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Table then compares direct numerical simulations of the corehd algorithm with the prediction that is obtained from the differential equations. The two are in excellent agreement, for the analysis higher precision can be obtained without much effort. When analyzing Erdős-Rényi graphs it is necessary to restrict the larg... |
Confident with this cross-check of our theory, we proceed to compare with other theoretical results. As stated in the introduction, Guggiola and Semerjian Guggiola and Semerjian ( 2015 have derived the size of the minimal contagious sets for random regular graphs using the cavity method. At the same time, several rigor... |
In table we quantify the gap between our upper bound and the results of Guggiola and Semerjian ( 2015 for larger values of [MATH] . Besides its simplicity, the corehd algorithm provides significantly better upper bounds than those known before. Clearly, we only consider a limited class of random graphs here and the bou... |
III Improving corehd The main focus of this paper has been, up until now, the analysis of corehd on random networks. Rather naturally the question of how to improve over it raises. In this section we evaluate the possibility of a simple local strategy that systematically improves over the corehd performance. We show th... |
The corehd algorithm (Alg. ) does not take into account any information of the neighborhood of a node. The theoretical analysis in the previous section owes its simplicity to this fact. However, the idea behind corehd can be extended to the next order by considering the structure of the nearest neighbors of a node. Onc... |
According to the previous section, selecting high degree nodes is a good strategy. Another natural approach is a greedy strategy that, in each step, selects a node such that the caused cascade of nodes dropping out of the core is maximized Guggiola and Semerjian ( 2015 ); Coja-Oghlan et al. 2015 ); Sen et al. 2017 In t... |
corehdld This approach selects high degree nodes, but then discriminates those that have many neighbors of high degree. The idea is that nodes that have neighbors of large degree might get removed in the trimming procedure preferentially and hence the degree of the node in question will likely decrease. More specifical... |
First select the set [MATH] and then update it according to [MATH] weak-neighbor The weak-neighbor strategy aims to remove first those nodes that have high degree and low average degree of the neighbors, thus causing a larger trimming step on average. There are different ways to achieve this. We tried two strategies th... |
The order in which the nodes are removed is according to [MATH] with [MATH] being the degree of node [MATH] and [MATH] the average degree of the neighbors of [MATH] |
We separate the two steps. First select the set [MATH] and then update it according to [MATH] Our implementation of the weak-neighbor algorithm is available in the open depository dem |
corehd-critical The coreHD-critical combines the corehd with the vanilla-greedy algorithm on the direct neighbors. Nodes are first selected according to their degree and subsequently among them we remove nodes first that have the largest number of direct neighbors that will drop out in the trimming process. |
First select the set [MATH] and then update it according to [MATH] Finally it is interesting to contrast the stated algorithms with a version in which the high degree selection step is left out, i.e. select [MATH] and then remove at random from this set. |
Let us summarize the results. First, we find that all above strategies improve over the corehd algorithm (at least in some regimes). Second, we find that among the different strategies the weak-neighbor algorithm performs best. |
While we have systematic numerical evidence that the weak-neighbor strategy performs best, it is not clear which are the effects responsible. What we can say for sure is that the generic locally greedy procedure of trying to reduce the size of the [MATH] -core at every step is not optimal. |
The most commonly considered greedy procedures do consider information not only from the direct neighborhood of a node. The vanilla-greedy approach removes nodes according to the size of cascade that is caused by their removal. Nodes are removed first that cause the largest cascade of nodes dropping out in the subseque... |
Next we contrast the corehd performance and the weak-neighbor performance with the performance of recently introduced Sen et al. 2017 algorithm citm [MATH] that uses neighborhood up to distance [MATH] and shown in Sen et al. 2017 to outperform a range of more basic algorithm. In figure we compare the performances of th... |
To discuss a little more the results observed in Figure , for small [MATH] the corehd algorithm outperforms the citm algorithm, but when [MATH] is increase the performance gap between them shrinks, and for large [MATH] (e.g. in Fig. part (a)) CIMT outperforms corehd . Both corehd and citm are outperformed by the weak-n... |
Finally, following up on Braunstein et al. 2016 we mention that in applications of practical interest it is possible to improve each of the mentioned algorithm by adding an additional random process that attempts to re-insert nodes. Consider the set [MATH] of nodes that, when removed, yield an empty [MATH] -core. Then ... |
IV Conclusion In this paper we study the problem of what is the smallest set of nodes to be removed from a graph so that the resulting graph has an empty [MATH] -core. The main contribution of this paper is the theoretical analysis of the performance of the corehd algorithm, proposed originally in Zdeborová et al. 2016... |
