Learning multifractal structure in large networks

Generating random graphs to model networks has a rich history. In this paper, we analyze and improve upon the multifractal network generator (MFNG) introduced by Palla et al. We provide a new result on the probability of subgraphs existing in graphs generated with MFNG. From this result it follows that we can quickly compute moments of an important set of graph properties, such as the expected number of edges, stars, and cliques. Specifically, we show how to compute these moments in time complexity independent of the size of the graph and the number of recursive levels in the generative model. We leverage this theory to a new method of moments algorithm for fitting large networks to MFNG. Empirically, this new approach effectively simulates properties of several social and information networks. In terms of matching subgraph counts, our method outperforms similar algorithms used with the Stochastic Kronecker Graph model. Furthermore, we present a fast approximation algorithm to generate graph instances following the multi- fractal structure. The approximation scheme is an improvement over previous methods, which ran in time complexity quadratic in the number of vertices. Combined, our method of moments and fast sampling scheme provide the first scalable framework for effectively modeling large networks with MFNG

Simplified Compute-and-Forward and Its Performance Analysis

The compute-and-forward (CMF) method has shown a great promise as an innovative approach to exploit interference toward achieving higher network throughput. The CMF was primarily introduced by means of information theory tools. While there have been some recent works discussing different aspects of efficient and practical implementation of CMF, there are still some issues that are not covered. In this paper, we first introduce a method to decrease the implementation complexity of the CMF method. We then evaluate the exact outage probability of our proposed simplified CMF scheme, and hereby provide an upper bound on the outage probability of the optimum CMF in all SNR values, and a close approximation of its outage probability in low SNR regimes. We also evaluate the effect of the channel estimation error (CEE) on the performance of both optimum and our proposed simplified CMF by simulations. Our simulation results indicate that the proposed method is more robust against CEE than the optimum CMF method for the examples considered.Comment: Submitted to IET Communications, 29 pages, 7 figures, 1 table, latex, The authors are with the Wireless Research Laboratory (WRL), Department of Electrical Engineering, Sharif University of Technology, Tehran, Ira

Metastability in a stochastic neural network modeled as a velocity jump Markov process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorprated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks

Sampling designs and robustness for the analysis of network data

This manuscript addresses three new practical methodologies for topics on Bayesian analysis regarding sampling designs and robustness on network data: / In the first part of this thesis we propose a general approach for comparing sampling designs. The approach is based on the concept of data compression from information theory. The criterion for comparing sampling designs is formulated so that the results prove to be robust with respect to some of the most widely used loss functions for point estimation and prediction. The rationale behind the proposed approach is to find sampling designs such that preserve the largest amount of information possible from the original data generating mechanism. The approach is inspired by the same principle as the reference prior, with the difference that, for the proposed approach, the argument of the optimization is the sampling design rather than the prior. The information contained in the data generating mechanism can be encoded in a distribution defined either in parameter’s space (posterior distribution) or in the space of observables (predictive distribution). The results obtained in this part enable us to relate statements about a feature of an observed subgraph and a feature of a full graph. It is proven that such statements can not be connected by invoking conditional statements only; it is necessary to specify a joint distribution for the random graph model and the sampling design for all values of fully and partially observed random network features. We use this rationale to formulate statements at the level of the sampling graph that help to make non-trivial statements about the full network. The joint distribution of the underlying network and the sampling mechanism enable the statistician to relate both type of conditional statements. Thus, for random network partially and fully observed features joint distribution is considered and useful statements for practitioners are provided. / The second general theme of this thesis is robustness on networks. A method for robustness on exchangeable random networks is developed. The approach is inspired by the concept of graphon approximation through a stochastic block model. An exchangeable model is assumed to infer a feature of a random networks with the objective to see how the quality of that inference gets degraded if the model is slightly modified. Decision theory methods are considered under model misspecification by quantifying stability of optimal actions to perturbations to the approximating model within a well defined neighborhood of model space. The approach is inspired by all recent developments across the context of robustness in recent research in the robust control, macroeconomics and financial mathematics literature. / In all topics, simulation analysis is complemented with comprehensive experimental studies, which show the benefits of our modeling and estimation methods