Completion Problems and Sparsity for Kemeny's Constant

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny's constant. We prove that for any partially specified stochastic matrix for which the problem is well-defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny's constant in two special cases: when the diagonal has been specified, and when all specified entries lie in a common row

The Case of Equality in the Dobrushin–Deutsch–Zenger Bound

Suppose that A=(ai,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin–Deutsch–Zenger (DDZ) bound on the eigenvalues of A other than μ is given by Z(A) = 1 2 max 1_s,t_n n Xr=1 |as,r − at,r| . When A a transition matrix of a finite homogeneous Markov chain so that μ = 1, Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, max{|_| | _ 2 _(A) \ {1}} , of the iteration xTi = xT i−1A, to the stationary distribution vector of the chain. In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which for some stochastic matrix A

Nonnegative alternating circulants leading to M-matrix group inverses

AbstractLet C be the set of all an n × n nonnegative irreducible alternating circulant matrices. We characterize a subset C of C such that if B ∈ C and the Perron root of B is r, then the group inverse (rI − B)# of the singular and irreducible M-matrixrI − B is also an M-matrix. This is equivalent to the fact that for each such B, the Perron root at B is a concave function in each of the off-diagonal entries. The characterization for the case when n is odd presents more difficulties than for the case when n is even, so the two cases are treated separately

Bounds on the subdominant eigenvalue involving group inverses with applications to graphs

summary:Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are $\lambda _1 > \lambda _2 \ge \ldots \ge \lambda _n$. In this paper we derive several lower and upper bounds, in particular on $\lambda _2$ and $\lambda _n$, but also, indirectly, on $\mu = \max _{2\le i \le n} |\lambda _i|$. The bounds are in terms of the diagonal entries of the group generalized inverse, $Q^{\#}$, of the singular and irreducible M-matrix $Q=\lambda _1 I-A$. Our starting point is a spectral resolution for $Q^{\#}$. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now $Q$ becomes $L$, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of $L^{\#}$ and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of $L$, by exploiting the diagonal entries of $L^{\#}$

Cross-Sectional Collaboration in Florida\u27s Emergency Management System

Florida faces a unique set of emergency management hazards prompted by the state’s geography, high volume of tourism, and position as a hub of international trade. The state has developed a highly adaptive emergency management system to deliver humanitarian assistance to Floridians affected by natural and other disasters. Despite the importance of collaboration in delivering humanitarian goods and services to Floridians in times of crisis, little is known as to how collaboration occurs, what impediments exist, and how organizations adapt to the dynamics of natural and other disasters. In this qualitative case study, the integrative framework for collaborative governance was applied to understand how voluntary organizations collaborated during hurricane response and relief efforts. Data were collected from the survey responses of nine voluntary organization emergency managers and after-action reports of county, state, and federal agencies. Data from survey responses and archival sources were analyzed and thematically coded. The findings showed that voluntary organizational collaboration resulted from teamwork, communication, and working towards the same purpose within a structured organizational framework. The key recommendations are that emergency management organizations should consistently provide all-hazards training and exercises to enhance voluntary organizations’ response to disasters and to study how collaboration occurs in other states with different emergency management constructs. This study may contribute to positive social change by providing emergency managers with the means to improve humanitarian responses to disasters through a deeper understanding of the collaborative processes involved

Cross-Sectional Collaboration in Florida\u27s Emergency Management System

Florida faces a unique set of emergency management hazards prompted by the state’s geography, high volume of tourism, and position as a hub of international trade. The state has developed a highly adaptive emergency management system to deliver humanitarian assistance to Floridians affected by natural and other disasters. Despite the importance of collaboration in delivering humanitarian goods and services to Floridians in times of crisis, little is known as to how collaboration occurs, what impediments exist, and how organizations adapt to the dynamics of natural and other disasters. In this qualitative case study, the integrative framework for collaborative governance was applied to understand how voluntary organizations collaborated during hurricane response and relief efforts. Data were collected from the survey responses of nine voluntary organization emergency managers and after-action reports of county, state, and federal agencies. Data from survey responses and archival sources were analyzed and thematically coded. The findings showed that voluntary organizational collaboration resulted from teamwork, communication, and working towards the same purpose within a structured organizational framework. The key recommendations are that emergency management organizations should consistently provide all-hazards training and exercises to enhance voluntary organizations’ response to disasters and to study how collaboration occurs in other states with different emergency management constructs. This study may contribute to positive social change by providing emergency managers with the means to improve humanitarian responses to disasters through a deeper understanding of the collaborative processes involved