Next to the theoretical analysis of corehd we proposed and investigated numerically several other simple strategies to improve over the corehd algorithm. All these strategies conserve the essential running time of [MATH] on graphs with maximum degree of [MATH] Among our proposals we observe the best to be the weak-neig... |
There are several directions that the present paper does not explore and that would be interesting project for future work. One is generalization of the differential equations analysis to the weak-neighbor algorithm. This should in principle be possible for the price of having to track the number of neigbors of a given... |
Acknowledgement We are thankful to the authors of Sen et al. 2017 for sharing their implementation of the citm algorithm with us. We would further like to thank Guilhem Semerjian for kind help, comments and indications to related work. LZ acknowledges funding from the European Research Council (ERC) under the European ... |
Appendix A Proofs A.1 Proof of Lemma Based on some worst-case assumptions, we can analyze the evolution of the number of degree- [MATH] edges. In particular, in the worst case, the initial removal of a degree- [MATH] node can generate [MATH] degree- [MATH] edges (i.e., all of its edges were attached to degree- [MATH] n... |
During the [MATH] -th trimming step, a random edge is chosen uniformly from the set of edges adjacent to nodes of degree [MATH] (i.e., degree less than [MATH] ). Let the random variable [MATH] equal the degree of the node adjacent to the other end and the random variable [MATH] equal the overall change in the number of... |
For [MATH] , this edge is distributed according to [EQUATION] where [MATH] denotes the number of edges of degree [MATH] before trimming. If [MATH] , then the edge connects two degree- [MATH] nodes and removal reduces the number of [MATH] edges by [MATH] If [MATH] , then the edge connects a degree- [MATH] node with a de... |
The crux of the analysis below is to make additional worst-case assumptions. One can upper bound the probability of picking degree- [MATH] edges because, after [MATH] steps of trimming, the number of degree- [MATH] edges can be at most [MATH] (i.e., if initial node removal generates [MATH] edges of degree- [MATH] and [... |
[EQUATION] In this formula, the [MATH] term represents initial removal of a degree- [MATH] node and the [MATH] term represents the edge removal associated with [MATH] steps of trimming. We note that, since edges attach two nodes of different degrees, all edges are counted twice in [MATH] Similarly, the [MATH] term repr... |
Now, we can upper bound the number of degree- [MATH] edges after [MATH] steps of trimming by the random sum [EQUATION] where each [MATH] is drawn independently according to the worst-case distribution ( 33 ). The term [MATH] represents the worst-case event that the initial node removal generates [MATH] edges of degree-... |
To complete the proof, we focus on the case of [MATH] Since the distribution of [MATH] gradually places more weight on the event [MATH] as [MATH] increases, we can also upper bound [MATH] by replacing [MATH] by i.i.d. copies of [MATH] From now on, let [MATH] denote a random variable with the same distribution as [MATH]... |
[EQUATION] Next, we observe that ( 33 ) converges in distribution (as [MATH] ) to a two-valued random variable [MATH] that places probability at most [MATH] on the point [MATH] and the remaining probability on [MATH] Since [MATH] , there is an [MATH] such that [MATH] for all [MATH] |
Finally, Hoeffding’s inequality implies that [EQUATION] Putting these together, we see that [EQUATION] for [MATH] As [MATH] , the last condition can be satisfied by choosing [MATH] Since this condition is trivial as [MATH] , the previous convergence condition will determine the minimum value for [MATH] when [MATH] is s... |
A.2 Proof Sketch for Lemma We note that graph processes like corehd often satisfy the conditions necessary for fluid limits of this type. The only challenge for the corehd process is that the [MATH] process may have unbounded jumps due to the trimming operation. To handle this, we use Lemma to show that, with high prob... |
To apply (Wormald, 1999 , Theorem 5.1) , we first fix some [MATH] and choose the valid set [MATH] to contain normalized degree distributions satisfying [MATH] and [MATH] This choice ensures that Lemma can be applied uniformly for all [MATH] Now, we describe how the constants are chosen to satisfy the necessary conditio... |
[MATH] applying Lemma with [MATH] to see that [MATH] To verify condition [MATH] of the theorem, we note that ( 21 ) is derived from the large-system limit of the drift and the error [MATH] can be shown to be [MATH] To verify condition [MATH] of the theorem, we note that that ( 21 ) is Lipschitz on [MATH] Finally, we ch... |
[EQUATION] these choices imply that the corehd process concentrates around [MATH] as stated. We only sketch this proof because very similar arguments have used previously for other graph processes Achlioptas ( 2000 2001 ); Cocco and Monasson ( 2001 |
# Source: arxiv 1806.03170 # Title: On Critical Threshold Value for Simple Games # Sections: all # Downloaded: 2026-03-02T09:22:57.740403+00:00 |
On Critical Threshold Value for Simple Games Abstract In this note, we show that for every simple game with [MATH] players the critical threshold value is at most [MATH] . This verifies the conjecture of Freixas and Kurz. |
