Statistical physics of complex information dynamics

Dissimilarity between synchronization processes on networks

In this study, we present a general framework for comparing two dynamical processes that describe the synchronization of oscillators coupled through networks of the same size. We introduce a measure of dissimilarity defined in terms of a metric on a hypertorus, allowing us to compare the phases of coupled oscillators. In the first part, this formalism is implemented to examine systems of networked identical phase oscillators that evolve with the Kuramoto model. In particular, we analyze the effect of the weight of an edge in the synchronization of two oscillators, the introduction of new sets of edges in interacting cycles, the effect of bias in the couplings, and the addition of a link in a ring. We also compare the synchronization of nonisomorphic graphs with four nodes. Finally, we explore the dissimilarities generated when we contrast the Kuramoto model with its linear approximation for different random initial phases in deterministic and random networks. The approach introduced provides a general tool for comparing synchronization processes on networks, allowing us to understand the dynamics of a complex system as a consequence of the coupling structure and the processes that can occur in it.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Optimizing parameter search for community detection in time-evolving networks of complex systems

Network representations have been effectively employed to analyze complex systems across various areas and applications, leading to the development of network science as a core tool to study systems with multiple components and complex interactions. There is a growing interest in understanding the temporal dynamics of complex networks to decode the underlying dynamic processes through the temporal changes in network structures. Community detection algorithms, which are specialized clustering algorithms, have been instrumental in studying these temporal changes. They work by grouping nodes into communities based on the structure and intensity of network connections over time, aiming to maximize the modularity of the network partition. However, the performance of these algorithms is highly influenced by the selection of resolution parameters of the modularity function used, which dictate the scale of the represented network, in both size of communities and the temporal resolution of the dynamic structure. The selection of these parameters has often been subjective and reliant on the characteristics of the data used to create the network. Here, we introduce a method to objectively determine the values of the resolution parameters based on the elements of self-organization and scale-invariance. We propose two key approaches: (1) minimization of biases in spatial scale network characterization and (2) maximization of scale-freeness in temporal network reconfigurations. We demonstrate the effectiveness of these approaches using benchmark network structures as well as real-world datasets. To implement our method, we also provide an automated parameter selection software package that can be applied to a wide range of complex systems.

Efficient simulations of epidemic models with tensor networks: Application to the one-dimensional susceptible-infected-susceptible model

The contact process is an emblematic model of a nonequilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible model, and it is widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of matrix product states (MPSs). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

Emergent information dynamics in many-body interconnected systems

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

Generalized network density matrices for analysis of multiscale functional diversity

The network density matrix formalism allows for describing the dynamics of information on top of complex structures and it has been successfully used to analyze, e.g., a system's robustness, perturbations, coarse-graining multilayer networks, characterization of emergent network states, and performing multiscale analysis. However, this framework is usually limited to diffusion dynamics on undirected networks. Here, to overcome some limitations, we propose an approach to derive density matrices based on dynamical systems and information theory, which allows for encapsulating a much wider range of linear and nonlinear dynamics and richer classes of structure, such as directed and signed ones. We use our framework to study the response to local stochastic perturbations of synthetic and empirical networks, including neural systems consisting of excitatory and inhibitory links and gene-regulatory interactions. Our findings demonstrate that topological complexity does not necessarily lead to functional diversity, i.e., the complex and heterogeneous response to stimuli or perturbations. Instead, functional diversity is a genuine emergent property which cannot be deduced from the knowledge of topological features such as heterogeneity, modularity, the presence of asymmetries, and dynamical properties of a system.

Boosting the Full Potential of PyMOL with Structural Biology Plugins

Over the past few decades, the number of available structural bioinformatics pipelines, libraries, plugins, web resources and software has increased exponentially and become accessible to the broad realm of life scientists. This expansion has shaped the field as a tangled network of methods, algorithms and user interfaces. In recent years PyMOL, widely used software for biomolecules visualization and analysis, has started to play a key role in providing an open platform for the successful implementation of expert knowledge into an easy-to-use molecular graphics tool. This review outlines the plugins and features that make PyMOL an eligible environment for supporting structural bioinformatics analyses.

