arXiv subjects

Ahad N. Zehmakan

Publications and source records attributed to Ahad N. Zehmakan.

50 records · Page 3Linked to original sources

arXiv2021

On the Oracle Complexity of Higher-Order Smooth Non-Convex Finite-Sum Optimization

We prove lower bounds for higher-order methods in smooth non-convex finite-sum optimization. Our contribution is threefold: We first show that a deterministic algorithm cannot profit from the finite-sum structure of the objective, and that simulating a pth-order regularized method on the whole function by constructing exact gradient information is optimal up to constant factors. We further show lower bounds for randomized algorithms and compare them with the best known upper bounds. To address some gaps between the bounds, we propose a new second-order smoothness assumption that can be seen as an analogue of the first-order mean-squared smoothness assumption. We prove that it is sufficient to ensure state-of-the-art convergence guarantees, while allowing for a sharper lower bound.

math.OC↗

arXiv2020

Majority Opinion Diffusion in Social Networks: An Adversarial Approach

We introduce and study a novel majority-based opinion diffusion model. Consider a graph $G$, which represents a social network. Assume that initially a subset of nodes, called seed nodes or early adopters, are colored either black or white, which correspond to positive or negative opinion regarding a consumer product or a technological innovation. Then, in each round an uncolored node, which is adjacent to at least one colored node, chooses the most frequent color among its neighbors. Consider a marketing campaign which advertises a product of poor quality and its ultimate goal is that more than half of the population believe in the quality of the product at the end of the opinion diffusion process. We focus on three types of attackers which can select the seed nodes in a deterministic or random fashion and manipulate almost half of them to adopt a positive opinion toward the product (that is, to choose black color). We say that an attacker succeeds if a majority of nodes are black at the end of the process. Our main purpose is to characterize classes of graphs where an attacker cannot succeed. In particular, we prove that if the maximum degree of the underlying graph is not too large or if it has strong expansion properties, then it is fairly resilient to such attacks. Furthermore, we prove tight bounds on the stabilization time of the process (that is, the number of rounds it needs to end) in both settings of choosing the seed nodes deterministically and randomly. We also provide several hardness results for some optimization problems regarding stabilization time and choice of seed nodes.

cs.DS↗

arXiv2020

Spread of Influence in Graphs

Consider a graph $G$ and an initial configuration where each node is black or white. Assume that in each round all nodes simultaneously update their color based on a predefined rule. One can think of graph $G$ as a social network, where each black/white node represents an individual who holds a positive/negative opinion regarding a particular topic. In the $r$-threshold (resp. $α$-threshold) model, a node becomes black if at least $r$ of its neighbors (resp. $α$ fraction of its neighbors) are black, and white otherwise. The $r$-monotone (resp. $α$-monotone) model is the same as the $r$-threshold (resp. $α$-threshold) model, except that a black node remains black forever. What is the number of rounds that the process needs to stabilize? How many nodes must be black initially so that black color takes over or survives? Our main goal in the present paper is to address these two questions

cs.DS↗

arXiv2020

Transition Operations over Plane Trees

The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set $\mathcal{T}(S)$ of noncrossing straight-line spanning trees on a finite point set $S$ in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations on point sets in both general position and convex position. In addition, we address variants of the problem where operations may be performed simultaneously or the edges are labeled. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.

math.CO↗

arXiv2019

Switches in Eulerian graphs

We show that the graph transformation problem of turning a simple graph into an Eulerian one by a minimum number of single edge switches is NP-hard. Further, we show that any simple Eulerian graph can be transformed into any other such graph by a sequence of 2-switches (i.e., exchange of two edge pairs), such that every intermediate graph is also Eulerian. However, finding the shortest such sequence also turns out to be an NP-hard problem.

cs.DS↗

arXiv2019

Target Set in Threshold Models

Consider a graph $G$ and an initial coloring, where each node is blue or red. In each round, all nodes simultaneously update their color based on a predefined rule. In a threshold model, a node becomes blue if a certain number or fraction of its neighbors are blue and red otherwise. What is the minimum number of nodes which must be blue initially so that the whole graph becomes blue eventually? We study this question for graphs which have expansion properties, parameterized by spectral gap, in particular the Erdős-Rényi random graph and random regular graphs.

cs.DS↗

arXiv2019

Fake news and rumors: a trigger for proliferation or fading away

The dynamics of fake news and rumor spreading is investigated using a model with three kinds of agents who are respectively the Seeds, the Agnostics and the Others. While Seeds are the ones who start spreading the rumor being adamantly convinced of its truth, Agnostics reject any kind of rumor and do not believe in conspiracy theories. In between, the Others constitute the main part of the community. While Seeds are always Believers and Agnostics are always Indifferents, Others can switch between being Believer and Indifferent depending on who they are discussing with. The underlying driving dynamics is implemented via local updates of randomly formed groups of agents. In each group, an Other turns into a Believer as soon as $m$ or more Believers are present in the group. However, since some Believers may lose interest in the rumor as time passes by, we add a flipping fixed rate $0<d<1$ from Believers into Indifferents. Rigorous analysis of the associated dynamics reveals that switching from $m=1$ to $m\ge2$ triggers a drastic qualitative change in the spreading process. When $m=1$ even a small group of Believers may manage to convince a large part of the community very quickly. In contrast, for $m\ge 2$, even a substantial fraction of Believers does not prevent the rumor dying out after a few update rounds. Our results provide an explanation on why a given rumor spreads within a social group and not in another, and also why some rumors will not spread in neither groups.

