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(1) Developing methods for the ** Data-Driven Analysis ** of ** Complex Systems ** in
nature and society. The theoretical frameworks I use for the purpose are ** Networks**,
** Nonlinear Dynamics ** and ** Applied Statistics ** (statistics of extreme/rare events,
statistical/machine learning, different flavors of regression).

(2) Mathematical and Computational ** Modeling ** of systems involving the
interplay of ** Networks, Nonlinear Dynamics, and Stochastic Processes.**

Below I list some of my specific research interests.

##### In my research, I have developed methods for the analysis of many different kinds of data sets—ranging from few hundred data points in a short time series to gigantic spatiotemporal data sets from various domains of study. Two of my favorite theoretical tools for the purpose has been the nonlinear time series analysis and the theory of complex networks.

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__Networks __epitomize many real world systems, where nodes represent different
elements of a system and edges represent connections between the elements.
The range of systems studied as networks is enormous. For example, the brain is a network of neurons;
society consists of networks of people; the network of web pages is what we call the world wide web.
The analysis of networks requires approaches drawn from many disciplines and the __Network Science__
is the name given to the compilation of these approaches.

__Networks__

__Network Science__

##### The straightforward application of network science to the problems in data analysis is when given data sets contain information about relationships between objects or agents. For instance, data sets of online social networks or networks of genes or proteins in biology. Another approach in the application of network science to data analysis is when underlying networks can be inferred from spatiotemporal or image data sets, for example in neuroscience using fMRI data. This approach is not only limited to cases where there is an underlying physical network but also include cases where the existence of a network is a theoretical abstraction. For example, climate networks or network representation of time series data into a network.

##### My research interests span both the above approaches. Please see following publications 1,2,3 for more detailed information. One of the significant examples worth stating here is the technique we developed to analyze large spatiotemporal rainfall data sets. We used this particular technique to analyze rainfall over South Asia during the summer monsoon. We constructed networks out of this data by first estimating nonlinear correlations between different grid points spread over the spatial region. For the purpose, we also introduced a new measure for correlations called the event synchronization (ES) [2,4]. This study provided some valuable insights into the spatial organization, scales, and structure of the extreme rainfall events during the summer monsoon.

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** **__ Nonlinear Time Series Analysis __ is the name given to the collection of
tools which use theories from nonlinear dynamical systems to solve
problems in time series analysis. This discipline is the one where
the theory of dynamical systems meets the data. The underlying
philosophy of nonlinear time series analysis is that nature is
inherently nonlinear and dynamical and data coming from observing
nature and its phenomena should be probed using the theories of
nonlinear dynamics. My primary interest in nonlinear time series
analysis is the following problem. Given a time series originating
from a nonlinear dynamical system (most natural and social systems
will satisfy this assumption);
then how to identify and predict dynamical transitions (qualitative shifts in dynamics) and instabilities in this system without constructing an explicit mathematical model for it.

__Nonlinear Time Series Analysis__

##### One of our attempts to solve this problem included proposing an approach based on the fluctuation of similarity (FLUS) [5,6]. The new similarity measure introduced in this method uses the recurrence properties of time points. We also developed a corresponding statistical test to estimate the significance of the identified transitions. To further verify the workings of the method we applied it to many known nonlinear models, and it was able to uncover bifurcation structures in them [5,6]. It even out performed the traditional Lyapunov exponents in some cases, revealing many complex transitions. In a real world situation, we applied this method to identify millennial-scale dynamical transitions in the Pleistocene proxy records of the Asian monsoon system [6].

##### We have also used above method FLUS (Fluctuation of Similarity), to identify transitions in rates of robberies and homicides in the US between 1975- 1993 and has been able to show that unemployment rate till the late 1980s controlled the dynamical complexity in robberies [5]. Furthermore, we identified a transition in homicide and robbery rates in the late 1980s and early 1990s [5].

##### I have always been fascinated by the patterns that emerge in computational simulations of nonlinear dynamical systems especially generating fractals using simple difference equations. Some of the images in this section of the page are created using similar equations. I have been attempting to use them in analyzing data and finding out patterns in the data. One of my favorite tools in nonlinear time series analysis is Recurrence plot (RP). Where one can convert a time series data into a binary image, spatial structures emerging in these RP's can be quantified using recurrence quantification analysis. In recent years methods from complex networks have also been used to analyze these spatial structures.

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** **__ Extreme or Rare Events __ such as floods, droughts, cyclones, or forest fires are of massive socioeconomic significance.
With changing climate an increase in frequency and variability of such events has been observed all over the planet.
My interest is in understanding and characterizing the new emerging patterns in extreme precipitation/rainfall events over monsoonal
regions of the world from historical climate data sets.

