15 research outputs found
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
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
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 tend to
appear an even number of times in the prime factorization of a given integer,
more so than for primes of the form . 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
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
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