Pairs Of Distinct Nodes Research Articles

Non-equilibrium random walks on multiplex networks

We introduce a non-equilibrium discrete-time random walk model on multiplex networks, in which at each time step the walker first undergoes a random jump between neighboring nodes in the same layer, and then tries to hop from one node to one of its replicas in another layer. We derive the so-called supra-Markov matrix that governs the evolution of the occupation probability of the walker. The occupation probability at stationarity is different from the weighted average over the counterparts on each layer, unless the transition probabilities between layers vanish. However, they are approximately equal when the transition probabilities between layers are very small, which is given by the first-order degenerate perturbation theory. Moreover, we compute the mean first passage time (MFPT) and the graph MFPT (GrMFPT) that is the average of the MFPT over all pairs of distinct nodes. Interestingly, we find that the GrMFPT can be smaller than that of any layer taken in isolation. The result embodies the advantage of global search on multiplex networks.

The mean and variance of the distribution of shortest path lengths of random regular graphs

The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of N nodes of degree c ⩾ 3, the DSPL, denoted by P(L = ℓ), follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form (CF) expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain CF expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by , where γ is the Euler–Mascheroni constant. While the leading term is known, this result includes a novel correction term, which yields very good agreement with the results obtained from direct numerical evaluation of ⟨L⟩ via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of ⟨L⟩ as a function of c or N. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance Var(L) of the DSPL, which captures the overall dependence of the variance on c but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as N is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.

K-pairwise disjoint paths routing in perfect hierarchical hypercubes

Hierarchical hypercubes (HHC), also known as cube-connected cubes, have been introduced in the literature as an interconnection network for massively parallel systems. Effectively, they can connect a large number of nodes while retaining a small diameter and a low degree compared to a hypercube of the same size. Especially (2 m +m)-dimensional hierarchical hypercubes ( $\mathit {HHC}_{2^{m}+m}$ ), called perfect HHCs, are popular as they are symmetrical, which is a critical property when designing routing algorithms. In this paper, we describe an algorithm finding, in an $\mathit{HHC}_{2^{m}+m}$ , mutually node-disjoint paths connecting k=?(m+1)/2? pairs of distinct nodes. This problem is known as the k-pairwise disjoint-path routing problem and is one of the important routing problems when dealing with interconnection networks. In an $\mathit{HHC}_{2^{m}+m}$ , our algorithm finds paths of lengths at most 2 m+1+m(2 m+1+1)+4 in O(25m ) time, where 2 m+1 is the diameter of an $\mathit{HHC}_{2^{m}+m}$ . Also, we have shown through an experiment that, in practice, the lengths of the generated paths are significantly lower than the worst-case theoretical estimations.

Embedding Cycles and Paths in Product Networks and Their Applications to Multiprocessor Systems

In this paper, we consider two embedding problems in Cartesian product networks: one is the pancycle problem, which involves embedding cycles of various lengths in the given product network; and the other is the panconnectivity problem, which involves embedding paths of various lengths between any pair of distinct nodes in the given product network. We then apply our technical lemmas and theorems to derive new topological properties of two multiprocessor systems, namely, generalized hypercubes and nearest neighbor mesh hypercubes.

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property Ak if, given k pairs of distinct nodes (s1, t1), …, (sk, tk) of V(G), there are k mutually node-disjoint paths, one connecting si and ti for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy Ak is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s1, t1), …, (sk, tk) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈logn⌉+1 in O(n2log*n) time.

An efficient algorithm for k-pairwise disjoint paths in star graphs

In this paper, we give an efficient algorithm for the following problem in the n-dimensional star graph G n : Given k = ⌈( n − l) 2 ⌉ pairs of distinct nodes ( s 1, t 1),…,( s k , t k ) in G n , find k node-disjoint paths S i → t i , 1 ⩽ i ⩽ k. Our algorithm finds the k disjoint paths of length at most d( G n ) + 5 in O( n 2) optimal time, where d(G n) = ⌊3( n − 1) 2 ⌋ is the diameter of G n . This improves the previous results of 4( n − 2) (path length) and O( n 4 log n) (time), respectively.

A Simple Optimal Parallel Algorithm for Reporting Paths in a Tree

We present optimal parallel solutions to reporting paths between pairs of nodes in an n-node tree. Our algorithms are deterministic and designed to run on an exclusive read exclusive write parallel random-access machine (EREW PRAM). In particular, we provide a simple optimal parallel algorithm for preprocessing the input tree such that the path queries can be answered efficiently. Our algorithm for preprocessing runs in O( log n) time using O(n/ log n) processors. Using the preprocessing, we can report paths between k node pairs in O( log n + log k) time using O(k + (n + S)/ log n) processors on an EREW PRAM, where S is the size of the output. In particular, we can report the path between a single pair of distinct nodes in O( log n) time using O(L/ log n) processors, where L denotes the length of the path.

A multi-commodity flow network design problem

There exist systems which can be usefully described by a network containingarcs through which a commodity of one type flows. This paper is concerned with finding a solution procedure for a particular multi-commodity flow network design problem. The problem is to identify a set of arcs in the network such that if travel is prohibited in them all flow travels by feasible paths and its total cost is minimal. The total flow in each arc may not exced its capacity, which is a known constant. Each arc and each node of the network has a non-negative constant unit traversal cost. Between each pair of distinct nodes there is a given non-negative rate of flow from the first vertex to the second which may be split up among a number of paths according to some constant traversal cost flow assignment process. The optimality criterion is the total traversal cost of all flow, which is to be minimized. Previous work on network design problems of this type is surveyed. The principal contribution of this paper is the presentation of a solution procedure for the above problem based on branch and bound enumeration. An illustrative numerical example is included. Computational experience gained in using the procedure with a FORTRAN IV program on an IBM 370 is favourable.

On maximal transitive subtournaments

A tournamentTn consists of a finite set of nodes1, 2, …, n such that each pair of distinct nodes i and j is joined by exactly one of the arcsij or ji. If the arc ij is in Tn we say that i beats j or j loses to i and write i→j. If each node of a subtournament A beats each node of a subtournament B we write A→B and let A + B denote the tournament determined by the nodes of A and B.

On Perfect Matchings

Introduction. A graph G = {N, E} is taken to be a finite set of nodes N together with a set of distinct edges E which are unordered pairs of distinct nodes. A matching M of the graph is a subset of the edges E with the property that no two edges of M are incident at a node. A matching M is perfect if every node is incident to an edge of M. For any finite set X, let IXI denote its cardinality. If S c N let G(N S) be the subgraph of G consisting of nodes N S (nodes of N but not S) and all edges of E which only join nodes of N S. For any subgraph H of G, let o(H) be the number of connected components of H each of which consists of an odd number of nodes. If, for some S c N, o(G(N S)) > ISI, we say that the set S is an imperfectable set of nodes of G.

Counting Coloured Graphs II

As in my earlier paper (2), a graph on n labelled nodes is a set of n objects called "nodes" distinguishable from each other and a set (possibly empty) of "edges," i.e. pairs of distinct nodes. Each edge is said to join its pair of nodes and no edge joins a node to itself. By a k-colouring of the nodes of such a graph we mean a mapping of the nodes onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. By a colouring of the edges of such a graph we mean a mapping of the edges onto a set of colours.