@conference {
	title = {Algebraic-Graphical Approach for Signed Dynamical Networks},
	author = {Shi, Guodong., Altafini, Claudio., Baras, John S.},
	year = {2017},
	month = {12/2017},
	abstract = {A signed network is a network with each link associated with a positive or negative sign. Models for nodes interacting over such signed networks, where two types of interactions are defined along the positive and negative links, respectively, arise from various biological, social, political, and economical systems. Starting from standard consensus dynamics, there are two basic types of negative interactions along negative links, namely state flipping or relative-state flipping. In this paper, we provide an algebraic-graphical method serving as a systematic tool of studying these dynamics over signed networks. Utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions, we show this method is useful to establish a series of basic convergence results for dynamics over signed networks.
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	address = {Melbourne, Australia},
	booktitle = {56th IEEE Conference on Decision and Control (CDC 2017)},
	pages = {pp. 2009-2014}
}
@article {
	title = {Dynamics over Signed Networks},
	author = {Shi, Guodong., Altafini, Claudio., Baras, John S.},
	year = {2018},
	month = {06/2018},
	abstract = {A signed network is a network with each link associated with a positive or negative sign. Models for nodes interacting over such signed networks, where two di erent types of interactions take place along the positive and negative links, respectively, arise from various biological, social, political, and economic systems. As modi cations to the conventional DeGroot dynamics for positive links, two basic types of negative interactions along negative links, namely the opposing rule and the repelling rule, have been proposed and studied in the literature. This paper reviews a few fundamental convergence results for such dynamics over deterministic or random signed networks under a uni ed algebraic-graphical method. We show that a systematic tool of studying node state evolution over signed networks can be obtained utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions.},
	journal = {SIAM}
}
@article {
	title = {The Evolution of Beliefs over Signed Social Networks},
	author = {Shi, Guodong., Proutiere, Alexandre., Johansson, Mikael., Baras, John S., Johansson, Karl Henrik.},
	year = {2016},
	month = {10/2016},
	abstract = {We study the evolution of opinions (or beliefs) over a social network modeled as a signed graph. The sign attached to an edge in this graph characterizes whether the corresponding individuals or end nodes are friends (positive links) or enemies (negative links). Pairs of nodes are randomly selected to interact over time, and when two nodes interact, each of them updates its opinion based on the opinion of the other node and the sign of the corresponding link. This model generalizes the DeGroot model to account for negative links: when two adversaries interact, their opinions go in opposite directions. We provide conditions for convergence and divergence in expectation, in mean-square, and in almost sure sense and exhibit phase transition phenomena for these notions of convergence depending on the parameters of the opinion update model and on the structure of the underlying graph. We establish a no-survivor theorem, stating that the difference in opinions of any two nodes diverges whenever opinions in the network diverge as a whole. We also prove a live-or-die lemma, indicating that almost surely, the opinions either converge to an agreement or diverge. Finally, we extend our analysis to cases where opinions have hard lower and upper limits. In these cases, we study when and how opinions may become asymptotically clustered to the belief boundaries and highlight the crucial influence of (strong or weak) structural balance of the underlying network on this clustering phenomenon},
	pages = {pp. 585-604},
	volume = {Vol. 64},
	number = {Issue 3},
	journal = {Operations Research}
}
@article {
	title = {Emergent Behaviors over Signed Random Dynamical Networks: Relative-State-Flipping Model},
	author = {Shi, Guodong., Proutiere, Alexandre., Johansson, Mikael., Baras, John S., Johansson, Karl Henrik.},
	year = {2016},
	month = {06/2016},
	abstract = {We study asymptotic dynamical patterns that emerge among a set of nodes interacting in a dynamically evolving signed random network, where positive links carry out standard consensus and negative links induce relative-state flipping. A sequence of deterministic signed graphs defines potential node interactions that take place independently. Each node receives a positive recommendation consistent with the standard consensus algorithm from its positive neighbors, and a negative recommendation defined by relative-state flipping from its negative neighbors. After receiving these recommendations, each node puts a deterministic weight to each recommendation and then encodes these weighted recommendations in its state update through stochastic attentions defined by two Bernoulli random variables. We establish a number of conditions regarding almost sure convergence and divergence of the node states. We also propose a condition for almost sure state clustering for essentially weakly balanced graphs, with the help of several martingale convergence lemmas. Some fundamental differences on the impact of the deterministic weights and stochastic attentions to the node state evolution are highlighted between the current relative-state- flipping model and the state-flipping model considered in Shi et al., IEEE Transaction on Control of Network Systems, 2015.},
	pages = {pp. 1-11},
	volume = {Vol. 3},
	number = {Issue 2},
	journal = {IEEE transactions on control of network systems}
}
@article {
	title = {Emergent Behaviors over Signed Random Dynamical Networks: State-Flipping Model},
	author = {Shi, Guodong., Proutiere, Alexandre., Johansson, Mikael., Baras, John S., Johansson, Karl Henrik.},
	year = {2015},
	month = {06/2015},
	abstract = {Recent studies from social, biological, and engineering network systems have drawn attention to the dynamics over signed networks, where each link is associated with a positive/negative sign indicating trustful/mistrustful, activator/ inhibitor, or secure/malicious interactions. We study asymptotic dynamical patterns that emerge among a set of nodes that interact in a dynamically evolving signed random network. Node interactions take place at random on a sequence of deterministic signed graphs. Each node receives positive or negative recommendations from its neighbors depending on the sign of the interaction arcs, and updates its state accordingly. Recommendations along a positive arc follow the standard consensus update. As in the work by Altafini, negative recommendations use an update where the sign of the neighbor state is flipped. Nodes may weight positive and negative recommendations differently, and random processes are introduced to model the time-varying attention that nodes pay to these recommendations. Conditions for almost sure convergence and divergence of the node states are established. We show that under this so-called state-flipping model, all links contribute to a consensus of the absolute values of the nodes, even under switching sign patterns and a dynamically changing environment. A no-survivor property is established, indicating that every node state diverges almost surely if the maximum network state diverges.},
	pages = {pp. 142-153},
	volume = {Vol. 2},
	number = {No. 2},
	journal = {IEEE transactions on control of network systems}
}
