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In weighted graphs the shortest path between two nodes is often reached through an indirect path, out of all possible connections, leading to structural redundancies which play key roles in the dynamics and evolution of complex networks. We…
Measuring similarity between complex objects is a fundamental task in many scientific fields. When objects are represented as graphs, graph similarity/distance measures offer a powerful framework for quantifying structural resemblance.…
Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years a multitude of diverse, ad hoc solutions to this problem have been introduced. Here we propose that simple and…
A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for…
Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…
In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our…
Evaluating similarity between graphs is of major importance in several computer vision and pattern recognition problems, where graph representations are often used to model objects or interactions between elements. The choice of a distance…
For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the…
The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
Graph operations or products, such as triangulation and Kronecker product have been extensively applied to model complex networks with striking properties observed in real-world complex systems. In this paper, we study hitting times and…
Resistance distance computation is a fundamental problem in graph analysis, yet existing random walk-based methods are limited to approximate solutions and suffer from poor efficiency on small-treewidth graphs (e.g., road networks). In…
We define and study two new kinds of "effective resistances" based on hubs-biased -- hubs-repelling and hubs-attracting -- models of navigating a graph/network. We prove that these effective resistances are squared Euclidean distances…
This Letter presents a unified approach for the fundamental relationship between structure and function in flow networks by solving analytically the voltages in a resistor network, transforming the network structure to an effective…
In a graph, nodes can be characterized locally (with their degree $k$) or globally (e.g. with their average length path $\xi$ to other nodes). Here we investigate how $\xi$ depends on $k$. Our earlier algorithm of the construction of the…
Effective Resistance (ER) is a fundamental tool in various graph learning tasks. In this paper, we address the problem of efficiently approximating ER on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ vertices and $m$ edges.…
An algorithm is introduced for predicting quantized resistances in graphene p-n junction devices that utilize more than a single entry and exit point for electron flow. Depending on the configuration of an arbitrary number of terminals,…
Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work we define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost…
In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance…