Network Density of States

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.Comment: 10 pages, 7 figure

The role of symmetry in neural networks and their Laplacian spectra

Human and animal nervous systems constitute complexly wired networks that form the infrastructure for neural processing and integration of information. The organization of these neural networks can be analyzed using the so-called Laplacian spectrum, providing a mathematical tool to produce systems-level network fingerprints. In this article, we examine a characteristic central peak in the spectrum of neural networks, including anatomical brain network maps of the mouse, cat and macaque, as well as anatomical and functional network maps of human brain connectivity. We link the occurrence of this central peak to the level of symmetry in neural networks, an intriguing aspect of network organization resulting from network elements that exhibit similar wiring patterns. Specifically, we propose a measure to capture the global level of symmetry of a network and show that, for both empirical networks and network models, the height of the main peak in the Laplacian spectrum is strongly related to node symmetry in the underlying network. Moreover, examination of spectra of duplication-based model networks shows that neural spectra are best approximated using a trade-off between duplication and diversification. Taken together, our results facilitate a better understanding of neural network spectra and the importance of symmetry in neural networks

Community Detection Boosts Network Dismantling on Real-World Networks

Network dismantling techniques have gained increasing interest during the last years caused by the need for protecting and strengthening critical infrastructure systems in our society. We show that communities play a critical role in dismantling, given their inherent property of separating a network into strongly and weakly connected parts. The process of community-based dismantling depends on several design factors, including the choice of community detection method, community cut strategy, and inter-community node selection. We formalize the problem of community attacks to networks, identify critical design decisions for such methods, and perform a comprehensive empirical evaluation with respect to effectiveness and efficiency criteria on a set of more than 40 community-based network dismantling methods. We compare our results to state-of-the-art network dismantling, including collective influence, articulation points, as well as network decycling. We show that community-based network dismantling significantly outperforms existing techniques in terms of solution quality and computation time in the vast majority of real-world networks, while existing techniques mainly excel on model networks (ER, BA) mostly. We additionally show that the scalability of community-based dismantling opens new doors towards the efficient analysis of large real-world networks.We acknowledge financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación/ SuMaEco Project (RTI2018-095441-B-C22) and the María de Maeztu Program for Units of Excellence in R&D (No. MDM-2017-0711). D.R.-R. also acknowledges the Fellowship No. BES-2016-076264 under the FPI program of MINECO, Spain.Peer reviewe

On the Origin of Biomolecular Networks

Biomolecular networks have already found great utility in characterizing complex biological systems arising from pairwise interactions amongst biomolecules. Here, we explore the important and hitherto neglected role of information asymmetry in the genesis and evolution of such pairwise biomolecular interactions. Information asymmetry between sender and receiver genes is identified as a key feature distinguishing early biochemical reactions from abiotic chemistry, and a driver of network topology as biomolecular systems become more complex. In this context, we review how graph theoretical approaches can be applied not only for a better understanding of various proximate (mechanistic) relations, but also, ultimate (evolutionary) structures encoded in such networks from among all types of variations they induce. Among many possible variations, we emphasize particularly the essential role of gene duplication in terms of signaling game theory, whereby sender and receiver gene players accrue benefit from gene duplication, leading to a preferential attachment mode of network growth. The study of the resulting dynamics suggests many mathematical/computational problems, the majority of which are intractable yet yield to efficient approximation algorithms, when studied through an algebraic graph theoretic lens. We relegate for future work the role of other possible generalizations, additionally involving horizontal gene transfer, sexual recombination, endo-symbiosis, etc., which enrich the underlying graph theory even further

Symptoms of complexity in a tourism system

Tourism destinations behave as dynamic evolving complex systems, encompassing numerous factors and activities which are interdependent and whose relationships might be highly nonlinear. Traditional research in this field has looked after a linear approach: variables and relationships are monitored in order to forecast future outcomes with simplified models and to derive implications for management organisations. The limitations of this approach have become apparent in many cases, and several authors claim for a new and different attitude. While complex systems ideas are amongst the most promising interdisciplinary research themes emerged in the last few decades, very little has been done so far in the field of tourism. This paper presents a brief overview of the complexity framework as a means to understand structures, characteristics, relationships, and explores the implications and contributions of the complexity literature on tourism systems. The objective is to allow the reader to gain a deeper appreciation of this point of view.Comment: 32 pages, 3 figures, 1 table; accepted in Tourism Analysi