"Graphlets give a spectral approach to graph limits for general graph sequences in a framework that unifies previous disparate approaches for dealing with dense graphs and sparse graphs. We will show that the con- vergence to graphlets under the appropriate spectral distance is equivalent to the convergence using the (normalized) cut distance. We then examine the geometry of graphlets, illustrated by examples of several families of graphlets and, in particular, graphlets with low ranks. We further dis- cuss a number of usages of graphlets, including universal scalable bases, universal embeddings vis heat kernels and the preservation of Cheeger cuts."

ETA: This is so not an easy read. I like what I understand, but I definitely have to make another attack on it.

"Randomizing networks using a naive “accept-all” edge-swap algorithm is generally biased. Building on recent results for nondirected graphs, we construct an ergodic detailed balance Markov chain with nontrivial acceptance probabilities for directed graphs, which converges to a strictly uniform measure and is based on edge swaps that conserve all in and out degrees. The acceptance probabilities can also be generalized to define Markov chains that target any alternative desired measure on the space of directed graphs in order to generate graphs with more sophisticated topological features. This is demonstrated by defining a process tailored to the production of directed graphs with specified degree-degree correlation functions. The theory is implemented numerically and tested on synthetic and biological network examples."

"We show how to sample acyclic digraphs uniformly at random through recursive enumeration. This provides an exact method which avoids the convergence issues of the alternative Markov chain methods. The limiting behaviour of the distribution of acyclic digraphs also allows us to sample arbitrarily large acyclic digraphs. Finally we discuss how to include various restrictions in the combinatorial enumeration for efficient uniform sampling of the corresponding graphs."

"The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs $G_N, Nge 1$ to some target graph $W$. The theory of dense graph convergence, including random dense graphs, is now well understood, but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the log-partition function limits, also known as free energy limits, appropriately normalized for the Gibbs distribution associated with $W$. In this paper we prove that the sequence of sparse ER graphs is right-converging when the tensor product associated with the target graph $W$ satisfies certain convexity property. We treat the case of discrete and continuous target graphs $W$. The latter case allows us to prove a special case of Talagrand's recent conjecture (more accurately stated as level III Research Problem 6.7.2 in his recent book), concerning the existence of the limit of the measure of a set obtained from $R^N$ by intersecting it with linearly in $N$ many subsets, generated according to some common probability law.
Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli and developed further in a series of papers. Specifically, Bayati et al establish the right-convergence property for Erdos-Renyi graphs for some special cases of $W$. In this paper most of the results in this paper follow as a special case of our main theorem."

"Graph theory and the fields of natural language processing and information retrieval are well-studied disciplines. Traditionally, these areas have been perceived as distinct, with different algorithms, different applications, and different potential end-users. However, recent research has shown that these disciplines are intimately connected, with a large variety of natural language processing and information retrieval applications finding efficient solutions within graph-theoretical frameworks. This book extensively covers the use of graph-based algorithms for natural language processing and information retrieval. It brings together topics as diverse as lexical semantics, text summarization, text mining, ontology construction, text classification, and information retrieval, which are connected by the common underlying theme of the use of graph-theoretical methods for text and information processing tasks. Readers will come away with a firm understanding of the major methods and applications in natural language processing and information retrieval that rely on graph-based representations and algorithms."

"We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [Song et al., Nature (London) 433 392 (2005)], RSR allows a more fine-grained analysis of the renormalization group (RG) flow and unravels new features that were not discussed in the previous analyses. In particular, we find that all networks exhibit a second-order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling behavior seen there. For critical trees it happens as N/N0→0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0→1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a starlike structure in agreement with the results of Radicchi et al. [ Phys. Rev. Lett. 101 148701 (2008)]. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering “supernodes” as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics."

In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schr"odinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.

"Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of $H_G$ in terms of the adjacency matrix of $G$ and we give a central limit theorem for $H_G$. Finally, we show that the mp-chart is easily tractable also for the complement graph." --- I have no idea what this means either.

"We study the realizability of scale-free networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse."

"In recent years many algorithms have been developed for finding patterns in graphs and networks. A disadvantage of these algorithms is that they use subgraph isomorphism to determine the support of a graph pattern; subgraph isomorphism is a well-known NP complete problem. In this paper, we propose an alternative approach which mines tree patterns in networks by using subgraph homomorphism. The advantage of homomorphism is that it can be computed in polynomial time, which allows us to develop an algorithm that mines tree patterns in arbitrary graphs in incremental polynomial time. Homomorphism however entails two problems not found when using isomorphism: (1) two patterns of different size can be equivalent; (2) patterns of unbounded size can be frequent. In this paper we formalize these problems and study solutions that easily fit within our algorithm."

The metric is not at all compelling, but it's nice to see this attempted at all.

"The relationship between a network's degree-degree correlation and a loose version of graph coloring is studied on networks with broad degree distributions. We find that, given similar conditions on the number of nodes, number of links, and clustering levels, fewer colors are needed to color disassortative than assortative networks. Since fewer colors create fewer independent sets, our finding implies that disassortative networks may have higher concurrency potential than assortative networks. This in turn suggests another reason for the disassortative mixing pattern observed in biological networks such as those of protein-protein interaction and gene regulation. In addition to the functional specificity and stability suggested by Maslov and Sneppen, a disassortative network topology may also enhance the ability of cells to perform crucial tasks concurrently..." Related to Judd/Kearns experiments on human graph coloring?

Less interesting than it sounded.

Limits of sequences of graphs; looks very cool and possibly statistically-applicable. (E.g., giving a reasonable meaning to the idea of "in the limit, as we get more data".)

Using the spectral graph theory to get new features, for visualization, clustering, &c. Also includes a good discussion of the local linear embedding algorithm. Will try to teach this in 350 but may make heads explode.

I notice that they don't even attempt to describe the problem, which has to do with the existence of synchronizing words for labeled directed graphs/stochastic automata.