Brain Functional Network Based on Small-Worldness and Minimum Spanning Tree for Depression Analysis

Since the outbreak and spread of corona virus disease 2019(COVID-19),the prevalence of mental disorders,such as depression,has continued to increase.To explore the abnormal changes of brain functional connections in patients with depression,this paper proposes a depression analysis method based on brain function network(BFN).To avoid the volume conductor effect,BFN was constructed based on phase lag index(PLI).Then the indicators closely related to depression were selected from weighted BFN based on small-worldness(SW)characteristics and binarization BFN based on the minimum spanning tree(MST).Differences analysis between groups and correlation analysis between these indicators and diagnostic indicators were performed in turn.The resting state electroencephalogram(EEG)data of 24 patients with depression and 29 healthy controls(HC)was used to verify our proposed method.The results showed that compared with HC,the information processing of BFN in patients with depression decreased,and BFN showed a trend of randomization.

MEAN-VARIANCE MODEL BASED ON FILTERS OF MINIMUM SPANNING TREE

This study aims to reduce the statistical uncertainty of the correlation coefficient matrix in the mean-variance model of Markowitz. A filtering algorithm based on minimum spanning tree （MST） is proposed. Daily data of the 30 stocks of the Hang Seng Index （HSI） and Dow Jones Index （DJI） from 2004 to 2009 are selected as the base dataset. The proposed algorithm is compared with the Markowitz method in terms of risk, reliability, and effective size of the portfolio. Results show that （1） although the predicted risk of portfolio built with the MST is slightly higher than that of Markowitz, the realized risk of MST filtering algorithm is much smaller; and （2） the reliability and the effective size of filtering algorithm based on MST is apparently better than that of the Markowitz portfolio. Therefore, conclusion is that filtering algorithm based on MST improves the mean-variance model of Markowitz.

ON THE INVERSE MINIMUM SPANNING TREE PROBLEM WITH MINIMUM NUMBER OF PERTURBED EDGES

Let G=<V, E, L> be a network with the vertex set V, the edge set E and the length vector L, and let T* be a prior determined spanning tree of G. The inverse minimum spanning tree problem with minimum number of perturbed edges is to perturb the length vector L to L+ , such that T* is one of minimum spanning trees under the length vector L+ and the number of perturbed edges is minimum. This paper establishes a mathematical model for this problem and transforms it into a minimum vertex covering problem in a bipartite graph G0, a path-graph. Thus a strongly polynomial algorithm with time complexity O(mn2) can be designed by using Hungarian method.

Efficient Minimum Spanning Tree Algorithms on the Reconfigurable Mesh

The reconfigurable mesh consists of an array of processors interconnected by a reconfigurable bus system. The bus system can be used to dynamically obtain various interconnection patterns among the processors. Recently, this model has attracted a lot of attention. In this paper, two efficient algorithms are proposed for computing the minimum spanning tree of an n-vertex undirected graph. One runs on an n×n reconfigurable mesh with time complexity of O(log^2 n). The other runs with time complexity of O(log n) on an n^(1+E)×n reconfigurable mesh, where < E < 1 is a constant. All these improve the previously known results on the reconfigurable mesh.

Investigating the Disparities of China's Insurance Market Based on Minimum Spanning Tree from the Viewpoint of Geography and Enterprise

In this paper, we investigate the disparities of China’s insurance market from the viewpoint of geography and enterprise by using the monthly data from January 2006 to December 2015. We divide the whole insurance market into two parts, namely property insurance and personal insurance.By constructing and analyzing minimum spanning trees of insurance market, we obtain the results as follows:(i) The connections between provinces are much closer than those of firms, and there are regional links between neighboring provinces in the minimum spanning tree(MST); and(ii) the domestic funded firms and foreign funded firms form two explicit clusters in the MSTs of property and personal insurance market.

