Longest Shortest Path Research Articles

Random Walks on Networks with Centrality-Based Stochastic Resetting

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.

Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter

Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. In case of incomplete data, the arrangements of known comparisons are crucial for the quality of results. We focus on decision problems where the set of pairwise comparisons can be chosen and it is designed completely before the decision making process, without any further prior information. The objective of this paper is to provide recommendations for filling patterns of incomplete pairwise comparison matrices based on their graph representation. The proposed graphs are regular and quasi-regular ones with minimal diameter (longest shortest path). Regularity means that each item is compared to others for the same number of times, resulting in a kind of symmetry. A graph on an odd number of vertices is called quasi-regular, if the degree of every vertex is the same odd number, except for one vertex whose degree is larger by one. We draw attention to the diameter, which is missing from the relevant literature, in order to remain the closest to direct comparisons. If the diameter of the graph of comparisons is as low as possible (among the graphs of the same number of edges), we can decrease the cumulated errors that are caused by the intermediate comparisons of a long path between two items. Contributions of this paper include a list containing (quasi-)regular graphs with diameter 2 and 3 up until 24 vertices. Extensive numerical tests show that the recommended graphs indeed lead to better weight vectors compared to various other graphs with the same number of edges. It is also revealed by examples that neither regularity nor small diameter is sufficient on its own, both properties are needed. Both theorists and practitioners can utilize the results, given in several formats in the appendix: plotted graph, adjacency matrix, list of edges, ‘Graph6’ code.

Finding the Exact Diameter of a Graph with Partial Breadth-First Searches

The diameter of a graph is the longest shortest path between two nodes. This paper presents an improved algorithm for finding the exact diameter of an undirected graph. Rather than performing complete breadth-first searches, these searches can be terminated early. The algorithm is readily parallelized, and is used to find the diameters of 4-peg Tower of Hanoi problem-space graphs with up to 18 discs. Performance improvements range from a factor of almost 2 to 5.88 over the previous state of the art.

Latency-optimal virtual backbone construction with acoustic communication in underwater sensor networks

The high attenuation of radio signals in water leaves acoustic waves the most viable communication media for underwater sensor networks. Nevertheless, acoustic communication suffers from significantly high latency because of its low propagation speed compared to radio communication. In this article, we consider the problem of constructing connected dominating sets as virtual backbones under acoustic communication. We abstract a wireless sensor network as a graph with weighted edges, where the weight of an edge represents the latency between the wireless nodes it links. Three approximation algorithms are proposed to optimize the latency of a connected dominating set. The first algorithm provides a two-approximation to the diameter of a connected dominating set, where the diameter is defined as the length of the longest shortest path in a graph. The second algorithm guarantees a six-approximation to the minimum latency between any pair of nodes and meanwhile has constant approximations to the connected domi...

English

Define the average path length in a connected graph G as the sum of the lengths of the shortest path between pairs of nodes is divided by the total number of pairs of nodes. Denoting the sum of the shortest path lengths between all pairs of nodes in an n x M rectangular lattice, having n > 2 fixed rows and m:?: 2 variable rows by Sm, we derive a first order linear but non-homogeneous recurrence relation for Sm, from which a closed-form expression for Sm is obtained. Using this explicit expression for Sm, one can then show that the average path length within this graph must be asymptotic to D/3, where D is the diameter, that is, the longest shortest path.

Systems Biology Analysis of the Endocannabinoid System Reveals a Scale-free Network with Distinct Roles for Anandamide and 2-Arachidonoylglycerol

We represented the endocannabinoid system (ECS) as a biological network, where ECS molecules are the nodes (123) and their interactions the links (189). ECS network follows a scale-free topology, which confers robustness against random damage, easy navigability, and controllability. Network topological parameters, such as clustering coefficient (i.e., how the nodes form clusters) of 0.0009, network diameter (the longest shortest path among all pairs of nodes) of 12, averaged number of neighbors (the mean number of connections per node) of 3.073, and characteristic path length (the expected distance between two connected nodes) of 4.715, suggested that molecular messages are transferred through the ECS network quickly and specifically. Interestingly, ∼75% of nodes are located on, or are active at the level of, the cell membrane. The hubs of ECS network are anandamide (AEA) and 2-arachidonoylglycerol (2-AG), which have also the highest value of betweeness centrality, and their removal causes network collapse into multiple disconnected components. Importantly, AEA is a ubiquitous player while 2-AG plays more restricted actions. Instead, the product of their degradation, arachidonic acid, and their hydrolyzing enzyme, fatty acid amide hydrolase, FAAH, have a marginal impact on ECS network, indeed their removal did not significantly affect its topology.

Hybrid switching and p-routing for optical burst switching networks

One promising switching technology for wavelength-division multiplexing optical networks is optical burst switching (OBS). However, there are major deficiencies of OBS. (1) The delay offset between a control message and its corresponding data burst is based on the diameter of a network. This affects network efficiency, quality-of-service, and network scalability.( 2) OBS adopts one-way resource reservation scheme, which causes frequent burst collision and, thus, burst loss. We address the above two important issues in OBS. In particular, we study how to improve the performance of delay and loss in OBS. To reduce the end-to-end delay, we propose a hybrid switching scheme. The hybrid switching is a combination of lightpath switching and OBS switching. A virtual topology design algorithm based on simulated annealing to minimize the longest shortest path through the virtual topology is presented. To minimize burst collision and loss, we propose a new routing algorithm, namely, p-routing, for OBS network. The p-routing is based on the wavelength available probability. A path that has higher available probability is less likely to drop bursts due to collision. The probability-based p-routing can reduce the volatility, randomness, and uncertainty of one-way resource reservation. Our studies show that hybrid switching and p-routing are complementary and both can dramatically improve the performance of OBS networks.