# Dense Subgraphs on Dynamic Networks

## Atish Das Sarma, Ashwin Lall, Danupon Nanongkai, and Amitabh Trehan

Abstract
In distributed networks, it is often useful for the nodes to be aware of dense subgraphs, e.g., such a dense subgraph could reveal dense substructures in otherwise sparse graphs (e.g. the World Wide Web or social networks); these might reveal community clusters or dense regions for possibly maintaining good communication infrastructure. In this work, we address the problem of self-awareness of nodes in a dynamic network with regards to graph density, i.e., we give distributed algorithms for maintaining dense subgraphs that the member nodes are aware of. The only knowledge that the nodes need is that of the dynamic diameter $D$, i.e., the maximum number of rounds it takes for a message to traverse the dynamic network. For our work, we consider a model where the number of nodes are fixed, but a powerful adversary can add or remove a limited number of edges from the network at each time step. The communication is by broadcast only and follows the CONGEST model. Our algorithms are continuously executed on the network, and at any time (after some initialization) each node will be aware if it is part (or not) of a particular dense subgraph. We give algorithms that $(2+\epsilon)$-approximate the densest subgraph and $(3 + \epsilon)$-approximate the at-least-k-densest subgraph (for a given parameter k). Our algorithms work for a wide range of parameter values and run in $O(D \log_{1+\epsilon} n)$ time. Further, a special case of our results also gives the first fully decentralized approximation algorithms for densest and at-least-k-densest subgraph problems for static distributed graphs.

Guest: Amitabh Trehan
Host: Shantanu Das

# Information Dissemination via Random Walks in d-Dimensional Space

## Henry Lam, Zhenming Liu, Michael Mitzenmacher, Xiaorui Sun and Yajun Wan

Abstract:
We study a natural information dissemination problem for multiple mobile agents in a bounded Euclidean space. Agents are placed uniformly at random in the d-dimensional space {-n, …, n}^d at time zero, and one of the agents holds a piece of information to be disseminated. All the agents then perform independent random walks over the space, and the information is transmitted from one agent to another if the two agents are sufficiently close. We wish to bound the total time before all agents receive the information (with high probability). Our work extends Pettarin et al’s work [10], which solved the problem for d = 2. We present tight bounds up to polylogarithmic factors for the case d = 3. (While our results extend to higher dimensions, for space and readability considerations we provide only the case d = 3 here.) Our results show the behavior when d >= 3 is qualitatively different from the case d = 2. In particular, as the ratio between the volume of the space and the number of agents varies, we show an interesting phase transition for three dimensions that does not occur in one or two dimensions.

Guest: Henry Lam
Host: Chen Avin

# Towards Robust and Efficient Computation in Dynamic Peer-to-Peer Networks

## John Augustine, Gopal Pandurangan, Peter Robinson and Eli Upfal

Abstract:
Motivated by the need for robust and fast distributed computation in highly dynamic Peer-to-Peer (P2P) networks, we study algorithms for the fundamental distributed agreement problem. P2P networks are highly dynamic networks that experience heavy node {\em churn} (i.e., nodes join and leave the network continuously over time). Our main contributions are randomized distributed algorithms that guarantee {\em stable almost-everywhere agreement} with high probability even under high adversarial churn in polylogarithmic number of rounds. In particular, we present the following results:
1. An $O(\log^2 n)$-round (n is the stable network size) randomized algorithm that achieves almost-everywhere agreement with high probability under up to {\em linear} churn {\em per round} (i.e., $\epsilon n$, for some small constant $\epsilon > 0$), assuming that the churn is controlled by an oblivious adversary (has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm).
2. An $O(\log m\log^3 n)$-round randomized algorithm that achieves almost-everywhere agreement with high probability under up to $\epsilon \sqrt{n}$ churn per round (for some small $\epsilon > 0$), where m is the size of the input value domain, that works even under an adaptive adversary (that also knows the past random choices made by the algorithm).
Our algorithms are the first-known, fully-distributed, agreement algorithms that work under highly dynamic settings (i.e., high churn rates per step). Furthermore, they are localized (i.e., do not require any global topological knowledge), simple, and easy to implement. These algorithms can serve as building blocks for implementing other non-trivial distributed computing tasks in dynamic P2P networks.

Guests: Gopal Pandurangan with John Augustine and Peter Robinson
Host: Chen Avin

# Coordinated Consensus in Dynamic Networks

## Fabian Kuhn, Yoram Moses and Rotem Oshman

Abstract:
We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and ∆-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution.

Guest: Rotem Oshman
Host: Chen Avin