Minimum Spanning Trees Across Well-Connected Cities and with Location-Dependent Weights

Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted length of the minimum spanning tree containing all the n nodes,where the edge length between nodes X_(i)and X_(j)is weighted by a factor that depends on the individual locations of X_(i)and X_(j).We use approximation methods to obtain variance estimates for MSTC_(n)and prove that if the cities are well connected in a certain sense,then MSTC_(n)appropriately centred and scaled converges to zero in probability.Using the above proof techniques we also study MST_(n),the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights.In this case,the variance of MST_(n)grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of MST_(n),appropriately centred and scaled.

Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

MINIMUM CONGESTION SPANNING TREES IN BIPARTITE AND RANDOM GRAPHS

The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.

THE DESIGN AND ANALYSIS OF ALGORITHM OF MINIMUM COST SPANNING TREE

This paper provides a method of producing a minimum cost spanning tree (MCST) using set operations. It studies the data structure for implementation of set operations and the algorithm to be applied to this structure and proves the correctness and the complexity of the algorithm. This algorithm uses the FDG (formula to divide elements into groups) to sort (the FDG sorts a sequence of n elements in expected tir O(n)) and uses the method of path compression to find and to unite. Therefore. n produces an MCST of an undirected network having n vertices and e edges in expected time O(eG(n)).

A Novel Binary Firefly Algorithm for the Minimum Labeling Spanning Tree Problem

Given a connected undirected graph G whose edges are labeled,the minimumlabeling spanning tree(MLST)problemis to find a spanning tree of G with the smallest number of different labels.TheMLST is anNP-hard combinatorial optimization problem,which is widely applied in communication networks,multimodal transportation networks,and data compression.Some approximation algorithms and heuristics algorithms have been proposed for the problem.Firefly algorithm is a new meta-heuristic algorithm.Because of its simplicity and easy implementation,it has been successfully applied in various fields.However,the basic firefly algorithm is not suitable for discrete problems.To this end,a novel discrete firefly algorithm for the MLST problem is proposed in this paper.A binary operation method to update firefly positions and a local feasible handling method are introduced,which correct unfeasible solutions,eliminate redundant labels,and make the algorithm more suitable for discrete problems.Computational results show that the algorithm has good performance.The algorithm can be extended to solve other discrete optimization problems.

NeuroPrim:An attention-based model for solving NP-hard spanning tree problems

Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios,often requiring intricate algorithmic design and exponential time.Recently,there has been growing interest in end-to-end deep neural networks for solving routing problems.However,such methods typically produce sequences of vertices,which make it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges,as in various spanning tree problems.In this paper,we propose NeuroPrim,a novel framework for solving various spanning tree problems by defining a Markov decision process for general combinatorial optimization problems on graphs.Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE.We apply our framework to three difficult problems on the Euclidean space:the degree-constrained minimum spanning tree problem,the minimum routing cost spanning tree problem and the Steiner tree problem in graphs.Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices.Additionally,we find that our model has strong generalization ability with no significant degradation observed on problem instances as large as 1,000.Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.

Simulating Transport Capacity, Delivery Speed, and Routing Efficiency to Predict Economic Growth

This study uses a simulation-based approach to investigate the impact of delivery delays due to constraints on transport capacity, transit speed, and routing efficiencies on an economy with various levels of interdependency among firms. The simulation uses object-oriented programming to create specialized production, consumption, and transportation classes. A set of objects from each class is distributed randomly on a 2D plane. A road network is then established between fixed objects using Prim’s MST (Minimum Spanning Tree) algorithm, followed by construction of an all-pair shortest path matrix using the Floyd Warshall algorithm. A genetic algorithm-based vehicle routing problem solver employs the all-pair shortest path matrix to best plan multiple pickup and delivery orders. Production units utilize economic order quantities (EOQ) and reorder points (ROP) to manage inventory levels. Hicksian and Marshallian demand functions are utilized by consumption units to maximize personal utility. The transport capacity, transit speed, routing efficiency, and level of interdependence serve as 4 factors in the simulation, each assigned 3 distinct levels. Federov’s exchange algorithm is used to generate an orthogonal array to reduce the number of combination replays from 34 to just 9. The simulation results of a 9-run orthogonal array on an economy with 6 mining facilities, 12 industries, 8 market centers, and 8 transport hubs show that the level of firm interdependence, followed by transit speed, has the most significant impact on economic productivity. The principal component analysis (PCA) indicates that interdependence and transit speed can explain 90.27% of the variance in the data. According to the findings of this research, a dependable and efficient regional transportation network among various types of industries is critical for regional economic development.

