Showing papers by "Steve Kirkland published in 2023"

A comparison of centrality measures and their role in controlling the spread in epidemic networks

Abstract: The ranking of nodes in a network according to their ``importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The current COVID-19 pandemic has recently rejuvenated the interest in this problem, as it is related to the selection of which individuals should be tested in a population of asymptomatic individuals, or which individuals should be vaccinated first. Motivated by the COVID-19 spreading dynamics, in this paper we review the most popular methods for node ranking in undirected unweighted graphs, and compare their performance in a benchmark realistic network, that takes into account the community-based structure of society. Also, we generalize a classic benchmark network originally proposed by Newman for ranking nodes in unweighted graphs, to show how ranks change in the weighted case.

Numerical ranges of cyclic shift matrices

Abstract: We study the numerical range of an $n\times n$ cyclic shift matrix, which can be viewed as the adjacency matrix of a directed cycle with $n$ weighted arcs. In particular, we consider the change in the numerical range if the weights are rearranged or perturbed. In addition to obtaining some general results on the problem, a permutation of the given weights is identified such that the corresponding matrix yields the largest numerical range (in terms of set inclusion), for $n \le 6$. We conjecture that the maximizing pattern extends to general $n\times n$ cylic shift matrices. For $n \le 5$, we also determine permutations such that the corresponding cyclic shift matrix yields the smallest numerical range.

Edge Addition and the Change in Kemeny's Constant

Abstract: Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures the average travel time for a random walk to reach a randomly selected vertex. It is known that when an edge is added to $G$, the value of Kemeny's constant may either decrease, increase, or stay the same. In this paper, we present a quantitative analysis of this behaviour when the initial graph is a tree with $n$ vertices. We prove that when an edge is added into a tree on $n$ vertices, the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$ while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We also identify the trees, and the edges to be added, that correspond to the maximum increase and maximum decrease. Throughout, both matrix theoretic and graph theoretic techniques are employed.

Powers of Karpelevic arcs and their Sparsest Realising matrices

Abstract: The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint arcs called the Karpelevic arcs. We provide a complete characterization of the Karpelevic arcs that are powers of some other Karpelevic arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevic arc of order n to be a power of another stochastic matrix.

Lead, trash, DDE, and young age of breeders linked to lower fertility in the first two decades of reintroduction for critically endangered California Condors in California

Abstract: In the first comprehensive assessment of the reproductive rates of critically endangered California Condors (Gymnogyps californianus) recovering from complete extirpation in the wild, we analyzed 20 years (1999–2018) of data from condor flocks in southern and central California. We found that several anthropogenic threats affected reproductive rates: (1) coastal space use by female condors was associated with lower hatch probability, presumably due to foraging on marine mammals and associated DDE exposure; (2) trash ingestion by chicks decreased fledging probability prior to implementation of trash management in 2007; and (3) all parent deaths during rearing resulted in chick or early fledgling deaths, and most parental deaths were due to lead poisoning. We also detected several effects on reproductive rates of the complex individual-based management of condors, which involves ongoing releases of captive-bred individuals and health interventions including treatment of lead poisoning. Recruitment rates were lower for new release sites, which we attribute to a lack of individual- and flock-level experience. In addition, the number of free-flying days in the wild in the year before first breeding and in the 8 weeks before subsequent breeding was positively associated with female and male recruitment and with female rebreeding probabilities, respectively, indicating that removing individuals from the wild may reduce their breeding success. Finally, probabilities of recruitment, rebreeding, and fledging all increased with age, and given the age distribution skew of the recovering flocks towards younger individuals, overall reproductive success was lower than would be expected at the stable age distribution. Thus, reproductive rates should increase over time as the mean age of California Condors increases if current and emerging threats to reproduction, including the loss of breeders due to lead poisoning, can be addressed.