We study variants of hierarchical modular network models suggested by Kaiser and Hilgetag [Frontiers in Neuroinformatics, 4 (2010) 8] to model functional brain connectivity, using extensive simulations and quenched mean-field theory (QMF), focusing on structures with a connection probability that decays exponentially with the level index. Such networks can be embedded in two-dimensional Euclidean space. We explore the dynamic behavior of the contact process (CP) and threshold models on networks of this kind, including hierarchical trees. While in the small-world networks originally proposed to model brain connectivity, the topological heterogeneities are not strong enough to induce deviations from mean-field behavior, we show that a Griffiths phase can emerge under reduced connection probabilities, approaching the percolation threshold. In this case the topological dimension of the networks is finite, and extended regions of bursty, power-law dynamics are observed. Localization in the steady state is also shown via QMF. We investigate the effects of link asymmetry and coupling disorder, and show that localization can occur even in small-world networks with high connectivity in case of link disorder.
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effe…
Renormalization group on hierarchical lattices is often considered a valuable tool to understand the critical behavior of more complicated statistical mechanical models. In presence of quenched disorder, however, in many model cases predictions obtained with the Migdal-Kadanoff bond removal approach fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In order to critically review this limitation we analyze the behavior of Ising and Blume-Emery-Griffiths models on more complicated hiera…
The Griffiths phase has been proposed to induce a stretched critical regime that facilitates self-organizing of brain networks for optimal function. This phase stems from the intrinsic structural heterogeneity of brain networks, such as the hierarchical modular structure. In this work, we extend this concept to modified hierarchical networks with small-world connections based on Hanoi networks. Through extensive simulations, we identify the role of an exponential distribution of the inter-moduli connectivity probability across hierarchies determining the emergence of the Griffiths phase. Numer…
In $d > 2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power-laws, thus it is called mixed-order or hybrid type. It has recently been shown that the introduction of quenched disorder rounds the discontinuity and second order phase transition and Griffiths phases appear. Here we provide numerical evidence, that even in case of high graph dimensional hierarchical modular networks the Griffiths phase of the $K=2$ threshold model is present below the hybrid phase transition. This is due to t…
We use an exact renormalization-group transformation to study the Ising model on a complex network composed of tightly-knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real-world networks. Varying the ratio K/J of of inter- to intra-community coupling, we obtain an unusual phase diagram: at high temperatures or small K/J we have a disordered phase with a Griffiths singularity in the free energy, due to the presence of rare large clusters, which we analyze through the Yang-Lee zeros in the complex magnetic field plane. As the temperature is low…
Hallmarks of criticality, such as power-laws and scale invariance, have been empirically found in cortical networks and it has been conjectured that operating at criticality entails functional advantages, such as optimal computational capabilities, memory, and large dynamical ranges. As critical behavior requires a high degree of fine tuning to emerge, some type of self-tuning mechanism needs to be invoked. Here we show that, taking into account the complex hierarchical-modular architecture of cortical networks, the singular critical point is replaced by an extended critical-like region which …
Many of the amazing functional capabilities of the brain are collective properties stemming from the interactions of large sets of individual neurons. In particular, the most salient collective phenomena in brain activity are oscillations, which require the synchronous activation of many neurons. Here, we analyse parsimonious dynamical models of neural synchronisation running on top of synthetic networks that capture essential aspects of the actual brain anatomical connectivity such as a hierarchical-modular and core-periphery structure. These models reveal the emergence of complex collective …
Power-law (PL) time dependent infection growth has been reported in many COVID-19 statistics. In simple SIR models the number of infections grows at the outbreak as $I(t) \propto t^{d-1}$ on $d$-dimensional Euclidean lattices in the endemic phase or follow a slower universal PL at the critical point, until finite sizes cause immunity and a crossover to an exponential decay. Heterogeneity may alter the dynamics of spreading models, spatially inhomogeneous infection rates can cause slower decays, posing a threat of a long recovery from a pandemic. COVID-19 statistics have also provided epidemic …
Connectivity patterns of relevance in neuroscience and systems biology can be encoded in hierarchical modular networks (HMNs). Moreover, recent studies highlight the role of hierarchical modular organization in shaping brain activity patterns, providing an excellent substrate to promote both the segregation and integration of neural information. Here we propose an extensive numerical analysis of the critical spreading rate (or "epidemic" threshold) --separating a phase with endemic persistent activity from one in which activity ceases-- on diverse HMNs. By employing analytical and computationa…
