## 1 Introduction

The formulation and analysis of the various models of the spread of influence such as disease and opinion in a population, virus in computer networks, adaptation of innovation and viral marketing in social networks have been the research subject of many authors in the recent years. Let a graph represent the underlying social network, where denotes the individuals or the elements of the network and represents the links or ties between them. These models and phenomena of the spread of influence have been studied using graph theory and in terms of (progressive) dynamic monopolies in graphs (or -conversion process and target set selection in some other articles). Let a social network be represented by a graph . We call such a graph social graph. By the social graph we only mean that the underlying graph corresponds to a social (or even virtual) network. We also assume that corresponding to any vertex of , there exists a threshold such that , where denotes the degree of in . A subset is said to be a dynamic monopoly if can be partitioned into such that for any , any vertex of has at least neighbors in . We say a vertex becomes active at time if belongs to . We denote by the threshold assignment for the vertices of . The smallest cardinality of any dynamic monopoly of is denoted by . Dynamic monopolies are modeling the spread of influence in , where is interpreted as the degree of susceptibility of the vertex . Dynamic monopolies were widely studied in the literature [1, 2, 5, 6, 9, 12, 13, 15, 18, 19], under the equivalent term “conversion sets” [7, 9] and also “target set selection” [1, 6, 16]. Dynamic monopolies have applications in viral marketing [8]. In modeling the phenomena of the spread of influence, it is assumed that when an element is activated, it remains active forever. In practice some models of spread of influence such as spread of a disease together with quarantination or the spread of virus in computer networks combined with decontamination process, do not match with the underlying conditions of dynamic monopolies [11]. From the other side, it was known that in some diseases the infection lasts only a limited period of time for each infected person [3]. Dynamic monopolies are not useful for the formulation of these latter phenomena. For this purpose, we introduce in this paper a new model for such diffusions of influence and call it weak dynamic monopoly. The following presents the formal definition and some related notations. For the graph theoretical terminology we refer the reader to [4].

###### Definition 1

. Assume that a graph together with a threshold assignment for its vertices is given. A subset is said to be a -weak dynamic monopoly (or -WDM) if can be partitioned into such that for any , any vertex of has at least neighbors in . By the size of the weak dynamic monopoly we mean the cardinality of . Let be a weak dynamic monopoly in as above. We call the value in the above definition, the processing time of . We denote the smallest size of any -WDM of by .

A special case of weak dynamic monopolies, the so-called monopolies were already defined and studied in the literature [10, 14]. In a graph , a subset is said to be a monopoly if each vertex has at least neighbors in . Denote the smallest size of any monopoly respect to the threshold function by . Also Flocchini et al. in [12] study the dynamic monopolies which activate the whole vertices of the graph in only one time step and call them stamos. Note that stamos, monopolies, and weak dynamic monopolies with the processing time one, are all equivalent concepts.

In the topics of monopolies or dynamic monopolies the following two special types of threshold assignments are mostly studied. In simple majority threshold we have for any vertex of the graph and in strict majority threshold, the threshold of any vertex is defined as .

The outline of the paper is as follows: We end this section by comparing three quantities , and (Theorem 1). In Section 2 we obtain some bounds for the size of weak dynamic monopolies in terms of the order and the processing time or the even-girth of graphs for general threshold assignments. Section 3 devotes to study the processing time of weak dynamic monopolies in terms of known graph parameters. In Section 3, an upper bound is given for the smallest size of weak dynamic monopolies in the Cartesian product of cycles and its processing time is determined. Finally, in Section 4 we show that for any , the smallest size of WDM can not be approximated within a factor of , unless , where is the order of the input graph.

The following theorem gives the comparison results between monopoly, dynamic monopoly and weak dynamic monopoly. For any two non-negative valued functions and , by we mean tends to zero as . Also we write if there exists a positive constant such that for any , .

###### Theorem 1

.

(i) Let be any graph. Then .

(ii) The equality may hold in each of the above inequalities.

(iii) There exists a sequence of graphs such that

(iv) There exists a sequence of graphs for which

Proof. The validity of the inequalities in (i) is clear from the definitions. To prove (ii), it is enough to consider the cycle graph on vertices with the strict majority threshold. We observe that . To prove (iii), replace any vertex of by (the complement of the complete graph on two vertices) and then join any two consecutive . Denote the resulting graph by and consider simple majority threshold for . It is easily seen that .

We now prove (iv). We obtain a sequence of graphs for which . Define , where is the join notation. Set for any vertex of . Denote the vertex of in by . A set consisting of and one vertex from forms a dynamic monopoly. Hence . We now show . Let be any WDM of . Assume first that does not belong to . In this case, since each vertex of has two neighbors in , then for and also is an independent set which implies . Assume now that . Let . In the following we show that the processing time of is at most two. Suppose partitions into , where and let . Denote the neighborhood of the vertex in by . Since , we have necessarily . And then . Since and , then and can not become active at time 2, which is a contradiction. Hence, the processing time is at most two. In the following we prove . Assume by the contrary that . Then there must be four consecutive vertices in which are not in . It is clear from the structure of that no three consecutive vertices in are activated at time 1 and also is independent. Then there are three possibilities:

1) and . This case clearly rules out the activation of and at time 1, a contradiction.

