Greedy Paths Research Articles

Greedy routing optimisation in hyperbolic networks

Finding the optimal embedding of networks into low-dimensional hyperbolic spaces is a challenge that received considerable interest in recent years, with several different approaches proposed in the literature. In general, these methods take advantage of the exponentially growing volume of the hyperbolic space as a function of the radius from the origin, allowing a (roughly) uniform spatial distribution of the nodes even for scale-free small-world networks, where the connection probability between pairs decays with hyperbolic distance. One of the motivations behind hyperbolic embedding is that optimal placement of the nodes in a hyperbolic space is widely thought to enable efficient navigation on top of the network. According to that, one of the measures that can be used to quantify the quality of different embeddings is given by the fraction of successful greedy paths following a simple navigation protocol based on the hyperbolic coordinates. In the present work, we develop an optimisation scheme for this score in the native disk representation of the hyperbolic space. This optimisation algorithm can be either used as an embedding method alone, or it can be applied to improve this score for embeddings obtained from other methods. According to our tests on synthetic and real networks, the proposed optimisation can considerably enhance the success rate of greedy paths in several cases, improving the given embedding from the point of view of navigability.

Hyperbolic Embeddings for Near-Optimal Greedy Routing

Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular, our embeddings guarantee that greedy routing always succeeds in reaching the target, and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real-world networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph [3], we achieve near-optimal results (i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths). In the case of the Internet graph, they are only 6% longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention. In addition to measuring the stretch, we empirically evaluate our algorithms regarding the size of the coordinates of the resulting embeddings and observe how it impacts the success rate when coordinates are not represented with very high precision. Since numerical difficulties are a major issue when performing computations in the hyperbolic plane, we consider variations of our algorithm that improve the success rate when using coordinates with lower precision.

Navigable maps of structural brain networks across species.

Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the structure of the connectomes of different species and their spatial layout. As a navigation strategy, we use greedy routing where nearest neighbors, in terms of geometric distance, are visited. We measure the fraction of successful greedy paths and their length as compared to shortest paths in the topology of connectomes. In Euclidean space, we find a striking difference between the navigability properties of mammalian and non-mammalian species, which implies the inability of Euclidean distances to fully explain the structural organization of their connectomes. In contrast, we find that hyperbolic space, the effective geometry of complex networks, provides almost perfectly navigable maps of connectomes for all species, meaning that hyperbolic distances are exceptionally congruent with the structure of connectomes. Hyperbolic maps therefore offer a quantitative meaningful representation of connectomes that suggests a new cartography of the brain based on the combination of its connectivity with its effective geometry rather than on its anatomy only. Hyperbolic maps also provide a universal framework to study decentralized communication processes in connectomes of different species and at different scales on an equal footing.

Every Schnyder drawing is a greedy embedding

In this paper, we show that every Schnyder drawing is a greedy embedding. Schnyder drawings are used to represent planar (maximal) graphs. It is a way of getting coordinates in R2 given a graph G=(V,E) such that the representation is planar. The Schnyder technique leads to a family of representations and previous results show that a particular representation may be chosen such that the drawing has additional properties like being greedy or monotone. In this article, we relax the definition of greediness to a definition that does not rely on the geometry and the Euclidean distance in R2, but rather on the combinatorial graph G. The construction of greedy paths valid for all Schnyder representations shows that, provided the relaxed definition, every Schnyder drawing is a greedy embedding.

Growing Homophilic Networks Are Natural Navigable Small Worlds.

Navigability, an ability to find a logarithmically short path between elements using only local information, is one of the most fascinating properties of real-life networks. However, the exact mechanism responsible for the formation of navigation properties remained unknown. We show that navigability can be achieved by using only two ingredients present in the majority of networks: network growth and local homophily, giving a persuasive answer how the navigation appears in real-life networks. A very simple algorithm produces hierarchical self-similar optimally wired navigable small world networks with exponential degree distribution by using only local information. Adding preferential attachment produces a scale-free network which has shorter greedy paths, but worse (power law) scaling of the information extraction locality (algorithmic complexity of a search). Introducing saturation of the preferential attachment leads to truncated scale-free degree distribution that offers a good tradeoff between these parameters and can be useful for practical applications. Several features of the model are observed in real-life networks, in particular in the brain neural networks, supporting the earlier suggestions that they are navigable.