cs.DM↗

arXiv2019

Phase Transition in Democratic Opinion Dynamics

Consider a community where initially, each individual is positive or negative regarding a reform proposal. In each round, individuals gather randomly in fixed rooms of different sizes, and all individuals in a room agree on the majority opinion in the room (with ties broken in favor of the negative opinion). The Galam model---introduced in statistical physics, specifically sociophysics---approximates this basic random process. We approach the model from a more mathematical perspective and study the threshold behavior and the consensus time of the model.

cs.DS↗

arXiv2019

Two Phase Transitions in Two-way Bootstrap Percolation

Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.

math.PR↗

arXiv2018

Tight Bounds on the Minimum Size of a Dynamic Monopoly

Assume that you are given a graph $G=(V,E)$ with an initial coloring, where each node is black or white. Then, in discrete-time rounds all nodes simultaneously update their color following a predefined deterministic rule. This process is called two-way $r$-bootstrap percolation, for some integer $r$, if a node with at least $r$ black neighbors gets black and white otherwise. Similarly, in two-way $α$-bootstrap percolation, for some $0<α<1$, a node gets black if at least $α$ fraction of its neighbors are black, and white otherwise. The two aforementioned processes are called respectively $r$-bootstrap and $α$-bootstrap percolation if we require that a black node stays black forever. For each of these processes, we say a node set $D$ is a dynamic monopoly whenever the following holds: If all nodes in $D$ are black then the graph gets fully black eventually. We provide tight upper and lower bounds on the minimum size of a dynamic monopoly.

cs.DS↗

arXiv2018

Opinion Forming in Erdos-Renyi Random Graph and Expanders

Assume for a graph $G=(V,E)$ and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the binomial random graph $\mathcal{G}_{n,p}$ and regular expanders. First we consider the behavior of the majority model in $\mathcal{G}_{n,p}$ with an initial random configuration, where each node is blue independently with probability $p_b$ and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely $\frac{\log n}{n}$. Furthermore, we discuss the majority model is a `good' and `fast' density classifier on regular expanders. More precisely, we prove if the second-largest absolute eigenvalue of the adjacency matrix of an $n$-node $Δ$-regular graph is sufficiently smaller than $Δ$ then the majority model by starting from $(\frac{1}{2}-δ)n$ blue nodes (for an arbitrarily small constant $δ>0$) results in fully red configuration in sub-logarithmically many rounds. As a by-product of our results, we show Ramanujan graphs are asymptotically optimally immune, that is for an $n$-node $Δ$-regular Ramanujan graph if the initial number of blue nodes is $s\leq βn$, the number of blue nodes in the next round is at most $\frac{cs}Δ$ for some constants $c,β>0$. This settles an open problem by Peleg.

cs.DS↗

arXiv2018

Dynamic Monopolies in Reversible Bootstrap Percolation

We study an extremal question for the (reversible) $r-$bootstrap percolation processes. Given a graph and an initial configuration where each vertex is active or inactive, in the $r-$bootstrap percolation process the following rule is applied in discrete-time rounds: each vertex gets active if it has at least $r$ active neighbors, and an active vertex stays active forever. In the reversible $r$-bootstrap percolation, each vertex gets active if it has at least $r$ active neighbors, and inactive otherwise. We consider the following question on the $d$-dimensional torus: how many vertices should be initially active so that the whole graph becomes active? Our results settle an open problem by Balister, Bollobás, Johnson, and Walters and generalize the results by Flocchini, Lodi, Luccio, Pagli, and Santoro.

cs.DS↗

arXiv2017

(Biased) Majority Rule Cellular Automata

Consider a graph $G=(V,E)$ and a random initial vertex-coloring, where each vertex is blue independently with probability $p_{b}$, and red with probability $p_r=1-p_b$. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex conserves its current color; this model is called majority model. If in case of a tie a vertex always chooses blue color, it is called biased majority model. We are interested in the behavior of these deterministic processes, especially in a two-dimensional torus (i.e., cellular automaton with (biased) majority rule). In the present paper, as a main result we prove both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, it is shown that for a two-dimensional torus $T_{n,n}$, there are two thresholds $0\leq p_1, p_2\leq 1$ such that $p_b \ll p_1$, $p_1 \ll p_b \ll p_2$, and $p_2 \ll p_b$ result in monochromatic configuration by red, stable coexistence of both colors, and monochromatic configuration by blue, respectively in $\mathcal{O}(n^2)$ number of steps

cs.FL↗

arXiv2017

Majority Model on Random Regular Graphs

Consider a graph $G=(V,E)$ and an initial random coloring where each vertex $v \in V$ is blue with probability $P_b$ and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random $d$-regular graph $\mathbb{G}_{n,d}$. It is shown that for all $ε>0$, $P_b \le 1/2-ε$ results in final complete occupancy by red in $\mathcal{O}(\log_d\log n)$ rounds with high probability, provided that $d\geq c/ε^2$ for a suitable constant $c$. Furthermore, we show that with high probability, $\mathbb{G}_{n,d}$ is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can take over in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg.

cs.DS↗