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Consensus of Interacting Particle Systems on Erdös-Rényi Graphs

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Consensus of Interacting Particle Systems on Erdös-Rényi Graphs Consensus of Interacting Particle Systems on Erd¨os-R´enyi Graphs Grant Schoenebeck∗ Fang-Yi Yuy Abstract Node Dynamics with Interacting Particle Systems—exemplified by the voter f(x) = x: model, iterative majority, and iterative k−majority processes|have found use in many disciplines including dis- This models has been extensively studied in math- tributed systems, statistical physics, social networks, and ematics [15, 22, 27, 28], physics [6, 9], and even in Markov chain theory. In these processes, nodes update their social networks [8, 34, 35, 36, 14]. A key question \opinion" according to the frequency of opinions amongst studied is how long it takes the dynamics to reach their neighbors. We propose a family of models parameterized by an consensus on different network typologies. update function that we call Node Dynamics: every node Iterative majority: In the iterative majority dynam- initially has a binary opinion. At each round a node is ics, in each round, a randomly chosen node updates uniformly chosen and randomly updates its opinion with the probability distribution specified by the value of the update to the opinion of the majority of its neighbors. This function applied to the frequencies of its neighbors' opinions. corresponds to the Node Dynamics where In this work, we prove that the Node Dynamics con- 8 1 if x > 1=2; verge to consensus in time Θ(n log n) in complete graphs and < f(x) = 1=2 if x = 1=2; dense Erd¨os-R´enyi random graphs when the update function : 0 if x < 1=2: is from a large family of \majority-like" functions. Our tech- nical contribution is a general framework that upper bounds Typically works about Majority Dynamics study 1) the consensus time. In contrast to previous work that relies when the dynamics converge, how long it takes the on handcrafted potential functions, our framework system- dynamics to converge, and whether they converge atically constructs a potential function based on the state space structure. to the original majority opinion|that is, does ma- jority dynamics successfully aggregate the original 1 Introduction opinion [25, 7, 23, 31, 37]. We propose the following stochastic process|that we Iterative k-majority: In this dynamics, in each call Node Dynamics|on a given network of n agents round, a randomly chosen node collects the opinion parameterized by an update function f : [0; 1] ! [0; 1]. of k randomly chosen (with replacement) neighbors In the beginning, each agent holds a binary \opinion", and updates to the opinion of the majority of those either red or blue. Then, in each round, an agent is k opinions. This corresponds to the Node Dynam- uniformly chosen and updates its opinion to be red with ics where probability f(p) and blue with probability 1−f(p) where k X k p is the fraction of its neighbors with the red opinion. f(x) = x`(1 − x)n−`: ` Node dynamics generalizes processes of interest in `=dk=2e many different disciplines including distributed systems, statistical physics, social networks, and even biology. A synchronized variant of this dynamics is pro- posed as a protocol for stabilizing consensus: col- Voter Model: In the voter model, at each round, lection of n agents initially hold a private opinion a random node chooses a random neighbor and and interact with the goal of agreeingp on one of updates to its opinion. This corresponds to the the choices, in the presence of O( n)-dynamic ad- versaries whichp can adaptively change the opinions of up to O( n) nodes at every round. In the syn- ∗University of Michigan, [email protected]. He is grate- chronized variant of this dynamics, Doerr et al. [17] fully supported by the National Science Foundation under Career prove 3-majority reaches \stabilizing almost" con- Award 1452915 and AitF Award 1535912. sensus on the complete graph in the presence of yUniversity of Michigan, [email protected]. He is gratefully p supported by the National Science Foundation under AitF Award O( n)-dynamic adversaries. Many works extend 1535912. this result beyond binary opinions [16, 13, 5, 1]. Copyright c 2018 by SIAM Unauthorized reproduction of this article is prohibited Iterative ρ-noisy majority model: [20, 21] In this and concentration property to control the behavior of dynamics, in each round, a randomly chosen node the process [5], or (3) using handcrafted potential func- updates the majority opinion of its neighbors with tions [31]. probability 1 − ρ and uniformly at random with Our results fill in a large gap that