arXiv subjects

Fern Y. Hunt

Publications and source records attributed to Fern Y. Hunt.

3 recordsLinked to original sources

arXiv2018

Using First Hitting Times to Find Sets that Maximize the Convergence Rate to Consensus

In a model of communication in a social network described by a simple consensus model, we pose the problem of finding a subset of nodes with given cardinality and fixed consensus values that enable the fastest convergence rate to equilibrium of the values of the remaining nodes. Given a network topology and a subset, called the stubborn nodes, the equilibrium exists and is a convex sum of the initial values of the stubborn nodes. The value at a non-stubborn node converges to its consensus value exponentially with a rate constant determined by the expected first hitting time of a random walker starting at the node and ending at the first stubborn node it visits. In this paper, we will use the sum of the expected first hitting times to the stubborn nodes as an objective function for a minimization problem. Its solution is a set with the fastest convergence rate. We present a polynomial time method for obtaining approximate solutions of the optimization problem for fixed cardinality less than that of a reference vertex cover. Under the assumption that the transition matrix for the random walk is irreducible and reversible, we also obtain an upper bound for the expected first hitting time and therefore an upper bound on the rate of convergence to consensus, using results from the mixing theory of Markov chains

cs.DM↗

arXiv2016

Optimal Spread in Network Consensus Models

In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enable the fastest spread of information? The dynamics of spread is described by a process dual to the movement from informed to uninformed nodes. In this setting, an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$, of random walks that start at nodes outside the set. In this paper,the problem is reformulated so that the search for solutions is restricted to a class of optimal and "near" optimal subsets of the graph. We introduce a submodular, non-decreasing rank function $ρ$, that permits some comparison between the solution obtained by the classical greedy algorithm and one obtained by our methods. The supermodularity and non-increasing properties of $F$ are used to show that the rank of our solution is at least $(1-\frac{1}{e})$ times the rank of the optimal set. When the solution has a higher rank than the greedy solution this constant can be improved to $(1-\frac{1}{e})(1+χ)$ where $χ>0$ is determined a posteriori. The method requires the evaluation of $F$ for sets of some fixed cardinality $m$, where $m$ is much smaller than the cardinality of the optimal set. When $F$ has forward elemental curvature $κ$, we can provide a rough description of the trade-off between solution quality and computational effort $m$ in terms of $κ$.

cs.DM↗

arXiv2014

The Structure of Optimal and Near Optimal Target Sets in Consensus Models

We consider the problem of identifying a subset of nodes in a network that will enable the fastest spread of information in a decentralized environment.In a model of communication based on a random walk on an undirected graph, the optimal set over all sets of the same or smaller cardinality minimizes the sum of the mean first arrival times to the set by walkers starting at nodes outside the set. The problem originates from the study of the spread of information or consensus in a network and was introduced in this form by V.Borkar et al. in 2010. More generally, the work of A. Clark et al. in 2012 showed that estimating the fastest rate of convergence to consensus of so-called leader follower systems leads to a consideration of the same optimization problem. The set function $F$ to be minimized is supermodular and therefore the greedy algorithm is commonly used to construct optimal sets or their approximations. In this paper, the problem is reformulated so that the search for solutions is restricted to optimal and near optimal subsets of the graph. We prove sufficient conditions for the existence of a greedoid structure that contains feasible optimal and near optimal sets. It is therefore possible we conjecture, to search for optimal or near optimal sets by local moves in a stepwise manner to obtain near optimal sets that are better approximations than the factor $(1-1/e)$ degree of optimality guaranteed by the use of the greedy algorithm. A simple example illustrates aspects of the method.

cs.DM↗