A Dilution-matching-encoding compaction of trajectories over road networks

Many devices nowadays record traveled routes as sequences of GPS locations. With the growing popularity of smartphones, millions of such routes are generated each day, and many routes have to be stored locally on the device or transmitted to a remote database. It is, thus, essential to encode the sequences, in order to decrease the volume of the stored or transmitted data. In this paper we study the problem of encoding routes over a vectorial road network (map), where GPS locations can be associated with vertices or with road segments. We consider a three-step process of dilution, map-matching and coding, which helps reducing the amount of transmitted data between the cellular device and remote servers. We present two methods to code routes. The first method represents the given route as a sequence of greedy paths. We provide two algorithms to generate a greedy-path code for a sequence of n vertices on the map. The first algorithm has O(n) time complexity, and the second one has O(n 2) time complexity, but it is optimal, meaning that it generates the shortest possible greedy-path code. Decoding a greedy-path code can be done in O(n) time. The second method encodes a route as a sequence of shortest paths. We provide algorithms to generate unidirectional and bidirectional optimal shortest-path codes. Encoding and decoding a shortest-path code can be done in O(k n 2 logn) time, where k is the length of the produced code, assuming the graph valency is bounded. Our experimental evaluation shows that shortest-path codes are more compact than greedy-path codes, justifying the larger time complexity.

On the choice of a spanning tree for greedy embedding of network graphs

Greedy embedding is a graph embedding that makes the simple greedy geometric packet forwarding successful for every source-destination pair. It is desirable that graph embeddings also yield low hop overhead (stretch) of the greedy paths over the corresponding shortest paths. In this paper we study how topological and geometric properties of embedded graphs influence the hop stretch. Based on the obtained insights, we design embedding heuristics that yield minimal hop stretch greedy embeddings and verify their effectiveness on models of synthetic graphs. Finally, we verify the effectiveness of our heuristics on instances of several classes of large, real-world network graphs.

On greedy network formation

Greedy navigability is a central issue in the theory of networks. However, the exogenous nature of network models do not allow for describing how greedy routable-networks emerge in reality. In turn, network formation games focus on the very emergence proess, but the applied shortest-path based cost functions exclude navigational aspects. This paper takes a frst step towards incorporating both emergence (missing in algorithmic network models) and greedy navigability (missing in network formation games) into a single framework, and proposes the Greedy Network Formation Game. Our first contribution is the game definition, where we assume a hidden metric space underneath the network, and, instead of usual shortest path metric, we use the length of greedy paths as the measure of communiation cost between players. Our main finding is that greedy-routable small worlds do not emerge on constant dimensional Eulidean grids. This simply means that the emergence of topologies on which w eunderstood the priniples of greedy forwarding cannot be explained endogenously. We also present a very brief outlook on how the situation hanges in the hyperbolic space.

Continuous first-passage percolation and continuous greedy paths model : linear growth

We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.

Continuous first-passage percolation and continuous greedy paths model: Linear growth

We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.

Extensions of a result on the synthesis of signals in the presence of inconsistent constraints

Image restoration in the presence of compatible convex constraints can be carried out by the method of convex projections [1]-[3]. In a recent interesting paper [4], Goldburg and Marks have used a modified version of the above technique to solve an optimization problem involving the synthesis of a signal subject to two inconsistent constraints. We complete this result and also show that their restriction to a real Hilbert space setting is unnecessary. A unique generalization of the above optimization problem to the case of more than two constraints does not seem possible. Nevertheless, considerations of symmetry have led us to a formulation which identifies minimizers as nodes on closed greedy paths and an important and potentially useful property of such paths is proven in Theorem 4.