Complex networks with scale-free nature and hierarchical modularity

Generative mechanisms which lead to empirically observed structure of networked systems from diverse fields like biology, technology and social sciences form a very important part of study of complex networks. The structure of many networked systems like biological cell, human society and World Wide Web markedly deviate from that of completely random networks indicating the presence of underlying processes. Often the main process involved in their evolution is the addition of links between existing nodes having a common neighbor. In this context we introduce an important property of the nodes, which we call mediating capacity, that is generic to many networks. This capacity decreases rapidly with increase in degree, making hubs weak mediators of the process. We show that this property of nodes provides an explanation for the simultaneous occurrence of the observed scale-free structure and hierarchical modularity in many networked systems. This also explains the high clustering and small-path length seen in real networks as well as non-zero degree-correlations. Our study also provides insight into the local process which ultimately leads to emergence of preferential attachment and hence is also important in understanding robustness and control of real networks as well as processes happening on real networks.Comment: 7 pages, 9 figure

Decisive role of fluctuations in the resource dependency networks

Individual components of many real-world complex networks produce and exchange resources among themselves. However, because the resource production in such networks is almost always stochastic, fluctuations in the production are unavoidable. In this paper, we study the effect of fluctuations on the resource dependencies in complex networks. To this end, we consider a modification of a threshold model of resource dependencies in networks that was recently proposed, where each vertex can either be in a fit or a degraded state. We study how the "network fitness" is affected as the fluctuation size is varied. We show that, the relative value of the average production with respect to the threshold, decides whether the fluctuations are beneficial or detrimental to the network fitness. We further show that the networks with a homogeneous degree distribution, such as the Erdos-Renyi network, perform better in terms of fitness and also produce lower wastage than the Scale-Free network. Our work shows that, in the study of resource dependencies in networks, the role of the fluctuations is as decisive as the average production.Comment: 9 pages, 7 figure

Importance of initial conditions in the polarization of complex networks

Currently used models of opinion formation use random initial conditions. In reality, most people in a social network, except for a small fraction of the population, are initially either unaware of, or indifferent to, the disputed issue. To explore the consequences of such specific initial conditions, we study the polarization of social networks when conflicting ideas arise on two different seed nodes and then spread according to a majority rule. Using the configuration model and the stochastic block model as examples, we show that this framework leads to substantially different outcomes than those which employ random initial conditions. Moreover, the empirically observed splits in the karate and the dolphins' networks naturally come out of this paradigm. Our work thus suggests that the existing opinion dynamics models should be reevaluated to incorporate the initial condition dependence.Comment: Removed 1 figure, the final versio

Biases in prime factorizations and Liouville functions for arithmetic progressions

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of P\'olya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.Comment: 25 pages, 6 figure

On partial information retrieval: the unconstrained 100 prisoner problem

We consider the classical 100 Prisoner problem and its variant, involving empty boxes, introduced by Gal and Miltersen. Unlike previous studies, here we focus on the winning probabilities for an arbitrary number of winners and attempts, which we call the unconstrained problem. We introduce general classes of strategies, applicable to different settings and quantify how efficient they are. We make use of Monte Carlo simulations, whenever analytic results are not available, to estimate with high accuracy the probabilities of winning. Finally, we also comment on the possible applications of our results in understanding processes of information retrieval, such as "memory" in living organisms

A random interacting network model for complex networks

This paper was developed within the scope of the DAAD-DST PPP-Indien project 55516784 (INT/FRG/DAAD/P-215) which funded exchange visits between the two participating institutes. B.G. was supported by the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP. J.K. acknowledges financial support from the Government of the Russian Federation (Agreement No. 14.Z50.31.0033). S.M.S. would like to thank University Grants Comission, New Delhi for the financial assistance as an SRF. B.G. and A.R. thank Niklas Boers for stimulating discussions and comments.Peer reviewedPublisher PD

Effect of money heterogeneity on resource dependency in complex networks

Exchange of resources among individual components of a system is fundamental to systems like a social network of humans and a network of cities and villages. For various reasons, the human society has come up with the notion of \emph{money} as a proxy for the resources. Here we extend the model of resource dependencies in networks, proposed in [\href{https://journals.aps.org/pre/abstract/10.1103/PhysRevE.102.062304}{Phys. Rev. E {\bf 102}, 062304 (2020)}], by incorporating the notion of money so that the vertices of a network can sell and buy required resources among themselves. We simulate the model using the configuration model as a substrate for homogeneous as well as heterogeneous degree distributions and using various exchange strategies. We show that a moderate amount of initial heterogeneity in the money on the vertices can significantly improve the survivability of Scale-free networks but not that of homogeneous networks like the Erd{\H o}s-R{\'e}nyi network.Comment: 8 pages, 6 Figure