Identification of Network Topology Variations Based on Spectral Entropy

Based on the fact that the traditional probability distribution entropy describing a local feature of the system cannot effectively capture the global topology variations of the network, some indicators constructed by the network adjacency matrix and Laplacian matrix come into being. Specifically, these measures are based on the eigenvalues of the scaled Laplace matrix, the eigenvalues of the network communicability matrix, and the spectral entropy based on information diffusion that has been proposed recently, respectively. In this article, we systematically study the dependence of these measures on the topological structure of the network. We prove from various aspects that spectral entropy has a better ability to identify the global topology than the traditional distribution entropy. Furthermore, the indicator based on the eigenvalues of the network communicability matrix achieves good results in some aspects while, overall, the spectral entropy is able to identify network topology variations from a global perspective.

Weighted simplicial complexes and their representation power of higher-order network data and topology

Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the adopted mathematical framework to describe higher-order networks starting from data of higher-order interactions. Unweighted simplicial complexes typically involve a loss of information of the data, though having the benefit to capture the higher-order topology of the data. In this work we show that weighted simplicial complexes allow one to circumvent all the limitations of unweighted simplicial complexes to represent higher-order interactions. In particular, weighted simplicial complexes can represent higher-order networks without loss of information, allowing one at the same time to capture the weighted topology of the data. The higher-order topology is probed by studying the spectral properties of suitably defined weighted Hodge Laplacians displaying a normalized spectrum. The higher-order spectrum of (weighted) normalized Hodge Laplacians is studied combining cohomology theory with information theory. In the proposed framework we quantify and compare the information content of higher-order spectra of different dimension using higher-order spectral entropies and spectral relative entropies. The proposed methodology is tested on real higher-order collaboration networks and on the weighted version of the simplicial complex model "Network Geometry with Flavor."

From the origin of life to pandemics: emergent phenomena in complex systems

When a large number of similar entities interact among each other and with their environment at a low scale, unexpected outcomes at higher spatio-temporal scales might spontaneously arise. This non-trivial phenomenon, known as emergence, characterizes a broad range of distinct complex systems-from physical to biological and social-and is often related to collective behaviour. It is ubiquitous, from non-living entities such as oscillators that under specific conditions synchronize, to living ones, such as birds flocking or fish schooling. Despite the ample phenomenological evidence of the existence of systems' emergent properties, central theoretical questions to the study of emergence remain unanswered, such as the lack of a widely accepted, rigorous definition of the phenomenon or the identification of the essential physical conditions that favour emergence. We offer here a general overview of the phenomenon of emergence and sketch current and future challenges on the topic. Our short review also serves as an introduction to the theme issue Emergent phenomena in complex physical and socio-technical systems: from cells to societies, where we provide a synthesis of the contents tackled in the issue and outline how they relate to these challenges, spanning from current advances in our understanding on the origin of life to the large-scale propagation of infectious diseases. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Grand Canonical Ensembles of Sparse Networks and Bayesian Inference

Maximum entropy network ensembles have been very successful in modelling sparse network topologies and in solving challenging inference problems. However the sparse maximum entropy network models proposed so far have fixed number of nodes and are typically not exchangeable. Here we consider hierarchical models for exchangeable networks in the sparse limit, i.e., with the total number of links scaling linearly with the total number of nodes. The approach is grand canonical, i.e., the number of nodes of the network is not fixed a priori: it is finite but can be arbitrarily large. In this way the grand canonical network ensembles circumvent the difficulties in treating infinite sparse exchangeable networks which according to the Aldous-Hoover theorem must vanish. The approach can treat networks with given degree distribution or networks with given distribution of latent variables. When only a subgraph induced by a subset of nodes is known, this model allows a Bayesian estimation of the network size and the degree sequence (or the sequence of latent variables) of the entire network which can be used for network reconstruction.

