Phase transitions with infinitely many absorbing states in complex networks

Abstract

We instigate the properties of the threshold contact process (TCP), a process showing an absorbing-state phase transition with infinitely many absorbing states, on random complex networks. The finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field (HMF) approximation and compared with extensive simulations, particularly in the case of heterogeneous scale-free networks. We observe that the TCP exhibits the same critical properties as the contact process (CP), which undergoes an absorbing-state phase transition to a single absorbing state. The accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a HMF theory is a universal feature of absorbing state phase transitions in complex networks, depending only on the locality of the interactions and independent of the number of absorbing states. The conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible-infected-susceptible epidemic spreading model, which in fact belongs to a different universality class in complex networks.

Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks

“…On the other hand, we observe that the critical point decreases for increasing migration rate, at odds with the constant HMF prediction λ c = 1/z. This observation is, however, reasonable since migration facilitates activity spreading to nonactive regions, and can be accounted for by means of a simple homogeneous pair approximation [36,37,39]. Within this approach (see Appendix B), we obtain a new threshold λ…”

“…[37,40]. To do so, let us define the probability Q(σ σ ) that a pair of neighbor sites in a homogeneous population are in the states (σ σ ), where σ = 0 (1) stands for an empty (occupied) node.…”

Heterogeneous pair-approximation for the contact process on complex networks

“…where Ω = N g, = g k k 2 2 , and f(x) is a scaling function independent of the degree distribution. It is easy to show [23] that…”

“…Notice that in the SIS dynamics an occupied vertex creates ('infects' in the epidemiological jargon) an offspring in each empty nearestneighbor at rate λ. Even being equivalent for strictly homogeneous graphs ( ≡ ∀ k k i i ), these models are very different for heterogeneous substrates (see the discussion in [23]). However, in both models the creation of particles is a catalytic process occurring exclusively in pairs of empty-occupied vertices, implying that the state devoid of particles is a fixed point of the dynamics and is called the absorbing state.…”

Finite size effects in epidemic spreading: the problem of overpopulated systems

“…However, there are also generalizations for countable state space Markov chains [32], continuoustime Markov chains [33] and general state spaces [34], along with more involved results for some particular examples like random walks [35], branching processes [36] and more-see [37] for a review. Although here we assume one absorbing state only, it should be noted that recently there has also been an effort put into investigation of systems with infinitely many absorbing states on complex networks [38]. Before we start with the quasi-stationary distribution theory, let us recall the Perron-Frobenius theorem (see e.g.…”