Dynamics over Signed Networks

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 different types of interactions take place along the positive and negative links, respectively, arise from various biological, social, political, and economic systems. As modifications 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 unified 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.Comment: In press, SIAM Revie

Spreading and structural balance on signed networks

Two competing types of interactions often play an important part in shaping system behavior, such as activatory or inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced or strictly balanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks

Spreading and structural balance on signed networks

Two competing types of interactions often play an important part in shaping system behavior, such as activatory or inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally balanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced or strictly balanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks

Graph partitions and cluster synchronization in networks of oscillators

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators

Spiking-YOLO: Spiking Neural Network for Energy-Efficient Object Detection

Over the past decade, deep neural networks (DNNs) have demonstrated remarkable performance in a variety of applications. As we try to solve more advanced problems, increasing demands for computing and power resources has become inevitable. Spiking neural networks (SNNs) have attracted widespread interest as the third-generation of neural networks due to their event-driven and low-powered nature. SNNs, however, are difficult to train, mainly owing to their complex dynamics of neurons and non-differentiable spike operations. Furthermore, their applications have been limited to relatively simple tasks such as image classification. In this study, we investigate the performance degradation of SNNs in a more challenging regression problem (i.e., object detection). Through our in-depth analysis, we introduce two novel methods: channel-wise normalization and signed neuron with imbalanced threshold, both of which provide fast and accurate information transmission for deep SNNs. Consequently, we present a first spiked-based object detection model, called Spiking-YOLO. Our experiments show that Spiking-YOLO achieves remarkable results that are comparable (up to 98%) to those of Tiny YOLO on non-trivial datasets, PASCAL VOC and MS COCO. Furthermore, Spiking-YOLO on a neuromorphic chip consumes approximately 280 times less energy than Tiny YOLO and converges 2.3 to 4 times faster than previous SNN conversion methods.Comment: Accepted to AAAI 202

Graph partitions and cluster synchronization in networks of oscillators

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators

The Evolution of Beliefs over Signed Social Networks

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 DeGroot model to account for negative links: when two enemies 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 {\it 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 {\it 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