Dynamic Monopolies Research Articles

Taxation and Durable Goods Monopoly*

This paper studies the role of taxation in durable good markets with dynamic monopolies. By conditioning the marginal tax rate on the volume of trade, the regulator can provide incentives for the monopolist to accelerate trade. When marginal cost pricing generates a loss for the monopolist, strategic delay cannot be avoided under regulatory budget constraint and the effects of tax policy depend on the monopolist's ability to commit. In the context of binary consumer types, we find a tax policy involving “back‐loaded subsidy” that achieves the second‐best outcome with commitment. In contrast, without commitment, a “front‐loaded subsidy” improves welfare.

Facets of the dynamic monopoly polytope: Linear ordering formulation

We study the polyhedral structure of the monopoly polytope which arises from an integer programming formulation of dynamic monopolies in graphs. This formulation extends from the linear ordering polytope formulation. Several families of valid and facet defining inequalities are investigated. We show that (anti-) arborescences lead to families of strong valid inequalities each of which contains exponentially many facet defining inequalities. We also generalize the k-fence and Möbius ladder inequalities for the monopoly polytope.

On irreversible spread of influence in edge-weighted graphs

Various kinds of spread of influence occur in real world social and virtual networks. These phenomena are formulated by activation processes and irreversible dynamic monopolies in combinatorial graphs representing the topology of the networks. In most cases, the nature of influence is weighted and the spread of influence depends on the weight of edges. The ordinary formulation and results for dynamic monopolies do not work for such models. In this paper we present a graph theoretical analysis for spread of weighted influence and mention a real world example realizing the activation model with weighted influence. Then we obtain some extremal bounds and algorithmic results for activation process and dynamic monopolies in directed and undirected graphs with weighted edges.

Bounds and extremal graphs for degenerate subsets, dynamic monopolies, and partial incentives

The famous lower bound α(G)≥∑u∈V(G)1dG(u)+1 on the independence number α(G) of a graph G due to Caro and Wei is known to be satisfied with equality if and only if the components of G are cliques. The bound has been generalized in the context of large degenerate subsets and small dynamic monopolies, and we extend Selkow’s strengthening of it in this way. Furthermore, we give a simple proof of a bound concerning partial incentives due to Cordasco, Gargano, Rescigno, and Vaccaro. Finally, we characterize the extremal graphs for these bounds as well as for a recent lower bound on the independence number due to Angel, Campigotto, and Laforest.

Dynamic monopolies in two-way bootstrap percolation

We study an extremal question for the (two-way) 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 two-way r−bootstrap percolation, each vertex gets active if it has at least r active neighbors, and inactive otherwise. We consider the following question on thed-dimensional torus: how many vertices must be initially active so that the whole graph becomes active? Our results settle an open problem by Balister et al. (2010) and generalize the results by Flocchini et al. (2004).

Dynamic monopolies for interval graphs with bounded thresholds

For a graph G and an integer-valued threshold function τ on its vertex set, a dynamic monopoly is a set of vertices of G such that iteratively adding to it vertices u of G that have at least τ(u) neighbors in it eventually yields the vertex set of G. It is known that the problem of finding a dynamic monopoly of minimum order is NP-hard for split graphs. We show that this problem can be solved in polynomial time for interval graphs with bounded threshold functions.

A polyhedral study of dynamic monopolies

Spread of influence in a network can be modeled and studied within the concept of dynamic monopolies in graphs. We give an integer programming formulation for finding a minimum dynamic monopoly in an undirected graph. The corresponding 0–1 polytope and its facets are studied and several families of facet defining inequalities are introduced. Computational experiments have been performed to show the strength of the IP formulation and its facet defining inequalities.

Tax and Welfare in Dynamic Monopoly

This paper studies the role of taxation in durable good markets with dynamic monopolies. By conditioning the marginal tax rate on the volume of trade, the regulator can provide incentives for the monopolist to accelerate trade. When marginal cost pricing is incentive-feasible, tax policy with back-loaded subsidy can induce the first-best allocation with budget neutrality. When marginal cost pricing generates a loss for the monopolist, strategic delay cannot be avoided under regulatory budget constraint. Reducing delay then requires different forms of tax policy depending on the monopolist's commitment power. In particular, without commitment, front-loaded subsidy improves welfare.

Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

Let G be a graph, and let ρ∈[0,1]. For a set D of vertices of G, let the set Hρ(D) arise by starting with the set D, and iteratively adding further vertices u to the current set if they have at least ⌈ρdG(u)⌉ neighbors in it. If Hρ(D) contains all vertices of G, then D is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function u↦⌈ρdG(u)⌉. Let hρ(G) be the minimum cardinality of such an irreversible dynamic monopoly.For a connected graph G of maximum degree at least 1ρ, Chang showed hρ(G)≤5.83ρn(G), which was improved by Chang and Lyuu to hρ(G)≤4.92ρn(G). We show that for every ϵ>0, there is some ρ(ϵ)∈(0,1) such that hρ(G)≤(2+ϵ)ρn(G) for every ρ in (0,ρ(ϵ)), and every connected graph G that has maximum degree at least 1ρ and girth at least 5. Furthermore, we show that hρ(T)≤ρn(T) for every ρ in (0,1], and every tree T that has order at least 1ρ.

An inclusion hierarchy of irreversible dynamos

In this paper we extend the setting of irreversible dynamic monopolies (or shortly, dynamos) from two to more than two colors. The distributed protocol can be described as follows: let G be a simple connected graph where every node has a color from a finite ordered set C={1,…,k} of colors. At each round, each node can increase its color by 1 according to the colors held by its neighbors. We are interested in the so called SCIM-dynamos, that is, subsets of nodes initially having color k leading all the nodes to recolor by k under our protocol. We show that there is a tradeoff between the size of a SCIM-dynamo and the size of C: if C is limited to two colors, our protocol coincides with the irreversible strong majority. Differently, increasing the number of colors employed permits to reduce the size of a SCIM-dynamo to that of a minimum simple majority dynamo. This gives rise to an inclusion hierarchy of dynamos from simple majority dynamos to strong majority dynamos. Our goal is to minimize the size of a SCIM-dynamo for a given G, and then to determine the minimum number of colors accordingly. In particular, given an m×n torus, we show that there exists a special class of minimum size SCIM-dynamos for k≈min(m,n)/2.

Triggering cascades on strongly connected directed graphs

Consider the following process of activation on a strongly connected directed graph G(V,E) with threshold function ϕ:V→N. In round zero, a set of vertices, called the seeds, are active. Thereafter, a vertex v∈V is activated in a round if it has at least ϕ(v) active in-neighbors in the previous round. Once a vertex is activated, it remains active. Sets of seeds activating all vertices after a finite number of rounds are known in the literature as irreversible dynamic monopolies (a.k.a. perfect target sets) corresponding to (G,ϕ). With ϕ(v)=⌈ρdeg−(v)⌉ for all v∈V, where deg−(v) denotes the indegree of v and ρ∈(0,1], this paper proves the existence of an irreversible dynamic monopoly of size no more than max⁡{4.92ρ|V|,1}.

On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold

AbstractLet G be a graph and let τ be an assignment of nonnegative integer thresholds to the vertices of G. A subset of vertices, D, is said to be a τ-dynamicmonopoly if V(G) can be partitioned into subsets D0 , D1, …, Dk such that D0 = D and for any i ∊ {0, . . . , k−1}, each vertex v in Di+1 has at least τ(v) neighbors in D0∪··· ∪Di. Denote the size of smallest τ-dynamicmonopoly by dynτ(G) and the average of thresholds in τ by τ. We show that the values of dynτ(G) over all assignments τ with the same average threshold is a continuous set of integers. For any positive number t, denote the maximum dynτ(G) taken over all threshold assignments τ with τ ≤ t, by Ldynt(G). In fact, Ldynt(G) shows the worst-case value of a dynamicmonopoly when the average threshold is a given number t. We investigate under what conditions on t, there exists an upper bound for Ldynt(G) of the form c|G|, where c < 1. Next, we show that Ldynt(G) is coNP-hard for planar graphs but has polynomial-time solution for forests.