2) and . In this case could not have two neighbors in , a contradiction.

3) and . In this case could not have two neighbors in , again a contradiction.

Therefore . In other words, , which completes the proof of (iv).

## 2 Some bounds for the size of weak dynamic monopolies

We first consider weak dynamic monopolies with strict majority threshold.

###### Theorem 2

. For any graph with strict majority threshold . Moreover the bound is tight.

Proof. It was shown in [10] that any graph contains a monopoly with strict majority threshold and no more than vertices. Then, by Theorem 1 we have .

We now obtain a graph satisfying . Consider vertex disjoint copies of . Add a new vertex to the graph and then connect to exactly one vertex from each copy of . Let be any minimum weak dynamic monopoly in , where the strict majority threshold is considered. We claim that consists of two vertices from each triangle of . Otherwise, let , and be vertices of a triangle and be adjacent to the central vertex of the graph. One of the vertices or has to be in , otherwise, neither nor becomes active by the other vertices of . Assume that . Since needs two active neighbors at the same time, then necessarily . We have therefore

It was proved in [18] that any strict majority dynamic monopoly in a cubic graph on vertices contains at least vertices. Hence by Theorem 1 we have the following.

###### Theorem 3

. Let be a cubic graph on vertices and for each vertex , . Then

In Proposition 1 we show that the bound of Theorem 3 is tight. In the following we obtain a lower bound for the size of any weak dynamic monopoly in terms of the processing time of the weak dynamic monopoly.

###### Theorem 4

. Let be a graph with the maximum degree . Assume that each vertex of has threshold such that . Suppose is any -WDM for with processing time . Then

Proof. Let be a partition for corresponding to WDM where . Let , , be the number of edges between and . Since each vertex in has at most neighbors in and each vertex in has at least neighbors in , then

From the other side, for each , each vertex in has at most neighbors in and every vertex in has at least neighbors in . Therefore

And hence

Using the latter inequality repetitively, we obtain the following inequality

It turns out that

which implies

In order to present Proposition 1 we need the following easy number theoretic fact.

###### Lemma 1

Proof. Let for some integer . The proof is easily obtained by the induction on .

The following proposition shows that in Theorem 3 and Theorem 4 the equality may hold for some graphs.

###### Proposition 1

. For any integers and with , there exists a cubic graph on vertices which contains a weak dynamic monopoly corresponding to the constant threshold assignment for the vertices of , such that the processing time of is and .

Proof. By Lemma 1 there are infinitely many for which there exists such that . Let and be two arbitrary integers such that . We construct a cubic graph on vertices in which each vertex has threshold 2 and

Note that in the above relation , since the graph is cubic. It can be easily shown that the second equality in the above relation holds for the values and . Now we explain the construction of . Consider a partition for in the form of such that and and for each , , . Note that . Put edges between and in such a way that each vertex in has exactly two neighbors in and each vertex in except one vertex say , has three neighbors in . The vertex of has only two neighbors in . For each , , put some edges between and in such a way that each vertex in has exactly one neighbor in and each vertex in has exactly two neighbors in . Now connect the only vertex of to the two vertices in and also to . Consider the constant threshold assignment for all the vertices of . It is easily seen that is a WDM with the desired cardinality. Therefore in the inequalities of Theorem 3 and Theorem 4, equality hold for the graph .

For -regular graphs we have the following analogous bound.

###### Theorem 5

.

(i) Let be a -regular graph on vertices and set for every vertex of . Let also be any -WDM with processing time . Then

(ii) If for each vertex , then

Proof.

(i) If then the graph is the cycle . Using Theorem 1 we have

For , the proof is similar to the proof of Theorem 4.

(ii) For this case too, the proof is similar to the proof of Theorem 4.

The following theorem presents a lower bound in terms of even-girth of the graph. By the even-girth, we mean the length of smallest even cycle in the graph. In the following by , for any two vertices and , we mean the distance between and in the graph.

###### Theorem 6

. Let be a graph with even-girth , where . Let be a threshold assignment for the vertices of such that for any vertex , and . Then

Proof. Let be the order of . To prove the theorem, we need to show the following inequality.

For this purpose, consider some vertex and let . Since then each vertex , , has at most one neighbor in and at most one neighbor in . Hence each , , has at least neighbors in . We have also for each vertex . Therefore and then .

Now suppose and , where is the neighborhood of in . Since then for any , and have no common neighbor. Let and , where . Let also . If is adjacent to some vertex in with , then none of have any neighbor in . Now suppose without loss of generality that is adjacent to some vertex from each , . Since for each we have , then each of these vertices has at least neighbors out of . Therefore we have

Now let be any WDM with the processing time . The set partitions as . There are two possibilities for :

1) .

In this case we consider a vertex say . The vertex has at least neighbors in . Each of these neighbors has at least neighbors in and in general for any any vertex in has at least neighbors in . Then

2) .

