Navigable Small-world Network Research Articles

LANNS

Nearest neighbor search (NNS) has a wide range of applications in information retrieval, computer vision, machine learning, databases, and other areas. Existing state-of-the-art algorithm for nearest neighbor search, Hierarchical Navigable Small World Networks (HNSW), is unable to scale to large datasets of 100M records in high dimensions. In this paper, we propose LANNS, an end-to-end platform for Approximate Nearest Neighbor Search, which scales for web-scale datasets. Library for Large Scale Approximate Nearest Neighbor Search (LANNS) is deployed in multiple production systems for identifying top-K (100 ≤ k ≤ 200) approximate nearest neighbors with a latency of a few milliseconds per query, high throughput of ~2.5k Queries Per Second (QPS) on a single node, on large (e.g., ~ 180M data points) high dimensional (50-2048 dimensional) datasets.

A Game Theoretic Model for the Formation of Navigable Small-World Networks—The Tradeoff between Distance and Reciprocity

Kleinberg proposed a family of small-world networks to explain the navigability of large-scale real-world social networks. However, the underlying mechanism that drives real networks to be navigable is not yet well understood. In this article, we present a game theoretic model for the formation of navigable small-world networks. We model the network formation as a game called the Distance-Reciprocity Balanced (DRB) game in which people seek for both high reciprocity and long-distance relationships. We show that the game has only two Nash equilibria: One is the navigable small-world network, and the other is the random network in which each node connects with each other node with equal probability, and any other network state can reach the navigable small world via a sequence of best-response moves of nodes. We further show that the navigable small-world equilibrium is very stable—(a) no collusion of any size would benefit from deviating from it; and (b) after an arbitrary deviations of a large random set of nodes, the network would return to the navigable small world as soon as every node takes one best-response step. In contrast, for the random network, a small group collusion or random perturbations is guaranteed to bring the network out of the random-network equilibrium and move to the navigable network as soon as every node takes one best-response step. Moreover, we show that navigable small-world equilibrium has much better social welfare than the random network, and we provide the price-of-anarchy and price-of-stability results of the game. Our empirical evaluation further demonstrates that the system always converges to the navigable network even when limited or no information about other players’ strategies is available, and the DRB game simulated on real-world networks leads to navigability characteristic that is very close to that of the real networks, even though the real-world networks have non-uniform population distributions different from Kleinberg’s small-world model. Our theoretical and empirical analyses provide important new insight on the connection between distance, reciprocity, and navigability in social networks.

Comparing local search paths with global search paths on protein residue networks: allosteric communication

Although proteins have been recognized as small-world networks and their small-world network properties of clustering and short paths have been exploited computationally to produce biologically relevant information, they have not been truly explored as such, i.e. as navigable small-world networks in the original spirit of Milgram's work. This research seeks to fill this gap by exploring local search on a network representation of proteins and to probe the source of navigability in proteins. Previously, we confirmed that proteins are navigable small-world networks and observed that local search paths exhibit different characteristics from global search paths. In this paper, we investigate the biological relevance of the differences in path characteristics on a type III receptor tyrosine kinase (KIT). A chief difference that works in favour of local search paths as intra-protein communication pathways is their weaker proclivity, compared to global search paths, for traversing long-range edges. Long-range edges tend to be less stable and their inclusion tends to decrease the communication propensity of a path. The source of protein navigability is traced to clustering provided by short-range edges. The majority of a protein's short-range edges reside within structures deemed important for long-range energy transport and modulation of allosteric communication in proteins. Therefore, the disruption of intra-protein communication as a result of the destruction of these structures via random rewiring is expected. A local search perspective leads us to this expected conclusion while a global search perspective does not. These findings initiate the compilation of a list of path properties that are characteristic of intra-protein pathways and could suggest fresh avenues for evolving and regulating navigable (small-world) networks.

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.

Navigable Small-World networks with few random bits

We study Small-World graphs in the perspective of their use in the development of efficient as well as easy to implement network infrastructures. Our analysis starts from the Small-World model proposed by Kleinberg: a grid network augmented with directed long-range random links. The choices of the long-range links are independent from one node to another. In this setting greedy routing and some of its variants have been analyzed and shown to produce paths of polylogarithmic expected length. We start from asking whether all the randomness, used in Kleinberg’s model for establishing the long-range contacts of the nodes, is indeed necessary to assure the existence of short paths. In order to deal with the above question, we impose (stringent) restrictions on the choice of long-range links and we show that such restrictions do not increase the average path length of greedy routing and its variations. We are able to decrease the number of random bits, required to establish each node’s long-range link, from Ω ( log n ) to O ( log log n ) on a network of size n . Diminishing the randomness in the choice of random links has several benefits; in particular, it implies an increase in the clustering of the graph, thus increasing the resilience of the network.