Finding Induced Subgraphs in Scale-Free Inhomogeneous Random Graphs

We describe a general approach of determining the distribution of the number of certain types of spanning subgraphs in the random graph ${\mathcal G}(n,p)$. Using this approach, we reprove the distribution of the number of Hamilton cycles with a proof that is much shorter than previously known proofs. We also achieve new results on determining the distribution of the number of spanning triangle-free subgraphs and the number of triangle-factors.

Dense subgraphs in random graphs

This paper first constructs a tiered network model of the interbank market. Then, from the perspective of contagion risk, it studies numerically the resilience of four types of interbank market network models to shocks, namely, tiered networks, random networks, small‐world networks, and scale‐free networks. This paper studies the interbank market with homogeneous and heterogeneous banks and analyzes random shocks and selective shocks. The study reveals that tiered interbank market networks and random interbank market networks are basically more vulnerable against selective shocks, while small‐world interbank market networks and scale‐free interbank market networks are generally more vulnerable against random shocks. Besides, the results indicate that, in the four types of interbank market networks, scale‐free networks have the highest stability against shocks, while small‐world networks are the most vulnerable. When banks are homogeneous, faced with selective shocks, the stability of the tiered interbank market networks is slightly lower than that of random interbank market networks, whereas, in other cases, the stability of the tiered interbank market networks is basically between that of random interbank market networks and that of scale‐free interbank market networks.

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if for a function that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least , and (ii) if , then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015

Scale-free networks, as one major class of complex networks, are characterized by a specific structural feature of power-law degree distributions. Examples of scale-free real-world networks include World Wide Web (WWW), email networks, citation and co-author scientific networks, social networks (Facebook, Twitter), networks of Internet routers, and protein-protein interaction networks. In this paper we model and evaluate the reliability of scale-free complex networks using a binary decision diagram (BDD)-based method. Two-terminal reliability is considered, which is defined as the probability that a specified pair of nodes can communication through at least one fault-tree path in the network. This reliability metric can measure, for example, how reliable the information can be shared between two parties or in general how reliable the information can be disseminated within a social network. The Barabasi-Albert model is used for generating the sample scale-free networks studied in this paper. The reliability performance of scale-free networks is compared to that of random networks under the Erdos-Renyi model. Degree distributions and clustering coefficients (average fraction of pairs of neighbors of a node which are also neighbors of each other) of sample scale-free and random networks are illustrated and compared. Their effects on the performance of the BDD-based method for the reliability analysis of both scale-free and random networks are also discussed.

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).

Coupling methods to establish threshold functions in random graphs

We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs.

If $m\geq2$ is constant and $0\leq r\leq\varepsilon\log\log n$ for a small positive constant $\varepsilon$, then whp a random walk with look-ahead $r$ on a scale-free graph $G=G_{(m,n)}$ has cover time $C_G(r)\sim(2/(m^{r-1}(m-1)))\;n\log n$.

We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph.

One of the classical questions in random graph theory is to understand the asymptotics of subgraph counts. In inhomogeneous random graph, this question has not been well studied. In this study, we investigate the asymptotic distribution of m‐cliques in a sparse inhomogeneous random graph. Under mild conditions, we prove that the number of m‐cliques converges in law to the standard normal distribution.

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

In this paper, we compare the efficiency of fault detection and diagnosis in networks having different topological properties, such as scale-free networks and Erdos-Renyi random graphs. Efficiency measures include both the number of tests (e.g., end-to-end network probes) necessary for diagnosis and the computational complexity of diagnosis. We observe that diagnosis in scale-free networks typically requires significantly larger number of tests than diagnosis in random networks. However, the computational complexity of diagnosis appears to be much lower for scale-free networks since the corresponding Bayesian network models used for probabilistic diagnosis tend to have much lower induced width - a topological parameter controlling the complexity of inference in Bayesian networks. We believe that our observations provide important insights for design and deployment of cost-efficient diagnostic methods in computer networks and distributed systems.

In this paper we assess the vulnerability of different synthetic complex networks by measuring the traffic performance in presence of intentional nodes and edge attacks. We choose which nodes or edges would be attacked by using several centrality measures, such as: degree, eigenvector and betweenness centrality. In order to obtain some information about the vulnerability of the four different complex networks (random, small world, scale-free and random geometric) we analyze the throughput of these networks when the nodes or the edges are attacked using some of the above mentioned strategies. When attack happens, the bandwidth is reallocated among the flows, which affects the traffic utility. One of the obtained results shows that the scale-free network gives the best flow performance and then comes random networks, small world, and the poorest performance is given by the random geometric networks. This changes dramatically after removing some of the nodes (or edges), giving the biggest performance drop to random and scale-free networks and smallest to random geometric and small world networks.