Here, we focus on the case when the edge weights are characterised by matrices, and investigate both structural and dynamical properties of the matrix-weighted networks (MWNs). Building on concepts from signed and complex-weighted graphs, we introduce the notion of coherence in MWNs, and illustrate the spectral properties and dynamical implications in consensus dynamics and random walks. The results are verified on synthetic networks with matrix weights.
Here, use Mittag-Leffler (ML) matrix functions to quantify the notion of balance of signed graphs. We show that the ML balance index can be derived from first principles on the basis of a nonconservative diffusion dynamic and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions. We also demonstrate the important information in the ML balance index with both artificial signed networks and real-world networks in various contexts, ranging from biological and ecological to social ones.
Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential approach is to start by identifying models of basal cognition, which give an abstract representation of a range organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner. We first show that our model connects resources in an efficient manner when the environment is constant. We then show that in an oscillatory environment our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We show that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. We identify connections between our model and basal cognition in biological systems and slime moulds, in particular, showing how oscillatory and problem-solving properties of these systems are captured by our model.
Here, we focus on the case when the weight matrix can contain complex numbers but is Hermitian, and investigate both structural and dynamical properties of the complex-weighted networks. Building on concepts from signed graphs, we introduce a classification of complex-weighted networks based on the notion of structural balance, and illustrate the shared spectral properties within each type. We then apply the results to characterise the dynamics of random walks on complex-weighted networks. Finally, we explore potential applications of our findings by generalising the notion of cut, and propose an associated spectral clustering algorithm. We also provide further characteristics of the magnetic Laplacian, associating directed networks to complex-weighted ones.
Here, we first introduce a classification of signed networks into balanced, antibalanced or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. Then we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance.
We unify the features from the classic models into a novel class of information diffusion model, and propose a general framework for the influence maximisation problem that is applicable to a broad range of functions describing the overall influence.
The complementarity and substitutability between products are essential concepts in retail and marketing.