This report provides an overview of recent development is structural and spectral properties of protein-protein interaction networks. PPI networks have scale free degree distribution, short diameter and very high clustering coefficient. All these have been known to contribute in functionally of networks. Adjacency matrix of PPI networks differs drastically from a random matrix, as former have only binary enters. Remarkably, short-range correlations in eigenvalues of such matrices exhibit Gaussian orthogonal ensemble statistics of RMT and thus bring them into universality class. Spectral rigidity provides measure of randomness in the underlying PPI networks.
The ample availability of data in functional genomic and proteomic information, and the development of high-throughput data-collection techniques, has resulted from basic gene-based traditional molecular biology to a systems approach of network biology . These complex biological systems and their information about genes, proteins, cells, tissues and organs require mathematical models and algorithms that can derive structural and functional relationship at all levels of biological organization .
In network biology, biological processes are considered as complex networks of interactions between the cell's numerous components rather than as independent interactions involving only a few molecules . With this development, some researchers have successfully constructed various biological networks like the gene- gene interaction network, protein- protein interaction network, the disease network, the phenotype network and the drug-target network, etc .
Proteins are known to be the structural elements of the biological system. They act as catalysts, messengers and form the basis of the functioning of the cell. And so the Protein'protein interactions (PPIs) are important in monitoring the events in a cell, in studying various signal and transcriptional pathways . PPI network are the networks in which proteins are the nodes and their connections are edge. Their connection can be physical or functional both. Research on PPI network has been done extensively in recent years. Analyzing the pattern of interaction of proteins in the cell can be used to understand how complex processes lead to diseases.
Proteins also known to be central workhorses also contribute in the development of new medicines by understanding the biological insights of a disease . The information can be utilized in discovering new molecular medicines, which can be designed to either cure the illness or alleviate its symptoms . PPI networks have paved the way in functional regulation revealing most of the biological activity in living cells. This has helped to analyze the binding sites of a protein and its mechanism to see the regulation in the functioning of the cell. . There are various other remarkable discoveries by which PPI networks are been exploited through the concept of network biology.
In this report, we review what actually a protein-protein interaction network is, why we use these networks, what potential area where these networks are being studied, structural implications of the network, their spectral properties and future prospects of studying the PPI networks.
2. Why study protein-protein interaction networks?
PPI networks give us an idea of the type of interactions proteins have and the mechanism involved with it. Exploiting these interactions gives us a direction to model cellular pathways like regulation , signaling  and metabolism . These pathways as a whole or even a part of it can be taken as the natural candidates for disease markers, and may also be the drug targets . Proteins form complexes and interact to regulate other proteins (up regulate or down regulate), to move other proteins within the cell. These interactions are the mechanisms by which the signaling pathways work .
Protein interaction networks (PINs) are widely considered as crucial contributors in two different ways: (a) characterization and understanding of biological processes and their abnormal functions in various diseases and (b) by providing a framework for the design of specific drugs  . Decades of focused analysis have revealed the significance of local PINs in biological processes. Some very important examples of such local PINs are protein complexes, where the multiple proteins combination give rise to functional units . However, the detailed knowledge about these protein complexes and interactions are still uncovered with respect to their relevance and functional consequences .
Networks, on the whole provide us an insight into identifying the various pathways involved in the diseases by taking into account the interactions between the vast arrays of proteins taking part in the processes . Analyzing the network parameters to deduce whether the PPI Networks taken into consideration exhibit scale-free nature or random nature is yet another aspect to be explored . Plotting the distribution of degree with respect to the number of nodes in the network might draw a clear picture of the underlying network properties and help us unravel the insights of these disease networks .
Research on PPI networks can broadly be characterized into two categories; the first one involving structural properties and second one, which deals and emphasize on the importance of spectral properties. In the following let us discuss these two frameworks in details.
3. Structural parameters used for PPI networks and important structural implications
Network theory provides a quantifiable description of networks. There are several network measures that enable the comparison and characterization of complex networks.
3.1 Connectivity (or) Degree: This is the most basic characteristic of a node. Degree (k) represents the number of links or connections the node has with the other nodes in the network. Degree distribution of PPI networks exhibits the power law behavior in most of the PPI networks. Power law behavior is known to arise from preferential attachment 
3.2 Degree distribution The degree distribution, P(k), gives the probability that a selected node has exactly k links. P(k) is obtained by counting the number of nodes N(k) with k = 1, 2, ... links and dividing by the number of nodes N. The degree distribution allows distinguishing between various network topologies  .
