Robust Graph Topologies for Networked Systems Waseem Abbas ∗ Magnus Egerstedt ∗ ∗ Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mails: wabbas@gatech.edu, magnus@gatech.edu). Abstract: Robustness of networked systems against noise corruption and structural changes in an underlying network topology is a critical issue for a reliable performance. In this paper, we investigate this issue of robustness in networked systems both from structural and functional viewpoints. Structural robustness deals with the effect of changes in a graph structure due to link or edge failures, while functional robustness addresses how well a system behaves in the presence of noise. We discuss that both of these aspects are inter-related, and can be measured through a common graph invariant. A graph process is introduced where edges are added to an existing graph in a step-wise manner to maximize robustness. Moreover, a relationship between the symmetry of an underlying network structure and robustness is also discussed. Keywords: Networked systems, Graph theoretic models, Robustness, Linear consensus. 1. INTRODUCTION Robustness in networked systems can be studied from two different perspectives. Firstly, how well a system behaves in the presence of noise, i.e. robustness against noise or functional robustness, and secondly what is the effect of change in network topology (due to edge or node failures) on the performance of such systems, i.e., structural robustness. Both of these aspects have been studied in the literature and various indices have been proposed to measure them. Edge (vertex) connectivity, alge- braic connectivity as introduced in Fiedler (1973), betweenness discussed in Freeman (1977), information centrality, toughness and other spectral measures (see Wu et al. (2011)) are some of the parameters that have been used to quantify structural robustness in graph structures. Robustness of networks where agents implement consensus protocols in the presence of noise has been addressed by providing various distributed algorithms and schemes to minimize corruption of noise in such systems. Examples include Xiao et al. (2007), Wang et al. (2009) and Young et al. (2010). Most of the studies on structural robustness and robustness against noise seem to be independent of each other, focusing either one of the aspects. Here, we show that both of these robustness viewpoints are in fact, related to each other and therefore, can be measured simultaneously by a same parameter. A network of agents can be modelled by an undirected graph where vertices represent agents and edges are the information exchange links among agents. Recently, Young et al. (2010) and Young et al. (2011) has shown that functional robustness of systems, where agents update their states by a linear con- sensus protocol in the presence of additive white noise, can be measured by a so called Kirchhoff index of a graph. On the other hand, Ellens et al. (2011) has shown that the effect of edge failures on the overall connectivity of a graph can be quantified by an effective graph resistance, which is equivalent to the Kirchhoff index of a graph (as shown in Klein and Randi´ c (93)). Thus, both aspects of robustness can be specified by an exactly same graph invariant. Klein and Randi´ c (93) introduced the Kirchhoff index of a graph through the notion of effective graph resistance. An electrical network can be obtained from a graph by replacing each edge with a unit resistance. The total electrical resistance between any two nodes in such a network is the effective resistance between the corresponding vertices of a graph. The sum of effective resistance between any two vertices is the Kirchhoff index, K f , or the effective resistance of a graph (see Ellens et al. (2011)). In this paper, we further explore this relationship between struc- tural robustness and functional robustness (robustness due to noise) in multiagent systems. The paper proposes to unify these two notions of robustness through the concept of Kirchhoff in- dex of the underlying network topology. We also investigate the role of various network topologies on the robustness property of these systems. In particular, Kirchhoff indices of some special families of graphs are computed, and these calculations are used to obtain a greedy algorithm for adding edges in a graph to maximize its robustness. Moreover, a relationship between the symmetry of a network structure and its robustness is also discussed. 2. ROBUSTNESS ISSUES IN NETWORKED SYSTEMS Agents exchange information with each other locally in dis- tributed systems. This exchange of information is possible through an interconnection network of agents that can be mod- elled by a graph structure. For example, agents agree on a com- mon value (that may be a sensor measurement) by implement- ing a linear consensus protocol. In fact, connectivity of the un- derlying graph structure is a necessary requirement for the con- sensus protocol to work (see Mesbahi and Egerstedt (2010) as an example). Moreover, the structure of the underlying network affects various properties of a system including convergence rates, connectivity of the network under edge (interconnection among agents) or vertex (agent) failures. A highly connected network is obviously less affected by an edge or vertex failures and is therefore, more robust to these deletions. Thus, the struc- ture of the interconnection infrastructure plays a key role when understanding the effects of edge or vertex failures.