In this case we have the following two subcases.

Subcase i) .

In this subcase we have

Hence

For , it is easily seen that . When we have

It is easily seen that for , . Then

Subcase ii) .

Let be the smallest one with this property. Consider the bipartite graph constructed on sets and and all edges of between them. Denote this new graph by . Then . Suppose is a minimal subset of with the property . It is easily seen that for each vertex , . Otherwise, let has only one neighbor in . Set . Then , a contradiction with the minimality of . On the other hand, every vertex in has at least neighbors in . Then we have

Hence, there is a vertex say in , such that . Consider induced subgraph of on and as and . Let . Now we have . Since and every vertex in has at least two neighbors in then . It is easily seen that and each vertex in has at least neighbor in . Then . By repeating this argument we obtain that for even , and . Since the even girth of is , then the girth of is at least . It implies that any vertex of has at most one neighbor in . Since is bipartite then there must be neighbors for the vertices of in . Therefore

For odd , and . Then we have

Hence . This completes the proof.

## 3 Bounds for processing time

In the following we consider the processing time of any WDM and obtain some upper bounds for it.

###### Theorem 7

. Let be a graph with a threshold assignment , where for each vertex and let be any WDM which partitions as . Then there are internally disjoint paths with length beginning from and containing only one vertex from each , .

Proof. By induction on , where , we show that for any set of distinct vertices, there are internally disjoint paths with length which begin from and end at , respectively and these paths contain only one vertex from each , . This is obvious when . Suppose , . Because each of has at least neighbors in , then there are distinct in such that , . By the induction hypothesis there are internally disjoint paths with length which begin from and end at , respectively and contain only one vertex from each , . Now, are the desired paths, where by we mean the extension of path by the edge .

The following corollaries are obtained from Theorem 7. By a starlike tree we mean any tree that is isomorphic to a subdivision of for some . Such a starlike tree contains a central vertex of degree and branches.

###### Corollary 1

. Let be a graph with threshold assignment , where for each vertex of , . Then there exists a starlike tree in with central vertex of degree and branches of length .

Proof. Let . The vertex has at least neighbors in . By Theorem 7, there are distinct paths of length that end at , respectively. A starlike tree can be easily obtained by adding vertex and edges to .

In the following corollary by we mean the maximum number of independent edges in .

###### Corollary 2

. Let be a graph with threshold assignment and . Then for any WDM with processing time we have

Proof. Since , by Corollary 1 there is a starlike tree with branches of length . By choosing independent edges from each branch we have a set containing independent edges. Therefore

This yields the desired result.

In the following corollary we present an upper bound for processing time in terms of the length of longest path in graphs.

###### Corollary 3

. Let be a graph with threshold assignment and let be the length of longest path in . Then for any WDM with processing time we have

i) If for each vertex , then ;

ii) If is the strict majority assignment, then .

Proof. i) By corollary 1, there is a path of length in . Hence .

ii) Let be a WDM which partitions as . Let also . Then has at least one neighbor in with . Since the activation process follows the strict majority rule, then has at least two neighbors in such as and whose degrees and thresholds in are at least two. By Theorem 7 there exist two internally disjoint paths and with length which end at and , respectively. Now is a path of length . Then .

We are going to show that in graphs with bounded maximum degree, the size of any WDM for and its processing time are not bounded by a constant value, i.e. one of them goes to infinity as .

###### Theorem 8

. Let be a graph on vertices such that for some constant . Let be any threshold assignment for and be any -WDM with the processing time . Then

Proof. Assume that partitions as . It is easily seen that for each , , . Then

Therefore

It can be shown that Theorem 8 is not valid when is not bounded. For example consider with any threshold assignment such that and . It is easily seen that any minimum WDM in this graph is contained in . Hence . From the other side, in such a minimum WDM, all the vertices are activated in at most two steps.

In the following theorem we determine the processing time in the Cartesian product of cycles denoted by . And also we obtain an upper bound for the smallest size of weak dynamic monopolies. Dynamic monopolies of this family of graphs were studied in [13].

###### Theorem 9

. Let , where the threshold of each vertex is 3. Then, the activation process for any WDM in ends after two steps and when .

Proof. Let be any WDM in and assume on the contrary that a vertex say becomes active at time 3. Then partitions as for some , . The vertex has three neighbors , and in . Let be a vertex which is adjacent to two neighbors of . The vertex should have three neighbors in but this is impossible. Hence any WDM of activates the whole graph in at most two time steps.

In Figure 3, a WDM for which is also an independent set is presented. Let be an independent WDM in . Let also the WDM , partitions the graph into three sets . Obviously, each vertex in has at least three neighbors in , then we have

We make the following claims concerning and .

Claim 1) is independent.

Otherwise, we obtain Figure 1, in which there are adjacent vertices in . This is a contradiction.

Claim 2) is independent and no vertex in has a neighbor in .

If we have two adjacent vertices in then the specified vertices in part (1) of Figure 2 can not have three neighbors in the previous time. And also we have . Otherwise, we have part (2) of Figure 2 which contradicts the independency of .

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