3.3 Clustering Coefficient The clustering coefficient was first coined by Watts and Strogatz . The clustering coefficient, C, for a particular node shows the pattern of connectivity with the neighbors of that given node, also known as cliquishness. The average clustering coefficient for all nodes in a network is taken to be the network-clustering coefficient. For an undirected graph, if a node vi has ki neighbors, the vertices can have ki (ki - 1)/2 edges within the neighborhood (Ni) . The clustering coefficient for an undirected graph G (V, E) (where V represents the set of vertices in the graph G and E represents the set of edges) can then be defined as:
The average clustering coefficient characterizes the overall tendency of nodes to form clusters or groups. C (k) is defined as the average clustering coefficient for all nodes with k links . PPI networks have very high cluster coefficient than the corresponding random networks . As cliques are known as building block for any complex systems, high value of CC in PPI networks provide clue about evolution of PPI networks in complex species 
3.4 Network Diameter: The network diameter d is the largest distance (shortest path, or geodesic path) between any two nodes in a network . It can also be viewed as the length of the 'longest' shortest path in the network, where dG(u, v) is the shortest path between u and v in G . A few authors have also used this term to denote the average geodesic distance in a network (which translates to the characteristic path length), although strictly the two measures are distinct . PPI networks are known to have small diameter, which is one of very important properties for information transmission in a networks. Small diameter in PPI networks suggests a fast communication between all the proteins, even when system size is very large .
3.5 Betweenness Centrality: Betweenness centrality is a measure of a vertex within a graph . For a graph G (V, E), with n vertices, the betweenness CB(v) of a vertex v is defined as where ??st is the number of shortest paths from s to t, and ??st(v) is the number of shortest paths from s to t that pass through a vertex v . A similar definition for 'edge betweenness' was given by Girvan and Newman . Nodes with a higher betweenness lie on a larger number of shortest paths in a network. PPI networks have been known to have few nodes with very high between ness centrality. Incidentally these proteins are also hubs, and have are involved in large number of interactions, and hence are very important .
4. Research work on structural analysis of PPI networks
Some novel tools in network analysis provide the possibility to count the important interacting proteins in huge size networks as well as proteins that connect them as neighbors . Genomic information plays a crucial role in understanding the severity of a disease by just analyzing the changes. Many diseases have been found to have aberrant protein-protein interactions, either due to the loss of essential and important interaction or by forming a protein complex at an inappropriate time or location . There exist experimentally validated PPI information for construction of networks of Multiple Sclerosis (MS) and Alzheimer disease (AD). The study aimed to assess whether the parameters of degree and betweenness, two fundamental measures in network theory, are properties that differentiate between node proteins and neighbors of these proteins in MS and AD. Specific features of these node proteins were revealed, whereby showing a lower average degree in both diseases and tissues, and a higher betweenness in AD and MS networks .
The analysis of hereditary disease revealed that the PPI network of hereditary disease-genes network are characterized by a larger degree, disease-genes, more common neighbors and quick communication to each other whereas these properties are not seen from the network identified from high-throughput yeast two-hybrid mapping approach and predicted interactions in PPIs network .
Molecular pathogenic mechanisms in Neurodegenerative disorders (NDDs) were investigated using PPI networks, the domain characteristics commonly identified in NDDs and correlation among NDDs based on domain information. They collected data on 201 interacting proteins and 13 compounds interactions from the literature. They found 19 proteins common to these six NDDs. These common proteins were mainly involved in the apoptosis and MAPK signaling pathways .
A novel method that combines protein structure information with protein interaction data to identify residues that form part of an interaction interface has been proposed. This prediction method can retrieve interaction hotspots. The method was applied to all mutations in the Online Mendelian Inheritance in Man (OMIM) database, predicting mutations to be related to an interaction defect. They concluded that mutations affect protein interactions in general .
There has been enormous research on applications of protein networks to disease in four major areas: identifying new disease genes; the study of their network properties; identifying disease-related sub networks; and network-based disease classification. For prediction of new diseases genes, they have concluded the idea, that proteins close to one another in a network cause similar diseases. This notion became an increasingly important factor in the hunt for disease genes .
