Using graph theory for automated electric circuit solving

Graph theory plays many important roles in modern physics and in many different contexts, spanning diverse topics such as the description of scale-free networks and the structure of the universe as a complex directed graph in causal set theory. Graph theory is also ideally suited to describe many concepts in computer science. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving

Multi-Terminal Graphs and Equilibrium Problem of Flows and Tensions

The notion of graph was generalized to make the abstract network theory applicable to economic networks, where multi-commodities are traded among productive sectors, intermediate sectors, and final demand sectors. The generalized graph, called the multi-terminal graph, has contributed to developing a new network theory, deriving that flows of multi-commodities correspond to electric currents in electric circuits. A well-known Leontief model has been reformulated in this generalized system. A notion of tension is defined to correspond the price in economics to the voltage in electric circuits. The equilibrium problem of flows and tensions is discussed on the basis of new network, and solved under a general hypothesis that characteristics of branches are represented by complete increasing curves, which express demand and supply curves.Article信州大学工学部紀要 54: 49-64 (1983)departmental bulletin pape

An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs

We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We analyze the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an $O(N^3\log N)$ upper bound for the expected convergence time on an arbitrary graph of size $N$, improving on the state of art bound of $O(N^5)$ for quantized consensus algorithms. Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology.Comment: to appear in IEEE Trans. on Automatic Control, January, 2015. arXiv admin note: substantial text overlap with arXiv:1208.078

Hybrid Algorithm based on Genetic Algorithm and Tabu Search for Reconfiguration Problem in Smart Grid Networks Using "R"

Reconfiguration of distribution networks aims to support the decision support, planning and/or real-time control of the operation of the electricity network. It is accomplished modifying the network structure of distribution feeders by changing the sectionalizing switches. Ensure higher levels of continuity and reliability to the electricity supply service are some of the requirements of consumers and electric power providers in the Smart Grid (SG) context. The goal of this paper is to propose a hybrid algorithm (Genetic and Tabu) for the reconfiguration problem based on " R " in order to better support the decision making process. Beyond that, " R " modeling of electricity networks improves the response time when handling issues of network reconfiguration using graph theory. The status of switches is decided according to graph theory subject to the radiality constraint of the distribution networks. The algorithm is presented and simulation results of IEEE 16-bus system, showing good results and computational efficiency

Effective graph resistance

AbstractThis paper studies an interesting graph measure that we call the effective graph resistance. The notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices. The objective of the paper is twofold. First, we survey known formulae of the effective graph resistance and derive other representations as well. The derivation of new expressions is based on the analysis of the associated random walk on the graph and applies tools from Markov chain theory. This approach results in a new method to approximate the effective graph resistance. A second objective of this paper concerns the optimisation of the effective graph resistance for graphs with given number of vertices and diameter, and for optimal edge addition. A set of analytical results is described, as well as results obtained by exhaustive search. One of the foremost applications of the effective graph resistance we have in mind, is the analysis of robustness-related problems. However, with our discussion of this informative graph measure we hope to open up a wealth of possibilities of applying the effective graph resistance to all kinds of networks problems

Power packet transferability via symbol propagation matrix

Power packet is a unit of electric power transferred by a power pulse with an information tag. In Shannon's information theory, messages are represented by symbol sequences in a digitized manner. Referring to this formulation, we define symbols in power packetization as a minimum unit of power transferred by a tagged pulse. Here, power is digitized and quantized. In this paper, we consider packetized power in networks for a finite duration, giving symbols and their energies to the networks. A network structure is defined using a graph whose nodes represent routers, sources, and destinations. First, we introduce symbol propagation matrix (SPM) in which symbols are transferred at links during unit times. Packetized power is described as a network flow in a spatio-temporal structure. Then, we study the problem of selecting an SPM in terms of transferability, that is, the possibility to represent given energies at sources and destinations during the finite duration. To select an SPM, we consider a network flow problem of packetized power. The problem is formulated as an M-convex submodular flow problem which is known as generalization of the minimum cost flow problem and solvable. Finally, through examples, we verify that this formulation provides reasonable packetized power.Comment: Submitted to Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science

An Upper Bound on the Convergence Time for Distributed Binary Consensus

The problem addressed in this paper is the analysis of a distributed consensus algorithm for arbitrary networks, proposed by B\'en\'ezit et al.. In the initial setting, each node in the network has one of two possible states ("yes" or "no"). Nodes can update their states by communicating with their neighbors via a 2-bit message in an asynchronous clock setting. Eventually, all nodes reach consensus on the majority states. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N4 logN) upper bound for the expected convergence time on an arbitrary graph of size N.Comment: 15th International Conference on Information Fusion, July 2012, 7 page