A CAPACITY EXPANSION PROBLEM WITH BUDGET CONSTRAINT AND BOTTLENECK LIMITATION

This paper considers a capacity expansion problem with budget constraint. Suppose each edge in the network has two attributes: capacity and the degree of difficulty. The difficulty degree of a tree T is the maximum. degree of difficulty of all edges in the tree and the cost for coping with the difficulty in a tree is a nondecreasing function about the difficulty degree of the tree. The authors need to increase capacities of some edges so that there is a spanning tree whose capacity can be increased to the maximum extent, meanwhile the total cost for increasing capacity as well as overcoming the difficulty in the spanning tree does not exceed a given budget D*. Suppose the cost for increasing capacity on each edge is a linear function about the increment of capacity, they transform this problem into solving some hybrid parametric spanning tree problems([1]) and propose a strongly polynomial algorithm.

MST-BASED CLUSTERING TOPOLOGY CONTROL ALGORITHM FOR WIRELESS SENSOR NETWORKS

In this paper, we propose a novel clustering topology control algorithm named Minimum Spanning Tree (MST)-based Clustering Topology Control (MCTC) for Wireless Sensor Networks (WSNs), which uses a hybrid approach to adjust sensor nodes' transmission power in two-tiered hi- erarchical WSNs. MCTC algorithm employs a one-hop Maximum Energy & Minimum Distance (MEMD) clustering algorithm to decide clustering status. Each cluster exchanges information between its own Cluster Members (CMs) locally and then deliveries information to the Cluster Head (CH). Moreover, CHs exchange information between CH and CH and afterwards transmits aggregated in- formation to the base station finally. The intra-cluster topology control scheme uses MST to decide CMs' transmission radius, similarly, the inter-cluster topology control scheme applies MST to decide CHs' transmission radius. Since the intra-cluster topology control is a full distributed approach and the inter-cluster topology control is a pure centralized approach performed by the base station, therefore, MCTC algorithm belongs to one kind of hybrid clustering topology control algorithms and can obtain scalability topology and strong connectivity guarantees simultaneously. As a result, the network topology will be reduced by MCTC algorithm so that network energy efficiency will be improved. The simulation results verify that MCTC outperforms traditional topology control schemes such as LMST, DRNG and MEMD at the aspects of average node's degree, average node's power radius and network lifetime, respectively.

General election effect on the network topology of Pakistan’s stock market: network-based study of a political event

To examine the interdependency and evolution of Pakistan’s stock market,we consider the cross-correlation coefficients of daily stock returns belonging to the blue chip Karachi stock exchange(KSE-100)index.Using the minimum spanning tree network-based method,we extend the financial network literature by examining the topological properties of the network and generating six minimum spanning tree networks around three general elections in Pakistan.Our results reveal a star-like structure after the general elections of 2018 and before those in 2008,and a tree-like structure otherwise.We also highlight key nodes,the presence of different clusters,and compare the differences between the three elections.Additionally,the sectorial centrality measures reveal economic expansion in three industrial sectors—cement,oil and gas,and fertilizers.Moreover,a strong overall intermediary role of the fertilizer sector is observed.The results indicate a structural change in the stock market network due to general elections.Consequently,through this analysis,policy makers can focus on monitoring key nodes around general elections to estimate stock market stability,while local and international investors can form optimal diversification strategies.

Gradient Gene Algorithm: a Fast Optimization Method to MST Problem

The extension of Minimum Spanning Tree(MST) problem is an NP hard problem which does not exit a polynomial time algorithm. In this paper, a fast optimization method on MST problem——the Gradient Gene Algorithm is introduced. Compared with other evolutionary algorithms on MST problem, it is more advanced: firstly, very simple and easy to realize; then, efficient and accurate; finally general on other combination optimization problems.

