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Spectral Properties of the Grounded Laplacian Matrix with Applications to Consensus in the Presence of Stubborn Agents

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Spectral Properties of the Grounded Laplacian Matrix with Applications to Consensus in the Presence of Stubborn Agents Spectral Properties of the Grounded Laplacian Matrix with Applications to Consensus in the Presence of Stubborn Agents Mohammad Pirani and Shreyas Sundaram Abstract— We study linear consensus and opinion dynamics convergence rate [12], [14]–[16]. Similarly, one can consider in networks that contain stubborn agents. Previous work has the problem of leader selection problem in networks in the shown that the convergence rate of such dynamics is given presence of noise [17], where the main goal is to minimize by the smallest eigenvalue of the grounded Laplacian induced by the stubborn agents. Building on this, we define a notion the steady state error covariance of the followers [18]. of centrality for each node in the network based upon the In this paper, we continue the investigation of convergence smallest eigenvalue obtained by removing that node from the rate in continuous-time linear consensus dynamics with stub- network. We show that this centrality can deviate from other born agents. Specifically, given the key role of the grounded well known centralities. We then characterize certain properties Laplacian in this setting, we provide a characterization of of the smallest eigenvalue and corresponding eigenvector of the grounded Laplacian in terms of the graph structure and the certain spectral properties of this matrix, adding to the expected absorption time of a random walk on the graph. literature on such matrices [12]–[14], [19]. We define a natural centrality metric, termed grounding centrality, based I. INTRODUCTION on the smallest eigenvalue of the grounded Laplacian in- Collective behavior in networks of agents has been studied duced by each node. This captures the importance of the in a variety of communities including sociology, physics, node in the network via the rate of convergence it would biology, economics, computer science and engineering [1], induce in the consensus dynamics if chosen as a stubborn [2]. A topic that has received particular interest is that of agent. We show that this centrality metric can deviate from opinion dynamics and consensus in networks, where the other common centrality metrics such as closeness centrality, agents repeatedly update their opinions or states via inter- betweenness centrality and degree centrality. We then provide actions with their neighbors [3]–[5]. For certain classes of bounds on the smallest eigenvalue of the grounded Laplacian, interaction dynamics, various conditions have been provided along with certain properties of the associated eigenvector. on the network topology that guarantee convergence to a Finally, we compare the smallest eigenvalue to the maximum common state [6]–[9]. expected absorption time for a random walk on the under- Aside from providing conditions under which convergence lying graph. We provide bounds based on the deviation in occurs, the question of what value the agents converge to components of the eigenvector for the smallest eigenvalue. is also of importance. In particular, the ability of certain Our results on the spectrum of the grounded Laplacian are individual agents in the network to excessively influence the of independent interest, with applications to various settings final value can be viewed as both a benefit and a drawback, [13], [19]. depending on whether those agents are viewed as leaders The structure of the paper is as follows. In Section III or adversaries. The effect of individual agents’ initial values we introduce the mathematical formulation of the consensus on the final consensus value has been studied in [10] [11]. problem in the presence of stubborn agents, and show the When a subset of agents is fully stubborn (i.e., they refuse role of the grounded Laplacian matrix. We introduce the to update their value), it has been shown that under a certain notion of grounding centrality in Section IV. Section V in- class of linear update rules, the values of all other agents troduces some spectral properties of the grounded Laplacian asymptotically converge to a convex combination of the matrix for general graphs as well as some unique properties stubborn agent’s values [12]. for special graphs such as trees. In Section VI we make a Given the ability of individuals to influence linear opinion connection to absorbing random walks on Markov chains and dynamics by keeping their values constant, a natural metric to provide bounds on the smallest eigenvalue of the grounded consider is the speed at which the population converges to the Laplacian in terms of the spread of the components in the final value for a given set of stubborn agents or leaders. The associated