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Small-world networks, which combine randomized and structured elements, are seen as prevalent in nature. Several random graph models have been given for small-world networks, with one of the most fruitful, introduced by Jon Kleinberg,…

Graphs are called navigable if one can find short paths through them using only local knowledge. It has been shown that for a graph to be navigable, its construction needs to meet strict criteria. Since such graphs nevertheless seem to…

Numerous studies show that most known real-world complex networks share similar properties in their connectivity and degree distribution. They are called small worlds. This article gives a method to turn random graphs into Small World…

We present an algorithm to grow a graph with scale-free structure of {\it in-} and {\it out-links} and variable wiring diagram in the class of the world-wide Web. We then explore the graph by intentional random walks using local…

The Small World phenomenon has inspired researchers across a number of fields. A breakthrough in its understanding was made by Kleinberg who introduced Rank Based Augmentation (RBA): add to each vertex independently an arc to a random…

There has been significant recent interest in graph-based nearest neighbor search methods, many of which are centered on the construction of navigable graphs over high-dimensional point sets. A graph is navigable if we can successfully move…

Dating back to two famous experiments by the social-psychologist, Stanley Milgram, in the 1960s, the small-world phenomenon is the idea that all people are connected through a short chain of acquaintances that can be used to route messages.…

We propose a dynamical process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable…

The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s, was theoretically explained by Kleinberg in 2000. However, from today's perspective his model has several severe…

Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs…

Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic…

The graph-navigability problem concerns how one can find as short paths as possible between a pair of vertices, given an incomplete picture of a graph. We study the navigability of graphs where the vertices are tagged by a number (between 1…

Continuing in the steps of Jon Kleinberg's and others celebrated work on decentralized search in small-world networks, we conduct an experimental analysis of a dynamic algorithm that produces small-world networks. We find that the algorithm…

We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small…

For various purposes and, in particular, in the context of data compression, a graph can be examined at three levels. Its structure can be described as the unlabeled version of the graph; then the labeling of its structure can be added; and…

In recent work, Jon Kleinberg considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power of the distance between…

This study introduces an algorithm that generates undirected graphs with three main characteristics of real-world networks: scale-freeness, short distances between nodes (small-world phenomenon), and large clustering coefficients. The main…

This paper mainly investigates why small-world networks are navigable and how to navigate small-world networks. We find that the navigability can naturally emerge from self-organization in the absence of prior knowledge about underlying…

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key…

Navigability of networks, that is the ability to find any given destination vertex starting from any other vertex, is crucial to their usefulness. In 2000 Kleinberg showed that optimal navigability could be achieved in small-world networks…