Spectral characteristics of network redundancy

The transition to synchronization of networked systems

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

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.

General optimization framework for accurate and efficient reconstruction of symmetric complex networks from dynamical data

The challenging problem of network reconstruction from dynamical data can in general be formulated as an optimization task of solving multiple linear equations. Existing approaches are of the two types: Point-by-point (PBP) and global methods. The local PBP method is computationally efficient, but the accuracies of its solutions are somehow low, while a global method has the opposite traits: High accuracy and high computational cost. Taking advantage of the network symmetry, we develop a novel framework integrating the advantages of both the PBP and global methods while avoiding their shortcomings: i.e., high reconstruction accuracy is guaranteed, but the computational cost is orders of magnitude lower than that of the global methods in the literature. The mathematical principle underlying our framework is block coordinate descent (BCD) for solving optimization problems, where the various blocks are determined by the network symmetry. The reconstruction framework is validated by numerical examples with a variety of network structures (i.e., sparse and dense networks) and dynamical processes. Our success is a demonstration that the general principle of exploiting symmetry can be extended to tackling the challenging inverse problem or reverse engineering of complex networks. Since solving a large number of linear equations is key to a plethora of problems in science and engineering, our BCD-based network reconstruction framework will find broader applications.

SPRDA: a link prediction approach based on the structural perturbation to infer disease-associated Piwi-interacting RNAs

piRNA and PIWI proteins have been confirmed for disease diagnosis and treatment as novel biomarkers due to its abnormal expression in various cancers. However, the current research is not strong enough to further clarify the functions of piRNA in cancer and its underlying mechanism. Therefore, how to provide large-scale and serious piRNA candidates for biological research has grown up to be a pressing issue. In this study, a novel computational model based on the structural perturbation method is proposed to predict potential disease-associated piRNAs, called SPRDA. Notably, SPRDA belongs to positive-unlabeled learning, which is unaffected by negative examples in contrast to previous approaches. In the 5-fold cross-validation, SPRDA shows high performance on the benchmark dataset piRDisease, with an AUC of 0.9529. Furthermore, the predictive performance of SPRDA for 10 diseases shows the robustness of the proposed method. Overall, the proposed approach can provide unique insights into the pathogenesis of the disease and will advance the field of oncology diagnosis and treatment.

Structural position vectors and symmetries in complex networks

Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.

Geometry and symmetry in biochemical reaction systems

Complex systems of intracellular biochemical reactions have a central role in regulating cell identities and functions. Biochemical reaction systems are typically studied using the language and tools of graph theory. However, graph representations only describe pairwise interactions between molecular species and so are not well suited to modelling complex sets of reactions that may involve numerous reactants and/or products. Here, we make use of a recently developed hypergraph theory of chemical reactions that naturally allows for higher-order interactions to explore the geometry and quantify functional redundancy in biochemical reactions systems. Our results constitute a general theory of automorphisms for oriented hypergraphs and describe the effect of automorphism group structure on hypergraph Laplacian spectra.

Estimating cellular redundancy in networks of genetic expression

Networks of genetic expression can be modeled by hypergraphs with the additional structure that real coefficients are given to each vertex-edge incidence. The spectra, i.e. the multiset of the eigenvalues, of such hypergraphs, are known to encode structural information of the data. We show how these spectra can be used, in particular, in order to give an estimation of cellular redundancy, a novel measure of gene expression heterogeneity, of the network. We analyze some simulated and real data sets of gene expression for illustrating the new method proposed here.

Dimension-reduction of dynamics on real-world networks with symmetry

We derive explicit formulae to quantify the Markov chain state-space compression, or lumping, that can be achieved in a broad range of dynamical processes on real-world networks, including models of epidemics and voting behaviour, by exploiting redundancies due to symmetries. These formulae are applied in a large-scale study of such symmetry-induced lumping in real-world networks, from which we identify specific networks for which lumping enables exact analysis that could not have been done on the full state-space. For most networks, lumping gives a state-space compression ratio of up to 10 7 , but the largest compression ratio identified is nearly 10 12 . Many of the highest compression ratios occur in animal social networks. We also present examples of types of symmetry found in real-world networks that have not been previously reported.

