Through first principles described in Section 2, the probability vector for the information diffusion model can be written as
$$\begin{array}{}{\displaystyle |p(t)\u3009=\sum _{S,K,I}P(S,K,I|N,t)\phantom{\rule{thinmathspace}{0ex}}|S,K,I\u3009,}\end{array}$$(12)

where *P*(*S*, *K*, *I*|*N*, *t*) denotes the probability that from a cohort of size *N* there are *S* susceptible agents, *K* known agents, and *I* informed agents at time *t*, leaving *N* – *S* – *K* – *I* refractory ones. |*S*, *K*, *I*〉 is a basis vector, linearly independent of other basis vectors with different susceptible, known, and informed counts. The state space 𝓢 is a finite set.

Let *X̂* with a hat on it represent an endomorphism of the vector space spanned by the above basis vectors. The Kolmogorov equation governing the information diffusion process can be written as
$$\begin{array}{}{\displaystyle \frac{\text{d}}{\text{d}t}|p(t)\u3009=H(t)|p(t)\u3009,}\end{array}$$(13)

with the time-evolution operator
$$\begin{array}{}H(t)=\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& k(t)(1-\theta (t))(1-{p}_{1}(t))(\hat{\tau}-\hat{S})\\ & \phantom{\rule{thinmathspace}{0ex}}+k(t)\theta (t)(\hat{\gamma}-\hat{S})+k(t)(1-\theta (t)){p}_{1}(t)(\hat{\sigma}-\hat{S})\\ & \phantom{\rule{thinmathspace}{0ex}}+k(t){p}_{3}(t)(\hat{\rho}-\hat{K})+{p}_{2}(t)(\hat{\delta}-\hat{I}),\end{array}$$(14)

where
$$\begin{array}{}\hat{\tau}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =S|S-1,K,I+1\u3009,\\ \hat{S}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =S|S,K,I\u3009,\\ \hat{\sigma}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =S|S-1,K+1,I\u3009,\\ \hat{I}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =I|S,K,I\u3009,\\ \hat{K}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =K|S,K,I\u3009,\\ \hat{\rho}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =K|S,K-1,I+1\u3009,\\ \hat{\gamma}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =S|S-1,K,I\u3009,\\ \hat{\delta}|S,K,I\u3009\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& =I|S,K,I-1\u3009.\end{array}$$(15)

Now we introduce another operator *η̂*, defined as
$$\begin{array}{}\hat{\eta}|S,K,I\u3009=K|S,K-1,I\u3009\end{array}$$(16)

so that the Lie algebra $\mathcal{L}=\text{span}\{\hat{\tau},\hat{S},\hat{\sigma},\hat{I},\hat{K},\hat{\rho},\hat{\gamma},\hat{\delta},\hat{\eta}\}$
is closed under the action of the Lie bracket. We display in the complete set of Lie brackets. The nine operators defined above are all linear operators acting on basis vectors. They are interpreted as follows: *τ̂* returns the number of susceptible agents, depletes the susceptible population by one, and increases the informed population by one; *Ŝ* returns the number of susceptible agents; *σ̂* returns the number of susceptible agents, depletes the susceptible population by one, and increases the known population by one; *Î* returns the number of informed agents; *K̂* returns the number of known agents; *ρ̂* returns the number of known agents, reduces the known population by one, and increases the informed population by one; *γ̂* returns the number of susceptible agents and depletes these by one; *δ̂* returns the number of informed agents and depletes these by one; *η̂* returns the number of known agents and depletes these by one.

Table 1 Values of [*X̂*, *Ŷ*] for the information diffusion model.

The remaining task is to apply the Wei-Norman method to Eq. (13). Namely, we want to look for a solution of the form
$$\begin{array}{}|p(t)\u3009=\\ {e}^{{g}_{1}(t)\hat{\eta}}{e}^{{g}_{2}(t)\hat{S}}{e}^{{g}_{3}(t)\hat{K}}{e}^{{g}_{4}(t)\hat{I}}{e}^{{g}_{5}(t)\hat{\tau}}{e}^{{g}_{6}(t)\hat{\sigma}}{e}^{{g}_{7}(t)\hat{\rho}}{e}^{{g}_{8}(t)\hat{\gamma}}{e}^{{g}_{9}(t)\hat{\delta}}|p(0)\u3009.\end{array}$$(17)

