We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes/volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degreẽ d. Hered denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degreed is at most 1. Examples are given to illustrate that both bounds are best possible.

Random graph theory is used to examine the ''small-world phenomenon''; any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n͞log d , where d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1͞k ␤ for some fixed exponent ␤. For the case of ␤ > 3, we prove that the average distance of the power law graphs is almost surely of order log n͞log d . However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < ␤ < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, which we call the core, having n c͞log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.

Preface vii Chapter 1. Graph Theory in the Information Age 1.1. Introduction 1.2. Basic definitions 1.3. Degree sequences and the power law 1.4. History of the power law 1.5. Examples of power law graphs 1.6. An outline of the book Chapter 2. Old and New Concentration Inequalities 2.1. The binomial distribution and its asymptotic behavior 2.2. General Chernoff inequalities 2.3. More concentration inequalities 2.4. A concentration inequality with a large error estimate 2.5. Martingales and Azuma's inequality 2.6. General martingale inequalities 2.7. Supermartingales and Submartingales 2.8. The decision tree and relaxed concentration inequalities Chapter 3. A Generative Model-the Preferential Attachment Scheme 3.1. Basic steps of the preferential attachment scheme 3.2. Analyzing the preferential attachment model 3.3. A useful lemma for rigorous proofs 3.4. The peril of heuristics via an example of balls-and-bins 3.5. Scale-free networks 3.6. The sharp concentration of preferential attachment scheme 3.7. Models for directed graphs Chapter 4. Duplication Models for Biological Networks 4.1. Biological networks 4.2. The duplication model 4.3. Expected degrees of a random graph in the duplication model 4.4. The convergence of the expected degrees 4.5. The generating functions for the expected degrees 4.6. Two concentration results for the duplication model 4.7. Power law distribution of generalized duplication models Chapter 5. Random Graphs with Given Expected Degrees 5.1. The Erdős-Rényi model 5.2. The diameter of G n,p iii iv CONTENTS 5.3. A general random graph model 5.4. Size, volume and higher order volumes 5.5. Basic properties of G(w) 5.6. Neighborhood expansion in random graphs 5.7. A random power law graph model 5.8. Actual versus expected degree sequence Chapter 6. The Rise of the Giant Component 6.1. No giant component if w < 1? 6.2. Is there a giant component ifw > 1? 6.3. No giant component ifw < 1? 6.4. Existence and uniqueness of the giant component 6.5. A lemma on neighborhood growth 6.6. The volume of the giant component 6.7. Proving the volume estimate of the giant component 6.8. Lower bounds for the volume of the giant component 6.9. The complement of the giant component and its size 6.10. Comparing theoretical results with the collaboration graph Chapter 7. Average Distance and the Diameter 7.1. The small world phenomenon 7.2. Preliminaries on the average distance and diameter 7.3. A lower bound lemma 7.4. An upper bound for the average distance and diameter 7.5. Average distance and diameter of random power law graphs 7.6. Examples and remarks Chapter 8. Eigenvalues of the Adjacency Matrix of G(w) 8.1. The spectral radius of a graph 8.2. The Perron-Frobenius Theorem and several useful facts 8.3. Two lower bounds for the spectral radius 8.4. An eigenvalue upper bound for G(w) 8.5. Eigenvalue theorems for G(w) 8.6. Examples and counterexamples 8.7. The spectrum of the adjacency matrix of power law graphs 170 Chapter 9. The SemiCircle Law for G(w) 9.1. Random matrices and Wigner's semicircle law 9.2. ...

