The Impact of Edge Correlations in Random Networks

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

Working capacity of deep networks with topology attacks

Deep learning plays an important role in the development of artificial intelligence (AI) technology. The security of deep networks has become the crucial thing to be considered. When the deep learning algorithms are implemented in the hardware platform, the interference for topology structure will appear because of cyber-attacks. We analyze the working capacity of acyclic deep networks under the topology attacks and injection attacks. Considering the topology structure of the deep network, the maximum working capacity is studied under the topology attacks and injection attacks. Furthermore, the robustness of the random networks is researched and the structural robustness index (SRI) is proposed to measure the toleration for the topology attacks. This work supplies some suggestions for building a robust deep network and improving the endogenous safety and security (ESS) of the deep networks.

Decomposition of Random Sequences into Mixtures of Simpler Ones and Its Application in Network Analysis

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti’s Theorem. In its original form, it applies to infinite 0–1 valued sequences with the special property that the distribution is invariant to permutations (called an exchangeable sequence). Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more detail than usual, with the intent of making it also accessible to readers who may not be closely familiar with the subject.

Energy-Aware Controllability of Complex Networks

Understanding the fundamental principles and limitations of controlling complex networks is of paramount importance across natural, social, and engineering sciences. The classic notion of controllability does not capture the effort needed to control dynamical networks, and quantitative measures of controllability have been proposed to remedy this problem. This article presents an introductory overview of the practical (i.e., energy-related) aspects of controlling networks governed by linear dynamics. First, we introduce a class of energy-aware controllability metrics and discuss their properties. Then, we establish bounds on these metrics, which allow us to understand how the structure of the network impacts the control energy. Finally, we examine the problem of optimally selecting a set of control nodes so as to minimize the control effort, and compare the performance of some simple strategies to approximately solve this problem. Throughout the article, we include examples of structured and random networks to illustrate our results. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 5 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The Value of Price Discrimination in Large Social Networks

We study the value of price discrimination in large social networks. Recent trends in industry suggest that, increasingly, firms are using information about social network to offer personalized prices to individuals based upon their positions in the social network. In the presence of positive network externalities, firms aim to increase their profits by offering discounts to influential individuals that can stimulate consumption by other individuals at a higher price. However, the lack of transparency in discriminative pricing may reduce consumer satisfaction and create mistrust. Recent research focuses on the computation of optimal prices in deterministic networks under positive externalities. We want to answer the question of how valuable such discriminative pricing is. We find, surprisingly, that the value of such pricing policies (increase in profits resulting from price discrimination) in very large random networks are often not significant. Particularly, for Erdös–Renyi random networks, we provide the exact rates at which this value decays in the size of the networks for different ranges of network densities. Our results show that there is a nonnegligible value of price discrimination for a small class of moderate-sized Erdös–Renyi random networks. We also present a framework to obtain bounds on the value of price discrimination for random networks with general degree distributions and apply the framework to obtain bounds on the value of price discrimination in power-law networks. Our numerical experiments demonstrate our results and suggest that our results are robust to changes in the model of network externalities. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.