Symmetries and cluster synchronization in multilayer networks

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure to the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling.

Next Generation Networks: Featuring the Potential Role of Emerging Applications in Translational Oncology

Nowadays, networks are pervasively used as examples of models suitable to mathematically represent and visualize the complexity of systems associated with many diseases, including cancer. In the cancer context, the concept of network entropy has guided many studies focused on comparing equilibrium to disequilibrium (i.e., perturbed) conditions. Since these conditions reflect both structural and dynamic properties of network interaction maps, the derived topological characterizations offer precious support to conduct cancer inference. Recent innovative directions have emerged in network medicine addressing especially experimental omics approaches integrated with a variety of other data, from molecular to clinical and also electronic records, bioimaging etc. This work considers a few theoretically relevant concepts likely to impact the future of applications in personalized/precision/translational oncology. The focus goes to specific properties of networks that are still not commonly utilized or studied in the oncological domain, and they are: controllability, synchronization and symmetry. The examples here provided take inspiration from the consideration of metastatic processes, especially their progression through stages and their hallmark characteristics. Casting these processes into computational frameworks and identifying network states with specific modular configurations may be extremely useful to interpret or even understand dysregulation patterns underlying cancer, and associated events (onset, progression) and disease phenotypes.

Symmetries in the time-averaged dynamics of networks: Reducing unnecessary complexity through minimal network models

Complex networks are the subject of fundamental interest from the scientific community at large. Several metrics have been introduced to characterize the structure of these networks, such as the degree distribution, degree correlation, path length, clustering coefficient, centrality measures, etc. Another important feature is the presence of network symmetries. In particular, the effect of these symmetries has been studied in the context of network synchronization, where they have been used to predict the emergence and stability of cluster synchronous states. Here, we provide theoretical, numerical, and experimental evidence that network symmetries play a role in a substantially broader class of dynamical models on networks, including epidemics, game theory, communication, and coupled excitable systems; namely, we see that in all these models, nodes that are related by a symmetry relation show the same time-averaged dynamical properties. This discovery leads us to propose reduction techniques for exact, yet minimal, simulation of complex networks dynamics, which we show are effective in order to optimize the use of computational resources, such as computation time and memory.

Stable Chimeras and Independently Synchronizable Clusters

Cluster synchronization is a phenomenon in which a network self-organizes into a pattern of synchronized sets. It has been shown that diverse patterns of stable cluster synchronization can be captured by symmetries of the network. Here, we establish a theoretical basis to divide an arbitrary pattern of symmetry clusters into independently synchronizable cluster sets, in which the synchronization stability of the individual clusters in each set is decoupled from that in all the other sets. Using this framework, we suggest a new approach to find permanently stable chimera states by capturing two or more symmetry clusters-at least one stable and one unstable-that compose the entire fully symmetric network.

Recovery of infrastructure networks after localised attacks

The stability of infrastructure network is always a critical issue studied by researchers in different fields. A lot of works have been devoted to reveal the robustness of the infrastructure networks against random and malicious attacks. However, real attack scenarios such as earthquakes and typhoons are instead localised attacks which are investigated only recently. Unlike previous studies, we examine in this paper the resilience of infrastructure networks by focusing on the recovery process from localised attacks. We introduce various preferential repair strategies and found that they facilitate and improve network recovery compared to that of random repairs, especially when population size is uneven at different locations. Moreover, our strategic repair methods show similar effectiveness as the greedy repair. The validations are conducted on simulated networks, and on real networks with real disasters. Our method is meaningful in practice as it can largely enhance network resilience and contribute to network risk reduction.

Complete characterization of the stability of cluster synchronization in complex dynamical networks

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based on the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. Understanding how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an optoelectronic experiment on a five-node network that confirms the synchronization patterns predicted by the theory.