Using (11) and the action of exponential operators shown in , we obtain the following linear relation
$$\begin{array}{}\phantom{\rule{1em}{0ex}}k(t)(1-\theta (t))(1-{p}_{1}(t))\hat{\tau}\\ \phantom{\rule{1em}{0ex}}+k(t)\theta (t)\hat{\gamma}+k(t)(1-\theta (t)){p}_{1}(t)\hat{\sigma}+k(t){p}_{3}(t)\hat{\rho}\\ \phantom{\rule{1em}{0ex}}-k(t){p}_{3}(t)\hat{K}+{p}_{2}(t)\hat{\delta}-{p}_{2}(t)\hat{I}-k(t)\hat{S}\\ ={\dot{g}}_{1}(t)\hat{\eta}+{\dot{g}}_{2}(t)\hat{S}+{\dot{g}}_{3}(t)(\hat{K}+{g}_{1}(t)\hat{\eta})+{\dot{g}}_{4}(t)\hat{I}\\ \phantom{\rule{1em}{0ex}}+{\dot{g}}_{5}(t){e}^{{g}_{4}(t)-{g}_{2}(t)}\hat{\tau}+{\dot{g}}_{6}(t){e}^{{g}_{3}(t)-{g}_{2}(t)}(\hat{\sigma}+{g}_{1}(t)\hat{\gamma})\\ \phantom{\rule{1em}{0ex}}+{\dot{g}}_{7}(t){e}^{{g}_{4}(t)}({e}^{-{g}_{3}(t)}\hat{\rho}-{g}_{6}(t){e}^{-{g}_{2}(t)}\hat{\tau})+{\dot{g}}_{8}(t){e}^{-{g}_{2}(t)}\hat{\gamma}\\ \phantom{\rule{1em}{0ex}}+{\dot{g}}_{9}(t)[{e}^{-{g}_{4}(t)}(\hat{\delta}+{g}_{1}(t)\hat{\gamma})-{g}_{5}(t){e}^{-{g}_{2}(t)}\hat{\gamma}\\ \phantom{\rule{1em}{0ex}}-{g}_{7}(t)({e}^{-{g}_{3}(t)}\hat{\eta}-{g}_{6}(t){e}^{-{g}_{2}(t)}\hat{\gamma})].\end{array}$$(18)

Table 2 Values of *e*^{g(adX̂)}*Ŷ* with a scalar *g* for the information diffusion model.

Solving the set of ordinary differential equations derived from (18) for each basis operator in 𝓛 yields
$$\begin{array}{}{g}_{1}(t)={\displaystyle {e}^{\mathit{\Psi}(t)}\underset{0}{\overset{t}{\int}}{p}_{2}(v){e}^{-\mathit{\Lambda}(v)}\underset{0}{\overset{v}{\int}}k(u){p}_{3}(u){e}^{\mathit{\Lambda}(u)-\mathit{\Psi}(u)}\text{d}u\text{d}v,}\\ {g}_{2}(t)={\displaystyle -\mathit{\Gamma}(t),{g}_{3}(t)=-\mathit{\Psi}(t),{g}_{4}(t)=-\mathit{\Lambda}(t),}\\ {g}_{5}(t)={\displaystyle \underset{0}{\overset{t}{\int}}k(u)(1-\theta (u))(1-{p}_{1}(u)){e}^{\mathit{\Lambda}(u)-\mathit{\Gamma}(u)}\text{d}u}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +\underset{0}{\overset{t}{\int}}k(v){p}_{3}(v){e}^{\mathit{\Lambda}(v)-\mathit{\Psi}(v)}\underset{0}{\overset{v}{\int}}k(u)(1-\theta (u)){p}_{1}(u){e}^{\mathit{\Psi}(u)-\mathit{\Gamma}(u)}\text{d}u\text{d}v,}\\ {g}_{6}(t)={\displaystyle \underset{0}{\overset{t}{\int}}k(u)(1-\theta (u)){p}_{1}(u){e}^{\mathit{\Psi}(u)-\mathit{\Gamma}(u)}\text{d}u,}\\ {g}_{7}(t)={\displaystyle \underset{0}{\overset{t}{\int}}k(u){p}_{3}(u){e}^{\mathit{\Lambda}(u)-\mathit{\Psi}(u)}\text{d}u,}\\ {g}_{9}(t)={\displaystyle \underset{0}{\overset{t}{\int}}{p}_{2}(u){e}^{-\mathit{\Lambda}(u)}\text{d}u,}\end{array}$$(19)