Recent analysis suggests that PPIs are essential to define the molecular networks that contribute to maintain homeostasis of an organism's body functions. Disruptions in PINs have been shown to result in diseases in both humans and animals. Diseases disrupting biochemical pathways such as hereditary coagulopathies (e.g. hemophilia), provided a deep insight in the biochemical pathways of acquired coagulopathies of complex diseases. A variety of complex liver diseases can lead to decreased synthesis of the same set of coagulation factors as in hemophilia .
Complex diseases such as different cancers have been shown to result from malfunctions of common proteins pathways. In order to discover, in high throughput, the molecular underpinnings of poorly characterized diseases, they have presented a statistical method to identify shared protein interaction network(s) between diseases. Significant correlations between diseases and shared protein networks were identified and evaluated in this study, demonstrating the high precision of the approach and correct non-trivial predictions, signifying the potential for discovery. In conclusion, they have demonstrated that the associations between diseases are directly correlated to their underlying protein-protein interaction networks, possibly providing insight into the underlying molecular mechanisms of phenotypes and biological processes disrupted in related diseases .
5. Spectral properties of protein-protien interaction networks
Recently focus of networks research is shifted to investigate spectral properties of the corresponding adjacency matrix. The adjacency matrix of a network is defined as A(i, j) where, i and j are the nodes of the network. When, A(i, j) = 1 nodes i and j are said to be connected and vice versa if A(i, j) = 0. Spectral properties of protein networks help to understand the functional and structural aspects of the underlying complex systems.
5.1 Nearest neighbor spacing distribution: Eigenvalues of the PPI networks are used to find short-range and long-range correlations between the nodes. The short-range correlations can be calculated by nearest neighbor spacing distribution (NNSD) by unfolding the eigenvalues and polynomial fitting the curve . We calculate Brody parameter, which decides weather the network, follows GOE statistics and predicts the randomness in the network. The Brody distribution is given as:
P (s) = As?? exp ('??s??+1
Where value of beta being one indicates GOE statistics. PPI interaction networks have been shown to follow GOE statistics of NNSD as beta value comes close to 1.
5.2 Long range spectral correlations: The long-range studies of eigenvalues are analyzed by Delta 3 statistics, for GOE statistics is given as:
'_3 (L)~1/??^2 ln'L
Which provides information about the amount of randomness in the underlying network.
Less is the value of L more random is the network. Study of eigenvalues helps in predicting the evolutionary pattern of various proteins between species . Delta_3 statistics of PPI networks are shown to follow RMT prediction till a certain value of L, afterwards, which it deviates from the prediction.
5.3 Eigenvector analysis: Eigenvector analysis is done to study the localization properties in a protein-protein interaction network. Inverse participation ratio is given as:
I^k=('_(l=1)^N '[E_l^k]'^4)/(('_(l=1)^N''''[E'_l^k]'^4)' 2)
This approach helps to identify certain specific proteins that play a significant role in analyzing structural and functional properties of the network. This approach is very recent and has been used to predict functional aberrations in the gene expression network of zebra fish under different toxic perturbation . This approach can further be extended to PPI networks in order to solve various medical mysteries in healthcare industry.
6. Conclusion and future plan
We find that the protein-protein interaction networks allow us to do a systems approach rather than the old reductionist approach. The PPI networks have degree distribution following power law behavior, which is one of the most common properties found with many other complex systems. In addition to this, PPI networks have small diameter, which has been emphasized to play crucial role in information transmission in the system. Furthermore spectral properties of PPI networks are shown to follow GOE statistics of RMT, which suggests that all PPI networks of different species have some minimal amount of randomness, which might be important for robustness. These properties and approach can be used to study the evolution, the system behavior (random or scale free or small world), interaction of different proteins, role in different proteins in chemical reactions and pathways, comparing normal and diseased state to find some important proteins, to designing potential drug targets to implement single drug therapy for various diseases.
Further, studying PPI networks is a burning area in which a lot of studies are done and in the near future with the wet and dry lab mixture it can solve various difficult problems, which are still not been overcome. My immediate plan to investigate protein-protein interaction network of disease networks by studying their structural and functional significance. Further calculating localization properties of various PPI to identify top contributing node (proteins) for their role in disease and normal state of a cell to detail study them and identify faulty pathways, potential sights in a protein by studying its structure for drug targeting and study the evolutionary pattern of the emergence of disease.