Resistance Distance, Kirchhoff Index, Foster's Theorems, and Generalizations

The emerging area of network science studies structural characteristics of networks and dynamical processes on networks such as spread of epidemics, vulnerability of power grids to cascading failures etc. In this area, several measures of network performance have been introduced and studied. In this dissertation, we study two measures, namely, resistance distance and Kirchhoff index. Treating each element of a graph as a resistance, resistance distance between two nodes u and v is the effective resistance across u and v. Kirchhoff index defined by the chemistry community is the sum of the effective resistances across all pairs of nodes of the graph. Kirchhoff index, also called network criticality, has been studied by the communication network community. Kirchhoff index has been studied using the graph Laplacian matrix which is the same as the indefinite admittance matrix of a resistance network. Our research is on reducing the computational effort in calculating the Kirchhoff index in networks. First a simpler formula for Kirchhoff index based on the properties of node-to-datum resistance matrix is presented. To avoid computational complexity and extraneous efforts of Moore-Penrose pseudoinverse, Kirchhoff index is calculated in terms of the inverse of the reduced Laplacian matrix. The notion of Laplacian matrix is then generalized using the fundamental cutset matrix of a graph. Two approaches to compute Kirchhoff index are presented: The first approach is based on a matrix transformation, and the second approach uses the concept of Kirchhoff polynomial of a graph. Kirchhoff polynomial of a graph introduced in this work is defined for each spanning tree of the graph. In 1949 and 1961 Foster established two theorems that give identities involving resistance distances. We introduce the concept of Weighted Kirchhoff index of a graph and study its relationship to Foster’s theorems. We present a generalization of Foster’s theorems that retains the circuit-theoretic flavor and elegance of Foster’s theorems, and develop a dual form of this theorem. Kirchhoff index captures the effect of topological structure on the performance of networks. It also captures the path diversity between nodes in a network. Kirchhoff index can be used to determine node betweenness in networks that are of interest in network vulnerability studies. In view of this, an efficient methodology to compute Kirchhoff index is required. For this purpose, we propose sequential and parallel algorithms. In addition, we introduce a novel 3-step approximation algorithm for calculation of resistance distance and Kirchhoff index

Ranking nodes in complex networks : a case study of the Gaubus

Abstract: Connecting points of interest through a well-planned, inter-connected network provides manifold benefits to commuters and service providers. In the South African context, traffic congestion has become of great concern. Given how the South Africa community is slowly developing towards the use of multi-modes of mobility, the Gautrain network can be used to promote the use of multi-modes of mobility, as the Gautrain has been identified as the backbone of mobility within the Gauteng province. Currently commuters have the option to board the Gaubus (a form of Bus Rapid Transit) at their origin points which will take them to the Gautrain station to board the Gautrain. The problem to be solved arises when a commuter wishes to traverse from any bus stop to the Gautrain station, currently he/she only has one option and if the bus network has a shutdown at any point in the network the commuter’s journey will not be possible. In solving this problem, we consider the problem of graph robustness (that is creating new alternative routes to increase node/bus stop connectivity). We initial use Strava data, to identify locations were cyclist prefer to cycle and at what time of day. In graph theory, the nodes with most spreading ability are called influential nodes. Identification of most influential nodes and ranking them based on their spreading ability is of vital importance. Closeness centrality and betweenness are one of the most commonly used methods to identify influential nodes in complex networks. Using the Gaubus network we identify the influential nodes/ bus stops, using the betweenness centrality measure. The results reveal the influential nodes with the highest connectivity as these have cross-connections in the network. Identification of the influential nodes presents an important implication for future planning, accessibility, and, more generally, quality of life

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations

Human brain anatomy and function display a combination of modular and hierarchical organization, suggesting the importance of both cohesive structures and variable resolutions in the facilitation of healthy cognitive processes. However, tools to simultaneously probe these features of brain architecture require further development. We propose and apply a set of methods to extract cohesive structures in network representations of brain connectivity using multi-resolution techniques. We employ a combination of soft thresholding, windowed thresholding, and resolution in community detection, that enable us to identify and isolate structures associated with different weights. One such mesoscale structure is bipartivity, which quantifies the extent to which the brain is divided into two partitions with high connectivity between partitions and low connectivity within partitions. A second, complementary mesoscale structure is modularity, which quantifies the extent to which the brain is divided into multiple communities with strong connectivity within each community and weak connectivity between communities. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom

