# On the Smallest Eigenvalue of

Grounded Laplacian Matrices

###### Abstract

We provide bounds on the smallest eigenvalue of grounded Laplacian matrices (which are obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between our upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set of rows and columns is removed from the Laplacian, and the probability of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches . We also show that for random -regular graphs with a single row and column removed, the smallest eigenvalue is . Our bounds have applications to the study of the convergence rate in consensus dynamics with stubborn or leader nodes.

## I Introduction

There has been a great deal of research over the past several decades dedicated to the study of the structure and dynamics of networks. These investigations span multiple disciplines and include combinatorial, probabilistic, game-theoretic, and algebraic perspectives [1, 2, 3, 4]. It has been recognized that the spectra of graphs (i.e., the eigenstructure associated with certain matrix representations of the network) provide insights into both the topological properties of the underlying network and dynamical processes occurring on the network [5, 6]. The eigenvalues and eigenvectors of the Laplacian matrix of the graph, for example, contain information about the connectivity and community structure of the network [7, 8, 9, 10], and dictate the convergence properties of certain diffusion dynamics [11].

A variant on the Laplacian that has attracted attention in recent years is the grounded Laplacian matrix, obtained by removing certain rows and columns from the Laplacian. The grounded Laplacian forms the basis for the classical Matrix Tree Theorem (characterizing the number of spanning trees in the graph), and also plays a fundamental role in the study of continuous-time diffusion dynamics where the states of some of the nodes in the network are fixed at certain values. The eigenvalues of the grounded Laplacian characterize the variance in the equilibrium values for noisy instances of such dynamics, and determine the rate of convergence to steady state [12, 13]. Optimization algorithms have been developed to select “leader nodes” in the network in order to minimize the steady-state variance or to maximize the rate of convergence [13, 14, 15, 16], and various works have studied the effects of the location of such leaders in distributed control and consensus dynamics [17, 18, 19, 20, 21].

In this paper, we provide a characterization of the smallest eigenvalue of grounded Laplacian matrices. Specifically, we provide graph-theoretic bounds on the smallest eigenvalue based on the number of edges leaving the grounded nodes, bottlenecks in the graph, and properties of the eigenvector associated with the eigenvalue. Our bounds become tighter as this eigenvector becomes more uniform; we provide graph properties under which this occurs. As a consequence of our analysis, we obtain the smallest eigenvalue of the grounded Laplacian matrix for Erdos-Renyi random graphs and random regular graphs.

## Ii Background and Notation

We use to denote an undirected graph where is the set of nodes (or vertices) and is the set of edges. We will denote the number of vertices by . The neighbors of node in graph are given by the set . The degree of node is , and the minimum and maximum degrees of the nodes in the graph will be denoted by and , respectively. If , the graph is said to be -regular. For a given set of nodes , the edge-boundary (or just boundary) of the set is given by . The isoperimetric constant of is given by [5]

Choosing to be the vertex with the smallest degree yields the bound .

### Ii-a Laplacian and Grounded Laplacian Matrices

The adjacency matrix for the graph is a matrix , where entry is if , and zero otherwise. The Laplacian matrix for the graph is given by , where is the degree matrix with . For an undirected graph , the Laplacian is a symmetric matrix with real eigenvalues that can be ordered sequentially as . The second smallest eigenvalue is termed the algebraic connectivity of the graph and satisfies the bound [5]

(1) |

We will designate a nonempty subset of vertices to be grounded nodes, and assume without loss of generality that they are placed last in the ordering of the nodes. We use to denote the number of grounded nodes that node is connected to (i.e., ). Removing the rows and columns of corresponding to the grounded nodes produces a grounded Laplacian matrix (also known as a Dirichlet Laplacian matrix) denoted by . When the set is fixed and clear from the context, we will simply use to denote the grounded Laplacian. For any given set , we denote the smallest eigenvalue of the grounded Laplacian by or simply .

When the graph is connected, the grounded Laplacian matrix is a positive definite matrix and all of the elements in its inverse are nonnegative [22]. From the Perron-Frobenius (P-F) theorem [23], the eigenvector associated with the smallest eigenvalue of the grounded Laplacian can be chosen to be nonnegative (elementwise). Furthermore, when the grounded nodes do not form a vertex cut, the eigenvector associated with the smallest eigenvalue is unique (up to normalization) can be chosen to have all elements positive.

