Heterogeneous mean-field theory for two-species symbiotic processes on networks

A simple model to study cooperation is the two-species symbiotic contact process (2SCP), in which two different species spread on a graph and interact by a reduced death rate if both occupy the same vertex, representing a symbiotic interaction. The 2SCP is known to exhibit a complex behavior with a rich phase diagram, including continuous and discontinuous transitions between the active phase and extinction. In this work, we advance the understanding of the phase transition of the 2SCP on uncorrelated networks by developing a heterogeneous mean-field (HMF) theory, in which the heterogeneity of contacts is explicitly reckoned. The HMF theory for networks with power-law degree distribution shows that the region of bistability (active and inactive phases) in the phase diagram shrinks as the heterogeneity level is increased by reducing the degree exponent. Finite-size analysis reveals a complex behavior where a pseudodiscontinuous transition at a finite size can be converted into a continuous one in the thermodynamic limit, depending on degree exponent and symbiotic coupling. The theoretical results are supported by extensive numerical simulations.

Figure 1

Transitions of the 2SCP model on networks where species are represented by red and blue dots while the nodes involved in the transition are depicted in green. (a) Individuals of species A or B in the node $i$ replicate a copy of themselves in a neighbor node if allowed (double occupation with the same species is forbidden). Spontaneous deaths on nodes that contain (b) one and (c) both species (symbiosis) happen with rates $\mu$ and ${\mu }_{\text{s}}$, respectively.

Figure 2

HMF theory for the 2SCP model for a power-law degree distributions with lower and upper cutoffs given by $k_{\text{min}}=3$ and $k_{\text{max}}=17320$, respectively, and degree exponent $\gamma$. (a) Example of hysteresis in the total prevalence curves for $\gamma =3.5$ and ${\mu }_{\text{s}}=0.2$. The hashed area indicates the bistability region. Phase diagram for 2SCP considering (b) $\gamma =3.5$, (c) $\gamma =2.7$, and (d) $\gamma =2.3$. The dotted lines represent the lower spinodal for the homogeneous theory ${\lambda }^{-}=\sqrt{4{\mu }_{\text{s}}(1-{\mu }_{\text{s}})}$.

Figure 3

Example of a (a) continuous and (b) discontinuous transition through transcritical and pitchfork bifurcations, respectively. The curves correspond to the solution of Eq. (15) with (a) ${\mu }_{\text{s}}=0.3$ and (b) ${\mu }_{\text{s}}=0.05$ for a degree exponent $\gamma =2.7$. The contagion rate $\lambda$ is increased from bottom to top. The blue curves are the critical ones.

Figure 4

Coefficient $\tilde{\beta }$, where $a_{\gamma -1}\propto \tilde{\beta }$, as function of symbiotic parameter ${\mu }_{\text{s}}$ on power-law networks with different values of the degree exponent $\gamma$ indicated in the legend. Vertical lines are roots of $\tilde{\beta }\left({\mu }_{\text{s}}\right)=0$ for $\gamma =2.7$ and 2.3.

Figure 5

Total prevalence (${\rho }_T=2\rho +{\rho }_{\text{AB}}$) as a function of $\lambda$ for $P\left(k\right)$ representing synthetic scale-free networks with ${\mu }_{\text{s}}=0.2$ and different sizes. (a) $\gamma =2.3$, (b) $\gamma =2.7$. (c) $\gamma =3.5$.

Figure 6

Finite-size scaling for (a) activation threshold and (b) discontinuity gap for 2SCP obtained with the HMF theory for power-law degree distributions with different values of $\gamma$ and ${\mu }_{\text{s}}=0.2$ fixed. Lower and upper degree cutoffs $k_{\text{min}}=3$ and $k_{\text{max}}=\sqrt{N}$ we adopted.

Figure 7

(a) Finite-size scaling (FSS) of discontinuity gap (${\mathrm{\Delta }}_{\text{p}}$) for 2SCP obtained with the HMF theory for power-law degree distributions with $\gamma =2.3$ and different values of ${\mu }_{\text{s}}$. Symbols are numerical data obtained from integration of HMF equations and solid lines nonlinear regression to perform the FSS; see main text. (b) Symbiosis coupling ${\mu }_{\text{s}}^{*}$ separating the discontinuous and continuous transitions for different values of the degree exponent, obtained theoretically (solid curve) and with the FSS of ${\mathrm{\Delta }}_{\text{p}}$ versus $N$.

Figure 8

(a) Total prevalence curves obtained with QS simulations of the 2SCP model on (a) annealed and (b) quenched networks with degree exponent $\gamma =2.3$ and different sizes (symbols) in comparison with HMF equations integration of the same degree distribution $P\left(k\right)$ (solid lines). The lower and upper degree cutoff are $k_{\text{min}}=3$ and $k_{\text{max}}=\sqrt{N}$ and the symbiosis parameter is ${\mu }_{\text{s}}=0.2$. Doted lines are guides to the eyes to visualize the discontinuity in simulation curves.

Figure 9

Comparison of the finite-size scaling of the (a) gap discontinuity ${\mathrm{\Delta }}_p$ and the (b) activation threshold ${\lambda }_{\text{c}}$ obtained in HMF theory and simulations on quenched and annealed networks with $\gamma =2.3$ (solid lines) and 2.7 (dashed lines). The color sequence is black for HMF theory, red for annealed, and blue for UCM networks. The lower and upper cutoffs are $k_{\text{min}}=3$ and $k_{\text{max}}=\sqrt{N}$, respectively, and the symbiosis parameter is ${\mu }_{\text{s}}=0.2$.