Cluster consensus in multi-agent networks with mutual information exchange

Abstract

The emergence of new technologies such as the Internet of things and the Cloud transforms the way we interact. Whether it be human to human interaction or human to machine interaction, the size of the networks keeps growing. As the networks get more complex nowadays with many interconnected components, it is necessary to develop distributed scalable algorithms so as to minimize the computation required in decision making in such large-scale systems. In this paper, we consider a setup where each agent in the network updates its opinion by relying on its neighbors’ opinions. The information exchange between the agents is assumed to be mutual. The cluster consensus problem is investigated for networks represented by static or time-varying graphs. Joint and integral connectivity conditions are utilized to determine the number of clusters that are formed, as the interactions among the agents evolve over time. Finally, some numerical examples are given to illustrate the theoretical results.

Appendix

Lemma 3

(Erkan and Akar 2016) Let ${W_{\text{1}}}, \ldots ,{W_j}$ be a finite set of row stochastic matrices that correspond to equivalent K connected graphs. Suppose that for each sequence ${W_{{i_{\text{1}}}}}, \ldots {\text{ }},{W_{{i_j}}}$ of positive length, the graph associated with the product is also equivalent K connected. Then, for each infinite sequence ${W_{{i_{\text{1}}}}}{\text{ }},{W_{{i_{\text{2}}}}}{\text{ }}, \ldots ,$ there exist vectors ${c_{\text{1}}}, \ldots ,{c_K}$ such that

$$\mathop {\lim }\limits_{{j \to \infty }} {W_i}_{{_{j}}}{W_i}_{{_{{j - 1}}}} \ldots {W_i}_{{_{1}}}={C_K}$$

(7)

where C _K is in the form

$${C_K} \triangleq {\text{diag}}\{ {{\mathbf{1}}_n}_{{_{1}}}{c_1}^{ \top }, \ldots ,{{\mathbf{1}}_n}_{{_{K}}}{c_K}^{ \top }\}$$

(8)

Lemma 4

(Erkan and Akar (2015) If the union of a set of undirected graphs $\{ {\mathcal{G}_{{t_1}}}, \ldots ,{\mathcal{G}_{{t_m}}}\}$ has K connected components and ${\mathcal{L}_{{t_j}}},\;j\,=\,1, \ldots ,m$ is the Laplacian matrix corresponding to each graph ${\mathcal{G}_{{t_j}}}$ in (3), then the matrix product satisfies

$$\mathop {{\text{lim}}}\limits_{{k \to \infty }} {({{\text{e}}^ - }^{{\mathcal{L}_{t_m}\Delta {t_m}}} \ldots {{\text{e}}^ - }^{{\mathcal{L}_{t_1}\Delta {t_1}}}~)^k}={C_K}$$

(9)

where C _K = ${\text{diag}}\{ {{\mathbf{1}}_n}_{{_{1}}}{c_1}^{ \top }, \ldots ,{{\mathbf{1}}_n}_{{_{K}}}{c_K}^{ \top }\}$and ∆t _j >0 are bounded.

Proof

Suppose that the union of the graphs ${\mathcal{G}_{{t_1}}}, \ldots ,{\mathcal{G}_{{t_m}}}$ has K connected components. Note that $- {\mathcal{L}_{{t_j}}}$, $j\,=\,{\text{1}}, \ldots ,m$ can be expressed as ${M_{{t_j}}} - {\eta _{{t_j}}}{I_n}$where ${\eta _{{t_j}}}$is the maximum value of diagonal entries of ${\mathcal{L}_{t_j}}$ and ${M_{{t_j}}}$is a matrix with non-negative elements. It can be written ${{\text{e}}^{ - \mathcal{L}{t_j}\Delta {t_j}}}={{\text{e}}^{{M_{{t_j}}}\Delta {t_j}}}$${{\text{e}}^{ - {\eta _{{t_j}}}\Delta {t_j}}}$, which implies that ${{\text{e}}^{ - {\mathcal{L}_t}_{{_{j}}}}}$ is a stochastic matrix with positive diagonal elements. Then following the proof of Lemma 6 in Erkan and Akar (2016), it can be shown that (9) holds.$\square$

Proof of Theorem 2

The set of all possible matrices ${{\text{e}}^{ - \mathcal{L}({t_i})}}$ can be chosen from the finite set $\bar {F}=\{ {{\text{e}}^{ - \mathcal{L}({t_i})}},{\tau _i} \in \bar {\tau }\}$. Suppose $[{t_{{i_j}+{l_j}}},{t_{{i_j}+1}})$ is uniformly bounded, i.e., there exists a constant B such that $\left| {{t_{{i_j}+1}} - {t_{{i_j}+{l_j}}}} \right|<B$ for all i, j. Consider the j-th time interval $[{t_{{i_j}}},{t_{{i_j}+1}})$ which is bounded due to fact that both $[{t_{{i_j}}},{t_{{i_j}+{l_j}}})$ and $[{t_{{i_j}+{l_j}}},{t_{{i_j}+1}})$are bounded. The union of the graphs across $[{t_{{i_j}}},{t_{{i_j}+1}}),$denoted as $\bar {G}({t_{{i_j}}}),$ has K connected components since the union of graphs across $[{t_{{i_j}}},{t_{{i_j}+{l_j}}})$ has K connected components. Let $\{ \mathcal{L}({t_{{i_j}}}),\mathcal{L}({t_{{i_j}+1}}), \ldots ,\mathcal{L}({t_{{i_{j+1}} - 1}})\}$ be the set of Laplacian matrices corresponding to each graph in the union $\bar {G}({t_{{i_j}}}).$

From Lemma 4, the matrix product ${{\text{e}}^{ - \mathcal{L}({t_{{i_{j+1}} - 1}})({\tau _{{i_{j+1}} - 1}})}} \ldots {{\text{e}}^{ - \mathcal{L}({t_{{i_{j+1}}}})({\tau _{{i_{j+1}}}})}}{{\text{e}}^{ - \mathcal{L}({t_{{i_j}}})({\tau _{{i_j}}})}},$ j ≥ 1 is K connected. Then by Lemma 3, we conclude that a network of agents utilizing consensus protocol (2) achieves K consensus equilibria.$\square$