Efficient modularity density heuristics in graph clustering and their applications

Modularity Density Maximization is a graph clustering problem which avoids the resolution limit degeneracy of the Modularity Maximization problem. This thesis aims at solving larger instances than current Modularity Density heuristics do, and show how close the obtained solutions are to the expected clustering. Three main contributions arise from this objective. The first one is about the theoretical contributions about properties of Modularity Density based prioritizers. The second one is the development of eight Modularity Density Maximization heuristics. Our heuristics are compared with optimal results from the literature, and with GAOD, iMeme-Net, HAIN, BMD- heuristics. Our results are also compared with CNM and Louvain which are heuristics for Modularity Maximization that solve instances with thousands of nodes. The tests were carried out by using graphs from the “Stanford Large Network Dataset Collection”. The experiments have shown that our eight heuristics found solutions for graphs with hundreds of thousands of nodes. Our results have also shown that five of our heuristics surpassed the current state-of-the-art Modularity Density Maximization heuristic solvers for large graphs. A third contribution is the proposal of six column generation methods. These methods use exact and heuristic auxiliary solvers and an initial variable generator. Comparisons among our proposed column generations and state-of-the-art algorithms were also carried out. The results showed that: (i) two of our methods surpassed the state-of-the-art algorithms in terms of time, and (ii) our methods proved the optimal value for larger instances than current approaches can tackle. Our results suggest clear improvements to the state-of-the-art results for the Modularity Density Maximization problem

Threshold Graphs Maximize Homomorphism Densities

Given a fixed graph $H$ and a constant $c \in [0,1]$, we can ask what graphs $G$ with edge density $c$ asymptotically maximize the homomorphism density of $H$ in $G$. For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximizing $G$ is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximization, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs $H$ and densities $c$ such that the optimizing graph $G$ is neither the quasi-star nor the quasi-clique, reproving a result of Day and Sarkar. We rederive a result of Janson et al. on maximizing homomorphism numbers, which was originally found using entropy methods. We also show that for $c$ large enough all graphs $H$ maximize on the quasi-clique, which was also recently proven by Gerbner et al., and in analogy with Kopparty and Rossman we define the homomorphism density domination exponent of two graphs, and find it for any $H$ and an edge

Optimizing spread dynamics on graphs by message passing

Cascade processes are responsible for many important phenomena in natural and social sciences. Simple models of irreversible dynamics on graphs, in which nodes activate depending on the state of their neighbors, have been successfully applied to describe cascades in a large variety of contexts. Over the last decades, many efforts have been devoted to understand the typical behaviour of the cascades arising from initial conditions extracted at random from some given ensemble. However, the problem of optimizing the trajectory of the system, i.e. of identifying appropriate initial conditions to maximize (or minimize) the final number of active nodes, is still considered to be practically intractable, with the only exception of models that satisfy a sort of diminishing returns property called submodularity. Submodular models can be approximately solved by means of greedy strategies, but by definition they lack cooperative characteristics which are fundamental in many real systems. Here we introduce an efficient algorithm based on statistical physics for the optimization of trajectories in cascade processes on graphs. We show that for a wide class of irreversible dynamics, even in the absence of submodularity, the spread optimization problem can be solved efficiently on large networks. Analytic and algorithmic results on random graphs are complemented by the solution of the spread maximization problem on a real-world network (the Epinions consumer reviews network).Comment: Replacement for "The Spread Optimization Problem

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model (HSCM), belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or $W$-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model (HSCM), belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or $W$-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit