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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