Random Walks with Look-Ahead in Scale-Free Random Graphs

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.

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.

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.

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.

Impact of social neighborhood on diffusion of innovation S-curve

The present thesis studies some important random walk-based algorithms, which are randomized rumor spreading and balanced allocation protocols on networks. In the rst part of the thesis, we study the Push and the Push-Pull protocols introduced by [DGG+87], which are basic randomized protocols for information dissemination on networks. In Chapter 2, we propose a new model where the number of calls of each node in every round is chosen independently according to a probability distribution R with bounded mean determined at the beginning of the process. In addition to the model being a natural extension of the standard protocols, it also serves as a more realistic model for rumor spreading in a network whose entities are not completely uniform and may have di erent levels of power. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance. While it is well-known that the standard protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes, we show that, if R follows a power law distribution with exponent β ∈ (2, 3), then the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, we show a runtime of Θ ( logn log logn ) . In Chapter 3, we analyze the behavior of the standard Push-Pull protocol on a class of random graphs, called random k-trees for every integer k > 2, that are suitable to model poorly connected, small-world and scale free networks. Here, we show that the Push-Pull protocol propagates a rumor from a randomly chosen informed node to almost all nodes of a random k-tree with n nodes in O((log n)1+ck) rounds with high probability, where 0 3 can be any integer. Let l be a given positive integer. In each round t, 1 6 t 6 n, ball t picks a node of G uniformly at random and performs a nonbacktracking random walk of length l from the chosen node. Then it deterministically selects a subset of the visited nodes as the potential choices and allocates itself on one of the choices with minimum load (ties are broken uniformly at random). Provided G has a su ciently large girth, we establish an upper bound for the maximum number of balls at any bin after allocating n balls by the algorithm. We also show that the upper bound is tight up to a O(log log n) factor. In particular, we show that if we set l = b(log n) 1+ 2 c, for any constant ∈ (0, 1], and G has girth at least ω(l), then the maximum load is bounded by O(1/ ) with high probability.

Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages, using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real-world graphs; for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees . Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.

We study the problem of finding a copy of a specific induced subgraph on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that finds a copy of any connected graph $H$ on a fixed number of $k$ vertices as an induced subgraph in a random graph with $n$ vertices. By exploiting the scale-free graph structure, the algorithm runs in $O(n k)$ time for small values of $k$. As a corollary, this shows that the induced subgraph isomorphism problem can be solved in time $O(nk)$ for the inhomogeneous random graph. We test our algorithm on several real-world data sets.

Scale-free networks generated by random walkers

Hubs play important roles in scale-free networks. Existing hub search algorithms mostly assume the availability of the global network structure and use a variety of centrality metrics to search the hubs in the network. However, when it is very difficult to obtain the network topology in large-scale networks, how we can search the hubs? In this paper, a hub search method based on sampling with biased algorithms is proposed and further four algorithms are compared, including improved MHRW (Metropolis-Hasting Random Walk), MDF (Maximum-Degree First), BFS (Breadth-First Search) and RW (Random Walk). The experiments on several datasets show that both MDF and improved MHRW algorithm can reach a higher HDR (Hub Detection Rate) than BFS and RW, and when the sampling rate goes above 10%, MDF and improved MHRW can find an average of more than 70% of hubs in scale-free network.

We study the critical behavior for percolation on inhomogeneous random networks on n vertices, where the weights of the vertices follow a power-law distribution with exponent τ∈(2,3). Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as n→∞. We identify the critical window for several scale-free random graph models, such as the Norros–Reittu model, Chung–Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which we see a sudden emergence of a “tiny” giant component. This is a novel behavior, which is in contrast with the critical behavior in other known universality classes with τ∈(3,4) and τ>4. Precisely, for edge-retention probabilities πn=λn−(3−τ)/2, there is an explicitly computable λc>0 such that the critical window is of the form λ∈(0,λc), where the largest clusters have size of order nβ with β=(τ2−4τ+5)/[2(τ−1)]∈[ 2−1,12) and have nondegenerate scaling limits, while in the supercritical regime λ>λc, a unique “tiny giant” component of size Θ(n) emerges, and its size concentrates. For λ∈(0,λ c), the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on Z+ studied in a seminal work by Durrett and Kesten (In A Tribute to Paul Erdős (1990) 161–176 Cambridge Univ. Press). For λ>λc, we use a relation to general inhomogeneous random graphs, as studied by Bollobás, Janson and Riordan (Random Structures Algorithms 31 (2007) 3–122), to prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight of order at least n.

Universal ‘winner-takes-it-all’ phenomenon in scale-free random graphs

formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements N--> infinity. We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scale-free networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scale-free biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells. (c) 2001 American Institute of Physics.

There has been a recent surge of interest in studying dynamical processes, including evolutionary optimization, on scale-free topologies. While various scaling parameters and assortativities have been observed in natural scale-free interaction networks, most previous studies of dynamics on scale-free graphs have employed a graph-generating algorithm that yields a topology with an uncorrelated degree distribution and a fixed scaling parameter. In this paper, we advance the understanding of selective pressure in scale-free networks by systematically investigating takeover times under local uniform selection in scale-free topologies with varying scaling exponents, assortativities, average degrees, and numbers of vertices. We demonstrate why the so-called characteristic path length of a graph is a nonlinear function of both scaling and assortativity. Neither the eigenvalues of the adjacency matrix nor the effective population size was sufficient to account for the variance in takeover times over the parameter space that was explored. Rather, we show that 97% of the variance of logarithmically transformed average takeover times, on all scale-free graphs tested, could be accounted for by a planar function of: 1) the average inverse degree (which captures the effects of scaling) and 2) the logarithm of the population size. Additionally, we show that at low scaling exponents, increasingly <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">positive</i> assortativities increased the variability between experiments on different random graph instances, while increasingly <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">negative</i> assortativities increased the variability between takeover times from different initial conditions on the same graph instances. We explore the mechanisms behind our sometimes counterintuitive findings, and discuss potential implications for evolutionary computation and other relevant disciplines.

While many scale-free (SF) networks have been introduced recently for complex systems, most of them are binary random graphs and the rate at which the node in the network increases its connectivity depends on the time it arrived. We propose a model of weighted scale-free networks incorporating a fit-gets-richer scheme which means the connectivity of the node depends on both the degree and fitness of the node. The topology and weights of links of the network evolve as time goes on. The combined numerical and analytical approach indicates that asymptotically the scaling behaviors of the total weight distribution and the connectivity distribution are identical. The asymptotical sameness has also been observed in real networks.