2.1. Equivalence in node-based and node-independent sampling

Throughout this paper, we have omitted the case where $\alpha_i+\beta_i=0$ or $\alpha_i+\beta_i = 2m_i$ , except for a specific case covered under Theorem 1. The case where $\alpha_i+\beta_i=0$ is when both values are zero, which leads to the reinforcement matrix

\begin{align*}\bigg(\begin{array}{c@{\quad}c}0 & m_i \\ m_i & 0\end{array}\bigg).\end{align*}

It is worth noting that preferential sampling with this reinforcement matrix is equivalent to de-preferential sampling with Pólya type reinforcement. However, as discussed later, this reinforcement scheme may not always lead to a deterministic limit. In this paper our focus is to analyse the cases where ${\mathbf{Z} }^t$ converges to a deterministic limit, so we do not address these specific cases.

More generally, for any node, linear sampling with parameter $q_i$ and reinforcement with $R_i$ is equivalent to uniform sampling with reinforcement using the matrix

\begin{equation*} \bigg( \begin{array} {l@{\quad}l} \alpha_i q_i + (m_i-\beta_i)(1-q_i) & (m_i-\alpha_i)q_i + \beta_i(1-q_i) \\ \alpha_i (1-q_i) + (m_i-\beta_i)q_i & (m_i-\alpha_i)(1-q_i) + \beta_i q_i \end{array}\bigg).\end{equation*}

Such node-dependent reinforcement models, where each node uses its own reinforcement scheme, have not been studied before. Despite equivalence through this coupling, we study the processes by separating node-based sampling and node-based reinforcement for clarity and application purposes. This distinction is important for extending existing models of de-preferential sampling (see [Reference Bandyopadhyay and Kaur4]) to interacting urns and for future exploration of non-linear sampling schemes. The non-linear sampling has been studied before in [Reference Crimaldi, Louis and Minelli7], but it is limited to complete graphs with sampling dependent on all the urns and a non-linear function of the proportion of balls of white colour in each urn. Our approach naturally extends this to linear node-based sampling on more general graphs, and we aim to explore non-linear node-based sampling in future work.

2.2. Exploration process on the graph

Suppose $q_i \in \{0, 1\}$ for all $i \in [N]$ and $V = {\mathcal{P} } \cup {\mathcal{D} }$ . We introduce an exploration process on the graph ${\mathcal{G} } =(V, {\mathcal{E} })$ that starts from an arbitrary node $v\in V$ and proceeds to explore its neighbours. In this process, nodes are categorised into subsets based on their sampling type and the types of nodes in their in-neighborhood. More specifically, ${\mathcal{P} }$ is partitioned into sets $P_1$ and $P_2$ , and ${\mathcal{D} }$ is partitioned into $D_1$ and $D_2$ , with $P_1$ , $P_2$ , $D_1$ , and $D_2$ initially empty. Depending on v’s sampling type, it is assigned to $P_1$ (if preferential) or $D_1$ (if de-preferential). In the subsequent steps, the sets $P_1, P_2, D_1, D_2$ are updated based on the sampling type of the newly explored node and their in-neighbours. If every node has a unique assignment, this results in a partition of V into these four disjoint subsets. The exploration process is illustrated in Figure 1. Detailed steps of the algorithm and examples are provided in the appendix (see Algorithm 1 in Appendix A). This approach thus classifies all finite directed graphs into two categories: graphs that admit partition via this exploration process and graphs that do not admit a partition.

Figure 1. The exploration process for the graph partition ${\mathcal{G} }(P_1, P_2,D_1, D_2)$ , as described in Steps 8 to 11 of Algorithm 1 (Appendix A). The arrows represent the directed edges where, for instance, an arrow from $D_1$ to $P_1$ means that there exist $u \in D_1$ and $v \in P_1$ such that $u \to v$ in ${\mathcal{G}}$ .

In Section 3, we state and prove the convergence and synchronisation results for ${\mathbf{Z} }^t$ . In particular, we show that when all $q_i \in \{0,1\}$ the limiting behaviour of the interacting urn process depends on whether the underlying graph admits a partition or not. In Section 4, we prove CLT type limit theorems for ${\mathbf{Z} }^t$ . Finally, in Section 5, we discuss some examples with simulations and applications in opinion dynamics.