### Ii-B Applications in Consensus with Stubborn Agents

Consider a multi-agent system described by the connected and undirected graph representing the structure of the system, and a set of equations describing the interactions between each pair of agents. In the study of consensus and opinion dynamics [11], each agent starts with an initial scalar state (or opinion) , which evolves over time as a function of the states of its neighbors. A commonly studied version of these dynamics involves a continuous-time linear update rule of the form

Aggregating the state of all of the nodes into the vector , the above equation produces the system-wide dynamical equation

(2) |

where is the graph Laplacian. When the graph is connected, the trajectory of the above dynamical system satisfies (i.e., all agents reach consensus on the average of the initial values), and the asymptotic convergence rate is given by [11].

Now suppose that there is a subset of agents whose opinions are kept constant throughout time, i.e., , such that . Such agents are known as stubborn agents or leaders (depending on the context) [13, 20]. In this case the dynamics (2) can be written in the matrix form

where and are the states of the non-stubborn and stubborn agents, respectively. Since the stubborn agents keep their values constant, the matrices and are zero. Thus, the matrix is the grounded Laplacian for the system, i.e., . It can be shown that the state of each follower asymptotically converges to a convex combination of the values of the stubborn agents, and that the rate of convergence is asymptotically given by , the smallest eigenvalue of the grounded Laplacian [13].

Similarly, one can consider discrete-time consensus dynamics (also known as DeGroot dynamics) with a set of stubborn nodes, given by the update equation , where is the state vector for the non-stubborn nodes at time-step , and is an nonnegative matrix given by , with constant [24]. Once again, each non-stubborn node will converge asymptotically to a convex combination of the stubborn nodes’ states. The largest eigenvalue of is given by , and determines the asymptotic rate of convergence. Thus, our bounds on the smallest eigenvalue of the grounded Laplacian will readily translate to bounds on the largest eigenvalue of .

There have been various recent investigations of graph properties that impact the convergence rate for a given set of stubborn agents, leading to the development of algorithms to find approximately optimal sets of stubborn/leader agents to maximize the convergence rate [13, 20, 19]. The bounds provided in this paper contribute to the understanding of consensus dynamics with fixed opinions by providing bounds on the convergence rate induced by any given set of stubborn or leader agents.

## Iii Bounds on the Smallest Eigenvalue of

The following theorem provides our core bounds on the smallest eigenvalue of the grounded Laplacian; in subsequent sections, we will characterize graphs where these bounds become tight.

###### Theorem 1

Consider a graph with a set of grounded nodes . Let denote the smallest eigenvalue of the grounded Laplacian and let be a corresponding nonnegative eigenvector, normalized so that the largest component is . Then

(3) |

where is the smallest eigenvector component in .

###### Proof:

From the Rayleigh quotient inequality [23], we have

for all with . Let be the subset of vertices for which is minimum, and assume without loss of generality that the vertices are arranged so that those in set come first in the ordering. The upper bound is then obtained by choosing , and noting that the sum of all elements in the top block of is equal to the sum of the number of neighbors each vertex in has outside (i.e., ). The upper bound readily follows by choosing the subset .

For the lower bound, we left-multiply the eigenvector equation by the vector consisting of all ’s to obtain

where is the number of grounded nodes in node ’s neighborhood. Using the fact that the eigenvector is nonnegative, this gives

Since , the lower bound is obtained. \qed

###### Remark 1

For the case that we have

where is the degree of the grounded node. Note that the smallest eigenvalue of the grounded Laplacian for a set of grounded nodes is always upper bounded by (since ), with equality if and only if all grounded nodes connect to all other nodes (it is easy to see that the smallest eigenvector component in this case).

###### Example 1

Consider the graph shown in Figure 1 consisting of two complete graphs on nodes, joined by a single edge. Suppose the black node in the figure is chosen as the grounded node. In this case, we have , and the extreme upper bound in (3) indicates that . Now, if we take to be the set of all nodes in the left clique, we have and , leading to by the intermediate upper bound in (3). for large