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## Applied Network Science

**Smart Grid**

**Project***: Novel approaches towards non-intrusive load monitoring*

A promising approach toward residential load management, non-intrusive load monitoring (NILM) is broadly described as extracting the power consumption profile of each appliance of a house from the aggregated power signal of all appliances. Depending on the availability of a dataset and appliances’ spec sheets, various machine learning algorithms and optimization-based methods have been used to address the NILM problem. Main challenges include chattering in the aggregated signal, the absence of a large training set, and the existence of appliances with close power consumption values. We have achieved state of the art results on multiple fronts of the NILM research.

We have proposed the first event-based optimization method for NILM. This method requires only a small dataset and outperforms existing optimization-based NILM methods, while having a lower computational complexity due its event-based mechanism as opposed to being sample-based. Furthermore, in our analysis, unlike the mainstream line of the NILM research, we have included state uncertainties, which is the likely reason for achieving the high accuracy. Diving deeper into the NILM problem, we have introduced novel learning-based algorithms for event detection and appliance mode extraction with 100% accuracy. The results have then been used in two different event-based classification NILM methods. Both proposed methods have proved to perform well in the presence of multi-mode appliances and the existence of various appliance modes with close power consumption values.

**Project:*** Distributed frequency control of microgrid*

Distributed cooperation and coordination within networked systems have become the focal point of research in a wide variety of scientific and engineering problems. Considering the problem of frequency and voltage synchronization in microgrids as a leader-follow consensus problem, the secondary control design can be conducted based upon distributed coordination theory in multi-agent systems. We have developed a distributed secondary frequency control scheme for an islanded ac microgrid under event-triggered communication.

We have proposed an integral type event-triggered mechanism by which each distributed generator (DG) asynchronously and periodically checks its triggering condition and determines whether to update its control inputs and broadcast its states to neighboring DGs. Event-checking time instants of the DGs is illustrated in the figure. In contrast to existing event-triggered strategies on secondary control of microgrids, under the proposed sampled-data based event-triggered mechanism, DGs need not be synchronized to a common clock and each individual DG checks its triggering condition periodically, relying on its own clock.

**Project:***Optimization of distribution networks via daily reconfiguration*

Details undisclosed. π

**Transportation Systems**

*Project: Traffic signal control in urban transportation networks*

Details undisclosed. π

*Project: Modeling disruption propagation in traffic networks*

With the high population density in urban areas, analysis and control of urban traffic networks have become a widely studied research topic. Traffic accidents are among the most important causes of disruption in traffic networks. Since the network is connected, an accident in a part of the network influences traffic flows in other parts of the network, which is reminiscent of notions of spread in network systems. In this project, influence of traffic accidents over various types of traffic flows is investigated both analytically and via simulations. Based on the findings, a node centrality metric will be introduced, which in essence quantifies disruption in the traffic network caused by an accident occurring at any given node.

**Opinion Dynamics**

*Project: Opinion dynamics in social networks*

One of the most active research areas of network science, opinion dynamics models aim to analyze the evolution of beliefs in a network by modeling how one’s belief is influenced by others. A carefully crafted model of opinion dynamics can help explain socio-psychological phenomena such as consensus or agreement, opinion polarization, and groupthink in a social network.

Inspired by the notion of agreement in opinion networks, e.g., in case of a jury deliberating on a verdict, a majority of articles on opinion dynamics models have been focused on explaining how a global agreement can be reached in a network of individuals. These models are based on the notion of conformity, which broadly refers to the tendency of an individual to act in such a way to fit in a group by adopting or touting what she perceives as the popular opinion. It is expected, as it happens in virtually every existing conformity-based model, that (conformist) individuals avoid extreme beliefs and agree on a middle ground position as time grows. Therefore, these models fail to explain some widely accepted social phenomena such as groupthink and opinion polarization in social networks, in which extreme beliefs can very well become popular.

Various types of cognitive biases, most notably those of the confirmation bias type, have been incorporated into conformity-based models to make them more realistic and better capture the evolution of opinions. The confirmation bias in a broad sense refers to the tendency of an individual to seek to confirm her preconceptions. This may be done via avoiding to be exposed to beliefs opposite to hers or selective perception or interpretation of the facts presented to her, among other means. The inclusion of the confirmation bias in conformity-based models has effectively broadened the research on the global agreement scenarios to more general scenarios where one or multiple consensus clusters could be reached. However, the emergence of popular extreme beliefs has still remained unjustified.

Some attempts have been made to modify classical conformity-based models to close this gap. A remarkable one is a model in which the notion of antagonism among individuals is incorporated. While antagonistic relationships can contribute to and justify shaping extreme beliefs, these beliefs have as well been observed to form in fully collaborative environments when groupthink occurs. In this project, we aim to develop a viable model of opinion dynamics capable of explaining all the aforementioned socio-psychological phenomena.