3.1. Conditions for synchronisation

We now explore the conditions for synchronisation, i.e. when the limiting fraction of balls of each colour is the same for every urn. Synchronisation occurs if and only if ${\mathbf{Z} }_F^\star = z^\star\mathbf{1}$ for some constant $z^\star$ , therefore from (5) we get

(6) \begin{equation} z^\star \big(\mathbf{1} - ({\mathcal{I} } B{\widetilde{A}})_F \mathbf{1}\big) = \big[{\mathbf{Z} }_S^0 ({\mathcal{I} } B {\widetilde{A}})_{SF} + ({\mathbf{a}} {\widetilde{A}})_F - ({\mathbf{q}} B{\widetilde{A}})_F\big].\end{equation}

This equality holds if each element of the vectors on both sides matches, i.e., for every $i\in F$ ,

\begin{align*}z^\star \Bigg(1 - \frac{1}{{\overline{m}}_i}\sum_{j \in F\cap N_i}(2q_j-1)r_j \Bigg)= \frac{1}{{\overline{m}}_i} \Bigg(\sum_{j \in S\cap N_i} Z_j^0 (2q_j-1) r_j + \sum_{j \in N_i} \alpha_j - q_j r_j\Bigg),\end{align*}

where $r_j = \alpha_j+\beta_j-m_j$ (which is also an eigenvalue of $R_j$ ). Therefore, the following are sufficient conditions for synchronisation.

Condition SC1. There exists a constant $\mu_F$ such that, for all $i \in F$ ,

\begin{align*}\frac{1}{{\overline{m}}_i} \sum_{j \in F\cap N_i} (2q_j-1)r_j =\mu_F.\end{align*}

Condition SC2. There exists a constant $\mu_0$ such that, for all $i \in F$ ,

\begin{align*}\frac{1}{{\overline{m}}_i}\Bigg(\sum_{j\in S\cap N_i}Z_j^0(2q_j-1)r_j+\alpha_j-q_j r_j\Bigg)=\mu_0.\end{align*}

These conditions ensure that different components of the vector in (6) are constant, leading to synchronisation within the framework of Theorem 1. Note that $\mu_F=1$ occurs only when $\alpha_j = \beta_j = m_j$ and $q_j=1$ for all j, i.e. when all nodes are preferential and of Pólya type – a case not considered fully in this paper but discussed briefly in Section 5.

Another way to understand synchronisation conditions is as follows. Let

\begin{align*}f_i({\mathbf{Z} }^t) = \frac{1}{{\overline{m}}_i} {\mathbb{E} }\big[W_i^{t+1} - W_i^t\mid {\mathcal{F} }_t\big]\end{align*}

be the average proportion of white balls added to urn i at time $t+1$ given ${\mathcal{F} }_t$ . Then, using (1) and (4), we find

\begin{align*} f_i({\mathbf{Z} }^t) & = \frac{1}{{\overline{m}}_i} {\mathbb{E} }\big[W_i^{t+1} - W_i^t\mid {\mathcal{F} }_t\big] \\ & = \frac{1}{{\overline{m}}_i} \sum_{j \in N_i} \alpha_j \big(q_j Z_j^t + (1-q_j)\big(1-Z_j^t\big)\big) + (m_j - \beta_j)\big(q_j \big(1-Z_j^t\big) + (1-q_j)Z_j\big) \\ & = \frac{1}{{\overline{m}}_i} \sum_{j \in N_i} Z_j^t (2q_j -1) r_j + \alpha_j - q_j r_j.\end{align*}

We can decompose $f_i$ into $f_i = f_i^\textrm{(fixed)} + f_i^\textrm{(random)}$ , where

\begin{align*} f_i^\textrm{(fixed)} & = \frac{1}{{\overline{m}}_i} \sum_{j \in N_i \cap S} Z_j^0 (2q_j -1) r_j + \alpha_j - q_j r_j, \\ f_i^\textrm{(random)} & = \frac{1}{{\overline{m}}_i} \sum_{j \in N_i \cap F} Z_j^t (2q_j -1) r_j.\end{align*}

The synchronisation occurs when the fixed part is the same for all i and the random part changes with the same rate in the direction $(1,1,\ldots, 1)$ , which is given by

\begin{align*}\big\langle \mathbf{1}, \nabla f_i^\textrm{(random)}({\mathbf{Z} }^t)\big\rangle =\frac{1}{{\overline{m}}_i} \sum_{j \in N_i \cap F} (2q_j -1) r_j.\end{align*}

Corollary 2. (Synchronisation.) Suppose the conditions of Theorem 1 hold. Then, under the synchronisation Conditions SC1 and SC2,

\begin{align*}Z_i^t \xrightarrow{\mathrm{a.s.}} z^\star = \frac{\mu_0}{1- \mu_F}\end{align*}

as $t \to \infty$ for every $i\in F$ .

