J. A. Rodriguez, E. Estrada, A. Gutierrez
In this paper we introduce the functional centrality as a generalization of the subgraph centrality. We propose a general method for characterizing nodes in the graph according to the number of closed walks starting and ending at the node. Closed walks are appropriately weighted according to the topological features that we need to measure.
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Betweenness centrality is a widely-used measure in the analysis of large complex networks. It measures the potential or power of a vertex to control the communication over the network under the assumption that information primarily flows over the shortest paths between them. In this paper we prove several results on betweenness centrality of Cartesian product of graphs.
The edge betweenness centrality of an edge is loosely defined as the fraction of shortest paths between all pairs of vertices passing through that edge. In this paper, we investigate graphs where the edge betweenness centrality of edges is uniform. It is clear that if a graph G is edge-transitive (its automorphism group acts transitively on its edges) then G has uniform edge betweenness centrality. However this sufficient condition is not necessary. Graphs that are not edge-transitive but have uniform edge betweenness centrality appear to be very rare. Of the over 11.9 million connected graphs…
There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest path between them. In this paper we present betweenness centrality of some important classes of graphs.
Eigenvector centrality is a linear algebra based graph invariant used in various rating systems such as webpage ratings for search engines. A generalization of the eigenvector centrality invariant is defined which is motivated by the need to design rating systems for bipartite graph models of time-sensitive and other processes. The linear algebra connection and some applications are described.
A Complete Characterization of the Inverse Eigenvector Centrality Problem for Undirected Graphs
研究无向连通图中给定正向量是否可通过适当边权赋值实现的问题，揭示稳定集及其邻域的全局约束条件。
The subgraph eigenvector centrality of graphs
本文提出基于子图结构的特征向量中心性，利用子图张量刻画顶点重要性，并证明其存在性，能区分正则图中经典中心性无法区分的顶点。
One of the more recent measures of centrality in social network analysis is the normalized harmonic centrality. A variant of the closeness centrality, harmonic centrality sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another. It is then normalized by dividing it by m-1, where m is the number of nodes of the graph. In this paper, we present notions regarding the harmonic centrality of some important classes of graphs.
Harmonic centrality calculates the importance of a node in a network by adding the inverse of the geodesic distances of this node to all the other nodes. Harmonic centralization, on the other hand, is the graph-level centrality score based on the node-level harmonic centrality. In this paper, we present some results on both the harmonic centrality and harmonic centralization of graphs resulting from some graph products such as Cartesian and direct products of the path $P_2$ with any of the path $P_m$, cycle $C_m$, and fan $F_m$ graphs.
Structure of betweenness uniform graphs with low values of betweenness centrality
本文反驳了关于介数均匀图介数中心性的猜想，通过补图视角刻画了介数中心性至多9/10的图，并猜想该刻画可推广至介数中心性小于1的所有BUG。
We give a new upper bound for the average graph distance in terms of the average Ollivier curvature. Here, the average Ollivier curvature is weighted with the edge betweenness centrality. Moreover, we prove that equality is attained precisely for the reflective graphs which have been classified as Cartesian products of cocktail party graphs, Johnson graphs, halved cubes, Schläfli graphs, and Gosset graphs.
In 2010, Joyce et. al defined the leverage centrality of vertices in a graph as a means to analyze functional connections within the human brain. In this metric a degree of a vertex is compared to the degrees of all it neighbors. We investigate this property from a mathematical perspective. We first outline some of the basic properties and then compute leverage centralities of vertices in different families of graphs. In particular, we show there is a surprising connection between the number of distinct leverage centralities in the Cartesian product of paths and the triangle numbers.
The discriminant power of centrality indices for the degree, eigenvector, closeness, betweenness and subgraph centrality is analyzed. It is defined by the number of graphs for which the standard deviation of the centrality of its nodes is zero. On the basis of empirical analysis it is concluded that the subgraph centrality displays better discriminant power than the rest of centralities. We also propose some new conjectures about the types of graphs for which the subgraph centrality does not discriminate among nonequivalent nodes.