Scalable Algorithms for the Analysis of Massive Networks

Die Netzwerkanalyse zielt darauf ab, nicht-triviale Erkenntnisse aus vernetzten Daten zu gewinnen. Beispiele für diese Erkenntnisse sind die Wichtigkeit einer Entität im Verhältnis zu anderen nach bestimmten Kriterien oder das Finden des am besten geeigneten Partners für jeden Teilnehmer eines Netzwerks - bekannt als Maximum Weighted Matching (MWM). Da der Begriff der Wichtigkeit an die zu betrachtende Anwendung gebunden ist, wurden zahlreiche Zentralitätsmaße eingeführt. Diese Maße stammen hierbei aus Jahrzehnten, in denen die Rechenleistung sehr begrenzt war und die Netzwerke im Vergleich zu heute viel kleiner waren. Heute sind massive Netzwerke mit Millionen von Kanten allgegenwärtig und eine triviale Berechnung von Zentralitätsmaßen ist oft zu zeitaufwändig. Darüber hinaus ist die Suche nach der Gruppe von k Knoten mit hoher Zentralität eine noch kostspieligere Aufgabe. Skalierbare Algorithmen zur Identifizierung hochzentraler (Gruppen von) Knoten in großen Graphen sind von großer Bedeutung für eine umfassende Netzwerkanalyse. Heutigen Netzwerke verändern sich zusätzlich im zeitlichen Verlauf und die effiziente Aktualisierung der Ergebnisse nach einer Änderung ist eine Herausforderung. Effiziente dynamische Algorithmen sind daher ein weiterer wesentlicher Bestandteil moderner Analyse-Pipelines. Hauptziel dieser Arbeit ist es, skalierbare algorithmische Lösungen für die zwei oben genannten Probleme zu finden. Die meisten unserer Algorithmen benötigen Sekunden bis einige Minuten, um diese Aufgaben in realen Netzwerken mit bis zu Hunderten Millionen von Kanten zu lösen, was eine deutliche Verbesserung gegenüber dem Stand der Technik darstellt. Außerdem erweitern wir einen modernen Algorithmus für MWM auf dynamische Graphen. Experimente zeigen, dass unser dynamischer MWM-Algorithmus Aktualisierungen in Graphen mit Milliarden von Kanten in Millisekunden bewältigt.Network analysis aims to unveil non-trivial insights from networked data by studying relationship patterns between the entities of a network. Among these insights, a popular one is to quantify the importance of an entity with respect to the others according to some criteria. Another one is to find the most suitable matching partner for each participant of a network knowing the pairwise preferences of the participants to be matched with each other - known as Maximum Weighted Matching (MWM). Since the notion of importance is tied to the application under consideration, numerous centrality measures have been introduced. Many of these measures, however, were conceived in a time when computing power was very limited and networks were much smaller compared to today's, and thus scalability to large datasets was not considered. Today, massive networks with millions of edges are ubiquitous, and a complete exact computation for traditional centrality measures are often too time-consuming. This issue is amplified if our objective is to find the group of k vertices that is the most central as a group. Scalable algorithms to identify highly central (groups of) vertices on massive graphs are thus of pivotal importance for large-scale network analysis. In addition to their size, today's networks often evolve over time, which poses the challenge of efficiently updating results after a change occurs. Hence, efficient dynamic algorithms are essential for modern network analysis pipelines. In this work, we propose scalable algorithms for identifying important vertices in a network, and for efficiently updating them in evolving networks. In real-world graphs with hundreds of millions of edges, most of our algorithms require seconds to a few minutes to perform these tasks. Further, we extend a state-of-the-art algorithm for MWM to dynamic graphs. Experiments show that our dynamic MWM algorithm handles updates in graphs with billion edges in milliseconds

Ranking Edges by their Impact on the Spectral Complexity of Information Diffusion over Networks

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is $\mathcal{O}(N^3)$ per edge for networks with $N$ nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and fast to compute, scaling as $\mathcal{O}(N)$ per edge. Focusing on a form of VNE that is associated with a transport operator $e^{-\beta{ L}}$, where ${ L}$ is a graph Laplacian matrix and $\beta>0$ is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales $\beta$ for diffusion.Comment: 24 pages, 7 figure