these results probability ρ. do not adequately cover. Mixing time is not well- 8 defined in non-reversible or reducible Markov chains, < 1 − ρ/2 if x > 1=2; and so does not apply to Markov chains with multiple f(x) = 1=2 if x = 1=2; absorption states, like in the consensus time question we : ρ/2 if x < 1=2: study. Reducing the dimension and using a mean field approximation fails for two reasons. First, summarizing Genetic Evolution Model: In biological systems, with a small set of parameters is not possible when the the chance of survival of an animal can depend process of interest has small imperfections (like in a on the frequencies of its kin and foes in the net- fixed Erd¨os-R´enyi graph). Second, the mean-field of work [3, 29]. Moreover, this frequency depend- our dynamics has unstable fixed points; in such cases ing dynamics is also known to model the dynamics the mean field does not serve as a useful proxy for the for maintaining the genetic diversities of a popula- Markov process. Handcrafting potential functions also tion [24, 32]. runs into several problems: the first is that because we Our Contribution We focus on a large set of consider dynamics on random graphs, the dynamic is update functions f that are symmetric, smooth, and not a priori well specified; so there is no specific dynamic satisfy a property well call \majority-like", intuitively to handcraft a potential function for. Secondly, we wish meaning that agents update to the majority opinion to solve the problem for a large class of update functions strictly more often than the fraction of neighbors hold- f, and so cannot individually hand-craft a potential ing the majority opinion. We obtain tight bounds for function for each one. Typically, the potential function the consensus time|the time that it takes the system to is closely tailored to the details of the process. reach a state where each node has an identical opinion| Additional Related Work Our model is similar on Erd¨os-R´enyi random graphs. to that of Schweitzer and Behera [33] who study a va- Our main technical tool is a novel framework for riety of update functions in the homogeneous setting upper bounding the hitting time for a general discrete- (complete graph) using simulations and heuristic argu- time homogeneous Markov chain (X ;P ), including non- ments. However, they leave a rigorous study to future reversible and even reducible Markov chains. This work. framework decomposes the problem so that we only need to upper bound two sets of parameters for all 2 Preliminaries x 2 X |the reciprocal of the probability of decreasing 2.1 Node Dynamics Given an undirected graph the distance to target 1=p+(x) and the ratio of the G = (V; E) let Γ(v) be the neighbors of node v and probability of decreasing the distance to the target and deg(v) = jΓ(v)j. the probability of increasing the distance to the target: We define a configuration x(G) : V ! f0; 1g to p−(x)/p+(x). Our technique can give much stronger assign the \color" of each node v 2 G to be x(G)(v) bounds than simply lower bounding p−(x) and upper so that x(G) 2 f0; 1gn. We will usually suppress the bounding p+(x). superscript when it is clear. We will use uppercase (e.g., Once we apply this decomposition to our consensus X(G)) when the configuration is a random variable. time problem, the problem becomes very manageable. Moreover we say v is red if x(v) = 1 and is blue if We show the versatility of our approach by extending x(v) = 0. We then write the set of red vertices as the results to a variant of the stabilizing consensus prob- x−1(1). We say that a configuration x is in consensus lem, where we show that all majority-like dynamics con- if x(·) is the constant function (so all nodes are red or vergence quickly to the \stabilizing almost" consensus all nodes are blue). Given a node v in configuration x on the complete graph in the presence of adversaries. jΓ(v)\X−1(1)j we define rx(v) = deg(v) to be its fraction of red A large volume of literature is devoted to bound- neighbors. ing the hitting time of different Markov process and achieving fast convergence. The techniques typically Definition 2.1. An update function is a mapping employed are (1) showing the Markov chain has fast f : [0; 1] 7! [0; 1] with the following properties: mixing time [30], (2) reducing the dimension of the pro- cess into small set of parameters (e.g. the frequency Monotone 8x; y 2 [0; 1], if x < y, then f(x) ≤ f(y). of each opinion) and using a mean field approximation Symmetric 8t 2 [0; 1=2], f(1=2 + t) = 1 − f(1=2 − t). Copyright c 2018 by SIAM Unauthorized reproduction of this article is prohibited Absorbing f(0) = 0 and f(1) = 1.
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