Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

Working with the market : a new approach to reducing urban slums in India

This paper examines the policy options for India as it seeks to improve living conditions of the poor on a large scale and reduce the population in slums. Addressing the problem requires first a diagnosis of the market at the city level and a recognition that government interventions, rather than thwarting the operations of the market, should seek to make it operate better. This can substantially reduce the subsidies required to assist low income households to attain decent living standards. The authors show that government programs that directly provide housing would cost, in conservative estimates, about of 20 to 30 percent of GDP, and cannot solve a problem on the scale of India's. Using two case studies, for Mumbai and Ahmedabad, the paper offers a critical examination of government policies that shape the real estate market and make formal housing unaffordable for a large part of the population. It illustrates how simple city level market diagnostics can be used to identify policy changes and design smaller assistance programs that can reach the poor. The linkage between chronic infrastructure backlogs and policies makes housing unnecessarily expensive. Increasing the carrying capacity of cities is essential for gaining acceptance of real estate policies suited to Indian cities. The authors propose approaches for funding major investments to achieve this.Housing&Human Habitats,Urban Housing,Public Sector Management and Reform,Regional Governance,Urban Governance and Management

Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

Metrics for network comparison using egonet feature distribution

Identifying networks with similar characteristics in a given ensemble, or detecting pattern discontinuities in a temporal sequence of networks, are two examples of tasks that require an effective metric capable of quantifying network (dis)similarity. Here we propose a method based on a global portrait of graph properties built by processing local nodes features. More precisely, a set of dissimilarity measures is defined by elaborating the distributions, over the network, of a few egonet features, namely the degree, the clustering coefficient, and the egonet persistence. The method, which does not require the alignment of the two networks being compared, exploits the statistics of the three features to define one- or multi-dimensional distribution functions, which are then compared to define a distance between the networks. The effectiveness of the method is evaluated using a standard classification test, i.e., recognizing the graphs originating from the same synthetic model. Overall, the proposed distances have performances comparable to the best state-of-the-art techniques (graphlet-based methods) with similar computational requirements. Given its simplicity and flexibility, the method is proposed as a viable approach for network comparison tasks

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference