Multiscale genesis of a tiny giant for percolation on scale-free 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 $C_G(r)\sim(2/(m^{r-1}(m-1)))\;n\log n$.

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.

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.

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

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, 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.

We adopt the concept of using pheromones to generate a set of static paths that can reach the performance of global dynamic routing strategy [Phys. Rev. E 81 (2010) 016113]. The path generation method consists of two stages. In the first stage, a pheromone is dropped to the nodes by packets forwarded according to the global dynamic routing strategy. In the second stage, pheromone static paths are generated according to the pheromone density. The output paths can greatly improve traffic systems' overall capacity on different network structures, including scale-free networks, small-world networks and random graphs. Because the paths are static, the system needs much less computational resources than the global dynamic routing strategy.

This paper applies graph theory metrics to network flow models, with the aim of assessing the possibility of using these metrics to identify vulnerable areas within infrastructure systems. To achieve this, a reduced complexity flow model that can be used to simulate flows in infrastructure networks is developed. The reason for developing this model is not to make the analysis easier, but to reduce the physical problem to its most basic level and therefore produce the most general flow model (i.e., applicable to the widest range of infrastructure networks). An initial assessment of the applicability of graph theory metrics to infrastructure networks is made by comparing the distribution of flows, calculated using this model, to the shortest average path length in three of the most recognized classes of network—scale-free networks, small-world networks, and random graph models—and it is demonstrated that for all three classes of network there is a strong correlation. This suggests that at least parts of graph theory may be used to inform one about the behavior of physical networks. The authors further demonstrate the utility of graph theory metrics by using them to improve their predictive skill in identifying vulnerable areas in a specific type of infrastructure system. This is done using a hydraulic model to calculate the flows in a sample water distribution network and then to show that using a combination of graph theory metrics and flow gives superior predictive skill over just one of these measures in isolation.

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.

Impact of social neighborhood on diffusion of innovation S-curve

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.

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.

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.

Collective decision making is crucial in human organizations and societies. When a collective is working on exploration of problem space and/or ideation for creative solutions, the evolutionary perspective is useful for conceptualizing and modeling collective decision making, where populations of ideas spread and evolve on a social network habitat via continual applications of evolutionary operators by human individuals (Sayama & Dionne 2015). Using an evolutionary approach to model collective decision making, we conducted agent-based simulations to investigate how collective decision making would be affected by the size and topology of social network structure (Sayama, Dionne & Yammarino 2013). In our model, each agent has its own utility function defined over a multi-dimensional problem space, which is marginally different from the “true” utility function that is not accessible from any agent. Each agent can memorize multiple ideas in mind, and iteratively applies evolutionary operators (e.g., replication, mutation, recombination, elimination) to the idea population it has. The outcomes of evolutionary operations are stored in the agent’s mind, and also sent to the neighbors to which the agent is connected. This allows the spread of ideas through social ties. Each simulation was run for a fixed number of iterations, and then the level of idea convergence at a social level and the true utility value of the most supported idea were measured as the performance metrics of collective decision making. The former metric characterizes the ability for the society to form consensus, while the latter characterizes its ability to find the true best solution. The size of the network (number of agents or nodes) was varied from 5 to 640 logarithmically. Three network topologies with the same average node degree were tested: Random, small-world and scale-free. For more details of the model and the simulation settings, see Sayama, Dionne & Yammarino (2013). Simulation results indicated that expanding the size of the social network generally improved the quality of ideas at the cost of decision convergence. Simulations with different social network topologies further revealed that collective decision making on small-world networks with high local clustering tended to achieve highest decision quality more often than on random or scale-free networks (Fig. 1). This can be understood in that local clustering helps agents in different regions in a network maintain their respective focus areas and engage in different local search, possibly enhancing the effective parallelism of collective decision making and therefore resulting in a greater number of successful decisions. In contrast, the links in random and scale-free networks are all “global”, mixing discussions prematurely and therefore reducing the effective parallelism of collective decision making. These observations have an interesting contrast with the fact that random and scale-free networks are highly efficient in information dissemination because of their global connectedness. Our results indicate that such efficiency of information dissemination may not necessarily imply the same for collective decision making. This work may also offer an evolutionary explanation for the nontrivial relationship between network clustering and problem solving, a problem that is actively debated in organizational science (Shore, Bernstein & Lazer 2014).