Sufficient Condition For Connectivity Research Articles

A sufficient connectivity condition for rigidity and global rigidity of linearly constrained frameworks in R2

We study the bar-and-joint frameworks in $\mathbb{R}^2$ such that some vertices are constrained to lie on some lines. The generic rigidity of such frameworks is characterised by Streinu and Theran (2010). Katoh and Tanigawa (2013) remarked that the corresponding matroid and its rank function can be characterised by using a submodular function. In this paper, we will transfer this characterisation of the rank function to the form of the value of a ``1-thin cover" and obtain a sufficient connectivity condition for rigidity and global rigidity of these frameworks analogous to the results of Lov\'asz and Yemini (1982).

Sufficient connectivity conditions for rigidity of symmetric frameworks

It is a famous result of Lovász and Yemini that 6-connected graphs are rigid in the plane (Lovász and Yemini, 1982). This was recently improved by Jackson and Jordán (2009) who showed that 6-mixed connectivity is also sufficient for rigidity. Here we give sufficient graph connectivity conditions for both ‘forced symmetric’ and ‘incidentally symmetric’ infinitesimal rigidity in the plane.

Connectivity, Traceability and Hamiltonicity

Let G be a simple, connected, triangle-free graph with minimum degree \(\delta \), order n and leaf number L(G). If G has a cut-vertex, we prove that \(L(G)\ge 4\delta -4\) and \(n\ge 4\delta -1\). Both lower bounds are sharp. The lower bound on the leaf number strengthens a result by Mukwembi for triangle-free graphs. As corollaries, we deduce sufficient conditions for connectivity, traceability and Hamiltonicity in triangle-free graphs. As an easy extension of a result by Goodman and Hedetiniemi, we show that a simple, connected, claw-free, paw-free graph G is Hamiltonian if and only if G is not a path. We consider only simple graphs, that is, graphs with neither loops nor multiple edges.

LP-based dual bounds for the maximum quasi-clique problem

A γ-quasi-clique is a simple and undirected graph with edge density of at least γ. Given a graph G, the maximum γ-quasi-clique problem (γ-QCP) consists of finding an induced γ-quasi-clique with the maximum number of vertices. γ-QCP generalizes the well-known maximum clique problem and its solution is useful for detecting dense subgraphs. After reviewing known integer linear programming formulations and dual bounds for γ-QCP, a new formulation obtained by decomposing star inequalities and combining edge inequalities is proposed. The model has an exponential number of variables but a linear number of constraints and its linear relaxation allows the computation by column generation of dual bounds for large and dense graphs. The connectivity of γ-quasi-cliques is also discussed and a new sufficient connectivity condition presented. An extensive computational experience shows the quality of the computed dual bounds and their performance in a branch-and-price framework, as well as the practical effectiveness of the connectivity condition.

Comparing brain connectivity metrics: a didactic tutorial with a toy model and experimental data

Objective. The objective of this paper is to didactically compare resting state connectivity networks computed using two different methods called phase locking value (PLV) and convergent cross-mapping (CCM). PLV is a ubiquitous measure of connectivity in electrophysiological research but is less often applied to fMRI BOLD timeseries since this model-based metric assumes that oscillatory coupling is a sufficient condition for connectivity. Alternatively, CCM is a model-free method, which detects potentially nonlinear causal influences based on the ability to estimate one timeseries with another and does not assume an oscillatory structure. Approach. We use a toy dataset to test the PLV and CCM algorithms under different known synchronization conditions. Additionally, experimental resting state EEG and fMRI datasets are used for comparison. Main results. The results show that the resting state brain networks computed using both algorithms produce similar results for both resting state EEG and fMRI datasets. For both neuroimaging datasets, the network characteristics follow the same trends and the similarity between the computed networks, for both algorithms, is highly significant. Significance. CCM is able to identify low or one-way connection strengths better than PLV but takes exponentially longer to compute. Based on these results, PLV provides a good metric for on-line network identification because it is both computationally fast and an excellent approximation of the network computed with CCM.

