Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the expected degree sequence, we show that with probability one, $\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of $F(\mathord{\cdot})$. We apply our results to well-known degree distributions, such as power-law and exponential. The asymptotic expressions of the spectral moments in each case provide significant insights about the bulk behavior of the eigenvalue spectrum

Detection and Isolation of Link Failures under the Agreement Protocol

In this paper a property of the multi-agent consensus dynamics that relates the failure of links in the network to jump discontinuities in the derivatives of the output responses of the nodes is derived and verified analytically. At the next step, an algorithm for sensor placement is proposed, which would enable the designer to detect and isolate any link failures across the network based on the observed jump discontinuities in the derivatives of the responses of a subset of nodes. These results are explained through elaborative examples.Comment: 6 pages, 3 figures, IEEE Conference on Decision and Control, 201

Learning And Decision Making In Groups

Many important real-world decision-making problems involve group interactions among individuals with purely informational interactions. Such situations arise for example in jury deliberations, expert committees, medical diagnoses, etc. We model the purely informational interactions of group members, where they receive private information and act based on that information while also observing other people\u27s beliefs or actions. In the first part of the thesis, we address the computations that a rational (Bayesian) decision-maker should undertake to realize her optimal actions, maximizing her expected utility given all available information at every decision epoch. We use an approach called iterated eliminations of infeasible signals (IEIS) to model the thinking process as well as the calculations of a Bayesian agent in a group decision scenario. Accordingly, as the Bayesian agent attempts to infer the true state of the world from her sequence of observations, she recursively refines her belief about the signals that other players could have observed and beliefs that they would have hold given the assumption that other players are also rational. We show that IEIS algorithm runs in exponential time; however, when the group structure is a partially ordered set the Bayesian calculations simplify and polynomial-time computation of the Bayesian recommendations is possible. We also analyze the computational complexity of the Bayesian belief formation in groups and show that it is NP-hard. We investigate the factors underlying this computational complexity and show how belief calculations simplify in special network structures or cases with strong inherent symmetries. We finally give insights about the statistical efficiency (optimality) of the beliefs and its relations to computational efficiency. In the second part, we propose the no-recall model of inference for heuristic decision-making that is rooted in the Bayes rule but avoids the complexities of rational inference in group interactions. Accordingly to this model, the group members behave rationally at the initiation of their interactions with each other; however, in the ensuing decision epochs, they rely on heuristics that replicate their experiences from the first stage and can be justified as optimal responses to simplified versions of their complex environments. We study the implications of the information structure, together with the properties of the probability distributions, which determine the structure of the so-called ``Bayesian heuristics\u27\u27 that the agents follow in this model. We also analyze the group decision outcomes in two classes of linear action updates and log-linear belief updates and show that many inefficiencies arise in group decisions as a result of repeated interactions between individuals, leading to overconfident beliefs as well as choice-shifts toward extreme actions. Nevertheless, balanced regular structures demonstrate a measure of efficiency in terms of aggregating the initial information of individuals. Finally, we extend this model to a case where agents are exposed to a stream of private data in addition to observing each other\u27s actions and analyze properties of learning and convergence under the no-recall framework

Long ties accelerate noisy threshold-based contagions

Network structure can affect when and how widely new ideas, products, and behaviors are adopted. In widely-used models of biological contagion, interventions that randomly rewire edges (generally making them "longer") accelerate spread. However, there are other models relevant to social contagion, such as those motivated by myopic best-response in games with strategic complements, in which an individual's behavior is described by a threshold number of adopting neighbors above which adoption occurs (i.e., complex contagions). Recent work has argued that highly clustered, rather than random, networks facilitate spread of these complex contagions. Here we show that minor modifications to this model, which make it more realistic, reverse this result: we allow very rare below-threshold adoption, i.e., rarely adoption occurs when there is only one adopting neighbor. To model the trade-off between long and short edges we consider networks that are the union of cycle-power-$k$ graphs and random graphs on $n$ nodes. Allowing adoptions below threshold to occur with order $1/\sqrt{n}$ probability along some "short" cycle edges is enough to ensure that random rewiring accelerates spread. Simulations illustrate the robustness of these results to other commonly-posited models for noisy best-response behavior. Hypothetical interventions that randomly rewire existing edges or add random edges (versus adding "short", triad-closing edges) in hundreds of empirical social networks reduce time to spread. This revised conclusion suggests that those wanting to increase spread should induce formation of long ties, rather than triad-closing ties. More generally, this highlights the importance of noise in game-theoretic analyses of behavior

Seeding with Costly Network Information

We study the task of selecting $k$ nodes in a social network of size $n$, to seed a diffusion with maximum expected spread size, under the independent cascade model with cascade probability $p$. Most of the previous work on this problem (known as influence maximization) focuses on efficient algorithms to approximate the optimal seed set with provable guarantees, given the knowledge of the entire network. However, in practice, obtaining full knowledge of the network is very costly. To address this gap, we first study the achievable guarantees using $o(n)$ influence samples. We provide an approximation algorithm with a tight (1-1/e){\mbox{OPT}}-\epsilon n guarantee, using $O_{\epsilon}(k^2\log n)$ influence samples and show that this dependence on $k$ is asymptotically optimal. We then propose a probing algorithm that queries ${O}_{\epsilon}(p n^2\log^4 n + \sqrt{k p} n^{1.5}\log^{5.5} n + k n\log^{3.5}{n})$ edges from the graph and use them to find a seed set with the same almost tight approximation guarantee. We also provide a matching (up to logarithmic factors) lower-bound on the required number of edges. To address the dependence of our probing algorithm on the independent cascade probability $p$, we show that it is impossible to maintain the same approximation guarantees by controlling the discrepancy between the probing and seeding cascade probabilities. Instead, we propose to down-sample the probed edges to match the seeding cascade probability, provided that it does not exceed that of probing. Finally, we test our algorithms on real world data to quantify the trade-off between the cost of obtaining more refined network information and the benefit of the added information for guiding improved seeding strategies