Migration in a Small World: A Network Approach to Modeling Immigration Processes

Existing theories of migration either focus on micro- or macroscopic behavior of populations; that is, either the average behavior of entire population is modeled directly, or decisions of individuals are modeled directly. In this work, we seek to bridge these two perspectives by modeling individual agents decisions to migrate while accounting for the social network structure that binds individuals into a population. Pecuniary considerations combined with the decisions of peers are the primary elements of the model, being the main driving forces of migration. People of the home country are modeled as nodes on a small-world network. A dichotomous state is associated with each node, indicating whether it emigrates to the destination country or it stays in the home country. We characterize the emigration rate in terms of the relative welfare and population of the home and destination countries. The time evolution and the steady-state fraction of emigrants are also derived

Degree Correlation in Scale-Free Graphs

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to $\beta$ existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution $p(\ell| k)$, we mean the degree distribution of nodes that are connected to a node of degree $k$. By the joint degree distribution $p(k,\ell)$, we mean the proportion of links that connect nodes of degrees $k$ and $\ell$. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution

Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variable and all other nodes receiving the broadcast update their state and companion variable. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.Comment: 30 pages, submitte

Efficient Distributed Online Prediction and Stochastic Optimization with Approximate Distributed Averaging

We study distributed methods for online prediction and stochastic optimization. Our approach is iterative: in each round nodes first perform local computations and then communicate in order to aggregate information and synchronize their decision variables. Synchronization is accomplished through the use of a distributed averaging protocol. When an exact distributed averaging protocol is used, it is known that the optimal regret bound of $\mathcal{O}(\sqrt{m})$ can be achieved using the distributed mini-batch algorithm of Dekel et al. (2012), where $m$ is the total number of samples processed across the network. We focus on methods using approximate distributed averaging protocols and show that the optimal regret bound can also be achieved in this setting. In particular, we propose a gossip-based optimization method which achieves the optimal regret bound. The amount of communication required depends on the network topology through the second largest eigenvalue of the transition matrix of a random walk on the network. In the setting of stochastic optimization, the proposed gossip-based approach achieves nearly-linear scaling: the optimization error is guaranteed to be no more than $\epsilon$ after $\mathcal{O}(\frac{1}{n \epsilon^2})$ rounds, each of which involves $\mathcal{O}(\log n)$ gossip iterations, when nodes communicate over a well-connected graph. This scaling law is also observed in numerical experiments on a cluster.Comment: 30 pages, 2 figure