Remarks on dynamic monopolies with given average thresholds

Remarks on dynamic monopolies with given average thresholds

Dynamic monopolies and feedback vertex sets in cycle permutation graphs, generalized Petersen graphs and torus cordalis

In this paper, we use the connection between dynamic monopolies and feedback vertex sets to establish explicit formulas and new bounds for decycling number $$\nabla (G)$$?(G) when $$G$$G is one of the following classes of graphs: cycle permutation graphs, generalized Petersen graphs, and torus cordalis. In the first part of this paper, we show that if $$G$$G is a cycle permutation graph or a generalized Petersen graph on $$2n$$2n vertices, then $$\nabla (G)=\lceil (n+1)/2\rceil $$?(G)=?(n+1)/2?. These results extend a recent result by Zaker (Discret Math 312:1136---1143, 2012) and partially answer a question of Bau and Beineke (Australas J Comb 25:285---298, 2002). Note that our definition of a generalized Petersen graph is more general than the one used in Zaker (Discret Math 312:1136---1143, 2012). The second and major part of this paper is devoted to proving new upper bounds and exact values on the size of the minimum feedback vertex set and minimum dynamic monopoly for torus cordalis. Our results improve the previous results by Flocchini in 2004.

ON DYNAMIC MONOPOLIES OF GRAPHS WITH PROBABILISTIC THRESHOLDS

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a graph and ${{\tau }}$ be an assignment of nonnegative thresholds to the vertices of $G$. A subset of vertices, $D$, is an irreversible dynamic monopoly of $(G, \tau )$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and, for all $i$ with $0 \leq i \leq k-1$, each vertex $v$ in $D_{i+1}$ has at least $\tau (v)$ neighbours in the union of $D_0, D_1, \ldots, D_i$. Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph $G$ represents the underlying network. The smallest cardinality of any dynamic monopoly of $(G,\tau )$ is denoted by $\mathrm{dyn}_{\tau }(G)$. In this paper we assume that the threshold of each vertex $v$ of the network is a random variable $X_v$ such that $0\leq X_v \leq \deg _G(v)+1$. We obtain sharp bounds on the expectation and the concentration of $\mathrm{dyn}_{\tau }(G)$ around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.

Dynamic monopolies in directed graphs: The spread of unilateral influence in social networks

Irreversible dynamic monopolies arise from the formulation of the irreversible spread of influence such as disease, opinion, adaptation of a new product, etc., in social networks. In some applications, the influence in the underlying network is unilateral or one-sided. In order to study the latter models we need to introduce the concept of dynamic monopolies in directed graphs. Let G be a directed graph such that the in-degree of any vertex of G is at least one. Let also τ:V(G)→N be an assignment of thresholds to the vertices of G. A subset M of vertices of G is called a dynamic monopoly for (G,τ) if the vertex set of G can be partitioned into D0∪⋯∪Dt such that D0=M and for any i≥1 and any v∈Di, the number of edges from D0∪⋯∪Di−1 to v is at least τ(v). One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment in which for any vertex v, τ(v)=⌈(deg−(v)+1)/2⌉, where deg−(v) stands for the in-degree of v. In this paper we first discuss some basic upper and lower bounds for the size of dynamic monopolies with general threshold assignments and then obtain some hardness results concerning the smallest size of dynamic monopolies in directed graphs. We prove that any strongly connected directed graph G admits a strict majority dynamic monopoly with at most ⌈|G|/2⌉ vertices. Next we show that any simple directed graph on n vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with at most n/2 vertices, where by a simple directed graph we mean any directed graph G=(V,E) such that (u,v)∈E implies (v,u)∉E for all u, v∈V. We show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the previous best known result. The final note of the paper deals with the possibility of the improvement of the latter n/2 bound.

Discrete and continuous dynamics in nonlinear monopolies

Dynamic monopolies are investigated with discrete and continuous time scales by assuming general forms of the price and cost functions. The existence of the unique profit maximizing output level is proved. The discrete model is then constructed with gradient adjustment. It is shown that the steady state is locally asymptotically stable if the speed of adjustment is small enough and it goes to chaos through period-doubling cascade as the speed becomes larger. The non-negativity condition that prevents time trajectories from being negative is derived. The discrete model is converted into the continuous model augmented with time delay and inertia. It is then demonstrated that stability can be switched to instability and complex dynamics emerges as the length of the delay increases and that instability can be switched to stability as the inertia coefficient becomes larger. Therefore the delay has the destabilizing effect while the inertia has the stabilizing effect.