Condition SSC1. There exist $\alpha^F,\alpha^S,\beta^F,\beta^S,m^F,m^S\in{\mathbb{Z}}_{\geq 0}$ with $\alpha^F+\beta^F < 2m^F+m^S$ such that, for every $i\in F$ , $\sum_{j \in N_i\cap S} m_j = m^S$ , $\sum_{j \in N_i\cap S} \beta_j = \beta^S$ , $\sum\nolimits_{j \in N_i\cap S} \alpha_j =\alpha^S$ , and

\begin{align*}\sum\limits_{j \in N_i\cap F} R_j = \begin{pmatrix} \alpha^F & \quad m^F -\alpha^F \\ m^F-\beta^F &\quad \beta^F \end{pmatrix}.\end{align*}

Condition SSC2. If $S\neq {\varnothing}$ , there exist $\alpha^{0,S},\beta^{0,S},m^{0,S} \in {\mathbb{Z}}_{\geq 0}$ such that, for every $i\in F$ ,

\begin{align*}\sum\limits_{j \in N_i\cap S} Z_j^0 R_j = \begin{pmatrix} \alpha^{0, S} & \quad m^{0, S} - \alpha^{0, S} \\ m^{0, S} -\beta^{0, S} & \quad \beta^{0, S} \end{pmatrix}.\end{align*}

Then:

1. (i) When there are no de-preferential nodes in the graph then, as $t \to\infty$ , for all $i\in F$ ,

   \begin{align*}Z_i^t \xrightarrow{\mathrm{a.s.}} z^\star = \frac{m^F+m^S-\beta^F-\beta^S-\big(m^{0, S} -\alpha^{0, S}-\beta^{0, S}\big)}{2m^F+m^S-\alpha^F-\beta^F}.\end{align*}
   In particular, if $S={\varnothing}$ and synchronisation Condition SSC1 holds then, for every $i \in V$ , as $t \to \infty$ ,
   \begin{align*}Z_i^t \xrightarrow{\mathrm{a.s.}} \frac{m^F-\beta^F}{2m^F-\alpha^F-\beta^F}.\end{align*}

2. (ii) When there are no preferential nodes in the graph,

   \begin{align*}Z_i^t \xrightarrow{\mathrm{a.s.}} z^\star = \frac{\alpha^F+\alpha^S + m^{0, S} -\alpha^{0, S}-\beta^{0, S}}{m^S+\alpha^F+\beta^F}\end{align*}
   as $t \to \infty$ for every $i\in F$ . In particular, if $S={\varnothing}$ , the fraction of white balls asymptotically synchronises to $c \in [0, 1]$ if, for all $i \in [N]$ , $(1-c) \sum_{j \in N_i} \alpha_j = c \sum_{j \in N_i} \beta_j$ .

Note that in both cases, when $S={\varnothing}$ the urns synchronise to $\frac12$ provided that $\alpha^F=\beta^F$ , i.e. for every $i \in [N]$ , $\sum_{j \in N_i} R_j$ is a classical Friedman type replacement matrix.

4.1. Proofs of fluctuation results

Proof of Theorem 2. From (8), $h({\mathbf{Z}}_F) = -{\mathbf{Z}}_F[I-{\mathbf{W}}_F] + {\mathbf{Z}}_S^0{\mathbf{W}}_{SF} + ({\mathbf{a}}{\widetilde{A}})_F - ({\mathbf{q}}B{\widetilde{A}})_F$ . Thus, ${\partial h(z)}/{\partial z} = -I + {\mathbf{W}}_F$ . Hence, when $\rho > \frac12$ , we apply [Reference Zhang15, Theorem 2.2] and get $\sqrt{t}\big({\mathbf{Z}}^t_F-{\mathbf{Z}}_F^\star\big) \xrightarrow{\mathrm{d}} {\mathcal{N}}(\mathbf{0},\Sigma)$ , where $\Sigma$ is defined as

\begin{align*}\Sigma = \int_0^\infty\big(\exp\big\{{-}\big(I-\mathbf{W}_F-\tfrac{1}{2}I\big)u\big\}\big)^\top\Gamma \exp\big\{{-}\big(I-\mathbf{W}_F-\tfrac{1}{2}I\big)u\big\}\mathrm{d}u.\end{align*}

