Assortative and disassortative mixing investigated using the spectra of graphs

Eigenvalue ratio statistics of complex networks: Disorder versus randomness

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (w_{c}) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate w_{c} with the time taken by maximum entropy random walker to reach the steady state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behavior.

Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes

Cooperative self-assembly is a ubiquitous phenomenon found in natural systems which is used for designing nanostructured materials with new functional features. Its origin and mechanisms, leading to improved functionality of the assembly, have attracted much attention from researchers in many branches of science and engineering. These complex structures often come with hyperbolic geometry; however, the relation between the hyperbolicity and their spectral and dynamical properties remains unclear. Using the model of aggregation of simplexes introduced by Šuvakov et al. [Sci. Rep. 8, 1987 (2018)2045-232210.1038/s41598-018-20398-x], here we study topological and spectral properties of a large class of self-assembled structures or nanonetworks consisting of monodisperse building blocks (cliques of size n=3,4,5,6) which self-assemble via sharing the geometrical shapes of a lower order. The size of the shared substructure is tuned by varying the chemical affinity ν such that for significant positive ν sharing the largest face is the most probable, while for ν<0, attaching via a single node dominates. Our results reveal that, while the parameter of hyperbolicity remains δ_{max}=1 across the assemblies, their structure and spectral dimension d_{s} vary with the size of cliques n and the affinity when ν≥0. In this range, we find that d_{s}>4 can be reached for n≥5 and sufficiently large ν. For the aggregates of triangles and tetrahedra, the spectral dimension remains in the range d_{s}∈[2,4), as well as for the higher cliques at vanishing affinity. On the other end, for ν<0, we find d_{s}≂1.57 independently on n. Moreover, the spectral distribution of the normalized Laplacian eigenvalues has a characteristic shape with peaks and a pronounced minimum, representing the hierarchical architecture of the simplicial complexes. These findings show how the structures compatible with complex dynamical properties can be assembled by controlling the higher-order connectivity among the building blocks.

Spectral properties of complex networks

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Understanding cancer complexome using networks, spectral graph theory and multilayer framework

Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer analysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine.

Evolution of correlated multiplexity through stability maximization

Investigating the relation between various structural patterns found in real-world networks and the stability of underlying systems is crucial to understand the importance and evolutionary origin of such patterns. We evolve multiplex networks, comprising antisymmetric couplings in one layer depicting predator-prey relationship and symmetric couplings in the other depicting mutualistic (or competitive) relationship, based on stability maximization through the largest eigenvalue of the corresponding adjacency matrices. We find that there is an emergence of the correlated multiplexity between the mirror nodes as the evolution progresses. Importantly, evolved values of the correlated multiplexity exhibit a dependence on the interlayer coupling strength. Additionally, the interlayer coupling strength governs the evolution of the disassortativity property in the individual layers. We provide analytical understanding to these findings by considering starlike networks representing both the layers. The framework discussed here is useful for understanding principles governing the stability as well as the importance of various patterns in the underlying networks of real-world systems ranging from the brain to ecology which consist of multiple types of interaction behavior.

Interplay of degree correlations and cluster synchronization

We study the evolution of coupled chaotic dynamics on networks and investigate the role of degree-degree correlation in the networks' cluster synchronizability. We find that an increase in the disassortativity can lead to an increase or a decrease in the cluster synchronizability depending on the degree distribution and average connectivity of the network. Networks with heterogeneous degree distribution exhibit significant changes in cluster synchronizability as well as in the phenomena behind cluster synchronization as compared to those of homogeneous networks. Interestingly, cluster synchronizability of a network may be very different from global synchronizability due to the presence of the driven phenomenon behind the cluster formation. Furthermore, we show how degeneracy at the zero eigenvalues provides an understanding of the occurrence of the driven phenomenon behind the synchronization in disassortative networks. The results demonstrate the importance of degree-degree correlations in determining cluster synchronization behavior of complex networks and hence have potential applications in understanding and predicting dynamical behavior of complex systems ranging from brain to social systems.

Reconstruction of evolved dynamic networks from degree correlations

We study the importance of local structural properties in networks which have been evolved for a power-law scaling in their Laplacian spectrum. To this end, the degree distribution, two-point degree correlations, and degree-dependent clustering are extracted from the evolved networks and used to construct random networks with the prescribed distributions. In the analysis of these reconstructed networks it turns out that the degree distribution alone is not sufficient to generate the spectral scaling and the degree-dependent clustering has only an indirect influence. The two-point correlations are found to be the dominant characteristic for the power-law scaling over a broader eigenvalue range.