*Project: Opinion dynamics in cyber-social networks*

Details undisclosed. π

**Biological Networks**

*Project: Analysis of the quorum sensing process in bacterial networks*

Quorum sensing is a “chemical language” used by bacteria to sense their population and coordinate gene expression accordingly. Sensing of the population is performed in a distributed fashion. We have developed a dynamical model for quorum sensing that explains how coordination occurs, taking different factors into consideration, e.g., the bacteria’s energy, molecule production rate, and growth rate. Predictions of the bacteria’s behavior based on the developed model are consistent with those derived from experimental data.

*Project: Centrality measures for genetic regulatory networks*

Details undisclosed. π

**Formation Control**

*Project: Distributed formation control based on complex-valued Laplacian*

Details undisclosed. π

**THEORETICAL NETWORK SCIENCE**

**Centrality**

It is crucial, whether for attacking or defensive purposes, in a network system to be able to compare the centrality of different agents. There exist numerous centrality measures in the literature, all based on the graph representing the agents’ interconnections, including degree centrality, eigenvector centrality, Katz centrality, PageRank, and cross-clique centrality.

*Project: Agent centrality in networks with structured uncertainties*

In this project, we have been interested in developing a novel type of centrality indices that quantify the contribution of an agent in the network performance in the presence of uncertainty. What sets apart our agent centrality indices from the existing ones in the literature is that they account for both the interconnection structure of the network and the uncertainty structure. Our results assert that agents can be ranked according to this centrality index and their rank can drastically change from the lowest to the highest, or vice versa, depending on the structure of the uncertainty. This fact gives rise to fundamental trade-offs on network centrality in the presence of multiple concurrent network uncertainties with different structures.

*Project: Input centrality*

Details undisclosed. π

**Structural Properties of Distributed Averaging Dynamics**

Distributed averaging dynamics are widely used in the areas of distributed optimization and distributed learning. They have numerous applications in designing technological networks system, such as in the field of robotics and control when coordination and cooperation of mobile agents such as sensors and robots are of interest, and are also believed to reliably capture the dynamics of real world network systems, such as biological, economical, and social networks.

Like other distributed systems, in a network with distributed averaging dynamics, every agent communicates data with its neighbors. In general, communication channels may or may not be bidirectional, an agent’s neighbor set may evolve over time, and time may be continuous or discrete. Influenced by its neighbors, in the discrete-time case, the agent updates its state to a weighted average of the current states of its neighbors and its own current state. As an illustrative example, in a social network, every individual communicates with, and is influenced by, her family members, friends, coworkers, etc., that all together form her neighbors. Some of the neighbors have greater influence on the individual while some have less influence, justifying higher or lower weights in the weighted average. Distributed averaging dynamics in the continuous time case allows for gradual updates of agents’ states, where each agent’s state at any given time is pulled toward states of the agent’s neighbors with different weights.

There are various problems of interest involving networks with distributed averaging dynamics including, but not limited to, the analysis of their asymptotic behavior and identifying key agents in the network. These problems can be defined for both continuous and discrete time domains, with separate sets of tools required to address them. Our relevant projects and findings are listed below.

*Project: Convergence analysis of distributed averaging systems*

A fundamental problem in the area of network systems is the consensus problem, also known as the rendezvous problem, which is to achieve overall system reliability when a number of faulty processes are present. This often requires processes to agree on a common value. Examples of applications of consensus in network systems include load balancing in power smart grids, coordination and cooperation in multi-robot systems, the clock synchronization in computer networks, and group decision-making in social networks.

Design and analysis of distributed time-varying averaging dynamics under which a consensus is reached among the agents has been the subject of extensive research for a number of decades. An equally important problem regarding the averaging dynamics is associated with the issue of the clustering of the agents. Researchers from different scientific disciplines have been interested in designing and analyzing dynamics that guarantee an agent clustering whether to one or multiple consensus clusters. Noticing that any distributed averaging dynamics is uniquely described by its underlying chain of row-stochastic matrices, consensus and clustering problems can be translated to ergodicity and class-ergodicity of backward Markov chains.

We introduce a large class of distributed averaging dynamics based on *balanced asymmetric chains* and derive succinct necessary and sufficient conditions for the ergodicity and class-ergodicity of these chains. Our basic conviction that the theory of inhomogeneous Markov chains could help understand the convergence properties of averaging dynamics led us to employ Sonin’s Decomposition-Separation (D-S) Theorem. The D-S Theorem together with the intuitions regarding the importance of Kolmogorov’s notion of absolute probability sequence eventually results in the discovery of the largest class of ergodic and class-ergodic chains of stochastic matrices known to this date for both discrete and continuous time cases.

*Project: Influential coalitions in distributed averaging systems*

In complex networks, where access to directly influence all or a large subset of agents is not granted, the ability to guide the agents’ formation towards a desired state by applying a few control inputs is a crucial requirement. For instance, in the case of social networks, finding an influential group of individuals is essential in designing effective marketing campaign for targeted advertising. This leads to the fundamental problem of identifying key agent coalitions in a network through which certain global specifics of the network can be controlled. We characterize agent coalitions through which certain global specifics of the network can be controlled, namely the *rendezvous point* and the *steady-state range*.