We propose a growing network model that consists of two tunable mechanisms: growth by merging modules which are represented as complete graphs and a fitness-driven preferential attachment. Our model exhibits the three prominent statistical properties are widely shared in real biological networks, for example gene regulatory, protein-protein interaction, and metabolic networks. They retain three power law relationships, such as the power laws of degree distribution, clustering spectrum, and degree-degree correlation corresponding to scale-free connectivity, hierarchical modularity, and disassor…
Generative mechanisms which lead to empirically observed structure of networked systems from diverse fields like biology, technology and social sciences form a very important part of study of complex networks. The structure of many networked systems like biological cell, human society and World Wide Web markedly deviate from that of completely random networks indicating the presence of underlying processes. Often the main process involved in their evolution is the addition of links between existing nodes having a common neighbor. In this context we introduce an important property of the nodes,…
In network science, a group of nodes connected with each other at higher probability than with those outside the group is referred to as a community. From the perspective that individual communities are associated with functional modules constituting complex systems described by networks, discovering communities is primarily important for understanding overall functions of these systems. Much effort has been devoted to developing methods to detect communities in networks since the early days of network science. Nevertheless, the method to reveal key characteristics of communities in real-world…
Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the pr…
An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organi…
In this paper we study two non-mean-field spin models built on a hierarchical lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: Given that such fluctuations are not negligible in non-mean-field models, we develop a n…
The quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks. However, it is not clear if such methods provide consistent, robust and meaningful results when considering hierarchies as a whole. Part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures. In this work we give a contribution by introducing the {\it hierarchical mutual information}, which is a generalization of the traditional mutual information, and allows to comp…
The spectral dimension is a generalization of the Euclidean dimension and quantifies the propensity of a network to transmit and diffuse information. We show that, in hierarchical-modular network models of the brain, dynamics are anomalously slow and the spectral dimension is not defined. Inspired by Anderson localization in quantum systems, we relate the localization of neural activity - essential to embed brain functionality - to the network spectrum and to the existence of an anomalous "Lifshitz dimension". In a broader context, our results help shedding light on the relationship between st…
In this paper I present a novel type of Topology and Weight Evolving Artificial Neural Network (TWEANN) system called Modular Discover & eXplore Neural Network (DXNN). Modular DXNN utilizes a hierarchical/modular topology which allows for highly scalable and dynamically granular systems to evolve. Among the novel features discussed in this paper is a simple and database friendly encoding for hierarchical/modular NNs, a new selection method aimed at producing highly compact and fit individuals within the population, a "Targeted Tunning" system aimed at alleviating the curse of dimensionality, a…
Hierarchical networks are attracting a renewal interest for modelling the organization of a number of biological systems and for tackling the complexity of statistical mechanical models beyond mean-field limitations. Here we consider the Dyson hierarchical construction for ferromagnets, neural networks and spin-glasses, recently analyzed from a statistical-mechanics perspective, and we focus on the topological properties of the underlying structures. In particular, we find that such structures are weighted graphs that exhibit high degree of clustering and of modularity, with small spectral gap…
Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large…
A major challenge in neuroscience is posed by the need for relating the emerging dynamical features of brain activity with the underlying modular structure of neural connections, hierarchically organized throughout several scales. The spontaneous emergence of coherence and synchronization across such scales is crucial to neural function, while its anomalies often relate to pathological conditions. Here we provide a numerical study of synchronization dynamics in the human connectome network. Our purpose is to provide a detailed characterization of the recently uncovered broad dynamic regime, in…
In this paper we study Markov processes and related first passage problems on a class of weighted, modular graphs which generalize the Dyson hierarchical model. In these networks, the coupling strength between two nodes depends on their distance and is modulated by a parameter $σ$. We find that, in the thermodynamic limit, ergodicity is lost and the "distant" nodes can not be reached. Moreover, for finite-sized systems, there exists a threshold value for $σ$ such that, when $σ$ is relatively large, the inhomogeneity of the coupling pattern prevails and "distant" nodes are hardly reached. The s…