Similarly, when $\rho = \frac12$ with multiplicity 1, using [Reference Zhang15, Theorem 2.2] we get

\begin{align*}\sqrt{\frac{t}{\log(t)}}\big({\mathbf{Z}}^t_F-{\mathbf{Z}}_F^\star\big)\xrightarrow{\mathrm{d}} {\mathcal{N}}(\mathbf{0},\Sigma),\end{align*}

where $\Sigma$ is defined as

\begin{align*} \Sigma = \lim_{t\to\infty}\int_0^{\log(t)}\big(\exp\big\{{-}\big(I-\mathbf{W}_F-\tfrac{1}{2}I\big)u\big\}\big)^\top \Gamma\exp\big\{{-}\big(I-\mathbf{W}_F-\tfrac{1}{2}I\big)u\big\}\mathrm{d}u. \end{align*}

Here, $\Gamma = \lim_{t\to\infty}{\mathbb{E}}\big[\big((\Delta\chi^{t+1}B{\widetilde{A}})_F\big)^\top \big(\big(\Delta\chi^{t+1}B{\widetilde{A}}\big)\big)_F \mid {\mathcal{F}}_t\big]$ . To compute $\Gamma$ , we use

\begin{equation*} \lim_{t\to\infty}{\mathbb{E}}\big[\big(\Delta\chi^{t+1}B{\widetilde{A}}\big)^\top \big(\Delta\chi^{t+1}B{\widetilde{A}}\big) \mid {\mathcal{F}}_t\big] = (B{\widetilde{A}})^{^{\top}}{\mathbb{E}}\big[\Delta\big(\chi^{t+1}\big)^\top\Delta\chi^{t+1} \mid {\mathcal{F}}_t\big]B{\widetilde{A}}. \end{equation*}

From the variance expression obtained in (3) we get

\begin{align*} \lim_{t\to\infty}{\mathbb{E}}\big[\big(\Delta\chi^{t+1}B{\widetilde{A}}\big)^\top \big(\Delta\chi^{t+1}B{\widetilde{A}}\big) \mid {\mathcal{F}}_t\big] & = \lim_{t\to\infty}(B{\widetilde{A}})^\top\mathop{\mathrm{Var}}\big(\Delta\chi^{t+1} \mid {\mathcal{F}}_t\big)B{\widetilde{A}} \\ & = ({\mathcal{I}}B{\widetilde{A}})^\top(-\Theta)({\mathcal{I}}B{\widetilde{A}}) + \frac{1}{4}{\widetilde{A}}^{^{\top}} B^2{\widetilde{A}} \\ & = -{\mathbf{W}}^\top\Theta{\mathbf{W}} + \frac{1}{4}{\widetilde{A}}^{^{\top}} B^2{\widetilde{A}}. \end{align*}

Thus, $\Gamma = \big({-}{\mathbf{W}}^\top\Theta{\mathbf{W}} + \frac{1}{4}{\widetilde{A}}^{^{\top}} B^2{\widetilde{A}}\big)_F$ . This completes the proof.

Proof of Corollary 4. Consider the following two cases:

1. (i) When ${\mathbf{q}} = \frac{1}{2}\mathbf{1}$ , we get ${\mathcal{I}} = {\mathbf{W}} = \mathbf{0}_{N\times N}$ , thus $\rho =1$ and $\Gamma = \frac{1}{4}({\widetilde{A}}^{^{\top}} B^2{\widetilde{A}})_F$ . Hence, $\sqrt{t}\big({\mathbf{Z}}^t_F - {\mathbf{Z}}_F^\star\big) \xrightarrow{\mathrm{d}} {\mathcal{N}}(\mathbf{0},\Sigma)$ , where $\Sigma = \Gamma\int_0^\infty\mathrm{e}^{-u}\,\mathrm{d}u = \Gamma = \frac{1}{4}({\widetilde{A}}^{^{\top}} B^2{\widetilde{A}})_F$ .

