Tree-like Network Research Articles

Leveraging percolation theory to single out influential spreaders in networks.

Among the consequences of the disordered interaction topology underlying many social, technological, and biological systems, a particularly important one is that some nodes, just because of their position in the network, may have a disproportionate effect on dynamical processes mediated by the complex interaction pattern. For example, the early adoption of a commercial product by an opinion leader in a social network may change its fate or just a few superspreaders may determine the virality of a meme in social media. Despite many recent efforts, the formulation of an accurate method to optimally identify influential nodes in complex network topologies remains an unsolved challenge. Here, we present the exact solution of the problem for the specific, but highly relevant, case of the susceptible-infected-removed (SIR) model for epidemic spreading at criticality. By exploiting the mapping between bond percolation and the static properties of the SIR model, we prove that the recently introduced nonbacktracking centrality is the optimal criterion for the identification of influential spreaders in locally tree-like networks at criticality. By means of simulations on synthetic networks and on a very extensive set of real-world networks, we show that the nonbacktracking centrality is a highly reliable metric to identify top influential spreaders also in generic graphs not embedded in space and for noncritical spreading.

A FRACTAL NETWORK MODEL FOR FRACTURED POROUS MEDIA

The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.

ANALYSIS OF CAPILLARY RISE IN ASYMMETRIC BRANCH-LIKE CAPILLARY

Transport in porous media is common in nature, attracting many attentions for a long time. Tree-like network model is often used as a simplification for porous space, expressing the complexity of pore spaces instead of capillary bundle. To investigate spontaneous imbibition characteristics in this network, a dynamic asymmetric branch-like capillary model is used to represent basic network structure, using fractal method to represent tortuosity. This work investigates the influence of parameters on imbibition process in the branch-like capillary model. An analytical equation for the imbibition mass versus time is derived. Parameters from capillary structures to liquid properties are taken into account and analyzed based on the numerical solution of the equation. It is found that the imbibition process in asymmetric branch-like capillary model can be recognized by four sections and brunching tubes are positive for imbibition process. Concomitantly, meniscus arrest event is simulated and discussed. Moreover, the influence of parameters on imbibition process is discussed. These parameters can be classified as static and dynamic. Static parameters mainly change the capillary force, which are related to the ultimate imbibition mass or imbibition ability, while dynamic parameters mainly have influence on resistance of flowing fluid, which are related to the imbibition speed in the imbibition process.

The entire mean weighted first-passage time on a family of weighted treelike networks.

In this paper, we consider the entire mean weighted first-passage time (EMWFPT) with random walks on a family of weighted treelike networks. The EMWFPT on weighted networks is proposed for the first time in the literatures. The dominating terms of the EMWFPT obtained by the following two methods are coincident. On the one hand, using the construction algorithm, we calculate the receiving and sending times for the central node to obtain the asymptotic behavior of the EMWFPT. On the other hand, applying the relationship equation between the EMWFPT and the average weighted shortest path, we also obtain the asymptotic behavior of the EMWFPT. The obtained results show that the effective resistance is equal to the weighted shortest path between two nodes. And the dominating term of the EMWFPT scales linearly with network size in large network.

Influence of weight heterogeneity on random walks in scale-free networks

Many systems are best described by weighted networks, in which the weights of the edges are heterogeneous. In this paper, we focus on random walks in weighted network, investigating the impacts of weight heterogeneity on the behavior of random walks. We study random walks in a family of weighted scale-free tree-like networks with power-law weight distribution. We concentrate on three cases of random walk problems: with a trap located at a hub node, a leaf adjacent to a hub node, and a farthest leaf node from a hub. For all these cases, we calculate analytically the global mean first passage time (GMFPT) measuring the efficiency of random walk, as well as the leading scaling of GMFPT. We find a significant decrease in the dominating scaling of GMFPT compared with the corresponding binary networks in all three random walk problems, which implies that weight heterogeneity has a significant influence on random walks in scale-free networks.

Arc Based Ant Colony Optimization Algorithm for optimal design of gravitational sewer networks

Abstract In this paper, constrained and unconstrained versions of a new formulation of Ant Colony Optimization Algorithm (ACOA) named Arc Based Ant Colony Optimization Algorithm (ABACOA) are augmented with the Tree Growing Algorithm (TGA) and used for the optimal layout and pipe size design of gravitational sewer networks. The main advantages offered by the proposed ABACOA formulation are proper definition of heuristic information, a useful component of the ant-based algorithms, and proper trade-off between the two conflicting search attributes of exploration and exploitation. In both the formulations, the TGA is used to incrementally construct feasible tree-like layouts out of the base layout. In the first formulation, unconstrained version of ABACOA is used to determine the nodal cover depths of sewer pipes while in the second formulation, a constrained version of ABACOA is used to determine the nodal cover depths of sewer pipes which satisfy the pipe slopes constraint. Three different methods of cut determination are also proposed to complete the construction of a tree-like network containing all base layout pipes, here. The proposed formulations are used to solve three test examples of different scales and the results are presented and compared with other available results in the literature. Comparison of the results shows that best results are obtained using the third cutting method in both the formulations. In addition, the results indicate the ability of the proposed methods and in particular the constrained version of ABACOA equipped with TGA to solve sewer networks design optimization problem. To be specific, the constrained version of ABACOA has been able to produce results 0.1%, 1% and 2.1% cheaper than those obtained by the unconstrained version of ABACOA for the first, second and the third test examples, respectively.

