In recent years, with the progress of micro sensors and embedded systems, it is desired to provide services which components are associated with each other. We call the system composed of elements which can autonomously perform decision-making as Multi-Agent System (MAS). We call each constituent element in MAS as Agent, and the behavior of the entire system is given by the activity that networked agents are influenced by each other. From the perspective of Control Engineering, it has been discussed the way of the cooperative control in MAS[1]. Above all, the consensus problem which argues convergence of every agentﾕs state value is one of the well known problem[2].In order to introduce communication network, we have to consider the security. In other words, we must maintain services normally even if there are attacks for the network. In this paper, the term ``communication" refers to getting state values from neighbors. Attacks to the network can be classified as either deception attacks or Denial-of-Service (DoS) attacks. The former affect the reliability of data by manipulating the packets transmitted over the links. See references[3],[4],[5]. In this paper, we apply the latter attack pattern, DoS attack. In general, DoS attack means that the web server becomes unavailable by excess load[6]. While in this paper, DoS attacks affect by links wholly, or respectively, and no information can be transmitted through links affected by attacks.There are many previous works about the consensus problem under DoS attack. For linear systems, many results are revealed byet al.,[7],[8],[9],[10],[11],[12]. Specifically, we focus on[10]. In this literature, the class of DoS attacks are given by the assumptions about duration, and frequency. At each sampling time, if communication is not possible by the attack, the control input is given by zero. Thus, to solve the consensus problem, systems have to persist communication.A significance result in the prior research is the formulation of DoS attacks and the system's evolution. However, we consider the class of attacks can be more relaxed by the other protocol deciding sampling times. Specifically, we don't assume the frequency of attacks.To sum up, our basic question is, how to formulate system protocols under DoS attacks which is not necessarily satisfied with the assumption about frequency. For that purpose, we introduce stochastic protocols deciding sampling times.

We consider the sampling times as random variables. That is, every sampling time is selected uniform randomly by the intervals which have the same length. In regards to the attack patterns, we consider three cases: the network-based attack, the edge-based attack, and the attack changing attack strategy. We prove that, for every attack cases we consider, consensus can be achieved using sampling times which are determined as above. For the network-based attack, Numerical examples are given as follows.

図1: The graph structure

図2: The evolution of states under DoS attacks

We investigate the consensus problem under DoS attack which is not necessarily satisfied with the assumption about frequency. As a result, for any attacks we assumed, we can propose the protocols which archive the consensus. However, we don't consider the performance of protocols, to be more specific, the convergence time of the system. If the time cost can be estimated, we can consider the attack trying to delay the convergence time. Moreover, we can consider the edge-based attack trying to manipulate converged state value. These extension are issues in the future.