2. (ii) When $q_i \neq \frac12$ for all i, since ${\mathcal{I} }$ is invertible we can write $\Gamma = -{\mathbf{W}}^\top\Theta{\mathbf{W}} + \frac{1}{4}{\mathbf{W}}^\top{\mathcal{I}}^{-2}{\mathbf{W}} = {\mathbf{W}}^\top\bar\Theta{\mathbf{W}}$ , where $\bar\Theta = -\Theta + \frac{1}{4}{\mathcal{I} }^{-2}$ . Assuming the decomposition for ${\mathbf{W}}$ , we have ${\mathbf{W}} = U\Lambda V$ with $V = U^{-1}$ . Therefore,

   \begin{align*} \Gamma = [{\mathbf{W}}^\top\bar\Theta{\mathbf{W}}]_F & = ({\mathbf{W}}_F)^\top\bar\Theta_F{\mathbf{W}}_F + ({\mathbf{W}}_{SF})^\top\bar\Theta_S{\mathbf{W}}_{SF} \\ & = V_F^\top\Lambda_F\big[U_F^\top\bar\Theta_F U_F + U_{SF}^\top\bar\Theta_S U_{SF}\big]\Lambda_F V_F \\ & = V_F^\top\Lambda_F(U^\top\bar\Theta U)_F\Lambda_F V_F. \end{align*}
   Let $\tilde\Sigma = \big(V_F^\top\big)^{-1} \Sigma V_F^{-1}$ . Then, for $\rho > \frac12$ ,
   \begin{equation*} \tilde\Sigma = \int_0^\infty\big(\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda_F\big)u\big\}\big)^\top\Lambda_F [U^\top\bar\Theta U]_F\Lambda_F\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda_F\big)u\big\}\,\mathrm{d}u. \end{equation*}
   For $i, j \in F$ ,
   \begin{align*} [\tilde\Sigma]_{ij} = \lambda_i\lambda_j\big[u_i^\top\bar\Theta u_j\big] \int_0^\infty\mathrm{e}^{-(1 - \lambda_i - \lambda_j)u}\,\mathrm{d}u = \frac{\lambda_i\lambda_j}{1-\lambda_i-\lambda_j}\big[u_i^\top \bar\Theta u_j\big]. \end{align*}
   Hence, $\Sigma = V_F^\top \tilde \Sigma V_F$ , where
   \begin{align*}[\Sigma]_{ij} = \sum_{k\in F}\sum_{\ell\in F}\frac{\lambda_k\lambda_\ell}{1-\lambda_k-\lambda_\ell} \big(u_k^\top\bar\Theta u_\ell\big)v_{ki}v_{lj}.\end{align*}

Now, for $\rho=\frac12$ , with $\lambda_{\max}({\mathbf{W}}_F)=\lambda_1=\frac12$ being simple, we have

\begin{equation*} \tilde\Sigma = \lim_{t\to\infty}\frac{1}{\log(t)}\int_0^{\log(t)} \big(\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda_F\big)u\big\}\big)^\top\Lambda_F \big[U^\top\Theta U\big]_F\Lambda_F\big(\exp\big\{{-}\big(\tfrac12 I-\Lambda_F\big)u\big\}\big)\,\mathrm{d}u. \end{equation*}

The (1,1) element is given by

\begin{align*} [\tilde\Sigma]_{11} & = \lambda_1\lambda_1\big(u_1^\top\Theta u_1\big)\lim_{t\to\infty}\frac{1}{\log(t)} \int_0^{\log(t)}\mathrm{e}^{-(1 - \lambda_1 - \lambda_1)u}\,\mathrm{d}u \\ & = \frac{1}{4}\big(u_1^\top\Theta u_1\big)\lim_{t\to\infty}\frac{1}{\log(t)} \int_0^{\log(t)}1\,\mathrm{d}u = \frac{1}{4}\big(u_1^\top\Theta u_1\big). \end{align*}

For every other $k,l \in F$ we have $\lambda_k+\lambda_l <1$ and thus

\begin{align*}\lim_{t\to\infty}\frac{1}{\log(t)}\int_0^{\log(t)}\mathrm{e}^{-(1 - \lambda_k - \lambda_l)u}\,\mathrm{d}u = 0.\end{align*}

Hence we get $[\Sigma]_{ij} = \frac{1}{4}\big(u_1^\top \Theta u_1 \big) v_{1i}v_{1j}$ .

