The physics of spreading processes in multilayer networks

The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent "multilayer" approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.Comment: 25 pages, 4 figure

A simple statistic for determining the dimensionality of complex networks

Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are computationally heavy and lack theoretical foundation. On the other hand, theoretical approaches do not apply to graphs with heterogeneous degree distributions, which is often the case for complex real-world networks. To address these drawbacks, we consider geometric inhomogeneous random graphs (GIRGs) as a random graph model, which captures a variety of properties observed in practice. These include a heterogeneous degree distribution and non-vanishing clustering coefficient, which is the probability that two random neighbours of a vertex are adjacent. In GIRGs, $n$ vertices are distributed on a $d$-dimensional torus and weights are assigned to the vertices according to a power-law distribution. Two vertices are then connected with a probability that depends on their distance and their weights. Our first result shows that the clustering coefficient of GIRGs scales inverse exponentially with respect to the number of dimensions, when the latter is at most logarithmic in $n$. This gives a first theoretical explanation for the low dimensionality of real-world networks observed by Almagro et. al. [Nature '22]. Our second result is a linear-time algorithm for determining the dimensionality of a given GIRG. We prove that our algorithm returns the correct number of dimensions with high probability when the input is a GIRG. As a result, our algorithm bridges the gap between theory and practice, as it not only comes with a rigorous proof of correctness but also yields results comparable to that of prior empirical approaches, as indicated by our experiments on real-world instances

Detection of L∞L_\infty Geometry in Random Geometric Graphs: Suboptimality of Triangles and Cluster Expansion

In this paper we study the random geometric graph $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ with $L_q$ distance where each vertex is sampled uniformly from the $d$-dimensional torus and where the connection radius is chosen so that the marginal edge probability is $p$. In addition to results addressing other questions, we make progress on determining when it is possible to distinguish $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ from the Endos-R\'enyi graph $\mathsf{G}(n,p)$. Our strongest result is in the extreme setting $q = \infty$, in which case $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^\infty_p,p)$ is the $\mathsf{AND}$ of $d$ 1-dimensional random geometric graphs. We derive a formula similar to the cluster-expansion from statistical physics, capturing the compatibility of subgraphs from each of the $d$ 1-dimensional copies, and use it to bound the signed expectations of small subgraphs. We show that counting signed 4-cycles is optimal among all low-degree tests, succeeding with high probability if and only if $d = \tilde{o}(np).$ In contrast, the signed triangle test is suboptimal and only succeeds when $d = \tilde{o}((np)^{3/4}).$ Our result stands in sharp contrast to the existing literature on random geometric graphs (mostly focused on $L_2$ geometry) where the signed triangle statistic is optimal

Análisis de conectividad funcional de la dinámica neuroenergética del TDAH = Functional Connectivity Analysis of Neuroenergetic Dynamics for ADHD

A fast and economic pilot study for measuring the neuroenergetic dynamics in an ADHD-diagnosed sample is performed. Based in a simplified connectome version, a graph theory application for neural connectivity, the performance and subjective states are linked through brain activity analysis during a behavioral attention test. ADHD is a neurobehavioral disorder related to a deficient filtering of stimuli, inefficacy performing in sustained activities and difficulties responding to unpredictable situations. There are two main strategies to evaluate this disorder: (1) behavioral tests and (2) neural biomarkers. Behavioral tests provide a criterion for classifying responses in a collection of tasks, looking for unstructured and inconsistent responses to given instructions or rules. Hyperactivity, inattention and impulsivity are some criteria analyzed. By the other hand, neural biomarkers are measurable indicators for particular states or diseases set up from EEG data. Since 2013, the theta/beta ratio was accepted as the ADHD biomarker, suggesting a misbalance of electrical brain activity. In this study, brain connectivity on sustained attention task performed by children between 7 to 13 years old from a public school. Ten participants were ADHD-diagnosed and five were selected for the control group to compare EEG signals collected with low-cost neuroheadset. Graphs show different connectivity dynamics in both groups for Theta (4-8 Hz), SMR (12-15 Hz) and Beta (15-20 Hz), indicating connectivity variations in brain regions according to the neuroenergetics theory. The connectivity in the ADHD group is reduced in lower frequencies first (Theta), then SMR and finally Beta. In contrast, the control graphs for Theta and SMR brainwaves are closer to the small-world networks and it can be noticed by comparing the measurements of the different graphs among themselves. The decay process corresponds to the bottom-up approach, where random stimuli trigger transitions from one state to the other, which is in this case the transition from attention to inattention. The declining of resources placed for disposal at the randomized SART stage might imply a limitation regulating the production of the required resources for the tasks fulfillment, as it has been reported in previous studies where other techniques are implemented