Stability of weighted spectral distribution in a pseudo tree-like network model**Project supported by the National Natural Science Foundation of China (Grant Nos. 61402485, 61303061, and 71201169).

The comparison of networks with different orders strongly depends on the stability analysis of graph features in evolving systems. In this paper, we rigorously investigate the stability of the weighted spectral distribution (i.e., a spectral graph feature) as the network order increases. First, we use deterministic scale-free networks generated by a pseudo tree-like model to derive the precise formula of the spectral feature, and then analyze the stability of the spectral feature based on the precise formula. Except for the scale-free feature, the pseudo tree-like model exhibits the hierarchical and small-world structures of complex networks. The stability analysis is useful for the classification of networks with different orders and the similarity analysis of networks that may belong to the same evolving system.

Global and local transport properties of steady and unsteady flow in a symmetrical bronchial tree

The branching structures and bifurcation flows in human lung are crucial factors for the functioning of the respiratory system. In this paper, both steady and unsteady flow in a symmetrical bronchial tree have been investigated, and the global and local transport properties are explored by fractal geometry and computational fluid dynamics method, respectively. Firstly, total flow properties for steady laminar flow and pulsatile flow in a fractal tree-like network are derived, and the global fluid dynamics behavior under volume constrain is discussed accordingly. And then, a mathematical model is developed for the steady and unsteady gas flow as well as gas-particle two-phase flow in a four-generation bifurcation in order to study the local transport characteristics of bronchial tree. The results indicate that the critical successive diameter ratio for the first few generations is below the prediction of Murray’s law while small airways follow Murray’s law. It has been also shown that the asymmetrical and non-uniform flow distribution can be realized through a symmetric branching structure with increased Reynolds number. The asymmetric ratio is found to be scaled with the Reynolds number as χ∼Re0.00124 in the steady respiratory condition. The effect of Reynolds number and respiratory frequency on the flow distribution and particle deposit efficiency are studied for different respiratory conditions, which show that the particle deposit efficiency can be increased with increased Stokes number. The present work is important for the morphology of the bronchial tree and understanding the physical mechanism of the bifurcation flow.

Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth, Pre-Existing Vessel Co-Option, Angiogenesis and Blood Perfusion.

We propose a coupled mathematical modelling system to investigate glioblastoma growth in response to dynamic changes in chemical and haemodynamic microenvironments caused by pre-existing vessel co-option, remodelling, collapse and angiogenesis. A typical tree-like architecture network with different orders for vessel diameter is designed to model pre-existing vasculature in host tissue. The chemical substances including oxygen, vascular endothelial growth factor, extra-cellular matrix and matrix degradation enzymes are calculated based on the haemodynamic environment which is obtained by coupled modelling of intravascular blood flow with interstitial fluid flow. The haemodynamic changes, including vessel diameter and permeability, are introduced to reflect a series of pathological characteristics of abnormal tumour vessels including vessel dilation, leakage, angiogenesis, regression and collapse. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. The simulation focuses on the avascular phase of tumour development and stops at an early phase of angiogenesis. The model is able to demonstrate the main features of glioblastoma growth in this phase such as the formation of pseudopalisades, cell migration along the host vessels, the pre-existing vasculature co-option, angiogenesis and remodelling. The model also enables us to examine the influence of initial conditions and local environment on the early phase of glioblastoma growth.

Optimal structure of damaged tree-like branching networks for the equivalent thermal conductivity

Damage of tree-like branching networks has being widely received considerable attention due to their wide existence in nature, engineering, cardiovascular system, lungs and cooling system of micro channels for electronic components. In the present work, we suppose the damaged number of channels of at some branching level of a tree-like branching network, and then the network becomes asymmetric. The transport properties of fluid flow and heat transfer properties in the damaged network will significantly differ from those in an undamaged network. The equivalent thermal conductivity of the damaged network is derived in this work. We find that the damaged channel number and branching levels have significant effect on optimal diameter ratio and the optimal thermal conductivity. It is found that the larger number of damaged channels results in the lower optimal thermal conductivity and the higher optimal diameter ratio. While the higher damaged branching levels lead to the lower optimal diameter ratio and higher optimal thermal conductivity. It is also found that the length ratios of the channels and total number of branching levels have no effect on optimal diameter ratio and the optimal thermal conductivity of damaged network. The present results may provide the important theoretical base for research and design of thermal transport systems.