Proof of Corollary 5. With the assumption ${\mathbf{W} }={\mathbf{W} }^\top$ , we get $S={\varnothing}$ and $U= V^\top$ . Thus, for $\rho>\frac12$ , we have

\begin{align*} \tilde\Sigma & = \int_0^\infty\big(\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda\big)u\big\}\big)^\top\Lambda [U^\top \bar\Theta U]\Lambda\big(\exp\big\{-\big(\tfrac{1}{2}I-\Lambda\big)u\big\}\big)\,\mathrm{d}u \\ & = c({\mathbf{q}},{\mathbf{Z}}^\star) \int_0^\infty\big(\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda\big)u\big\}\big)\Lambda^2 \big(\exp\big\{{-}\big(\tfrac{1}{2}I-\Lambda\big)u\big\}\big)\,\mathrm{d}u \\ & = c({\mathbf{q}},{\mathbf{Z}}^\star)\Lambda^2(I-2\Lambda)^{-1}. \end{align*}

This implies that $\Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star)V^\top\Lambda^2(I-2\Lambda)^{-1}V = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2(I-2{\mathbf{W}})^{-1}$ . Now, for part (ii), i.e. when $\rho=\frac12$ , we have

\begin{align*}\tilde\Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star)\Lambda^2\lim_{t\to\infty}\frac{1}{\log(t)} \int_0^{\log(t)}\mathrm{e}^{-(I-2\Lambda)u}\,\mathrm{d}u = c({\mathbf{q}},{\mathbf{Z}}^\star)\Lambda^2 \left( \begin{array} {c@{\quad}c}1 & \mathbf{0}\\ \mathbf{0} &\mathbf{0} \end{array}\right).\end{align*}

This implies that

\begin{align*} \Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2 V^\top \left(\begin{array} {c@{\quad}c}1 & \mathbf{0}\\ \mathbf{0} &\mathbf{0} \end{array}\right) V.\end{align*}

Thus, for $\rho>\frac12$ we get $\Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2(I-2{\mathbf{W}})^{-1}$ and for $\rho=\frac12$ we get

\begin{align*}\Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2 V^\top \left(\begin{array}{c@{\quad}c} 1 & \mathbf{0}\\ \mathbf{0} &\mathbf{0} \end{array}\right) V.\end{align*}

Further, under Condition SC1, $\sum_{i=1}^N[{\mathbf{W}}]_{ij} = \frac{1}{{\overline{m}}_j}\sum_{i\in N_j} {\mathcal{I}}_{i,i}(\alpha_i+\beta_i-m_i) = \mu_F= \frac{1}{2}$ is the maximal eigenvalue of ${\mathbf{W}}$ and the corresponding normalised eigenvector is $({1}/{\sqrt{N}}) \mathbf{1}$ . Hence we get

\begin{equation*} \Sigma = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2 V^\top \left( \begin{array} {c@{\quad}c} 1 & \mathbf{0}\\ \mathbf{0} &\mathbf{0} \end{array}\right)V = c({\mathbf{q}},{\mathbf{Z}}^\star){\mathbf{W}}^2\frac{1}{N}J = \frac{c({\mathbf{q}},{\mathbf{Z}}^\star)}{4N} J. \end{equation*}

This completes the proof.

Remark 6. When all nodes are preferential, under Condition SSC1,

\begin{align*}{\mathbf{Z}}^\star=\frac{m^F-\beta^F}{2m^F-\alpha^F-\beta^F}\end{align*}

(note that under the conditions of Corollary 5, $S={\varnothing}$ ). Thus, for $\rho>\frac12$ we get

\begin{align*}\Sigma = \frac{(m^F-\beta^F)(m^F-\alpha^F)}{(2m^F-\alpha^F-\beta^F)^2}{\mathbf{W}}^2 (I-2{\mathbf{W}})^{-1}\end{align*}

and for $\rho=\frac12$ we get

\begin{align*}\Sigma = \frac{\big(m^F-\beta^F\big)\big(m^F-\alpha^F\big)}{(2m^F-\alpha^F-\beta^F)^2} {\mathbf{W}}^2 V^\top \left(\begin{array} {c@{\quad}c} 1 & \mathbf{0}\\ \mathbf{0} &\mathbf{0} \end{array}\right)V.\end{align*}

Under Condition SSC1 with $S={\varnothing}$ , $\mu_F=({\alpha^F +\beta^F -m^F})/{m^F}$ . Thus, $({\alpha^F +\beta^F -m^F})/{m^F} = \frac{1}{2}$ is the maximal eigenvalue of ${\mathbf{W}}$ with the corresponding normalised eigenvector $({1}/{\sqrt{N}})\mathbf{1}$ . Hence, we get

\begin{align*}\Sigma = \frac{\big(m^F-\beta^F\big)\big(m^F-\alpha^F\big)}{N (m^F)^2} J.\end{align*}

Similarly, when all nodes are de-preferential we get

\begin{align*}\Sigma = \frac{\alpha^F \beta^F}{N\big(\alpha^F + \beta^F\big)^2}J = \frac{4\alpha^F \beta^F}{9N(m^F)^2}J.\end{align*}

5.1. Simulation results

In this section, we present the simulation results for a cycle graph with four nodes, where all the nodes are of Pólya type and $q_i\in \{0,1\}$ for all i. We explore three specific cases for this graph.