Design principles for cancer therapy guided by changes in complexity of protein-protein interaction networks

Background

The ever-increasing expanse of online bioinformatics data is enabling new ways to, not only explore the visualization of these data, but also to apply novel mathematical methods to extract meaningful information for clinically relevant analysis of pathways and treatment decisions. One of the methods used for computing topological characteristics of a space at different spatial resolutions is persistent homology. This concept can also be applied to network theory, and more specifically to protein-protein interaction networks, where the number of rings in an individual cancer network represents a measure of complexity.

Results

We observed a linear correlation of R = −0.55 between persistent homology and 5-year survival of patients with a variety of cancers. This relationship was used to predict the proteins within a protein-protein interaction network with the most impact on cancer progression. By re-computing the persistent homology after computationally removing an individual node (protein) from the protein-protein interaction network, we were able to evaluate whether such an inhibition would lead to improvement in patient survival. The power of this approach lied in its ability to identify the effects of inhibition of multiple proteins and in the ability to expose whether the effect of a single inhibition may be amplified by inhibition of other proteins. More importantly, we illustrate specific examples of persistent homology calculations, which correctly predict the survival benefit observed effects in clinical trials using inhibitors of the identified molecular target.

Conclusions

We propose that computational approaches such as persistent homology may be used in the future for selection of molecular therapies in clinic. The technique uses a mathematical algorithm to evaluate the node (protein) whose inhibition has the highest potential to reduce network complexity. The greater the drop in persistent homology, the greater reduction in network complexity, and thus a larger potential for survival benefit. We hope that the use of advanced mathematics in medicine will provide timely information about the best drug combination for patients, and avoid the expense associated with an unsuccessful clinical trial, where drug(s) did not show a survival benefit.

Reviewers

This article was reviewed by Nathan J. Bowen (nominated by I. King Jordan), Tomasz Lipniacki, and Merek Kimmel.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-015-0058-5) contains supplementary material, which is available to authorized users.

Partial synchronization and partial amplitude death in mesoscale network motifs

We study the interplay between network topology and complex space-time patterns and introduce a concept to analytically predict complex patterns in networks of Stuart-Landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology. These patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes. The underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks. In addition it directly links the stability of complete in-phase synchronization to only a small subset of topological eigenvalues of a network.

Dimensionality reduction and spectral properties of multilayer networks

Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered systems include multiple subsystems and layers of connectivity. This new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically. One way to address this question is to study the spectral properties of such networks. Here we apply the framework of graph quotients, which occurs naturally in this context, and the associated eigenvalue interlacing results to the adjacency and Laplacian matrices of undirected multilayer networks. Specifically, we describe relationships between the eigenvalue spectra of multilayer networks and their two most natural quotients, the network of layers and the aggregate network, and show the dynamical implications of working with either of the two simplified representations. Our work thus contributes in particular to the study of dynamical processes whose critical properties are determined by the spectral properties of the underlying network.

Symmetry in critical random Boolean network dynamics

Using Boolean networks as prototypical examples, the role of symmetry in the dynamics of heterogeneous complex systems is explored. We show that symmetry of the dynamics, especially in critical states, is a controlling feature that can be used both to greatly simplify analysis and to characterize different types of dynamics. Symmetry in Boolean networks is found by determining the frequency at which the various Boolean output functions occur. There are classes of functions that consist of Boolean functions that behave similarly. These classes are orbits of the controlling symmetry group. We find that the symmetry that controls the critical random Boolean networks is expressed through the frequency by which output functions are utilized by nodes that remain active on dynamical attractors. This symmetry preserves canalization, a form of network robustness. We compare it to a different symmetry known to control the dynamics of an evolutionary process that allows Boolean networks to organize into a critical state. Our results demonstrate the usefulness and power of using the symmetry of the behavior of the nodes to characterize complex network dynamics, and introduce an alternative approach to the analysis of heterogeneous complex systems.

Topological Strata of Weighted Complex Networks

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and -more recently- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.