Maximum Likelihood Associative Memories

Associative memories are structures that store data in such a way that it can later be retrieved given only a part of its content -- a sort-of error/erasure-resilience property. They are used in applications ranging from caches and memory management in CPUs to database engines. In this work we study associative memories built on the maximum likelihood principle. We derive minimum residual error rates when the data stored comes from a uniform binary source. Second, we determine the minimum amount of memory required to store the same data. Finally, we bound the computational complexity for message retrieval. We then compare these bounds with two existing associative memory architectures: the celebrated Hopfield neural networks and a neural network architecture introduced more recently by Gripon and Berrou

Reconstructing a Graph from Path Traces

This paper considers the problem of inferring the structure of a network from indirect observations. Each observation (a "trace") is the unordered set of nodes which are activated along a path through the network. Since a trace does not convey information about the order of nodes within the path, there are many feasible orders for each trace observed, and thus the problem of inferring the network from traces is, in general, illposed. We propose and analyze an algorithm which inserts edges by ordering each trace into a path according to which pairs of nodes in the path co-occur most frequently in the observations. When all traces involve exactly 3 nodes, we derive necessary and sufficient conditions for the reconstruction algorithm to exactly recover the graph. Finally, for a family of random graphs, we present expressions for reconstruction error probabilities (false discoveries and missed detections)

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