences.”Note that this is exactly what we did on a network in Eq. (16.3)! So, essentially, the Laplacianmatrix of a graph is a discrete equivalent of the Laplacian operator for continuousspace. The only difference, which is quite unfortunate in my opinion, is that they are definedwith opposite signs for historical reasons; compare the first term of Eq. (13.17) andEq.(16.4), and you will see that the Laplacian matrix did not absorb the minus sign insideit. So, conceptually,∇ 2 ⇔ −L. (16.6)I have always thought that this mismatch of signs was so confusing, but both of these“Laplacians” are already fully established in their respective fields. So, we just have to livewith these inconsistent definitions of “Laplacians.”Now that we have the mathematical equations for diffusion on a network, we candiscretize time to simulate their dynamics in Python. Eq. (16.3) becomes[c i (t + ∆t) = c i (t) +α ∑ j∈N i(cj (t) − c i (t) )] ∆t (16.7)= c i (t) + α[( ∑j∈N ic j (t))− c i (t) deg(i)]∆t. (16.8)Or, equivalently, we can also discretize time in Eq. (16.4), i.e.,c(t + ∆t) = c(t) − αLc(t)∆t (16.9)= (I − αL∆t) c(t), (16.10)where c is now the state vector for the entire network, and I is the identity matrix. Eqs. (16.8)and (16.10) represent exactly the same diffusion dynamics, but we will use Eq. (16.8) forthe simulation in the following, because it won’t require matrix representation (which couldbe inefficient in terms of memory use if the network is large and sparse).We can reuse Code 16.4 for this simulation. We just need to replace the updatefunction with the following:Code 16.7: net-diffusion.pyalpha = 1 # diffusion constantDt = 0.01 # Delta t