The Optimal Deployment, Coverage, and Connectivity Problems in Wireless Sensor Networks: Revisited

Finding an optimal node deployment strategy in wireless sensor networks (WSNs) that would reduce cost, be robust to node failures, reduce computation, and communication overhead, and guarantee a high level of coverage along with network connectivity is a difficult problem. In fact, sensing coverage and network connectivity are two of the most fundamental problems in WSNs as they can directly impact the network lifetime and operation. In this paper, we consider deriving optimal conditions for connectivity with coverage in WSNs. Most versions of this problem are (NP-complete), while approximation algorithms cannot be developed for some versions of polynomial time, unless P = NP. Hence, we also develop a heuristic for some versions of the problem and the efficacy of the heuristic will be evaluated through extensive simulations. We are also interested in determining the probability of finding a path between a given pair of nodes over a given topology of WSNs. This will serve as a measure of connectivity with coverage of the network. Hence, we derive necessary and sufficient conditions for connectivity with coverage over a clustered structure in WSNs. Then, employing queuing networks modeling techniques, we present a dynamic programming study of the connectivity with coverage of clustered structure and its effect on routing in generalized WSNs. The performance evaluation of the proposed schemes shows that availability of nodes, sensor node coverage, and the connectivity were sufficiently enhanced to maximize network lifetime.

Inferring Link Metrics From End-To-End Path Measurements: Identifiability and Monitor Placement

We investigate the problem of identifying individual link metrics in a communication network from end-to-end path measurements, under the assumption that link metrics are additive and constant. To uniquely identify the link metrics, the number of linearly independent measurement paths must equal the number of links. Our contribution is to characterize this condition in terms of the network topology and the number/placement of monitors, under the constraint that measurement paths must be cycle-free. Our main results are: 1) it is generally impossible to identify all the link metrics by using two monitors; 2) nevertheless, metrics of all the interior links not incident to any monitor are identifiable by two monitors if the topology satisfies a set of necessary and sufficient connectivity conditions; 3) these conditions naturally extend to a necessary and sufficient condition for identifying all the link metrics using three or more monitors. We show that these conditions not only facilitate efficient identifiability tests, but also enable an efficient algorithm to place the minimum number of monitors in order to identify all link metrics. Our evaluations on both random and real topologies show that the proposed algorithm achieves identifiability using a much smaller number of monitors than a baseline solution.

Sustainable network dynamics

We propose a dynamic graph-theoretic model for ecosystem management as a control over networked system composed of target nodes and unmarked nodes. The network is represented by a complete graph, in which all vertices are connected by a unique edge. Target nodes are attracted by the objective function issued from the ecosystem-based management. They pull the network toward the objective position. The management policy is considered successful if the graph remains connected in time, that is, target nodes attain their objectives and unmarked nodes stay in the convex hull. At the time of the ecosystem network transfer, the model yields a separation theorem as well as a sustainability criterion to maintain full connectivity of the ecosystem. Our definition of sustainability can be easily linked to the general definition of sustainability as ecosystem integrity preservation. At last, we identify three management rules, dependent on the nature of connections, to ensure the maintenance of connectivity in time. The necessary and sufficient conditions for connectivity are (1) synchronous time-delays; (2) coequal magnitudes of increasing objective functions; and (3) nonnegativity of the objective function.

Connectivity of Heterogeneous Wireless Networks

We address the percolation-based connectivity of large-scale ad hoc heterogeneous wireless networks, where secondary users exploit channels temporarily unused by primary users and the existence of a communication link between two secondary users depends on not only the distance between them but also the transmitting and receiving activities of nearby primary users. We introduce the concept of connectivity region defined as the set of density pairs-the density of secondary users and the density of primary transmitters - under which the secondary network is connected. Using theories and techniques from continuum percolation, we analytically characterize the connectivity region of the secondary network and reveal the tradeoff between proximity (the number of neighbors) and the occurrence of spectrum opportunities. Specifically, we establish three basic properties of the connectivity region-contiguity, monotonicity of the boundary and uniqueness of the infinite connected component, where the uniqueness implies the occurrence of a phase transition phenomenon in terms of the almost sure existence of either zero or one infinite connected component; we identify and analyze two critical densities which jointly specify the profile as well as an outer bound on the connectivity region; we study the impacts of secondary users' transmission power on the connectivity region and the conditional average degree of a secondary user and demonstrate that matching the interference ranges of the primary and the secondary networks maximizes the tolerance of the secondary network to the primary traffic load. Furthermore, we establish a necessary condition and a sufficient condition for connectivity, which lead to an outer bound and an inner bound on the connectivity region.