eigenvector. Section VII concludes the paper. convergence rate is dictated by spectral properties of certain II. DEFINITIONS AND NOTATION matrices; in continuous-time dynamics, this is the grounded Laplacian matrix [13]. There are various recent works that We use G(V; E) to denote a weighted and undirected graph investigate the leader selection problem, where the goal is to where V is the set of vertices (or nodes) and E ⊆ V × select a set of leaders (or stubborn agents) to maximize the V is the set of edges. The neighbors of vertex vi 2 V in graph G are given by the set Ni = fvj 2 V j (vi; vj) 2 This material is based upon work supported by the Natural Sciences Eg. The weight associated with an edge connecting vertex and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering at the University of vi 2 V to vertex vj 2 V is denoted by wij. We take the Waterloo. E-mail: fmpirani, [email protected] weights to be nonnegative and symmetric (i.e., wij = wji). We define d = P w ; in unweighted graphs where agents, and that the rate of convergence is asymptotically i vj 2Ni ij each wij 2 f0; 1g, di simply denotes the degree of vertex vi. given by the smallest eigenvalue of L11 [12]. The matrix L11 We will be considering a nonempty subset of vertices S ⊂ V is called the grounded Laplacian formed by removing the to be stubborn, and assume without loss of generality that the rows and columns corresponding to the stubborn agents from stubborn agents are placed last in an ordering of the agents. L. In the rest of the paper, the grounded Laplacian formed We say a vertex vi 2 V nS is an α-vertex if Ni \S 6= ;, and by removing the rows and columns of L corresponding to say vi is a β-vertex otherwise. Vertices in V n S will also be the vertices in set S ⊂ V will be denoted by LgS, and Lgs called followers. if S = fvsg. When the set S is fixed and clear from the context, we will simply use the notation Lg to denote the III. BASIC MODELS grounded Laplacian. For any given set S, we denote the A. Consensus Model smallest eigenvalue of the grounded Laplacian matrix by Consider a network of agents described by the connected λ(LgS) or simply λ. and undirected graph G(V; E), representing the structure of Remark 1: When the graph G is connected, the grounded the system, and a set of differential equations describing the Laplacian matrix is a positive definite matrix and its inverse interactions between each pair. In the study of consensus and is a nonnegative matrix (i.e., a matrix whose elements are continuous-time opinion dynamics [7], each agent vi 2 V nonnegative) [20]. From the Perron-Frobenius (P-F) theorem, starts with an initial scalar state (or opinion) yi(t), which the eigenvector associated with the smallest eigenvalue of it continuously updates as a function of the states of its the grounded Laplacian can be chosen to be nonnegative neighbors. A commonly studied version of these dynamics (elementwise). Furthermore, when the stubborn agent is not involves a linear update rule of the form a cut vertex, the eigenvector associated with the smallest X eigenvalue can be chosen to have all elements positive. y_ (t) = w (y (t) − y (t)); (1) i ij j i There have been various recent investigations of graph vj 2Ni properties that impact the convergence rate for a given set of where wij is the weight assigned by node vi to the state of its stubborn agents, leading to the development of algorithms neighbor vj. Aggregating the state of all of the nodes into to find approximately optimal sets of stubborn agents to T the vector Y (t) = y1(t) y2(t) ··· yn(t) , equation maximize the convergence rate [12], [14], [16]. Motivated (1) can be written as by the fact that the smallest eigenvalue of the grounded Laplacian dominates the convergence rate, in the next section Y_ = −LY; (2) we define a notion of grounding centrality for each node where L is the weighted graph Laplacian defined as L = in the network, corresponding to how quickly that node D − A with degree matrix D = diag(d1; d2; : : : ; dn) and would lead the rest of the network to steady state if chosen weighted adjacency matrix A containing the weight wij in to be stubborn. We show via an example that grounding entry (i; j). centrality can deviate from other common centrality metrics For an undirected graph G with symmetric weights (i.e., in networks, and then in subsequent sections, we provide wij = wji for all i 6= j), L is a symmetric matrix with real bounds on the grounding centrality based on graph theoretic eigenvalues that can be ordered sequentially as 0 = λ1 ≤ properties. λ ≤ · · · ≤ λ ≤ 2∆ where ∆ = max P w . When 2 n i vj 2Ni ij the graph G is connected, the second smallest eigenvalue λ2 IV. GROUNDING CENTRALITY is positive and all nodes asymptotically reach consensus on cT Y (0) There are various metrics for evaluating the importance of the value yeq = T where c is the left eigenvector of L c 1 individual nodes in a network.
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