Reaction-diffusion processes and meta-population models in heterogeneous networks Supplementary information

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1 Reaction-diffusion processes and meta-population models in heterogeneous networs Supplementary information Vittoria Colizza 1,2, Romualdo Pastor-Satorras 3, Alessandro Vespignani 2,1 January 19, Complex Networs Lagrange Laboratory, Institute for Scientific Interchange (ISI), Torino, Italy 2 School of Informatics and Department of Physics, Indiana University, Bloomington IN 3 Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, Barcelona, Spain 1 Diffusion of bosonic particles Here we simply consider the diffusion of a single type of bosonic particles - total number equal to N - on a complex networs. We denote with N the total number of particles in nodes with degree and with n the number of nodes with degree. The quantity ρ = N n (1) 1

2 represents the average number of particles in each node with degree. The behavior in time of N (t) is thus given by: t N = N + n P ( ) 1 ρ. (2) Dividing by n and assuming an uncorrelated networ, for which P ( ) = P ( ) /, we have: t ρ = ρ + P ( )ρ. (3) Denoting by ρ = P ( )ρ the average of ρ in the system, we simply obtain for the stationary state: ρ = ρ, (4) i.e. the density of particles on each node is proportional to its degree. This result readily shows that, as a net effect of the diffusion, particles accumulate on nodes with high degree. The density of particles in each node is therefore self-consistently determined by the networ heterogeneity and the diffusion process. 2 Solution of the reaction-diffusion equations Bosonic reaction-diffusion processes in heterogeneous networs, identified by reaction eqs. (1) and (2) of the manuscript, can be described in terms of the average densities ρ A,, ρ B, of A and B particles, respectively, in nodes of degree class. This mean-field approach thus assumes that all nodes having the same degree are equivalent. The dynamical rate equations for any diffusion probabilities D A and D B, under the assumption of no correlations, are given by t ρ B, = ρ B, + (1 D B ) [(1 )ρ B, + Γ ] + D B [(1 )ρ B + Γ], (5) t ρ A, = ρ A, + (1 D A ) [ρ B, + ρ A, Γ ] + D A [ρ B + ρ A Γ], (6) 2

3 with ρ B = P ()ρ B,, ρ A = P ()ρ A, and Γ = P ()Γ, where the reaction ernel Γ taes the form Γ = ρ A, ρ B,, for type I processes, (7) Γ = ρ A,ρ B, ρ, for type II processes. (8) The stationary state is obtained by imposing the conditions t ρ B, = 0 and t ρ A, = 0, which result in the following equations: ρ B, = (1 D B ) [(1 )ρ B, + Γ ] + D B [(1 )ρ B + Γ], (9) ρ A, = (1 D A ) [ρ B, + ρ A, Γ ] + D A [ρ B + ρ A Γ]. (10) Multiplying this last equations by P () and summing over, we obtain the general result Using this last expression, eqs. (9) and (10) are simplify to: ρ B = Γ. (11) ρ B, = (1 D B ) [(1 )ρ B, + Γ ] + D B ρ B, (12) ρ A, = (1 D A ) [ρ B, + ρ A, Γ ] + D A ρ A. (13) Let us consider the different possibilities regarding the values of the diffusion coefficients. Diffusing B and nondiffusing A particles (D B = 1, D A = 0) From eqs. (12) and (13), we have ρ B, = ρ B, ρ A, = ρ B, + ρ A, Γ, (14) the last equation translating into ρ B, = Γ. (15) In type I processes, the definition of Γ yields, from eq. (15), ρ A, = ρ A =, ρ B = ρ. (16) 3

4 The condition for the existence of an active phase is thus given by ρ >, (17) so that in type I processes the transition depends on the total density of particles in the system. For type II processes, on the other hand, eq. (15) gives ρ A, = ( ρ, ρ B, = 1 ) ρ. (18) Averaging this last expresion over the system yields ( ρ B = ρ 1 ), (19) and the active phase exists for > 1, (20) recovering an epidemic threshold that depends only on the rection rates, regardless of the value of the density. Diffusing B and A particles (D B = D A = 1) In this case, we have ρ B, = ρ B, ρ A, = ρ A, ρ = ρ. (21) For the type I processes, inserting in the definition of Γ the above values, we have yielding ρ A = ρ B = Γ = The condition for the presence of an active phase is thus given by 2 2 ρ Aρ B, (22) 2 2, ρ B = ρ 2 2. (23) ρ > 2 2. (24) 4

5 For the type II processes, instead, we have ρ B = Γ = ρ A ρ B ρ, (25) yielding ρ A = ρ, ( ρ B = ρ 1 ), (26) with the threshold condition > 1. (27) Thus, while the threshold in type II processes is left invariant, the critical point in type I processes in strongly affected by the topological fluctuations of the underlying networ, suppressing the phase transition in the infinite networ size limit. General diffusing B and A particles (D B = 1, 0 < D A < 1) We have now ρ B, = ρ B, ρ A, = (1 D A ) [ρ B, + ρ A, Γ ] + D A ρ A. (28) Substituting the value of ρ B, into the expression for ρ A, and rearranging terms, we obtain for type I processes Using this last expression, we can write ρ A, = (1 D A) ρ B + D A (ρ ρ B) D A + (1 D A ) ρ. (29) B ρ B = ρ P ()ρ A, = ρ P () (1 D A) ρ B + D A (ρ ρ B) D A + (1 D A ) ρ. (30) B We can thus write this equation in the form ρ B = f(ρ B ), so that a solution exists if the first derivative of the function f(ρ B ) in the origin in greater than one. Imposing the condition f (ρ B = 0) > 1, we recover the condition for the presence of an active phase, ρ > 2 2 (31) independent of D A. 5

6 For type II processes, on the other hand, introducing the expression of the reaction ernel, we have ρ B, = [ ρ B, ρ A, = (1 D A ) ρ B, + ρ A, ρ ] A,ρ B, + D A ρ A, + ρ B, ρ A. (32) It is easy to see that the solution of this set of equations taes the form ρ B, = ρ B, ρ A, = ρ A. (33) Inserting this expressions into eq. (32), and simplifying factors /, we are led to from where we obtain, as in the case D A = 1, ρ A = ρ, 0 = ρ B ρ Aρ B ρ, (34) ρ B = ρ ( 1 ), (35) with threshold > 1. (36) Thus, the presence of a diffusion of A particles D A < 1 does not affect the value of the critical threshold. It has, however, an important effect on the functional form of ρ A,, which, in the case of type I processes, saturates to a constant value for large degrees. 6