Spectral analysis of normalized graph Laplacian can reveal inheritance structural properties of a network and can be an important feature of the landmark to solve the structural problem. From spectra of normalized Laplacian matrix, I am developing a tool that helps to understand the network structure with deep perception so that we could recognize the source of the network. I am exploring the information about different topological properties of a graph carried by complete spectra of normalized graph Laplacian and investigating how and why structural properties are reflected by the spectra and how spectra change according to different networks from different sources. For large network, not only the particular eigenvalue carries information about the structure but also the density of the eigenvalues at particular points carry many information about the same. This study reflects that spectral distribution is an important characteristic of a network and an excellent diagnostic to categorize the different networks from different sources.
What could be the evolutionary process behind the formation of similar structures? There is interplay between dynamics of the network and inheritance structure. So evolutionary processes that are responsible for the construction of the network could be studied from spectra of connectivity matrix. Different graph operation related to evolution of a network produce specific eigenvalue. Construction with those operation describe certain processes of graph formation that leave characteristic traces in the spectrum. So useful plausible hypothesis about evolutionary process could be made and it would be easy to take decision about the evolutionary assumption that is more relevance for the evolution of that system by investigating the spectra of a graph constructed from actual data. Based on this idea any network could be reconstructed with very similar structure.
Applying a meaningful distance measure, we show that network structures are more similar within the same class than between classes. We analyze metabolic networks from different species and find a separation of the three domains, Bacteria, Archaea and Eukarya. Additionally, by using our network-based method, we aim at the understanding of evolutionary relationships between species that could complement other methods based on rRNA sequence and enzyme content.
Many networks in nature are directed. So far no appropriate tool is available to study systematically the structure of large directed networks. I am exploring the architecture of directed graphs using the eigenvalues of the normalized graph Laplacian. The information or mass flow in different biological systems are not the same. I am trying to understand the biological significance of having specific pattern of the flow in a network constructed from a particular domain by dividing the topology in different layers and parts each of those has different structural properties and plays diverse roles in fulfilling the purposes of the network. I am investigating how the eigenvalue of the Laplacian can be used to locate the vertices in the different layers and to explore the connectivities between those layers. Using different classes of biological networks, I am showing that the spectra of the networks are similar within the same class, but have significant differences between the classes. Consequently, the spectrum graph Laplacian can be used as a good classifier for different derected networks.
The functional connectivity pattern in human brain for diverse memory task is not fully explored yet. The way these brain functional networks evolve when a human get older is more interesting to analyze. I am investigating the changes in the connectivity pattern of the human brain memory functional network in the effect of aging. For this purpose I am exploiting the fMRI data of the co-activating areas (Voxel of size 8 cubic mm and measured by BLOD signal) of human brain of several young, mid-age and old persons for different memory tasks.
Last modified: Feb. 23, 2011
Anirban Banerjee, Department of Mathematics & Statistics and Department of Biological Sciences, Indian Institute of Science Education and Research, Kolkata