Generalized degeneracy, dynamic monopolies and maximum degenerate subgraphs

A graph G is said to be k-degenerate if any subgraph of G contains a vertex of degree at most k. The degeneracy of graphs has many applications and was widely studied in graph theory. We first generalize k-degeneracy by introducing κ-degeneracy of graphs, where κ is any non-negative function on the vertex set of the graph. We present a polynomial time algorithm to determine whether a graph is κ-degenerate. Let τ:V(G)→Z be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a τ-dynamic monopoly of G, if the vertices of G can be partitioned into subsets D0,D1,…,Dk such that D0=D and for any i∈{0,…,k−1}, each vertex v in Di+1 has at least τ(v) neighbors in D0∪⋯∪Di. The concept of dynamic monopolies is used for the formulation and analysis of spread of influence such as disease or opinion in social networks and is the subject of active research in recent years. We obtain a relationship between degeneracy and dynamic monopoly of graphs and show that these two concepts are dual of each other. Using this relationship, we introduce and study dynt(G), which is the smallest cardinality of any τ-dynamic monopoly among all threshold assignments τ with average threshold τ̄=t. We give an explicit formula for dynt(G), and obtain some lower and upper bounds for it. We show that dynt(G) is NP-complete but for complete multipartite graphs and some other classes of graphs it can be solved by polynomial time algorithms. For the regular graphs, dynt(G) can be approximated within a ratio of nearly 2. Finally we consider the problem of determining the maximum size of κ-degenerate (or k-degenerate) induced subgraphs in any graph and obtain some upper and lower bounds for the maximum size of such subgraphs.

Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades

Consider the following reversible cascade on a simple directed graph G=(V,E). In round zero, a set of vertices, called the seeds, are active. In round k∈Z+, a vertex v∈V is activated (deactivated) if at least (resp., fewer than) ϕ(v) of its in-neighbors are active in round k−1, where ϕ:V→N. An irreversible cascade is defined similarly except that active vertices cannot be deactivated. Two specific candidates for the threshold function ϕ are ϕmaj(v)strict≡⌈(deg−(v)+1)/2⌉ and ϕρ(v)≡⌈ρ⋅deg−(v)⌉, where deg−(v) denotes the indegree of v∈V and ρ∈(0,1]. An irreversible dynamic monopoly is a set of seeds that leads all vertices to activation after finitely many rounds of the irreversible cascade with ϕ=ϕmajstrict. A set of vertices, S, is said to be convergent if no vertex will ever change its state, from active to inactive or vice versa, once the set of active vertices equals S. This paper shows that an irreversible dynamic monopoly of size at most ⌈|V|/2⌉ can be found in polynomial (in |V|) time if G is strongly connected. Furthermore, we show that for any constant ϵ>0, all ρ∈(0,1], ϕ=ϕρ, p∈[(1+ϵ)(ln(e/ρ))/n,1] and with probability 1−n−Ω(1), every convergent set of the Erdős–Rényi random graph G(n,p) has size O(⌈ρn⌉) or n−O(⌈ρn⌉). Our result on convergent sets holds for both the reversible and irreversible cascades.

Reversible iterative graph processes

Given a graph G, a function f:V(G)→Z, and an initial 0/1-vertex-labelling c1:V(G)→{0,1}, we study an iterative 0/1-vertex-labelling process on G where in each round every vertex v changes its label if and only if at least f(v) neighbours have a different label. For special choices of the values of f, such processes model consensus issues and have been studied under names such as local majority processes or iterative polling processes in a large variety of contexts especially in distributed computing. Our contributions concern computational aspects related to the minimum cardinality rf(G) of sets of vertices with initial label 1 such that during the process on G all vertices eventually change their label to 1. Such sets are known as dynamic monopolies or dynamos for short. We establish a hardness result and describe efficient algorithms for restricted instances on paths and cycles.