Spanning tree trajectory optimization in the galaxy space

The 10th edition of the Global Trajectory Optimization Competition considered the problem of the galaxy settlement wherein competitors from all over the world were expected to design the trajectories of different settler vessels to maximize the given multi-faceted merit function.The synthesis methods used by the winning team,led jointly by the National University of Defense Technology(NUDT)and Xi’an Satellite Control Center(XSCC),are described along with a greedy search method and the improved solution obtained by University of Jena.Specifically,we presented a layout-first topology-second approach that allows an efficient settlement tree search guided by the pre-specified ideal spatial distribution.We also explained how the problem of constructing settlement trees can be modeled as the widely studied minimum spanning tree problem.Furthermore,University of Jena explored the possibility that a greedy search can generate even better settlement trees,based on the same initial conditions,when compared to that of the winning solution.

A Fast Algorithm for Support Vector Clustering

Support Vector Clustering (SVC) is a kernel-based unsupervised learning clustering method. The main drawback of SVC is its high computational complexity in getting the adjacency matrix describing the connectivity for each pairs of points. Based on the proximity graph model [3], the Euclidean distance in Hilbert space is calculated using a Gaussian kernel, which is the right criterion to generate a minimum spanning tree using Kruskal's algorithm. Then the connectivity estimation is lowered by only checking the linkages between the edges that construct the main stem of the MST (Minimum Spanning Tree), in which the non-compatibility degree is originally defined to support the edge selection during linkage estimations. This new approach is experimentally analyzed. The results show that the revised algorithm has a better performance than the proximity graph model with faster speed, optimized clustering quality and strong ability to noise suppression, which makes SVC scalable to large data sets.

A Location-Based Clustering Topology Control Algorithm in WSN

Aiming at the existing problems in Leach algorithm,which has short network survival time and high energy consumption,a new location-based clustering topology control algorithm is proposed.Based on Leach algorithm,improvements have been done.Firstly,when selecting cluster head,node degree,remaining energy,and the number of being cluster head,these three elements are taken into consideration.Secondly,by running the minimum spanning tree algorithm,the tree routing is constructed.Finally,selecting the next hop between clusters is done by MTE algorithm.Simulation results show that the presented control algorithm has not only a better adaptability in the large-scale networks,but also a bigger improvement in terms of some indicators of performance such as network lifetime and network energy consumption.

Energy-Delay Efficient Unified Routing Protocol for Wireless Sensor Networks

One of the most important challenges in the Wireless Sensor Networks is to improve the performance of the network by extending the lifetime of the sensor nodes. So the focus is on obtaining a trade-off between minimizing the delay involved and reducing the energy consumption of the sensor nodes which directly translate to an extended lifetime of the sensor nodes. An effective Sleep-wake scheduling mechanism can prolong the lifetime of the sensors by eliminating idle power listening, which could result in substantial delays. To counter this, an anycast forwarding scheme that could forward the packet opportunistically to the first awaken node may result in retransmissions as if the chosen node falls in resource constraints. The algorithm, namely Prim’s-Dual is proposed to solve the said problem. The algorithm considers five crucial parameters, namely the residual energy of the nodes, transmission power, receiving power, packet loss rate, interference from which the next hop is determined to extend the lifetime of the sensor node. Since the proposed work is framed keeping critical event monitoring in mind, the sleep-wake scheduling is modified as low-power, high-power scheduling where all nodes are in low-power and the nodes needed for data transmission are respectively turned on to high-power mode. The integrated framework provides several opportunities for performance enhancement for conflict-free transmissions. The aim of our algorithm is to show reliable, energy efficient transfer without compromising on lifetime and delay. The further effectiveness of the protocol is verified. The results demonstrate that the proposed protocol can efficiently handle network scalability with acceptable latency and overhead.