New centrality measure: ksi-centrality
本文提出ksi-中心性及其归一化形式，通过拉普拉斯矩阵刻画节点及其邻居的重要性，并证明其与聚类系数、代数连通度和Cheeger数相关，可有效区分真实网络与人工小世界网络。
In graph-based applications, a common task is to pinpoint the most important or ``central'' vertex in a (directed or undirected) graph, or rank the vertices of a graph according to their importance. To this end, a plethora of so-called centrality measures have been proposed in the literature. Such measures assess which vertices in a graph are the most important ones by analyzing the structure of the underlying graph. A family of centrality measures that are suited for graph databases has been recently proposed by relying on the following simple principle: the importance of a vertex in a graph …
The goal of this paper is to present a centrality measurement for the nodes of a hypergraph, by using existing literature which extends eigenvector centrality from a graph to a hypergraph, and literature which give a general centrality measurement for a graph. We will use this measurement to say more about the number of communications in a hypergraph, to implement a learning mechanism, and to construct certain networks.
The graph invariant examined in this paper is the largest eigenvalue of the adjacency matrix of a graph. Previous work demonstrates the tight relationship between this invariant, the birth and death rate of a contagion spreading on the graph, and the trajectory of the contagion over time. We begin by conducting a simulation confirming this and explore bounds on the birth and death rate in terms of well-known graph invariants. As a result, the change in the largest eigenvalue resulting from removal of a vertex in the network is the best measure of effectiveness of interventions that slow the sp…
In this paper some results about the controllability of spectral centrality in a complex network are presented. In particular, the inverse problem of designing an unweigthed graph with a prescribed centrality is considered, by showing that for every possible ranking eventually with ties, an unweighted directed/undirected complex network can be found whose PageRank or eigenvector centrality gives the ranking considered. Different families of networks are presented in order to analytically solve this problem either for directed and undirected graphs with and without loops.
Efficient computation of Katz centrality for very dense networks via negative parameter Katz
针对极稠密网络，提出通过补图高效计算Katz中心性排序的方法，并解释负参数Katz中心性，同时为加权图提供近似方法及充分条件。
New results and open problems on subgraph centrality
本文针对子图中心性及其他基于图游走的节点中心性度量，呈现了若干新结果与一系列开放问题，该度量是网络分析中广泛应用的中心性度量，具有多领域应用价值且关联多个数学分支。
Bounds on the closeness centrality of a graph
本文研究连通图的归一化接近中心性及其与平均距离的乘积，证明基本界 $1\leq\bar l\bar{\mathsf C}_C<2$，导出十个新下界并给出特定图族的渐近结果。
Due to shorter range communication becoming more prevalent with the development of multiple-input, multiple-output antennas (MIMO) and millimeter wave communications, multi-hop, intra-cell communication is anticipated to play a major role in 5G. This is developed in this thesis. Our analysis involves a stochastic spatial network model called a random geometric graph, which we use to model a network of interconnected devices communicating wirelessly without any separate, pre-established infrastructure. [The remaining abstract is available in the front matter of this thesis.]
On the Connectivity and the Diameter of Betweenness-Uniform Graphs
本文研究介数均匀图的性质，证明连通介数均匀图要么是环要么是3-连通图，且高最大度数的介数均匀图直径较小。
Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $β$-subgraph centrality, which is based on the exponential of the matrix $βA$, where $A$ is the adjacency matrix of the graph and $β$ is a real parameter ("inverse temperature"). We prove that for algebraic $β$, two vertices with equal $β$-subgraph centrality are necessarily cospectral. We further show that two such vertices must have the same degree and eigenvector centralities. O…
We exhibit a one-to-one correspondence between $3$-colored graphs and subarrangements of certain hyperplane arrangements denoted $\mathcal J_n$, $n \in \mathbb N$. We define the notion of centrality of $3$-colored graphs which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial $χ_{\mathcal J_n}$ of $\mathcal J_n$ can be expressed in terms of the number of central $3$-colored graphs, and we compute $χ_{\mathcal J_n}$ for $n = 2, 3$.
Finite-sample certification and operating envelopes for spectral clustering and graph centrality
本文提出了一种基于有限样本的谱聚类和图中心性推断方法，通过显式构造确信区域，提供非渐近的覆盖保证和稳定性区域。
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