Hierarchical Models for Independence Structures of Networks

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erd\"os-R\'enyi and beta-models to create hierarchical Erd\"os-R\'enyi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.Comment: 19 pages, 7 figure

Counting independent sets in triangle-free graphs

Ajtai, Koml\'os, and Szemer\'edi proved that for sufficiently large $t$ every triangle-free graph with $n$ vertices and average degree $t$ has an independent set of size at least $\frac{n}{100t}\log{t}$. We extend this by proving that the number of independent sets in such a graph is at least $$2^{(1/2400)\frac{n}{t}\log^2{t}}.$$ This result is sharp for infinitely many $t,n$ apart from the constant. An easy consequence of our result is that there exists $c'>0$ such that every $n$-vertex triangle-free graph has at least $$2^{c'\sqrt n \log n}$$ independent sets. We conjecture that the exponent above can be improved to $\sqrt{n}(\log{n})^{3/2}$. This would be sharp by the celebrated result of Kim which shows that the Ramsey number $R(3,k)$ has order of magnitude $k^2/\log k$

2.5K-Graphs: from Sampling to Generation

Understanding network structure and having access to realistic graphs plays a central role in computer and social networks research. In this paper, we propose a complete, and practical methodology for generating graphs that resemble a real graph of interest. The metrics of the original topology we target to match are the joint degree distribution (JDD) and the degree-dependent average clustering coefficient ($\bar{c}(k)$). We start by developing efficient estimators for these two metrics based on a node sample collected via either independence sampling or random walks. Then, we process the output of the estimators to ensure that the target properties are realizable. Finally, we propose an efficient algorithm for generating topologies that have the exact target JDD and a $\bar{c}(k)$ close to the target. Extensive simulations using real-life graphs show that the graphs generated by our methodology are similar to the original graph with respect to, not only the two target metrics, but also a wide range of other topological metrics; furthermore, our generator is order of magnitudes faster than state-of-the-art techniques

Note on the smallest root of the independence polynomial

One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real

Relaxation dynamics of maximally clustered networks

We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics---the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erd\H{o}s--R\'enyi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erd\H{o}s--R\'enyi phenomenology