Let us assume that the state of an agent is its position in a multi-dimensional Euclidean space. One of our main contributions in this project is to dictate an arbitrary rendezvous point by controlling only a coalition of agents. By dictating an arbitrary rendezvous point, we mean making the entire system converge to an arbitrary point in the space. To dictate an arbitrary rendezvous point by controlling an agent coalition, we have developed two different methods, one via an *Γminence Grise Coalition* (EGC) and the other one via a *rendezvous set*. We have in fact achieved a more general objective, that is arbitrary shaping of the steady-state range via an agent coalition named a *range-deciding set*. The steady-state range, or simply the range, is defined as the smallest convex subset of the Euclidean space containing all the agents’ states as time grows. Intuitively speaking, after a sufficiently large time, every agent will always be located within or arbitrarily close to the range.

**MISCELLANEOUS**

**Fractional-Order Systems**

Fractional calculus is a mathematical area introduced as an extension of traditional calculus in the eighteenth century. In a broad sense, it deals with derivatives and integrals of both integer and non-integer orders. Over the past few decades, fractional calculus has been employed increasingly within various research disciplines to model and design dynamical systems, leading to the emergence of the field of fractional-order systems. In fact, many real world processes have proved to be of fractional nature. To name a few, fractional calculus can be applied to modeling heat conduction, finance systems, and the dynamics of viscoelastic materials, transmission lines, and electrical capacitors. A dynamical system involving fractional calculus is referred to as a fractional-order system.

*Project: Stability analysis of fractional-order systems*

Among the most important properties of any dynamical system is their stability. In this project, we exclusively deal with the stability properties of fractional-order systems. Our goal is to characterize the stability criteria, also known as the safe zone, of different fractional-order systems based on the system parameters. Our specific contributions are threefold as described below.

(i) In practice, to computationally analyze fractional-order operators, we must approximate them by their integer-order counterparts. Thus, a comprehensive comparison between the stability criteria of a fractional-order system and its integer-order approximation is crucial. To achieve our goal, we consider a general commensurate fractional-order system, with the derivative order between 0 and 1, and an approximation of it via a well-known frequency-based method. We argue that for small frequencies, i.e., those close to 0, the stability criterion of the original system is a subset of that of the approximating one, which means an unstable system may become stable through approximation. We further discuss explicitly the intermediate and high frequencies and show particularly that the two stability criteria in these cases do not yield an inclusion relation.

(ii) It was previously observed that the fractional-order Van der Pol system could demonstrate oscillating behavior. However, the oscillations range of the fractional-order Van der Pol system remained to be determined. We fully address this problem by obtaining a succinct inequality, depending on the system parameters, that precisely determines the oscillation range of the system. Furthermore, we show that if the oscillations of the fractional-order Van der Pol system are of regular type, the resulting limit cycle is not unique and indeed varies according to the initial conditions. This characteristic of fractional-order Van der Pol system is of particular interest as it opposes that of its integer-order counterpart where the limit cycle is unique. We finally verified the results using numerical examples.

(iii) For a general LTI fractional-order system in the Pseudo state space form where the fractional derivatives are rational numbers between 0 and 1, we investigate the maximum number of generated frequencies given the inner dimension of the system. More specifically, inspired by the restricted difference bases concept and the Golomb ruler, a length 6 of which shown in the figure, we develop a novel multi-frequency oscillatory fractional order system that led to a non-trivial lower bound on the maximum of number of generated frequencies given the inner dimension of the system. We further establish a number of other lower and upper bounds on that number and verified the effectiveness of the results via numerical examples.

*Project: Suppressing chaos using fractional controllers*

Chaos in dynamical systems refers to a type of undesirable behavior where the output of the system becomes highly sensitive to the initial conditions. Chaos results in a complicated and unpredictable wandering of the system over longer periods of time. This undesirable behavior of dynamical systems underscores the crucial need to design controllers to suppress chaotic behaviors, which constitutes the main objective of this project. In this project, we design fractional controllers to suppress chaos in a class of dynamical systems. More specifically, for a general three-dimensional nonlinear integer-order system, we take two approaches toward suppressing chaos, i.e., making the closed loop system locally stable at its fixed point.

In one of our designed controllers, the control input is a fractional integrator of a linear combination of the states in the system when linearized around its fixed point. In the other one, the control input is a combination of an integer-order derivative of one state and a fractional-order derivative of another state of the system. Our controllers can be applied to various well-known three-dimensional system such as the Lorenz system, Rossler system, Chua system, Chen system, and LΓΌ system. The performance of our first controller is examined for the Chen system and a chaotic oscillator of canonical structure, while our second controller is shown to perform well for a chaotic circuit.