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This paper presents a novel approach for clustering, which is based on quasi-consensus of dynamical linear high-order multiagent systems. The graph topology is associated with a selected multiagent system, with each agent corresponding to one vertex. In order to reveal the cluster structure, the agents belonging to a similar cluster are expected to aggregate together. To establish the theoretical foundation, a necessary and sufficient condition is given to check the achievement of group consensus. Two numerical instances are furnished to illustrate the results of our approach.

With a rapid increase in the scale of massive data and information, the probing of potential knowledge in big data, such as data structures and certain unknown correlations, has been attracting more and more scholars. As an essential technique of data mining, clustering has been widely applied in various practical fields, such as linguistics [

Clustering is about assigning certain set of data points into different clusters, for the purpose of highlighting the similitude of data points being organized in the same cluster, while simultaneously reflecting the distinctions between different clusters [

In addition to the abovementioned literature, some clustering algorithms have also been presented which are built on the study of certain dynamical behaviors of networks, for example, the random walk algorithms and the network synchronization algorithms. Random walk implies shifting along a random route, with each step to a new nearest vertex. By random walk algorithms [

In this research article, a novel method for clustering is presented, which is rooted upon observation to the motions of dynamical multiagent systems. Concretely speaking, for the purpose of discerning the different affiliations of vertices in a given network, the selected dynamical multiagent system should be linked with the concerned network topology, and there should be a one-to-one correspondence between each pair of dynamical agent and vertex. The agents keep on moving inside a common space, either physical or abstract. Those corresponding to the vertices that are affiliated with any specific cluster should aggregate as time elapses. Finally, the clustering structure would be uncovered automatically, according to the formation of agent positions.

Analogical topics as discussed above are usually referred to as group consensus in the field of control theory. Yu and Wang earlier were concerned with this phenomenon in [

The novelty of the current work compared with the aforementioned results in the literature lies in several perspectives. As far as we know, in the existing researches, the expected clustering formation is prescribed in advance, paying primary attention to synthesis of specific information transfer protocol for each group, so that the agents assigned to the group could ultimately aggregate. In contrast, in the framework of the current paper, the cluster distribution should completely be determined by the topology of network itself, without the need of any prerequisite knowledge aforehand. In this regard, the advantage of the current work is explicit, especially in its potential practicability.

Besides introducing a novel method for clustering, a major contribution of this paper is proposing a criterion for checking whether or not a high-order LTI (Linear Time-Invariant) multiagent system can reach group consensus, which generalizes the existing well-known necessary and sufficient condition for consensus achievement [

In addition to the above, the current paper provides an exemplification of the usage of unstable dynamical systems, whereas, in contrast, unstable systems are conventionally regarded as being insignificant in control theory.

Theoretical studies on dynamical multiagent systems have already been extensive in the area of control theory, especially on the consensus problem. However, the application instances corresponding to these theories that can well support them are still scarce. Our exploration attempts to introduce a practical scenario from the field of data analysis, under a motivation to facilitate applying, verifying, or enriching certain relevant researches.

The remaining part of the current paper holds the following organization. The preliminaries and model formulation are introduced in Section

The topology of the network for clustering can be expressed by

For such a purpose, a procedure rooted in observing the motions of dynamical multiagent systems is put forward, to figure out the clustering formation automatically. The procedure is composed of the following two fundamental steps.

Define a dynamical multiagent system as follows and attach with it the network topology concerned, with each agent associated with one particular vertex.

Determine appropriate initial values

As time elapses, group consensus or quasi-consensus would manifest, as long as the overall setup of both the structure and parameters of dynamical multiagent system (

The definition below formulates the concept of group consensus discussed in this paper.

For the dynamical multiagent system (

The assumption that the graph is both undirected and connected is merely due to practical requirements of clustering and is not due to any technical limitations. This can be understood through later discussions.

In this section, the detail of the clustering process is elaborated by employing high-order LTI multiagent systems, which are described as

To this end, some previous results on consensus are reviewed first.

For the dynamical LTI high-order system (

The Laplacian matrix

The Laplacian matrix

For the dynamical multiagent system (

the graph topology

all the matrices

Lemma

Consider the dynamical system (

If the stack vector of the states of agents is defined as

According to (

Due to the structure of

Equation (

If

Theorem

For the sake of the clustering application, only the undirected graph should be concerned. However, Theorem

Matrix

Both the matrices

Theorem

This section will exemplify the presented technique of clustering by two simple and typical simulation instances.

Consider a graph in Figure

The weighted adjacency matrix is

Instance with group consensus. Default edge weight is 1.

Group consensus motions with

In the second example, consider the graph illustrated in Figure

The adjacency matrix is

Although not all agents could achieve a precise group consensus in this case, from Figure

Second graph where default edge weight is 1.

Quasi-consentaneous state trajectories of Example

Variation of difference between states of agents 2 and 3.

From Figure

Let us inspect the dynamical equation of agent 3; that is,

A novel method for clustering is introduced in the current paper, based on group consensus or quasi-consensus of the motions of dynamical systems. For the sake of categorizing the vertex set into disparate clusters, a specific dynamical multiagent system is attached to the graph addressed, with each vertex associated with an agent. If the dynamical system is appropriately configured, then the agents affiliated with the similar cluster should gradually assemble together in their common state space; meanwhile the different clusters diverge away mutually. In addition to the method of clustering, a primary contribution is the presentation of a criterion for checking whether or not a group consensus could be reached by high-order linear multiagent systems, which is a necessary and sufficient condition generalizing the previous well-known condition of consensus. In fact, the scope of possible utility of the condition for group consensus is broader, beyond the clustering scenario being dealt with here. Two typical simulation instances are exhibited to exemplify the process of clustering method. As to the potential future extensions, there are several explicit major directions: (

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by National Natural Science Foundation (NNSF) of China (Grants 61374054 and 61263002), by Fundamental Research Funds for the Central Universities (Grants 31920160003 and 31920170141), and by Program for Young Talents of State Ethnic Affairs Commission (SEAC) of China (Grant 2013-3-21).