Consider first the case when all nodes are preferential except node 4 (see Figure 2), i.e. ${\mathcal{I}} = \mathop{\mathrm{Diag}}(1,1,1,-1)$ . We observe that this case satisfies condition Theorem 1(iii), as it does not have a valid graph partition. Thus by Theorem 1, ${\mathbf{Z}}^t$ has a deterministic limit $\frac12\mathbf{1}$ , which is independent of the initial vector ${\mathbf{Z}}^0$ . Figure 3 illustrates the convergence of $Z^t_1, \ldots, Z_4^t$ . Note that, in this case, the eigenvalues of the matrix $I-{\mathbf{W}}$ are

\begin{align*}1 + \frac{1}{\sqrt{2}\,} + \frac{\mathrm{i}}{\sqrt{2}\,}, \quad1 + \frac{1}{\sqrt{2}\,} - \frac{\mathrm{i}}{\sqrt{2}\,}, \quad1 - \frac{1}{\sqrt{2}\,} + \frac{\mathrm{i}}{\sqrt{2}\,}, \quad1 - \frac{1}{\sqrt{2}\,} - \frac{\mathrm{i}}{\sqrt{2}\,}.\end{align*}

Therefore, $\rho = 1-({1}/{\sqrt{2}}) <\frac12$ and $\Re(\lambda_{\max}) = {1}/{\sqrt{2}}$ ; thus, from (22) we get $\mathop{\mathrm{Var}}({\mathbf{Z}}^t) \preccurlyeq t^{\sqrt{2}-2}$ .

Figure 2. A graph with four nodes, with ${\mathcal{P}} =\{1,2,3\}$ , ${\mathcal{D}} = \{4\}$ .

Figure 3. Convergence of $Z^t_1, \ldots, Z_4^t$ in six different simulations. In this case, the limit is deterministic: 0.5 for all urns.

We now consider two examples of cycle graphs with four vertices where Theorem 1 does not apply. The first graph has all preferential nodes, i.e. ${\mathcal{I}} = \mathop{\mathrm{Diag}}(1,1,1,1)$ (see Figure 4(a)). The second graph has alternate preferential and de-preferential nodes, i.e. ${\mathcal{I}} = \mathop{\mathrm{Diag}}(1,1,-1,-1)$ (see Figure 4(b)). Since a valid graph partition exists according to Algorithm 1 (see Appendix A) for both cases, condition Theorem 1(iii) is not satisfied. Therefore, the urn configuration in these graphs does not converge to a deterministic limit.

Figure 4. Graphs that do not satisfy the conditions of Theorem 1.

The first case corresponds to a specific instance of Pólya type reinforcement at each node in a d-regular graph (where $d_i^{\mathrm{in}} = d_i^{\mathrm{out}} = d$ for all i) for $d=2$ , which was previously studied in [Reference Kaur and Sahasrabudhe10]. The authors showed that synchronisation occurs, i.e. there exists a random variable $Z^\infty$ such that ${\mathbf{Z}}_F^\star = Z^\infty \mathbf{1}$ (as illustrated through simulations in Figure 5).

Figure 5. Convergence of $Z^t_1, \ldots, Z_4^t$ in six different simulations.

The simulations in Figure 6 suggest that in the second case, the limit is of the form $(Z^\infty, 1-Z^\infty, 1-Z^\infty, Z^\infty)$ . This is consistent with Remark 2.

Figure 6. Convergence of $Z^t_1, \ldots, Z_4^t$ in six different simulations.

For a graph that can be partitioned using Algorithm 1 (Appendix A), the fraction of balls of either colour in each urn tends to a random limit. Specifically, from our simulations (see Figure 7), we conjecture that in a cycle graph with alternating preferential and de-preferential nodes, the limiting behaviour results in the fractions of balls of either colour in $P_1, D_2$ (or $P_2, D_1$ ) converging to the same limit. Further analysis of these cases, with a more general sampling scheme, is left as future work.

Figure 7. Histogram of $Z^t_i$ in 100 different simulations, at $t=100\,000$ , for the four interacting urns placed on the nodes of the graph as in Figure 4(b).

