Abstract:

We often come across many networks in our daily lives, for example, the electronic communication network, surface and air transport networks etc. In addition, international trade among different countries defines the economic trade networks, biologists define the protein interaction network and gene regulatory network as the examples of biological networks etc. Therefore, quite expectedly, ’Study of Complex Networks’ has been recognized as a multidisciplinary topic.

During the last decade, extensive research efforts have been devoted towards the study of structure, function and activities of different complex networks. Many new information and properties of these networks have been known. Though these networks are random in nature, yet it has become increasingly apparent that the well-known model of Graph Theory like Random Graphs (RG)’ is no more appropriate to describe these complex networks. In particular, unlike RG, many networks have been observed to have power law degree distributions, (degree of a nodes is the number of links meeting at that node) and consequently few very large degree nodes, called as ’hubs’. Such networks are called ’Scale-free Networks’ since they lack a characteristic value for the nodal degrees in the asymptotic limit of very large sizes. Further, it has also been realized that the efficiency of transport networks has been hidden in their ’Small-world’ properties. In the present thesis, we report the study of complex networks and related phenomena from the point of view of Statistical Physics from different disciplines of science.

A. Econophysics:

i. Modeling the structure and properties of a Trade Network

The evolution of economic status of a society takes place in terms of mutual trades among its different members, they may be individuals or corporates. To understand the intricacies of the trade dynamics it is necessary to understand the underlying network of mutual trades among different trading members. When a pair of traders take part in a mutual business, a trade relationship is established between them. We have studied a model of trade network where each individual trader or corporate is a node of the network and when two such members take part in a mutual business, a link is established between the corresponding nodes. Using a model of wealth distribution, where traders are characterized by their individual quenched random saving propensities and trade among themselves by bipartite transactions, we mimic the enhanced rates of trading of the rich by introducing the preferential selection rule using a pair of continuously tunable parameters. The bipartite trading defines a growing trade network of traders linked by their mutual trade relationships. With the preferential selection rule this network appears to be highly heterogeneous characterized by the scale-free nodal degree and the link weight distributions and presents signatures of nontrivial strength-degree correlations (Phys. Rev. E, 81,016111 (2010)).

ii. Modeling the self-organized critical evolution of the wealth distribution in a society

In a society, all the individual members tend to improve their economic status. However, poorer the member more is the social pressure felt to uplift its economic condition and consequently the poorest agent feels the strongest pressure. Using the framework of the Pianegonda et. al. model, we have studied a conservative self-organized extremal model based on the above observation with a stochastic bipartite trading rule. More precisely, in a bipartite trade one agent must be the poorest one and the other one is selected randomly from the neighbors of the first agent. The two agents then randomly reshuffle their entire amount of wealth without saving. This model is one of the few examples of non-dissipative self-organized critical systems where the entire wealth of the society is strictly conserved. We estimate a number of critical exponents which indicate this model is likely to be in a new universality class, different from the well established models of Self-organized Criticality. How long a typical agent has to wait to get a chance for a mutual trade? The time interval between two successive updates of an agent is referred as the ‘Persistence Time’ and it has been observed for the first time that in the stationary state it follows a non-trivial power law distribution (Fractals, 21, 163 (2012)).

B. Spreading Phenomena in Networks:

iii. SIS and SIR type disease spreading models with partial isolation on networks

Spreading of an infectious disease from an infected person to other susceptible individuals depends on the existing number of people in the contact neighborhood of the infected person in a population. If it is possible to maintain that the infected individuals are completely isolated, the disease would not spread in the society. However, in all practical cases, this kind of isolation is not perfect but only partial. Here, we studied the effect of partial isolation in disease spreading processes using the well-known models of susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models where individuals are located at the nodes of several graphs representing the contact networks in a society. In this model we impose a restriction: each infected individual can probabilistically infect only up to a maximum number n of his susceptible neighbors. Numerical study of this model shows that the critical values of the spreading rates for endemic states are non-zero in both models and decreases as 1/n with n, on all graphs including scale-free graphs. In particular, the SIR model on square lattice with n=2 found to be special case, characterized by a new bond percolation threshold (Fractals, 21, 1350015 (2013))

C. Long Range Connectivity in a System of Growing Discs:

iv. Network of growing discs in a plane leading to the Space-filling Percolation

Nature of transition in Explosive Percolation (EP) is studied extensively in recent years. Though wide class of EP models show very sharp change in their order parameters for finite size systems and appear to exhibit discontinuous transition are actually turned out to be continuous transition in the asymptotic limit of the large system sizes. We propose and study a variant of the continuum percolation (CP) model to exhibit a similar discontinuous-like continuous transition. A pattern of circular discs inside a square is generated by filling a large number of growing circular discs one by one at random position with ‘slight’ overlapping. More elaborately, every disc grows from a nucleation centre that is selected at a random location within the uncovered region. The growth rate δ is a fixed parameter of the model, which is continuously tunable. When a growing disc overlaps with at least another disc, it stops growing and is called to be ‘frozen’. Numerical simulation of the model shows the signature of a discontinuous-like-continuous transition similar to Achlioptas process. Critical area coverage at the transition point approaches to unity at δ → 0, implying the limiting pattern is space-filling. Fractal dimension of the pore space is found to be 1.42(10) and the contact network of the discs is found to be a scale-free network (Phys. Rev. E, 89, 032103 (2014)).

D. Properties of the Earthquake Network:

v. Weighted network analysis of earthquake seismic data

We have used the method of Abe et. al. to generate a weighted earthquake network associated with the time series of occurrences of the tremors over a long duration and the positions of their epicenters. Here, the entire earthquake region is digitized into a grid, where a cell represents a node if and only if at least one tremor occurs within this cell. In addition, a bond is drawn between every pair of successive events. In our analysis, the number of bonds between a pair of nodes is defined as the weight of the link connecting the nodes. Weighted network is useful to gain better insights about the structural properties and correlations present in the network. It is observed that different properties of the weighted network are quite different from those of their un-weighted counterparts (Physica A 433, 336 (2015)).