Resilience of networks formed of interdependent modular networks

Many infrastructure networks have a modular structure and are also interdependent with other infrastructures. While significant research has explored the resilience of interdependent networks, there has been no analysis of the effects of modularity. Here we develop a theoretical framework for attacks on interdependent modular networks and support our results through simulations. We focus, for simplicity, on the case where each network has the same number of communities and the dependency links are restricted to be between pairs of communities of different networks. This is particularly realistic for modeling infrastructure across cities. Each city has its own infrastructures and different infrastructures are dependent only within the city. However, each infrastructure is connected within and between cities. For example, a power grid will connect many cities as will a communication network, yet a power station and communication tower that are interdependent will likely be in the same city. It has previously been shown that single networks are very susceptible to the failure of the interconnected nodes (between communities) (Shai et al 2014 arXiv:1404.4748) and that attacks on these nodes are even more crippling than attacks based on betweenness (da Cunha et al 2015 arXiv:1502.00353). In our example of cities these nodes have long range links which are more likely to fail. For both treelike and looplike interdependent modular networks we find distinct regimes depending on the number of modules, m. (i) In the case where there are fewer modules with strong intraconnections, the system first separates into modules in an abrupt first-order transition and then each module undergoes a second percolation transition. (ii) When there are more modules with many interconnections between them, the system undergoes a single transition. Overall, we find that modular structure can significantly influence the type of transitions observed in interdependent networks and should be considered in attempts to make interdependent networks more resilient.

Popping Balloons: A Case Study of Dynamical Fragmentation.

Understanding the physics of fragmentation is important in a wide range of industrial and geophysical applications. Fragmentation processes involve large strain rates and short time scales that take place during crack nucleation and propagation. Using rubber membranes, we develop an experimental analysis that enables us to track the fragmentation process in situ in both time and space. We find that bursting a highly stretched membrane yields a treelike fragmentation network that originates at a single seed crack, followed by successive crack tip-splitting events. We show that a dynamic instability drives this branching mechanism. Fragmentation occurs when the crack tip speed attains a critical velocity for which tip splitting becomes the sole available mechanism of releasing the stored elastic energy. Given the general character of the fragmentation processes, this framework should be applicable to other crack networks in brittle materials.

Calculations of first passage time of delayed tree-like networks

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.

A Waterfilling Algorithm for Multiple Access Point Connectivity With Constrained Backhaul Network

The 4G/5G paradigm offers user equipment (UE) simultaneous connectivity to a plurality of wireless access points (APs). We consider a UE communicating with its destination through multiple uplinks operating on orthogonal wireless channels with unequal bandwidth. The APs are connected via a tree-like backhaul network with destination at the root and non-ideal link capacities. We develop a power allocation scheme that achieves a near-optimal rate without explicit knowledge of the backhaul network topology at transmitter side. The proposed algorithm waterfills a dynamically selected subset of uplinks using a low-overhead backhaul load feedback.

Universality at Breakdown of Quantum Transport on Complex Networks.

We consider single-particle quantum transport on parametrized complex networks. Based on general arguments regarding the spectrum of the corresponding Hamiltonian, we derive bounds for a measure of the global transport efficiency defined by the time-averaged return probability. For treelike networks, we show analytically that a transition from efficient to inefficient transport occurs depending on the (average) functionality of the nodes of the network. In the infinite system size limit, this transition can be characterized by an exponent which is universal for all treelike networks. Our findings are corroborated by analytic results for specific deterministic networks, dendrimers and Vicsek fractals, and by Monte Carlo simulations of iteratively built scale-free trees.

Mathematical modeling of complex contagion on clustered networks

The spreading of behavior, such as the adoption of a new innovation, is influenced bythe structure of social networks that interconnect the population. In the experiments of Centola (Science, 2010), adoption of new behavior was shown to spread further and faster across clustered-lattice networks than across corresponding random networks. This implies that the “complex contagion” effects of social reinforcement are important in such diffusion, in contrast to “simple” contagion models of disease-spread which predict that epidemics would grow more efficiently on random networks than on clustered networks. To accurately model complex contagion on clustered networks remains a challenge because the usual assumptions (e.g. of mean-field theory) regarding tree-like networks are invalidated by the presence of triangles in the network; the triangles are, however, crucial to the social reinforcement mechanism, which posits an increased probability of a person adopting behavior that has been adopted by two or more neighbors. In this paper we modify the analytical approach that was introduced by Hebert-Dufresne et al. (Phys. Rev. E, 2010), to study disease-spread on clustered networks. We show how the approximation method can be adapted to a complex contagion model, and confirm the accuracy of the method with numerical simulations. The analytical results of the model enable us to quantify the level of social reinforcement that is required to observe—as in Centola’s experiments—faster diffusion on clustered topologies than on random networks.

