Algebraic Connectivity Control in Distributed Networks by Using Multiple Communication Channels

Abstract

The effectiveness of collaboration in distributed networks, such as sensor networks and multi-agent systems, relies on nodes’ ability to exchange information. The availability of various communication protocols with different technical properties opens the possibility to guarantee connectivity during a system’s operation in any condition. A communication network can be represented by a graph on which connectivity can be expressed by a well-known algebraic connectivity value or Fiedler value. It is one of the most important tools used in many applications where connectivity preservation is required. In this paper, a trust-based consensus algorithm for algebraic connectivity estimation has been implemented. To guarantee the accomplishment of the global objective and the system’s performance, our contributions include: (i) a novel decentralized framework for combining multiple communication channels in a resulting channel and (ii) a decentralized algebraic connectivity control law that dynamically changes the number of agents in the system during operation. The proposed algebraic connectivity control strategy has been evaluated in simulations and in a real multi-robot system using two channels with different properties and initial topologies.

Social Graphs and Their Applications to Robotics

“…In addition, in distributed networks, the effectiveness of collaboration like networks and multi-agent systems is based on vertices' capability to exchange information. The availability of different communication protocols with diverse technical characteristics opens the possibility to guarantee connectivity during a system's operation in any condition [116].…”

“…It is one of the most substantial tools employed in several applications that require maintaining connectivity. This value is determined by the graph topology and the parameters of a graph, for instance, the number of vertices n, number of edges m, minimal degree 𝑑 , etc [116]. Also, for many common graph topologies, such as cycle or cube graphs, and complete graphs, the algebraic connectivity is known and can be defined by the number of vertices n in the graph, such as, the algebraic connectivity for a cycle graph: 𝜆 = 2(1 − 𝑐𝑜𝑠 ), whilst for a complete graph 𝜆 = 𝑛 [116].…”

“…This value is determined by the graph topology and the parameters of a graph, for instance, the number of vertices n, number of edges m, minimal degree 𝑑 , etc [116]. Also, for many common graph topologies, such as cycle or cube graphs, and complete graphs, the algebraic connectivity is known and can be defined by the number of vertices n in the graph, such as, the algebraic connectivity for a cycle graph: 𝜆 = 2(1 − 𝑐𝑜𝑠 ), whilst for a complete graph 𝜆 = 𝑛 [116]. However, the graph topology can be changed by allowing processes on the graph, which may involve the addition or removal of vertices and edges.…”

“…However, the graph topology can be changed by allowing processes on the graph, which may involve the addition or removal of vertices and edges. Thus, in an incomplete graph, the maximal value of the algebraic connectivity is upper bounded by the graph parameters (i.e., the upper-bound of for 𝜆 an incomplete graph with n vertices is defined by 𝜆 ≤ 𝑛 -2, whereas the bound related to the minimal degree is indicated by the following inequality: 2𝑑 − 𝑛 + 2 ≤ 𝜆 ≤ 𝑑 ) [116]. Furthermore, second smallest eigenvalue is usually used to catch connectivity of dynamic networks.…”

“…Here the tasks can represent objectives such as tracking a target, collisions avoidance and reaching a desired position [163] [164]. Maintaining connectivity during the operation of an MRS can be executed utilising two approaches: (1) maintaining the local connectivity approach, and (2) maintaining the global connectivity approach [116], the last one is considered in this thesis. The global connectivity approach depended on mathematical formalism; in this, the communication network of an MRS is represented by utilising graph theory [116].…”