Extremal Properties of Complex Networks

We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph qualities for any arbitrary number of nodes and edges and analytically derive the form and properties of such networks

A Bag-of-Paths Node Criticality Measure

This work compares several node (and network) criticality measures quantifying to which extend each node is critical with respect to the communication flow between nodes of the network, and introduces a new measure based on the Bag-of-Paths (BoP) framework. Network disconnection simulation experiments show that the new BoP measure outperforms all the other measures on a sample of Erdos-Renyi and Albert-Barabasi graphs. Furthermore, a faster (still O(n^3)), approximate, BoP criticality relying on the Sherman-Morrison rank-one update of a matrix is introduced for tackling larger networks. This approximate measure shows similar performances as the original, exact, one

Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and it closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known it current betweenness and resistance closeness (information) centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Non-monotonic betweenness behaviors are found to characterize nodes that lie close to inter-community boundaries in the studied networks

Graph measures and network robustness

Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness

Transport Processes on Homogeneous Planar Graphs with Scale-Free Loops

We consider the role of network geometry in two types of diffusion processes: transport of constant-density information packets with queuing on nodes, and constant voltage-driven tunneling of electrons. The underlying network is a homogeneous graph with scale-free distribution of loops, which is constrained to a planar geometry and fixed node connectivity $k=3$. We determine properties of noise, flow and return-times statistics for both processes on this graph and relate the observed differences to the microscopic process details. Our main findings are: (i) Through the local interaction between packets queuing at the same node, long-range correlations build up in traffic streams, which are practically absent in the case of electron transport; (ii) Noise fluctuations in the number of packets and in the number of tunnelings recorded at each node appear to obey the scaling laws in two distinct universality classes; (iii) The topological inhomogeneity of betweenness plays the key role in the occurrence of broad distributions of return times and in the dynamic flow. The maximum-flow spanning trees are characteristic for each process type.Comment: 14 pages, 5 figure

Incorporating Betweenness Centrality in Compressive Sensing for Congestion Detection

This paper presents a new Compressive Sensing (CS) scheme for detecting network congested links. We focus on decreasing the required number of measurements to detect all congested links in the context of network tomography. We have expanded the LASSO objective function by adding a new term corresponding to the prior knowledge based on the relationship between the congested links and the corresponding link Betweenness Centrality (BC). The accuracy of the proposed model is verified by simulations on two real datasets. The results demonstrate that our model outperformed the state-of-the-art CS based method with significant improvements in terms of F-Score

Not quite what’s on paper? Comparison between theoretical and actual information-sharing networks in the Ugandan rural water service sector

Under Uganda’s decentralised system, rural water service delivery remains to some extent problematic. Several studies attribute the possible causes of deficiencies in the water sector to governance issues. This article applies social network analysis to map upward and downward water-related information flows between the actors of local government from village to district level. Comparing the actual information-sharing network with what’s on paper reveals a less reciprocal and more centralised network than that theoretically envisaged. Some actors, such as the district water officer, are more central than expected in terms of sending and receiving information, while others seem to underperform. Our findings show, however, that it is not the political–administrative information exchange which is the biggest obstacle, but rather information flows between higher (district and sub-county) and lower (parish and village) levels of the local governance structure. Adding water users to the analysis reveals the village chairperson as the most crucial broker of information upward to duty bearers at district level. The limited role of water user committees also becomes apparent. The authors conclude that information communication technology holds potential to overcome some of the bottlenecks (eg distance) hindering the flow of water-related information between actors at different levels