A Novel Monolithic MILP Framework for Lot-Sizing and Scheduling of Multiproduct Treelike Pipeline Networks

Detailed scheduling of long-distance multiproduct pipelines has received growing attention in the past few years. It helps the planner to reduce the number of pump and segment switches between active and idle conditions to obtain savings on pump operating and maintenance costs. Most contributions on the detailed scheduling of multiproduct pipelines concern networks with a single straight line. Large-scale pipeline networks, however, usually have a treelike configuration, featuring a mainline and several secondary lines, transporting smaller volumes of refined petroleum products over shorter distances. This work addresses the scheduling of a multiproduct treelike pipeline through a continuous-time mixed integer linear programming (MILP) model that allows the execution of simultaneous deliveries from a unique refinery to multiple downstream terminals so as to get a substantial increase in transportation capacity. Contrary to previous contributions on treelike pipeline systems, the new model solves batch siz...

A tree-like Bayesian structure learning algorithm for small-sample datasets from complex biological model systems.

BackgroundThere are increasing efforts to bring high-throughput systems biology techniques to bear on complex animal model systems, often with a goal of learning about underlying regulatory network structures (e.g., gene regulatory networks). However, complex animal model systems typically have significant limitations on cohort sizes, number of samples, and the ability to perform follow-up and validation experiments. These constraints are particularly problematic for many current network learning approaches, which require large numbers of samples and may predict many more regulatory relationships than actually exist.ResultsHere, we test the idea that by leveraging the accuracy and efficiency of classifiers, we can construct high-quality networks that capture important interactions between variables in datasets with few samples. We start from a previously-developed tree-like Bayesian classifier and generalize its network learning approach to allow for arbitrary depth and complexity of tree-like networks. Using four diverse sample networks, we demonstrate that this approach performs consistently better at low sample sizes than the Sparse Candidate Algorithm, a representative approach for comparison because it is known to generate Bayesian networks with high positive predictive value. We develop and demonstrate a resampling-based approach to enable the identification of a viable root for the learned tree-like network, important for cases where the root of a network is not known a priori. We also develop and demonstrate an integrated resampling-based approach to the reduction of variable space for the learning of the network. Finally, we demonstrate the utility of this approach via the analysis of a transcriptional dataset of a malaria challenge in a non-human primate model system, Macaca mulatta, suggesting the potential to capture indicators of the earliest stages of cellular differentiation during leukopoiesis.ConclusionsWe demonstrate that by starting from effective and efficient approaches for creating classifiers, we can identify interesting tree-like network structures with significant ability to capture the relationships in the training data. This approach represents a promising strategy for inferring networks with high positive predictive value under the constraint of small numbers of samples, meeting a need that will only continue to grow as more high-throughput studies are applied to complex model systems.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0194-7) contains supplementary material, which is available to authorized users.

Functional Analysis and Characterization of Differential Coexpression Networks.

Differential coexpression analysis is emerging as a complement to conventional differential gene expression analysis. The identified differential coexpression links can be assembled into a differential coexpression network (DCEN) in response to environmental stresses or genetic changes. Differential coexpression analyses have been successfully used to identify condition-specific modules; however, the structural properties and biological significance of general DCENs have not been well investigated. Here, we analyzed two independent Saccharomyces cerevisiae DCENs constructed from large-scale time-course gene expression profiles in response to different situations. Topological analyses show that DCENs are tree-like networks possessing scale-free characteristics, but not small-world. Functional analyses indicate that differentially coexpressed gene pairs in DCEN tend to link different biological processes, achieving complementary or synergistic effects. Furthermore, the gene pairs lacking common transcription factors are sensitive to perturbation and hence lead to differential coexpression. Based on these observations, we integrated transcriptional regulatory information into DCEN and identified transcription factors that might cause differential coexpression by gain or loss of activation in response to different situations. Collectively, our results not only uncover the unique structural characteristics of DCEN but also provide new insights into interpretation of DCEN to reveal its biological significance and infer the underlying gene regulatory dynamics.

Improvement of constructal tree-like network for “volume-point” heat conduction with variable cross-section conducting path and without the premise of optimal last-order construct

The constructal “tree-like” network has been proved an effective way for “volume-point” heat conduction. This paper aims to further improve the constructal “tree-like” network with the optimization objectives of minimizations of maximum thermal resistance and entransy dissipation rate respectively. By removing the constraint that the last-order construct must be optimal, and using variable cross-section conducting path, the maximum thermal resistance and entransy dissipation rate can be greatly reduced. It is also found that the optimal construct for minimum peak temperature is similar to the optimal one for minimum entransy dissipation rate. They hold the same shape and configuration, and the main difference of them is the shape of high-conductivity path.