Michael Zargham

Optimal Resource Allocation for Network Protection Against Spreading Processes

We study the problem of containing spreading processes in arbitrary directed networks by distributing protection resources throughout the nodes of the network. We consider that two types of protection resources are available: 1) preventive resources able to defend nodes against the spreading (such as vaccines in a viral infection process) and 2) corrective resources able to neutralize the spreading after it has reached a node (such as antidotes). We assume that both preventive and corrective resources have an associated cost and study the problem of finding the cost-optimal distribution of resources throughout the nodes of the network. We analyze these questions in the context of viral spreading processes in directed networks. We study the following two problems: 1) given a fixed budget, find the optimal allocation of preventive and corrective resources in the network to achieve the highest level of containment and 2) when a budget is not specified, find the minimum budget required to control the spreading process. We show that both the resource allocation problems can be solved in polynomial time using geometric programming (GP) for arbitrary directed graphs of nonidentical nodes and a wide class of cost functions. We illustrate our approach by designing optimal protection strategies to contain an epidemic outbreak that propagates through an air transportation network.

Optimal vaccine allocation to control epidemic outbreaks in arbitrary networks

We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary contact network by distributing vaccination resources throughout the network. We analyze a networked version of the Susceptible-Infected-Susceptible (SIS) epidemic model when individuals in the network present different levels of susceptibility to the epidemic. In this context, controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the optimal distribution of vaccines throughout the network to control the spread of an epidemic outbreak. We propose a convex framework to find cost-optimal distribution of vaccination resources when different levels of vaccination are allowed.We illustrate our approaches with numerical simulations in a real social network.

A distributed newton method for network optimization

Most existing work uses dual decomposition and subgradient methods to solve network optimization problems in a distributed manner, which suffer from slow convergence rate properties. This paper proposes an alternative distributed approach based on a Newton-type method for solving minimum cost network optimization problems. The key component of the method is to represent the dual Newton direction as the solution of a discrete Poisson equation involving the graph Laplacian. This representation enables using an iterative consensus-based local averaging scheme (with an additional input term) to compute the Newton direction based only on local information. We show that even when the iterative schemes used for computing the Newton direction and the stepsize in our method are truncated, the resulting iterates converge superlinearly within an explicitly characterized error neighborhood. Simulation results illustrate the significant performance gains of this method relative to subgradient methods based on dual decomposition.

Accelerated dual descent for network optimization

Dual descent methods are commonly used to solve network optimization problems because their implementation can be distributed through the network. However, their convergence rates are typically very slow. This paper introduces a family of dual descent algorithms that use approximate Newton directions to accelerate the convergence rate of conventional dual descent. These approximate directions can be computed using local information exchanges thereby retaining the benefits of distributed implementations. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits superlinear convergence within a neighborhood of the optimal value. Numerical analysis corroborates that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods.

Accelerated Dual Descent for Network Flow Optimization

We present a fast distributed solution to the convex network flow optimization problem. Our approach uses a family of dual descent algorithms that approximate the Newton direction to achieve faster convergence rates than existing distributed methods. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits super-linear convergence within a neighborhood of the optimal value. Numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods. A connection with recent developments that use consensus to compute approximate Newton directions is also presented.

Traffic optimization to control epidemic outbreaks in metapopulation models

We propose a novel framework to study viral spreading processes in metapopulation models. Large subpopulations (i.e., cities) are connected via metalinks (i.e., roads) according to a metagraph structure (i.e., the traffic infrastructure). The problem of containing the propagation of an epidemic outbreak in a metapopulation model by controlling the traffic between subpopulations is considered. Controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. Based on this spectral condition, we propose a convex optimization framework to find cost-optimal approaches to traffic control in epidemic outbreaks.

Accelerated backpressure algorithm

We develop an Accelerated Back Pressure (ABP) algorithm using Accelerated Dual Descent (ADD), a distributed approximate Newton-like algorithm that only uses local information. Our construction is based on writing the backpressure algorithm as the solution to a network feasibility problem solved via stochastic dual subgradient descent. We apply stochastic ADD in place of the stochastic gradient descent algorithm. We prove that the ABP algorithm guarantees stable queues. Our numerical experiments demonstrate a significant improvement in convergence rate, especially when the packet arrival statistics vary over time.

A distributed line search for network optimization

Dual descent methods are used to solve network optimization problems because descent directions can be computed in a distributed manner using information available either locally or at neighboring nodes. However, choosing a stepsize in the descent direction remains a challenge because its computation requires global information. This work presents an algorithm based on a local version of the Armijo rule that allows for the computation of a stepsize using only local and neighborhood information. We show that when our distributed line search algorithm is applied with a descent direction computed according to the Accelerated Dual Descent method [18], key properties of standard backtracking line search using the Armijo rule are recovered. We use simulations to demonstrate that our algorithm is a practical substitute for its centralized counterpart.

Network optimization under uncertainty

Network optimization problems are often solved by dual gradient descent algorithms which can be implemented in a distributed manner but are known to have slow convergence rates. The accelerated dual descent (ADD) method improves this convergence rate by distributed computation of approximate Newton steps. This paper shows that a stochastic version of ADD can be used to solve network optimization problems with uncertainty in the constraints as is typical of communication networks. We prove almost sure convergence to an error neighborhood when the mean square error of the uncertainty is bounded and give a more restrictive sufficient condition for exact almost sure convergence to the optimal point. Numerical experiments show that stochastic ADD converges in two orders of magnitude fewer iterations than stochastic gradient descent.

On dual convergence of the distributed Newton method for Network Utility Maximization

The existing distributed algorithms for Network Utility Maximization (NUM) problems mostly rely on dual decomposition and first-order (gradient or subgradient) methods, which suffer from slow rate of convergence. Recent works [17] and [18] proposed an alternative distributed Newton-type second-order algorithm for solving NUM problems with self-concordant utility functions. This algorithm is implemented in the primal space and involves for each primal iteration computing the dual variables using a finitely terminated iterative scheme obtained through novel matrix splitting techniques. These works presented a convergence rate analysis for the primal iterations and showed that if the error level in the Newton direction (resulting from finite termination of dual iterations) is below a certain threshold, then the algorithm achieves local quadratic convergence rate to an error neighborhood of the optimal solution. This paper builds on these works and presents a convergence rate analysis for the dual iterations that enables us to explicitly compute at each primal iteration the number of dual steps that can satisfy the error level. This yields for the first time a fully distributed second order method for NUM problems with local quadratic convergence guarantee. Simulation results demonstrate significant convergence rate improvement of our algorithm, even when only one dual update is implemented per primal iteration, relative to the existing first-order methods based on dual decomposition.