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Our digital technology depends on mathematics to compute current flow and design its devices. Mathematics describes current flow by an idealization, Kirchhoff's current law. All the electrons that flow into a node flow out. This…
Identifying influential nodes and edges in directed networks remains a fundamental challenge across domains from social influence to biological regulation. Most existing centrality measures face a critical limitation: they either discard…
The effective resistance between a pair of nodes in a weighted undirected graph is defined as the potential difference induced between them when a unit current is injected at the first node and extracted at the second node, treating edge…
We solve the problem of identifying (reconstructing) network topology from steady state network measurements. Concretely, given only a data matrix $\mathbf{X}$ where the $X_{ij}$ entry corresponds to flow in edge $i$ in configuration…
We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected…
Many networked systems such as electric networks, the brain, and social networks of opinion dynamics are known to obey conservation laws. Examples of this phenomenon include the Kirchoff laws in electric networks and opinion consensus in…
We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair.…
The way in which electric power depends on the topology of circuits with mixed voltage and current sources is examined. The power flowing in any steady-state DC circuit is shown to depend on a minimal set of key variables called fundamental…
To describe the flow of a miscible quantity on a network, we introduce the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. This is a natural description of an array of inductances and…
Improving the controllability of power networks is crucial as they are highly complex networks operating in synchrony; even minor perturbations can cause desynchronization and instability. To that end, one needs to assess the criticality of…
We consider general, steady, balanced flows of a commodity over a network where an instance of the network flow is characterized by edge flows and nodal potentials. Edge flows in and out of a node are assumed to be conserved, thus…
We study infinite resistor networks perturbed by line defects, in which the resistances are periodically modified along a single line. Using the Sherman-Morrison identity applied to the reciprocal-space representation of the lattice Green's…
Photonic computing has recently become an interesting paradigm for high-speed calculation of computing processes using light-matter interactions. Here, we propose and study an electromagnetic wave-based structure with the ability to…
We show that if the sum of the resistances of an electrical network $N$ is finite, then there is a unique electrical current in $N$ provided we do not allow, in a sense, any flow to escape to infinity.
A probabilistic method is derived for solution of ohmic circuit problems. It is compared to the standard approach, which is construction and solution of a set of coupled, linear equations manifesting Kirchhoff's laws. An example is made of…
We consider a special form of parametric generalized equations arising from electronic circuits with AC sources and study the effect of perturbing the input signal on solution trajectories. Using methods of variational analysis and strong…
We calculate effective impedances of infinite $LC$- and $CL$- ladder networks as limits of effective impedances of finite network approximations, using a new method, which involves precise mathematical concepts. These concepts are related…
Bach et al. [1] recently presented an algorithm for constructing confluent drawings, by leveraging power graph decomposition to generate an auxiliary routing graph. We identify two issues with their method which we call the node split and…
We investigate by random-walk simulations and a mean-field theory how growth by biased addition of nodes affects flow of the current through the emergent conducting graph, representing a digital circuit. In the interior of a large network…
Studies of models of current flow behaviour in Electrical Impedance Tomography (EIT) have shown that the current density distribution varies extremely rapidly near the edge of the electrodes used in the technique. This behaviour imposes…