Statistical physics of exchangeable sparse simple networks, multiplex networks, and simplicial complexes

Exchangeability is a desired statistical property of network ensembles requiring their invariance upon relabeling of the nodes. However, combining sparsity of network ensembles with exchangeability is challenging. Here we propose a statistical physics framework and a Metropolis-Hastings algorithm defining exchangeable sparse network ensembles. The model generates networks with heterogeneous degree distributions by enforcing only global constraints while existing (nonexchangeable) exponential random graphs enforce an extensive number of local constraints. This very general theoretical framework to describe exchangeable networks is here first formulated for uncorrelated simple networks and then it is extended to treat simple networks with degree correlations, directed networks, bipartite networks, and generalized network structures including multiplex networks and simplicial complexes. In particular here we formulate and treat both uncorrelated and correlated exchangeable ensembles of simplicial complexes using statistical mechanics approaches.

Multiscale Information Propagation in Emergent Functional Networks

Complex biological systems consist of large numbers of interconnected units, characterized by emergent properties such as collective computation. In spite of all the progress in the last decade, we still lack a deep understanding of how these properties arise from the coupling between the structure and dynamics. Here, we introduce the multiscale emergent functional state, which can be represented as a network where links encode the flow exchange between the nodes, calculated using diffusion processes on top of the network. We analyze the emergent functional state to study the distribution of the flow among components of 92 fungal networks, identifying their functional modules at different scales and, more importantly, demonstrating the importance of functional modules for the information content of networks, quantified in terms of network spectral entropy. Our results suggest that the topological complexity of fungal networks guarantees the existence of functional modules at different scales keeping the information entropy, and functional diversity, high.

Persistence of information flow: A multiscale characterization of human brain

Information exchange in the human brain is crucial for vital tasks and to drive diseases. Neuroimaging techniques allow for the indirect measurement of information flows among brain areas and, consequently, for reconstructing connectomes analyzed through the lens of network science. However, standard analyses usually focus on a small set of network indicators and their joint probability distribution. Here, we propose an information-theoretic approach for the analysis of synthetic brain networks (based on generative models) and empirical brain networks, and to assess connectome's information capacity at different stages of dementia. Remarkably, our framework accounts for the whole network state, overcoming limitations due to limited sets of descriptors, and is used to probe human connectomes at different scales. We find that the spectral entropy of empirical data lies between two generative models, indicating an interpolation between modular and geometry-driven structural features. In fact, we show that the mesoscale is suitable for characterizing the differences between brain networks and their generative models. Finally, from the analysis of connectomes obtained from healthy and unhealthy subjects, we demonstrate that significant differences between healthy individuals and the ones affected by Alzheimer's disease arise at the microscale (max. posterior probability smaller than 1%) and at the mesoscale (max. posterior probability smaller than 10%).

pyProGA-A PyMOL plugin for protein residue network analysis

The field of protein residue network (PRN) research has brought several useful methods and techniques for structural analysis of proteins and protein complexes. Many of these are ripe and ready to be used by the proteomics community outside of the PRN specialists. In this paper we present software which collects an ensemble of (network) methods tailored towards the analysis of protein-protein interactions (PPI) and/or interactions of proteins with ligands of other type, e.g. nucleic acids, oligosaccharides etc. In parallel, we propose the use of the network differential analysis as a method to identify residues mediating key interactions between proteins. We use a model system, to show that in combination with other, already published methods, also included in pyProGA, it can be used to make such predictions. Such extended repertoire of methods allows to cross-check predictions with other methods as well, as we show here. In addition, the possibility to construct PRN models from various kinds of input is so far a unique asset of our code. One can use structural data as defined in PDB files and/or from data on residue pair interaction energies, either from force-field parameters or fragment molecular orbital (FMO) calculations. pyProGA is a free open-source software available from https://gitlab.com/Vlado_S/pyproga.