In recent years, there has been a surge of research interest in networks with scale-free topologies, partly due to the fact that they are prevalent in scientific research and real-life applications. In this paper, we study random-walk issues on a family of two-parameter scale-free networks, called $(x, y)$-flowers. These networks, which are constructed in a deterministic recursive fashion, display rich behaviors such as the small-world phenomenon and pseudofractal properties. We derive analytically the mean commute times for random walks on $(x, y)$-flowers and show that the mean commute times scale with the network size as a power-law function with exponent governed by both parameters $x$ and $y$. We also determine the mean effective resistance and demonstrate that it changes sharply between different choices of $x$ and $y$. Furthermore, we compare mean commute times for $(x, y)$-flowers with those for Erdős–Rényi random graphs. Our theoretical results are verified by numerical studies.

Several network models have been proposed to explain the link structure observed in online social networks. This paper addresses the problem of choosing the model that best fits a given real-world network. We implement a model-selection method based on unsupervised learning. An alternating decision tree is trained using synthetic graphs generated according to each of the models under consideration. We use a broad array of features, with the aim of representing different structural aspects of the network. Features include the frequency counts of small subgraphs (graphlets) as well as features capturing the degree distribution and small-world property. Our method correctly classifies synthetic graphs, and is robust under perturbations of the graphs. We show that the graphlet counts alone are sufficient in separating the training data, indicating that graphlet counts are a good way of capturing network structure. We tested our approach on four Facebook graphs from various American universities. The models that best fit these data are those that are based on the principle of preferential attachment.

By extending Heider's and Cartwright–Harary's theory of balance in deterministic social structures, we study the problem of balance in social structures in which relations among individuals are random. An appropriate model for representing such structures is that of random signed graphs $G_{n,p,q}$, defined as follows. Given a set of $n$ vertices and fixed numbers $p$ and $q$, $0 \lt p + q \lt 1$, then between each pair of vertices, there exists a positive edge, a negative edge, or no edge with respective probabilities $p, q, 1 − p − q$.

We first show that almost always (i.e., with probability tending to 1 as $n \to \infty$), the random signed graph $G_{n,p,q}$ is unbalanced. Subsequently we estimate the maximum order of a balanced induced subgraph in $G_{n,p,p}$ and show that its order achieves only a finite number of values. Next, we study the asymptotic behavior of the degree of balance and give upper and lower bounds for the line index of balance. Finally, we study the threshold function of balance, e.g., a function $p_0 (n)$ such that if $p \gg p_0 (n)$, then the random signed graph $G_{n,p,p}$ is almost always unbalanced, and otherwise, it is almost always balanced.

In this paper we extend and generalize the standard spectral graph theory (or random-walk theory) on undirected graphs to digraphs. In particular, we introduce and define a normalized digraph Laplacian (Diplacian for short) $\Gamma$ for digraphs, and prove that (1) its Moore–Penrose pseudoinverse is the discrete Green’s function of the Diplacian matrix as an operator on digraphs, and (2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive a new formula for computing hitting and commute times in terms of the Moore–Penrose pseudoinverse of the Diplacian, or equivalently, the singular values and vectors of the Diplacian.

Furthermore, we show that the Cheeger constant defined in “Laplacians and the Cheeger Inequality for Directed Graphs,” is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric, the largest singular value of the skewed Laplacian $\nabla = (\Gamma − \Gamma^T )/2$, to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound than that in “Laplacians and the Cheeger Inequality for Directed Graphs,” on the Markov chain mixing rate, and a bound on the second-smallest singular value of $\Gamma$.

We propose and analyze a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, online social behavior, and information processing in neuroscience. We model the evolving network as a discrete-time Markov chain and study a very general framework in which edges conditioned on the current state appear or disappear independently at the next time step. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cellphone data, and find support for a block model in which individuals are assigned to one of two distinct groups having different within-group and across-group dynamics.

We study a strategic game in which every node of a graph is owned by a player who has to choose a color. A player’s payoff is 0 if at least one neighbor selected the same color; otherwise, it is the number of players who selected the same color. The social cost of a state is defined as the number of distinct colors that the players use. It is ideally equal to the chromatic number of the graph, but it can substantially deviate because every player cares about his own payoff, however bad the social cost may be. Following previous work in Panagopoulou and Spirakis, “A Game Theoretic Approach for Efficient Graph Coloring,” on the Nash equilibria of the coloring game, we give worst-case bounds on the social cost of stable states. Our main contribution is an improved (tight) bound for the worst-case social cost of a Nash equilibrium, as well as the study of strong equilibria, their existence, and how far they are from social optima.

We devise methods for finding approximations of the generalized inverse of the graph Laplacian matrix, which arises in many graph-theoretic applications. Finding this matrix in its entirety involves solving a matrix inversion problem, which is resourcedemanding in terms of consumed time and memory and hence impractical whenever the graph is relatively large. Our approximations use only a few eigenpairs of the Laplacian matrix and are parametric with respect to this number, so that the user can compromise between effectiveness and efficiency of the approximate solution. We apply the devised approximations to the problem of computing current-flow betweenness centrality on a graph. However, given the generality of the Laplacian matrix, many other applications can be sought. We experimentally demonstrate that the approximations are effective already with a constant number of eigenpairs. These few eigenpairs can be stored with a linear amount of memory in the number of nodes of the graph, and in the realistic case of sparse networks, they can be efficiently computed using one of the many methods for retrieving a few eigenpairs of sparse matrices that abound in the literature.