Ching-Lueh Chang

Spreading messages

We model a network in which messages spread by a simple directed graph G=(V,E) and a function @a:V->N mapping each v@?V to a positive integer less than or equal to the indegree of v. The graph G represents the individuals in the network and the communication channels between them. An individual v@?V will be convinced of a message when at least @a(v) of its in-neighbors are convinced. Suppose we are to convince a message to the individuals by first convincing a subset of individuals, called the seeds, and then let the message spread. We study the minimum number min-seed (G,@a) of seeds needed to convince all individuals at the end. In particular, we prove a lower bound on min-seed (G,@a) and the NP-completeness of computing min-seed (G,@a). We also analyze the special case, called the strict-majority scenario, where each individual is convinced of a message when more than half of its in-neighbors are convinced. For the strict-majority scenario, we prove three results. First, we show that with high probability over the Erdos-Renyi random graphs G(n,p), @W(min{n,1/p}) seeds are needed to convince all individuals at the end. Second, if G=(V,E) is undirected, then a set of s uniformly random samples from V convinces no more than an expected s(2|E|+2|V|)|V| individuals at the end. Third, in a digraph G=(V,E) with a positive minimum indegree, one can find in polynomial (in |V|) time a set of at most (23/27)|V| seeds convincing all individuals.

Bounding the number of tolerable faults in majority-based systems

Consider the following coloring process in a simple directed graph G(V,E) with positive indegrees. Initially, a set S of vertices are white. Thereafter, a black vertex is colored white whenever the majority of its in-neighbors are white. The coloring process ends when no additional vertices can be colored white. If all vertices end up white, we call S an irreversible dynamic monopoly (or dynamo for short). We derive upper bounds of 0.7732|V| and 0.727|V| on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. When G is an undirected connected graph without isolated vertices, upper bounds of ⌈|V|/2 ⌉ and

A lower bound for metric 1-median selection

\lfloor |V|/2 \rfloor

A deterministic sublinear-time nonadaptive algorithm for metric 1-median selection

are given on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. Let e>0 be any constant. We also show that, unless

Spreading Messages

\text{NP}\subseteq \text{TIME}(n^{O(\ln \ln n)}),

On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs

no polynomial-time, ((1/2−e)ln |V|)-approximation algorithms exist for finding a minimum irreversible dynamo.

Triggering cascades on strongly connected directed graphs

Consider the problem of finding a point in an n-point metric space with the minimum average distance to all points. We show that this problem has no deterministic o ( n 2 ) -query ( 4 - ź ) -approximation algorithms for any constant ź 0 . Given oracle access to an n-point metric space, let metric 1-median be the problem of finding a point with the minimum average distance to the other points.We show that metric 1-median has no deterministic o ( n 2 ) -query ( 4 - ź ) -approximation algorithms for any constant ź 0 .

SETS OF K-INDEPENDENT STRINGS

We give a deterministic O ( h n 1 + 1 / h ) -time ( 2 h ) -approximation nonadaptive algorithm for 1-median selection in n-point metric spaces, where h ? Z + ? { 1 } is arbitrary. Our proof generalizes that of Chang 2.

Triggering Cascades on Strongly Connected Directed Graphs

We model a network in which messages spread by a simple directed graph G= (V,E) [8] and a function i¾?: Vi¾?i¾? mapping each vi¾? Vto a positive integer less than or equal to the indegree of v. The graph Grepresents the individuals in the network and the communication channels between them. An individual vi¾? Vwill be convinced of a message when at least i¾?(v) of its in-neighbors are convinced. Suppose we are to convince a message to all individuals by convincing a subset Si¾? Vof individuals at the beginning and then let the message spread. We study minimum-sized sets Sneeded to convince all individuals at the end. In particular, our results include a lower bound on the size of a minimum Sand the NP-completeness of computing a minimum S. Our lower bound utilizes a technique in [9]. Finally, we analyze the special case where each individual is convinced of a message when more than half of its in-neighbors are convinced.

Stable sets of threshold-based cascades on the erdős-rényi random graphs

Consider the following cascading process on a simple undirected graph G(V,E) with diameter Δ. In round zero, a set S⊆V of vertices, called the seeds, are active. In round i+1, i∈ℕ, a non-isolated vertex is activated if at least a ρ∈(0,1] fraction of its neighbors are active in round i; it is deactivated otherwise. For k∈ℕ, let min-seed(k)(G,ρ) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed(k)(G,ρ). In particular, if G is connected and there exist constants C>0 and γ>2 such that the fraction of degree-k vertices in G is at most C/kγ for all k∈ℤ+, then min-seed(Δ)(G,ρ)=O(⌈ργ−1|V|⌉). Furthermore, for n∈ℤ+, p=Ω((ln(e/ρ))/(ρn)) and with probability 1−exp(−nΩ(1)) over the Erdős-Rényi random graphs G(n,p), min-seed(1)(G(n,p),ρ)=O(ρn).