Symmetry in Complex Networks

A multi-tissue metabolome atlas of primate pregnancy

Pregnancy induces dramatic metabolic changes in females; yet, the intricacies of this metabolic reprogramming remain poorly understood, especially in primates. Using cynomolgus monkeys, we constructed a comprehensive multi-tissue metabolome atlas, analyzing 273 samples from 23 maternal tissues during pregnancy. We discovered a decline in metabolic coupling between tissues as pregnancy progressed. Core metabolic pathways that were rewired during primate pregnancy included steroidogenesis, fatty acid metabolism, and arachidonic acid metabolism. Our atlas revealed 91 pregnancy-adaptive metabolites changing consistently across 23 tissues, whose roles we verified in human cell models and patient samples. Corticosterone and palmitoyl-carnitine regulated placental maturation and maternal tissue progenitors, respectively, with implications for maternal preeclampsia, diabetes, cardiac hypertrophy, and muscle and liver regeneration. Moreover, we found that corticosterone deficiency induced preeclampsia-like inflammation, indicating the atlas's potential clinical value. Overall, our multi-tissue metabolome atlas serves as a framework for elucidating the role of metabolic regulation in female health during pregnancy.

SMGen: A Generator of Synthetic Models of Biochemical Reaction Networks

Several software tools for the simulation and analysis of biochemical reaction networks have been developed in the last decades; however, assessing and comparing their computational performance in executing the typical tasks of computational systems biology can be limited by the lack of a standardized benchmarking approach. To overcome these limitations, we propose here a novel tool, named SMGen, designed to automatically generate synthetic models of reaction networks that, by construction, are characterized by relevant features (e.g., system connectivity and reaction discreteness) and non-trivial emergent dynamics of real biochemical networks. The generation of synthetic models in SMGen is based on the definition of an undirected graph consisting of a single connected component that, generally, results in a computationally demanding task; to speed up the overall process, SMGen exploits a main–worker paradigm. SMGen is also provided with a user-friendly graphical user interface, which allows the user to easily set up all the parameters required to generate a set of synthetic models with any number of reactions and species. We analysed the computational performance of SMGen by generating batches of symmetric and asymmetric reaction-based models (RBMs) of increasing size, showing how a different number of reactions and/or species affects the generation time. Our results show that when the number of reactions is higher than the number of species, SMGen has to identify and correct a large number of errors during the creation process of the RBMs, a circumstance that increases the running time. Still, SMGen can generate synthetic models with hundreds of species and reactions in less than 7 s.

The Use of a Game Theory Model to Explore the Emergence of Core/Periphery Structure in Networks and Its Symmetry

In network systems characterized by complex interactions of various types, core-periphery structures can be found. In this paper, we deal with such questions as what processes can lead to the emergence of core-periphery formation, whether this structure is symmetric, and to what extent. Namely, the question of symmetry in a complex network is still the subject of intense research interest. Symmetry can relate to network topology, network relationships, and other processes on networks. To answer these questions, we modified the model of the classic social dilemma called the repeated prisoner’s dilemma (or repeated PD game) by adding the cost of maintaining relationships between the pairs of players (partners) and especially by adding the possibility of ending some relationships. We present the results of simulations that suggest that the players’ network strategy (i.e., partner selection or termination of relationships with some partners) is the driving force behind the emergence of a core-periphery structure in networks rather than the player’s strategy in PD. Our results also suggest that the formed core is symmetric, and this symmetry is a result of the symmetric interactions of core players. Our outcomes can help understand various economic or social questions related to creating centers or peripheries, including their symmetry in different network systems.

A Structured Table of Graphs with Symmetries and Other Special Properties

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly discovered, may find wide applications, for example, in design of network topologies.

Characterization of Symmetry of Complex Networks

Recently, symmetry in complex network structures has attracted some research interest. One of the fascinating problems is to give measures of the extent to which the network is symmetric. In this paper, based on the natural action of the automorphism group Aut ( Γ ) of Γ on the vertex set V of a given network Γ = Γ ( V , E ) , we propose three indexes for the characterization of the global symmetry of complex networks. Using these indexes, one can get a quantitative characterization of how symmetric a network is and can compare the symmetry property of different networks. Moreover, we compare these indexes to some existing ones in the literature and apply these indexes to real-world networks, concluding that real-world networks are far from vertex symmetric ones.

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.

Package Network Model: A Way to Capture Holistic Structural Features of Open-Source Operating Systems

Open-source software has become a powerful engine for the development of the software industry. Its production mode, which is based on large-scale group collaboration, allows for the rapid and continuous evolution of open-source software on demand. As an important branch of open-source software, open-source operating systems are commonly used in modern service industries such as finance, logistics, education, medical care, e-commerce and tourism, etc. The reliability of these systems is increasingly valued. However, a self-organizing and loosely coupled development approach complicates the structural analysis of open-source operating system software. Traditional methods focus on analysis at the local level. There is a lack of research on the relationship between internal attributes and external overall characteristics. Consequently, conventional methods are difficult to adapt to complex software systems, especially the structural analysis of open-source operating system software. It is therefore of great significance to capture the holistic structure and behavior of the software system. Complex network theory, which is adequate for this task, can make up for the deficiency of traditional software structure evaluation methods that focus only on local structure. In this paper, we propose a package network model, which is a directed graph structure, to describe the dependency of open-source operating system software packages. Based on the Ubuntu Kylin Linux Operating system, we construct a software package dependency network of each distributed version and analyze the structural evolution through the dimensions of scale, density, connectivity, cohesion, and heterogeneity of each network.

Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs

The generalization of Farey graphs and extended Farey graphs all originate from Farey graphs. They are simultaneously scale-free and small-world. A labeling of the vertices for them are proposed here. All of the shortest paths between any two vertices in these two graphs can be determined only on their labels. The number of shortest paths between any two vertices is the product of two Fibonacci numbers; it is increasing almost linearly with the order or size of the graphs. However, the label-based routing algorithm runs in logarithmic time O(logn). Our efficient routing protocol for Farey-type models should help contribute toward the understanding of several physical dynamic processes.

How Symmetric Are Real-World Graphs? A Large-Scale Study

The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

Thermodynamic characterization of synchronization-optimized oscillator networks

We consider a canonical ensemble of synchronization-optimized networks of identical oscillators under external noise. By performing a Markov chain Monte Carlo simulation using the Kirchhoff index, i.e., the sum of the inverse eigenvalues of the Laplacian matrix (as a graph Hamiltonian of the network), we construct more than 1,000 different synchronization-optimized networks. We then show that the transition from star to core-periphery structure depends on the connectivity of the network, and is characterized by the node degree variance of the synchronization-optimized ensemble. We find that thermodynamic properties such as heat capacity show anomalies for sparse networks.

On the existence of wavelet symmetries in archaea DNA

This paper deals with the complex unit roots representation of archea DNA sequences and the analysis of symmetries in the wavelet coefficients of the digitalized sequence. It is shown that even for extremophile archaea, the distribution of nucleotides has to fulfill some (mathematical) constraints in such a way that the wavelet coefficients are symmetrically distributed, with respect to the nucleotides distribution.