5.2. Application to opinion dynamics

Our model is motivated by the network-based opinion dynamics model discussed in [Reference Kaur and Sahasrabudhe10]. This model uses urns to represent opinions in a network, with white and black balls indicating positive and negative views, respectively. An individual’s opinion $O_i^t$ can be represented either as a fraction $Z_i^t$ , which is supported on [0,1], or as a sign, $\mathrm{Sign}\big(Z_i^t - \frac12\big) \in \{-1, 0, 1\}$ . In this model, stubborn nodes are treated as bots, with $Z^0_i$ being the bot’s power to influence towards the ‘positive/favourable’ opinion.

At each time step, every individual reveals their true opinion with probability $q_i$ and reinforces their opinion based on the type of reinforcement applied: Pólya type reinforcement reinforces only the revealed opinion, whereas non-Pólya type reinforcement adds a mix of both types of views. Our main results show that on a strongly connected network if there is at least one individual with $q_i \in (0, 1)$ , all individuals’ opinions converge to a deterministic limit. In the case when all $q_i \in \{0, 1\}$ , the existence of a deterministic limiting opinion depends on the reinforcement type as well as the graph structure. We also obtain conditions for asymptotic consensus.

We briefly discuss the implications of our results for the opinion dynamics model. Consider a cycle graph on four nodes with edges $i \to i+1$ for $1 \leq i \leq 3$ and $4 \to 1$ . Note that for directed cycles, ${\widetilde{A}}=A$ and therefore the $m_i$ do not contribute to the limiting opinion. Let $x_i = (2q_i-1)r^\prime_i$ , where $r^\prime_i=(a_i + b_i-1)$ . The limiting opinion of node 1 is given by

\begin{align*}Z_1^\star = \frac{1}{1-x_1 x_2 x_3 x_4} [a_4-q_4r^\prime_4+(a_1-q_1r^\prime_1)x_2x_3x_4 + (a_2-q_2r^\prime_2) x_3x_4+(a_3-q_3 r^\prime_3)x_4 ]. \end{align*}

Suppose, for $i \in [N]$ , that $a_i=a$ and $b_i=b$ . Then, $r_i^\prime = r^\prime$ (say) for all $1 \leq i \leq 4$ . Further assume that $q_1 = \frac12$ . We consider two cases:

Case I: When all the other nodes are preferential, i.e. $2, 3, 4 \in {\mathcal{P} }$ , we get

\begin{align*} Z_1^\star(\mathrm{I}) & = 1-b+\frac12(1+a-b)(r^\prime)^3+ (1-b) (r^\prime)^2+(1-b)r^\prime \\ & = \frac{a(r^\prime)^3}{2} + (1-b) \bigg(1+r^\prime+(r^\prime)^2 + \frac{(r^\prime)^3}{2}\bigg).\end{align*}

Case II: With $2, 3 \in {\mathcal{P} }$ and $4 \in {\mathcal{D} }$ , we get

\begin{align*}Z_1^\star(\mathrm{II})= a - \frac{a(r^\prime)^3}{2} - (1-b)\bigg(r^\prime+(r^\prime)^2 + \frac{(r^\prime)^3}{2}\bigg).\end{align*}

Here, $Z_1^\star(\mathrm{I})$ and $Z_1^\star(\mathrm{II})$ denote the limiting configuration of urn 1 in the two cases. Note that $Z_1^\star(\mathrm{II}) = Z_1^\star(\mathrm{I})$ when $r^\prime=0$ , and $Z_1^\star(\mathrm{II}) < Z_1^\star(\mathrm{I})$ when $r^\prime > 0$ . Now consider a bot (or a stubborn vertex s) attached to node 1, with $2, 3 \in {\mathcal{P} }$ and $4 \in {\mathcal{D} }$ (as shown in Figure 8).

Figure 8. A cycle graph with a stubborn node s attached.

In this case, the fraction of balls of white colour in urn 1 converges to $Z_1^\star(s)=Z_1(\mathrm{II}) + f\big(Z_s^0, r, {\mathbf{m}}\big)$ , where $f\big(Z_s^0, r, {\mathbf{m}}\big)>0$ for $r>0$ . Thus, a bot can be used to mitigate the effect of the de-preferential node attached to 1. Further, our results provide explicit expressions that can determine the optimal ‘strength’ (given by $Z_s^0$ and the reinforcement matrix) of the bot(s) required to obtain a specific limiting opinion profile. We remark that for a more complicated graph, the optimal positions of the bots (with varying strengths) on the network is an interesting problem in this context.