__Extreme or Rare Events__

##### Monsoons are not only among the most prominent and dynamic phenomena of the climate system manifesting over large parts of the tropics but also forms the lifeline of several regions of the world. It has been argued that the basic origin of the monsoons lies in the differential heating of the land and the sea during the summer season, which results in setting of a positive moisture advection feedback leading to widespread rainfall. The release of latent heat in the processes of precipitation over the land provides the feedback for maintaining this temperature gradient thus sustaining monsoonal circulation. One of the region of interest in my research has been South Asia, which receive roughly 80% of the total annual rainfall during summer monsoon months [2,4,7].

##### In one of our recent work employing quantile regression we quantified spatiotemporal patterns and trends that have emerged during the recent decades over South Asia under changing climatic conditions [7]. Our analysis helped in explaining previously contradicting results of trends in average monsoonal rainfall over South Asia.

__References:__

[1] I. Barnett, N. Malik, P. Mucha, J.P. Onnela Feature-Based Classification of Networks. arxiv:1610.05868

[2] N. Malik, B. Bookhagen, N. Marwan, J. Kurths: Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks, Climate Dynamics, 39, 3-4, 971-987, (2012).

[3] R. M. McGranaghan, A. Mannucci, O. Verkhoglyadova , N. Malik Finding Multi-scale Correlations in our Geospace Observational System: Network Analysis of Total Electron Content. Journal of Geophysical Research: Space Physics, 122 (2017)

[4] N. Malik, N. Marwan, J. Kurths : Spatial structures and directionalities in Monsoonal precipitation over South Asia, Nonlinear Processes in Geophysics , 17(5), 371–381, (2010).

[5] N. Malik, N. Marwan, Y. Zou, P. Mucha, J. Kurths: Fluctuation of similarity (FLUS) to detect transitions between distinct dynamical regimes in short time series, Physical Review E , 89, 062908 (2014).

[6] N. Malik, Y. Zou, N. Marwan, J. Kurths: Dynamical regimes and transitions in Plio-Pleistocene Asian monsoon, Europhysics Letters, 97, 40009, (2012).

[7] N. Malik, B. Bookhagen, P. Mucha: Spatiotemporal patterns and trends of Indian monsoonal rainfall extremes, Geophysical Research Letters, 43 (4), 1710-1717, (2016)

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__ Complex Systems __ are one of the most pervasive types of systems in nature and society.
In most general case complex systems consist of three ingredients: nonlinearity, stochasticity,
and intricate interactions among individual units of the systems. One of the primary challenges
in modeling complex systems is the fact that __ Nonlinear Dynamical Systems __ interacting
through an intricate web of connections tend to behave differently than when observed
as individual units [1,2,3]. Hence, analytical tools used in the analysis of low
dimensional nonlinear dynamical systems are of limited usefulness in studying complex systems.
One of the significant development in studying complex systems has been the __ New Science of
Networks__. Network theory provides the powerful mathematical framework to represent and analyze
the intricate connections among individual units of a complex system. Another ubiquitous component in
complex systems with far reaching consequences is noise. A complete and accurate study of complex systems
must take into account noise and include methods of __Stochastic Processes__ in the analysis.

__Complex Systems__

__Nonlinear Dynamical Systems__

__New Science of Networks__

__Stochastic Processes__

##### One problem within the realms of complex systems that excites me a lot is the role of network topology or structure in altering various types of dynamics, both in the noisy and noiseless environments. An answer to this problem can lead to a proper mathematical understanding of the effect of different macro and micro level network structures on nonlinear dynamical systems. In search of the answer, with my colleagues, I have modeled processes involved in the spread of biological and social contagions and noise induced phenomena in networks [3,4,5,6].

##### The most significant work I have done in this direction is the modeling of collective opinion formation on co-evolving networks [3,4]. The dynamic model I have used for the purpose is the well known voter model. In my work, I have shown that transitivity/clustering in a network can play a critical role in altering the dynamics of voter model simulated on a co-evolving network. I have also been attempting to use a similar model to analyze online social network data coming from Facebook and Twitter.

__References:__

[1] N. Malik, B. Ashok, J. Balakrishnan Noise-induced synchronization in bidirectionally coupled Type-I neurons European Physical Journal B , 74, 177, (2010)

[2] N. Malik, B. Ashok, J. Balakrishnan Complete synchronization in coupled Type-I neurons Pramana : Journal of Physics , 74, 2, 189-205, (2010)

[3] N. Malik, P. Mucha Role of social environment and social clustering in spread of opinions in co-evolving networks Chaos, 23, 043123, (2013)

[4] N. Malik, F. Shi, H-W. Lee, P. Mucha Transitivity reinforcement in the coevolving voter model, Chaos 26, 123112, (2016)

[5] H-W. Lee, N. Malik, P. Mucha Evolutionary Prisoner’s Dilemma Games Coevolving on Adaptive Networks Journal of Complex Networks, cnx018, (2017)

[6] H-W. Lee, N. Malik, F. Shi, P. Mucha Social Clustering in Epidemic Spread on Coevolving Networks. arxiv:1707.04933 (2017)