Sustainable Network Dynamics

We propose a dynamic graph-theoretic model for ecosystem management as a control over networked system composed of target nodes and unmarked nodes. The network is represented by a complete graph, in which all vertices are connected by a unique edge. Target nodes are attracted by the objective function issued from the ecosystem-based management. They pull the network toward the objective position. The management policy is considered successful if the graph remains connected in time, that is, target nodes attain their objectives and unmarked nodes stay in the convex hull. At the time of the ecosystem network transfer, the model yields a separation theorem as well as a sustainability criterion to maintain full connectivity of the ecosystem. Our definition of sustainability can be easily linked to the general definition of sustainability as ecosystem integrity preservation. At last, we identify three management rules, dependent on the nature of connections, to ensure the maintenance of connectivity in time. The necessary and sufficient conditions for connectivity are (1) synchronous time-delays; (2) coequal magnitudes of increasing objective functions; and (3) nonnegativity of the objective function.

Nonholonomic Systems and Sub-Riemannian Geometry

This paper presents several classical mechanical systems with nonholonomic con- straints from the point of view of sub-Riemannian geometry. For those systems that satisfy the bracket generating condition the system can move continuously between any two given states. How- ever, the paper provides a counterexample to show that the bracket generating condition is not also a sufficient condition for connectivity. All possible motions of the system correspond to curves tan- gent to the distribution defined by the nonholonomic constraints. Among the connecting curves we distinguish an optimal one which minimizes a certain energy induced by a natural sub-Riemannian metric on the non-integrable distribution. The paper discusses several classical problems such as the knife edge, the skater, the rolling disk and the nonholonomic bicycle.

A sufficient connectivity condition for generic rigidity in the plane

A graph G = ( V , E ) is said to be 6-mixed-connected if G − U − D is connected for all sets U ⊆ V and D ⊆ E which satisfy 2 | U | + | D | ≤ 5 . In this note we prove that 6-mixed-connected graphs are (redundantly globally) rigid in the plane. This improves on a previous result of Lovász and Yemini.

On the connectivity and diameter of small-world networks

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.

On the connectivity and diameter of small-world networks

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.

Unreliable sensor grids: coverage, connectivity and diameter

We consider an unreliable wireless sensor grid network with n nodes placed in a square of unit area. We are interested in the coverage of the region and the connectivity of the network. We first show that the necessary and sufficient conditions for the random grid network to cover the unit square region as well as ensure that the active nodes are connected are of the form p( n) r 2( n) ∼ log( n)/ n, where r( n) is the transmission radius of each node and p( n) is the probability that a node is “active” (not failed). This result indicates that, when n is large, even if each node is highly unreliable and the transmission power is small, we can still maintain connectivity with coverage. We also show that the diameter of the random grid (i.e., the maximum number of hops required to travel from any active node to another) is of the order n / log ( n ) . Finally, we derive a sufficient condition for connectivity of the active nodes (without necessarily having coverage). If the node success probability p( n) is small enough, we show that connectivity does not imply coverage.

On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces

In this paper we study the following aspects of the open-shell coupled-cluster (CC) theories: (a) we examine the current theoretical status regarding the existence or non-existence of a linked-cluster theorem, ensuring the connectedness of the cluster amplitudes and the effective Hamiltonian; (b) we lay down the necessary and sufficient conditions for connectivity for a general (incomplete) model space, involving valence particles as well as valence holes; and (c) critically re-assess the earlier theoretical works from a comprehensive and unifying view point.

The Determination of Frequency in Systems of Mutually Synchronized Oscillators

The synchronization of large systems of geographically separated oscillators is of considerable practical interest for pulse code modulation (PCM) switching. This study examines the factors that determine the frequency at which such a system operates, considering both the procedure by which it is set up and the topology of system interconnections. A necessary and sufficient connectivity condition is established.