and
$$\begin{array}{}{\displaystyle {g}_{8}(t)=\underset{0}{\overset{t}{\int}}{e}^{-\mathit{\Gamma}(u)}[k(u)\theta (u)}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-k(u)(1-\theta (u)){p}_{1}(u){g}_{1}(u)-{p}_{2}(u){g}_{1}(u)]\text{d}u\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +\underset{0}{\overset{t}{\int}}{p}_{2}(u){e}^{-\mathit{\Lambda}(u)}({g}_{5}(u)-{g}_{6}(u){g}_{7}(u))\text{d}u,}\end{array}$$(20)

where *Γ*(*t*) := ${\int}_{0}^{t}$*k*(*u*)d*u*, *Ψ*(*t*) := ${\int}_{0}^{t}$*k*(*u*)*p*_{3}(*u*)d*u*, and *Λ*(*t*) := ${\int}_{0}^{t}$*p*_{2}(*u*)d*u*.

In order to recover the final-size equation [16, Eq. (6)] for the SKIR model, we take |*p*(0)〉 = |*N* −1, 0, 1 〉, and assume that *k*(*t*) ≡ *k*, *θ*(*t*) ≡ *θ*, and *p*_{i}(*t*) ≡ *p*_{i} for *i* = 1, 2, 3. Set |𝓢(*t*)〉 = ∑_{S,K,I}*S*|*S*, *K*, *I*〉. Thus, *s*(*t*) := *S*(*t*)/*N* = 〈𝓢(*t*)|*p*(*t*)〉/*N*^{2} is the density of susceptible agents in the population at time *t*. Since *R*(∞) (or *S*(∞)) is independent of *p*_{1} and *p*_{3} [16], we take *p*_{1} = *p*_{3} = 0 in (19), (20), and let *t* tends to infinity, which gives $|p(\mathrm{\infty})\u3009={e}^{-k(1-\theta )\hat{\tau}/{p}_{2}}{e}^{\hat{\delta}}|p(0)\u3009.$ Finally, we obtain
$$\begin{array}{}s(\mathrm{\infty})=\u3008\mathcal{S}(\mathrm{\infty})|p(\mathrm{\infty})\u3009/{N}^{2}\approx {e}^{-\frac{k(1-\theta )}{{p}_{2}}}(1-s(\mathrm{\infty})).\end{array}$$(21)

Note that Eq. (21) is essentially in line with Eq. (6) in [16] as desired.

Finally, we perform simulations on an Erdős-Rényi network and a small-world network with network size *N* = 5000 and average degree *k* = 10, respectively. For each data point, 1000 independent dynamical realizations are used to calculate the pertinent average values, which are averaged over 50 network realizations. In Figure 2, we illustrate the dynamical behavior of the information diffusion model which agrees with the theoretical prediction under normalized time span. A key feature of our model as compared to that of [16] is that known agents are allowed to be non-vanishing (c.f. Figure 2(b)) since known agents cannot become informed automatically, *i*.*e*., *p*_{4} = 0 [16]. On the other hand, under the choice of parameters as shown in Figure 2(a), we observed that all the agents have heard the information in the end, *i*.*e*. *R*(*t*) tends to *N* as *t* grows.

Figure 2 Densities of four kinds of agents, susceptible *S*(*t*)/*N*, known *K*(*t*)/*N*, informed *I*(*t*)/*N*, and refractory *R*(*t*)/*N*, as a function of *t* with *N* = 5000 agents and |*p*(0)〉 = |0.99⋅ *N*, 0, 0.01 ⋅ *N* 〉. Information dissemination is shown in (a) under parameters *k* = 10, *θ* = 0.6, *p*_{1} = 0.3, *p*_{2} = 0.4, and *p*_{3} = 0.6 over ER networks (main panel) and small-world networks (inset), and in (b) under parameters *k* = 10, *θ* = 0.8, *p*_{1} = 0.3, *p*_{2} = 0.8, and *p*_{3} = 0.6 over ER networks (main panel) and small-world networks (inset). Curves are numerical solutions and symbols are averages from simulations with error bars indicating the standard deviations.

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