Random walk based distributed algorithms make use of a token that circulates in the system according to a random walk scheme to achieve their goal. To study their efficiency and compare it to one of the deterministic solutions, one is led to compute certain quantities, namely the hitting times and the cover time. Until now, only bounds on these quantities were defined. First, this paper presents two generalizations of the notions of hitting and cover times to weighted graphs. Indeed, the properties of random walks on symmetrically weighted graphs provide interesting results on random walk based distributed algorithms, such as local load balancing. Both of these generalization are proposed to precisely represent the behaviour of these algorithms, and to take into account what the weights represent. Then, we propose an algorithm to compute the n^2 hitting times on a weighted graph of n vertices, which we improve to obtain a O(n^3) complexity. This complexity is the lowest up to now. This algorithm computes both of the generalizations that we propose for the hitting times on a weighted graph. Finally, we provide the first algorithm to compute the cover time (in both senses) of a graph. We improve it to achieve a complexity of O(n^3 2^n). The algorithms that we present are all robust to a topological change in a limited number of edges. This property allows us to use them on dynamic graphs.
We study the problem of testing conductance in the setting of distributed computing and give a two-sided tester that takes $\mathcal{O}(\log(n) / (εΦ^2))$ rounds to decide if a graph has conductance at least $Φ$ or is $ε$-far from having conductance at least $Φ^2 / 1000$ in the distributed CONGEST model. We also show that $Ω(\log n)$ rounds are necessary for testing conductance even in the LOCAL model. In the case of a connected graph, we show that we can perform the test even when the number of vertices in the graph is not known a priori. This is the first two-sided tester in the distributed …
Honest cooperation among individuals in a network can be achieved in different ways. In online networks with some kind of central authority, such as Ebay, Airbnb, etc. honesty is achieved through a reputation system, which is maintained and secured by the central authority. These systems usually rely on review mechanisms, through which agents can evaluate the trustworthiness of their interaction partners. These reviews are stored centrally and are tamper-proof. In decentralized peer-to-peer networks, enforcing cooperation turns out to be more difficult. One way of approaching this problem is b…
This paper demonstrates and proves that the coordination of actions in a distributed swarm can be enhanced by using quantum entanglement. In particular, we focus on - Global and local simultaneous random walks, using entangled qubits that collapse into the same (or opposite) direction, either random direction or totally controlled simultaneous movements. - Identifying eavesdropping from malicious eavesdroppers aimed at disturbing the simultaneous random walks by using entangled qubits that were sent at random or with predefined bases. - Identifying Byzantine robots or malicious robots that are…
The growing number of applications for distributed ledger technologies is driving both industry and academia to solve the limitations of blockchain, particularly its scalability issues. Recent distributed ledger technologies have replaced the blockchain linear structure with a more flexible directed acyclic graph in an attempt to accommodate a higher throughput. Despite the fast-growing diffusion of directed acyclic graph based distributed ledger technologies, researchers lack a basic understanding of their behavior. In this paper we analyze the Tangle, a directed acyclic graph that is used (w…
Designing effective algorithms for community detection is an important and challenging problem in {\em large-scale} graphs, studied extensively in the literature. Various solutions have been proposed, but many of them are centralized with expensive procedures (requiring full knowledge of the input graph) and have a large running time. In this paper, we present a distributed algorithm for community detection in the {\em stochastic block model} (also called {\em planted partition model}), a widely-studied and canonical random graph model for community detection and clustering. Our algorithm call…
Personalized PageRank (PPR) has enormous applications, such as link prediction and recommendation systems for social networks, which often require the fully PPR to be known. Besides, most of real-life graphs are edge-weighted, e.g., the interaction between users on the Facebook network. However, it is computationally difficult to compute the fully PPR, especially on large graphs, not to mention that most existing approaches do not consider the weights of edges. In particular, the existing approach cannot handle graphs with billion edges on a moderate-size cluster. To address this problem, this…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion of mixing of a random walk on a (undirected) graph, called local mixing. Informally, the local mixing with respect to a given node $s$, is the mixing of a random walk probability distribution restricted to a large enough subset of nodes --- say, a subset of size at least $n/β$ for a given parameter $β$ --- containing $s$. The time to mix over such a subset by…
Many ant species employ distributed population density estimation in applications ranging from quorum sensing [Pra05], to task allocation [Gor99], to appraisal of enemy colony strength [Ada90]. It has been shown that ants estimate density by tracking encounter rates -- the higher the population density, the more often the ants bump into each other [Pra05,GPT93]. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuri…
We consider global problems, i.e. problems that take at least diameter time, even when the bandwidth is not restricted. We show that all problems considered admit efficient solutions in low-treewidth graphs. By ``efficient'' we mean that the running time has polynomial dependence on the treewidth, a linear dependence on the diameter (which is unavoidable), and only a polylogarithmic dependence on $n$, the number of nodes in the graph. We present the algorithms solving the following problems in the CONGEST model which all attain $\tilde{O(τ^{O(1)}D)}$-round complexity (where $τ$ and $D$ denote …