We study the elemental abundances of C, N, O, Al, Si, S, Cr, Mn, Fe, Ni, and Zn in a sample of 14 damped Lyα systems (galaxies) with H I column density N (H I)≥ 10 20 cm −2 , using high quality spectra of quasars obtained with the 10m Keck telescope. To ensure accuracy, only weak, unsaturated absorption lines are used to derive ion column densities and elemental abundances. Combining these abundance measurements with similar measurements in the literature, we investigate the chemical evolution of damped Lyα galaxies based on a sample of 23 systems in the redshift range 0.7 < z < 4.4. The main conclusions are as follows.1. The damped Lyα galaxies have (Fe/H) in the range of 1/10 to 1/300 solar, clearly indicating that these are young galaxies in the early stages of chemical evolution. The N (H I)-weighted mean metallicity of the damped Lyα galaxies between 2 < z < 3 is (Fe/H)=0.028 solar. There is a large scatter, about a factor of 30, in (Fe/H) at z < 3, which we argue probably results from the different formation histories of the absorbing galaxies or a mix of galaxy types.2. Comparisons of the distribution of (Fe/H) vs redshift for the sample of damped Lyα galaxies with the similar relation for the Milky Way disk indicate that the damped Lyα galaxies are much less metal-enriched than the Galactic disk in its past. Since there is evidence from our analyses that depletion of Fe by dust grains in the sample galaxies is relatively unimportant, the difference in the enrichment level between the sample of damped Lyα galaxies and the Milky Way disk suggests that damped Lyα galaxies are probably not high-redshift spiral disks in the traditional sense. Rather, they could represent a thick disk phase of galaxies, or more likely the spheroidal component of galaxies, or dwarf galaxies.3. The mean metallicity of the damped Lyα galaxies is found to increase with decreasing redshift, as is expected. All four of the damped Lyα galaxies at z > 3 in our sample have (Fe/H) around 1/100 solar or less. In comparison, a large fraction of the damped Lyα galaxies at z < 3 have reached ten times higher metallicity. This suggests that the time around z = 3 may be the epoch of galaxy formation in the sense that galaxies are beginning to form the bulk of their stars. Several other lines of evidence appear to point to the same conclusion, including the evolution of the neutral baryon 1 Hubble Fellow

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In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent ␤, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains. E igenvalues of graphs are useful for controlling many graph properties and consequently have numerous algorithmic applications including low rank approximations, ‡ information retrieval (1), and computer vision. § Of particular interest is the study of eigenvalues for graphs with power-law degree distributions (i.e., the number of vertices of degree j is proportional to j Ϫ␤ for some exponent ␤). It has been observed by many research groups (2-8, ¶) that many realistic massive graphs including Internet graphs, telephone-call graphs, and various social and biological networks have power-law degree distributions.For the classical random graphs based on the Erdös-Rényi model, it has been proved by Füredi and Komlós that the spectrum of the adjacency matrix follows the Wigner semicircle law (9). Wigner's theorem (10) and its extensions have long been used for the stochastic treatment of complex quantum systems that lie beyond the reach of exact methods. The semicircle law has extensive applications in statistical and solid-state physics (21,22). Here we intend to reconcile these two schools of thought on eigenvalue distributions. To begin with, there are in fact several ways to associate a matrix to a graph. The usual adjacency matrix A associated with a (simple) graph has eigenvalues quite sensitive to the maximum degree (which is a local property). The combinatorial Laplacian D Ϫ A, with D denoting the diagonal degree matrix, is a major tool for enumerating spanning trees and has numerous applications (13, 14). Another matrix associated with a graph is the, which controls the expansion͞isoperimetrical properties (which are global) and essentially determines the mixing rate of a random walk on the graph. The traditional random matrices and random graphs are regular or almost regular, thus the spectra of all the above three matrices are basically the same (with possibly a scaling factor or a linear shift). However, for graphs with uneven degrees, the above three matrices can h...

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs. Introduction.The Ricci curvature plays a very important role on geometric analysis on Riemannian manifolds. Many results are established on manifolds with non-negative Ricci curvature or on manifolds with Ricci curvature bounded below.The definition of the Ricci curvature on metric spaces was first from the well-known Bakry and Emery notation. Bakry and Emery [1] found a way to define the "lower Ricci curvature bound" through the heat semigroup (P t ) t ≥0 on a metric measure space M. There are some recent works on giving a good notion for a metric measure space to have a "lower Ricci curvature bound", see [21], [18] and [19]. Those notations of Ricci curvature work on so called length spaces. In 2009, Ollivier [20] gave a notion of coarse Ricci curvature of Markov chains valid on arbitrary metric spaces, such as graphs.Graphs and manifolds are quite different in their nature. But they do share some similar properties through Laplace operators, heat kernels, and random walks, etc. Many pioneering works were done by Chung, Yau, and their coauthors [3,

We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.