Related papers: Random Electrical Networks on Complete Graphs II: …
In Part I of this work we defined a generalization of the concept of effective resistance to directed graphs, and we explored some of the properties of this new definition. Here, we use the theory developed in Part I to compute effective…
The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability.
This paper extends the definitions of effective resistance and effective conductance to characterize the overall relation (positive coupling or antagonism) between any two disjoint sets of nodes in a signed graph. It generalizes the…
We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…
In the field of computer science, the network reliability problem for evaluating the network failure probability has been extensively investigated. For a given undirected graph $G$, the network failure probability is the probability that…
We perform a massive evaluation of neural networks with architectures corresponding to random graphs of various types. We investigate various structural and numerical properties of the graphs in relation to neural network test accuracy. We…
The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this…
Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The resistance distance $R_G(x,y)$ between two vertices $x,y$ of $G$ is defined to be the effective resistance between the two vertices in the corresponding…
We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective…
We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…
Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we…
Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When…
Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes.…
The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices…
We completely characterize when the free effective resistance of an infinite graph can be expressed in terms of simple hitting probabilities of the graphs random walk.
Assessing and improving the robustness of a graph $G$ are critical steps in network design and analysis. To this end, we consider the optimisation problem of removing $k$ edges from $G$ such that the resulting graph has minimal robustness,…
We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high…
Consider a rooted infinite Galton-Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi_e$ indexed by all the edges in the tree. We assign the resistance $m^d \xi_e$ to each edge $e$ at…
Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of…
In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an…