Random Deletion in a Scale-Free Random Graph Process

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.

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.

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.

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.

It is now known that the complexity of network topology has a great impact on the stabilization of complex dynamical networks. In this work, we study the control of random networks and scale-free networks. Conditions are investigated for globally or locally stabilizing such networks. Our strategy is to apply local feedback control to a small fraction of network nodes. We propose the concept of virtual control for microscopic dynamics throughout the process with different pinning schemes for both random networks and scale-free networks. We explain the main reason why significantly less local controllers are required by specifically pinning the most highly connected nodes in a scale-free network than those required by the randomly pinning scheme, and why there is no significant difference between specifically and randomly pinning schemes for controlling random dynamical networks. We also study the synchronization phenomenon of controlled dynamical networks in the stabilization process, both analytically and numerically.

Activity at rest provides valuable information about functional organization of brain networks. However, lack of cellular resolution in human brain imaging techniques precludes analysis of some of the key properties of resting state networks. Here we used a large-scale c-Fos imaging of resting state neuronal activity in the mouse brain combined with graph analysis methods to get deeper insights into structure and functional significance of resting state brain networks. We compared experimentally identified networks with model networks having the same parameters of edges and nodes: random, scale free and small world networks, using parameters of clustering and global efficiency. Clustering of experimental networks was at the same level as in scale free network – the number of clusters in experimental networks exceeded the random level. However, global efficiency analysis revealed that these clusters have weak interactions with each other or even no interaction at all – thus the global efficiency of experimental networks was at the same level as of a random network. We also found that a single strong traumatic experience can lead to global and long-term changes in the structure of resting state networks. The resting state network of mice which underwent traumatic experience had greater entropy than that of naive mice, especially in the zone of increased influence of strong correlations. We suggest that these changes reflect replay of neuronal assemblies involved in the states of past experience.

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number,m, of edges are added at each step.Letf(t) be the number of edges added at stept. In the standard scale‐free model,f(t) =mconstant, whereas in this paper we considerf(t) = [tc],c> 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth.We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior.For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. Whenf(t) =mconstant, this is the standard scale‐free model, and the power law of the degree sequence is 3. Whenf(t) = [tc],c< 1, the degree sequence of the process exhibits a power law with parameterx= (3 −c)/(1 −c). Asc→ 0,x→ 3, which gives a value ofx= 3, as in standard scale‐free model. Thus no more slowly growing monotone functionf(t) alters the power law of this model away fromx= 3. Whenc= 1, so thatf(t) =t, the expected degree of all vertices ist, the vertex degree is concentrated, and the degree sequence does not have a power law.For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process whenf(t) =mis constant, the power law of the degree sequence isx= 2 + 1/m. Whenf(t) = [tc],c> 0, the power law becomesx= 1 + 1/(1 +c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011

Impact of social neighborhood on diffusion of innovation S-curve

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is 1 + omega (n) n^{-1/3}, with omega (n) tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as n rightarrow infty . As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the ‘sparse’ regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

Cyclic Preferential Attachment in Complex Networks

Revealing corresponding identities of a dedicated individual in several different complex systems is a common task in many areas, and this task is transferred to a node matching problem among complex networks in this paper. A feasible node matching algorithm based on network structure is proposed. Through solving node matching problems on different types of networks by our algorithm, it is revealed that the structure of the networks under study may significantly influence the final matching results. For example, it is found that higher matching precision can be obtained on random networks with moderate density of links, and the results on small-world networks are always better than those on random or regular networks. Moreover, in scale-free networks, it seems that hub nodes play dominant roles, i.e., better matching results can be expected by selecting nodes with larger degrees as the revealed matched nodes. These findings will help us design more efficient node matching algorithm in the future.

Uncorrelated random scale-free (URSF) networks are useful null models for checking the effects of scale-free topology on network-based dynamical processes. Here, we present a comparative study of the jamming level of gradient networks based on URSF networks and Erdős–Rényi (ER) random networks. We find that the URSF networks are less congested than ER random networks for the average degree ⟨k⟩>kc (kc ≈ 2 denotes a critical connectivity). In addition, by investigating the topological properties of the two kinds of gradient networks, we discuss the relations between the topological structure and the transport efficiency of the gradient networks. These findings show that the uncorrelated scale-free structure might allow more efficient transport than the random structure.

We study the random planar graph process introduced by Gerke et al. [Random Structures Algorithms, 32 (2008), pp. 236–261]: Begin with an empty graph on vertices, consider the edges of the complete graph one by one in a random ordering, and at each step add an edge to a current graph only if the graph remains planar. They studied the number of edges added up to step for “large" . In this paper we extend their results by determining the asymptotic number of edges added up to step in the early evolution of the process when . We also show that this result holds for a much more general class of graphs, including outerplanar graphs, planar graphs, and graphs on surfaces.

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.