Structural and spectral analysis of complex biological and social networks

Abstract

Our life revolves around systems that are awfully complicated. Consider for example the society that we live in, is an agglomerate of billions of individuals associated with each other in diverse intricate relationships. The biological existence of each individual is further rooted in complex interactions between thousands of genes and proteins within the cells. Coordinated activity of numerous neurons in our brain govern our ability to reason and comprehend the world around us. At the technological front, mobile communication integrates billions of cell phones from distant parts of the world linked through computers, base stations and satellites. Thus, integrated with complexity of these systems also arises the accumulation of a huge volume of data that multiples every single day. Our every movement, may it be a quick email check, a credit card swipe, navigation through global positioning system (GPS), and so on, leaves behind traces of digital records. Moreover, with the outburst in scientific advancements, there has been accumulation of massive amounts of data emerging from various fields ranging from the cosmology, meteorology, transportation system, society, biology to language, which remains unattended. This led to the increasing need of a framework that can study the collective behavior of individual components locked up in complex interaction patterns. The emergence of network science at the dawn of 21st century, as a means to map these interaction patterns, proved to be a vivid description of scientific excellence which transformed our understanding of the systems around us. When pulled together under the framework of networks, the evidently complex data offered comprehensive pictures of both individuals and groups, in a fashion that was barely conceivable earlier.Network science has shown tremendous success in modeling and understanding complex phenomena in various model and real world systems for the past few decades and thus has turned out to be an emerging field of research. Complexnetworks appear in extremely diverse contexts, for instance, in understanding the spread of behavior across a population, assessing organizational performance, outbreak detection of disease propagation, etc. Yet many of these networks appear to share certain nontrivial, similar patterns in connections between their elements. Understanding the origins of these patterns and identifying and characterizing new ones is one of the main driving forces for research in complex networks. The dominant objective of network science has been on developing universal theories and models that transcend syst em-specific details and describe the different systems in a meaningful yet statistical sense.This field has not only provided a wide range of tools and techniques for analyzing the data but has also enabled us to draw meaningful interpretations out of the visually complex data, with prime focus on the interaction behavior among the individual units. The success of network theory in the past few decades lies in providing an in-depth understanding about the structural and dynamical properties of various real world complex systems. However, most of the studies performed so far have extensively used various structural attributes of networks, with very little emphasis being laid on spectra. The set of eigenvalues of a network, also termed as network spectrum is known to provide information about some basic topological properties of the underlying network, which is beyond the purview of structural analysis of networks alone. For instance, degeneracy at zero eigenvalue exhibits a direct relation with the presence of duplication (either complete or partial) in the corresponding networks.Further, we probe for characterizing systems by analyzing spectral fluctuations of the corresponding networks under random matrix theory (RMT) framework. RMT, initially proposed byWigner to study statistical properties of nuclear spectra, has later shown wide applicability, for instance in stock-market indices, atmospheric statistics, human EEG, etc. On one hand, at the fundamental level, RMT helps in relating the spectral fluctuations with the associated structural changes, for example while attaining a random network from a regular lattice through edge rewiring (Fig. 1), transition from Poisson to GOE statistics happens exactly at the small world transition, a state known for its distinctive structural signature. On the other hand, tools of RMT help in capturing and quantifying the extent of randomness present in real systems, such as in case of Alzheimer’s disease. Research work pertaining to this thesis aims at simplifying the complexity of various real-world biological and social systems under the conjoined framework of network theory, spectral graph theory and RMT. The results obtained through our systematic investigations is expected to enhance the understanding pertaining to the impact of interaction behavior on the network architecture and node attributes, as well as enable us to characterize different systems based on their structural and spectral signatures.