Motif-based mean-field approximation of interacting particles on clustered networks

Interacting particles on graphs are routinely used to study magnetic behavior in physics, disease spread in epidemiology, and opinion dynamics in social sciences. The literature on mean-field approximations of such systems for large graphs typically remains limited to specific dynamics, or assumes cluster-free graphs for which standard approximations based on degrees and pairs are often reasonably accurate. Here, we propose a motif-based mean-field approximation that considers higher-order subgraph structures in large clustered graphs. Numerically, our equations agree with stochastic simulations where existing methods fail.

Figure 1

Schematic illustration of the model. (a) A finite number of network motifs occurring in the network. (b) An example ECM vertex with two motif participations as vertex 2 in $G^{\left(1\right)}$ and one each as vertices $3,1,3$ in $G^{\left(2\right)},G^{\left(3\right)},G^{\left(4\right)}$ respectively, i.e., $d^1\equiv (d^{1,1},d^{1,2})=(0,2), d^2=(0,0,1), d^3=(1,0,0,0), d^4=(0,0,1,0)$. (c) Simplicial SIS dynamics [14] as an example of general neighborhood-dependent dynamics. Susceptible vertices are infected at rate $\tau$ by infected neighbors, and additionally at rate ${\tau }_{▵}$ for each shared neighbor.

Figure 2

Schematic illustration of the MMF equations for two states. (a) A fixed, finite number of network motifs with associated motif colorings. (b) Common, shared jump rates between motif colorings. (c) A visualization of all jumps from and to configuration $z$ (center), where the jump rates are given by the number of motif configurations times their shared motif jump rates. Here, $d^1=(2,2), d^2=(0,0,1)$ and $z^{1,1}=(1,1,0,0), z^{1,2}=(0,0,2,0), z^{2,3}=(0,1,0,0,0,0,0,0)$.

Figure 3

Mean-field approximations in the SIS model using the edge and triangle graphs as motifs. (a)–(c) Results for $\tau =0.3, \gamma =0.9$. (a) $P^{\mathrm{a},3}, P_0\left(I\right)=0.2$, (b) $P^{\mathrm{u},3}, P_0\left(I\right)=0.3$, (c) $P^{\mathrm{d},2}, P_0\left(I\right)=0.4$. (d)–(f) Results for $P^{\mathrm{a},3}$. (d) $\tau =0.3, \gamma =0.9, P_0\left(I\right)=0.2$, (e) $\tau =0.5, \gamma =0.7, P_0\left(I\right)=0.6$, (f) $\tau =0.65, \gamma =0.55, P_0\left(I\right)=0.75$.

Figure 4

Mean-field approximations in the Ising Glauber model using the edge and triangle graphs as motifs. (a)–(c) Results for $T{J}^{-1}=3$. (a) $P^{\mathrm{a},3}, P_0\left(D\right)=0.2$, (b) $P^{\mathrm{u},3}, P_0\left(D\right)=0.3$, (c) $P^{\mathrm{d},2}, P_0\left(D\right)=0.4$. (d)–(f) Results for $P^{\mathrm{a},3}, P_0\left(D\right)=0.33$. (d) $T{J}^{-1}=1$, (e) $T{J}^{-1}=3$ (f) $T{J}^{-1}=4$.

Figure 5

Mean-field approximations in the SIS model using the edge and square graphs as motifs and $P^{\mathrm{a},2}$. (a) $\tau =0.3, \gamma =0.9, P_0\left(I\right)=0.2$, (b) $\tau =0.5, \gamma =0.7, P_0\left(I\right)=0.6$, (c) $\tau =0.65, \gamma =0.55, P_0\left(I\right)=0.75$.

Figure 6

A comparison between MMF and numerical Gillespie simulation in the simplicial SIS model using ${\tau }_{▵}=\tau$, the edge and triangle graphs as motifs and $\tau =0.3, \gamma =0.9$. (a) $P^{\mathrm{a},3}, P_0\left(I\right)=0.2$, (b) $P^{\mathrm{u},3}, P_0\left(I\right)=0.2$, (c) $P^{\mathrm{d},2}, P_0\left(I\right)=0.4$.

Figure 7

The MMF approximation is almost indiscernible from the numerical simulation in the SIR model using the edge and triangle graphs as motifs and $P^{\mathrm{a},2}$. (a) $\tau =0.3, \gamma =0.9, 1-P_0\left(S\right)=P_0\left(I\right)=0.2$, (b) $\tau =0.5, \gamma =0.7, 1-P_0\left(S\right)=P_0\left(I\right)=0.6$, (c) $\tau =0.6, \gamma =0.5, 1-P_0\left(S\right)=P_0\left(I\right)=0.5$.