Introduction Let [MATH] be a finite set of players. We call a function [MATH] monotone if [MATH] for all [MATH] such that [MATH] . The pair [MATH] is called a simple game if [MATH] [MATH] and [MATH] is a monotone [MATH] function. We refer the reader to |
for further reading on simple games. A simple game [MATH] partitions the collection of all possible player coalitions into two collections: the collection of winning coalitions |
[MATH] and the collection of losing coalitions [MATH] Weighted voting games are a natural family of simple games. A weighted voting game is defined by a finite set of players [MATH] and a vector [MATH] [MATH] [MATH] , where |
[EQUATION] Here, we use the notation [MATH] for a vector [MATH] and [MATH] Clearly, every weighted voting game is a simple game. To show that the reverse is not true let us consider the following example from |
Example 1 Let [MATH] for some even [MATH] and the value function [MATH] be as follows [EQUATION] Obviously, coalitions [MATH] [MATH] are winning while the two coalitions [MATH] [MATH] are losing. |
If the desired vector [MATH] exists for Example , then on one side [MATH] and on the other side [MATH] , showing that for [MATH] this game is not a weighted voting game. |
To understand whether a simple game is a weighted voting game, we could use the critical threshold value introduced in . Before we define the critical threshold value of a simple game, let us define the following polyhedron |
[EQUATION] The critical threshold value can be defined as [EQUATION] Observe, that [MATH] if and only if the simple game [MATH] is a weighted voting game. |
The example in shows that [MATH] can be as large as [MATH] , because [MATH] lies in the convex hull of the characteristic vectors of winning coalitions while [MATH] lies in the convex hull of the characteristic vectors of losing coalitions. Freixas and Kurz |
conjectured that there is no simple game with a larger value of [MATH] . Here, we state the variant of the conjecture of Freixas and Kurz from |
Conjecture 2 (Conjecture of Freixas and Kurz) For a simple game with [MATH] players, the collection of winning coalitions [MATH] and the collection of losing coalitions [MATH] , we have |
[EQUATION] In the conjecture of Freixas and Kurz was verified for simple games with all minimal wining coalitions of size [MATH] and for simple games with no minimal winning coalitions of size [MATH] . In |
it was shown that [MATH] for general simple games. Before going to the proof, we would like to say that our approach is inspired by the work of Ahmad Abdi, Gérard Cornuéjols and Dabeen Lee on identically self-blocking clutters |
(Section 3). Proof To prove the conjecture we reformulate, strengthen and only then verify it. A coalition [MATH] [MATH] is called a cover of [MATH] if [MATH] has at least one common player with every coalition in [MATH] . We call the collection of covers of [MATH] the blocker of [MATH] and denote it by [MATH] |
. Due to the definition of simple games, we have [EQUATION] Hence, the critical threshold value can be reformulated as follows [EQUATION] |
Here, [MATH] stands for the scalar product of two vectors [MATH] and [MATH] Conjecture 3 (Reformulation of Conjecture of Freixas and Kurz) |
For a simple game with [MATH] players and the collection of winning coalitions [MATH] , we have [EQUATION] Next, we prove Theorem , which is a strengthening of Conjecture . For the proof we need the following straightforward remark, which we leave as an exercise. |
Remark 4 Let [MATH] be a polyhedron and let [MATH] be the optimal solution of the program [MATH] . Then [MATH] is an optimal solution of the linear program [MATH] |
Theorem 5 (Strengthening of Conjecture of Freixas and Kurz) For a simple game with [MATH] players and the collection of winning coalitions [MATH] , we have |
[EQUATION] In particular, if [MATH] is the optimal solution for the program [EQUATION] then [EQUATION] Proof. Let us consider the unique optimal solution [MATH] for the program [MATH] . By Remark [MATH] is an optimal solution for the program [MATH] . Thus, [MATH] is an optimal solution for the program [MATH] . Thus, we... |
[EQUATION] finishing the proof. To finish the note, let us discuss when Conjecture provides a tight upper bound for the critical threshold value. The next theorem shows that if the upper bound in Conjecture is tight, then this fact can be certified in the same way as in Example |
Theorem 6 For a simple game with [MATH] players and the collection of winning coalitions [MATH] and the collection of losing coalitions [MATH] , we have |
[EQUATION] if and only if [MATH] lies in the convex hull of the characteristic vectors of winning coalitions and [MATH] lies in the convex hull of the characteristic vectors of losing coalitions. |
Proof. Clearly, if [MATH] lies in the convex hull of the characteristic vectors of winning coalitions and [MATH] lies in the convex hull of the characteristic vectors of losing coalitions, then for every [MATH] we have |
[EQUATION] showing that [MATH] and hence [MATH] by Theorem On the other hand, from the proof of Theorem we know that if [MATH] then [MATH] is an optimal solution for [MATH] with value [MATH] . Let us show that [MATH] lies in the convex hull of the characteristic vectors of winning coalitions. To do that consider an opt... |
In the same way as the proof of Theorem , we could show that [EQUATION] where [MATH] is the optimal solution for the program [EQUATION] |
Thus, if [MATH] equals [MATH] , then [MATH] and [MATH] lies in [MATH] . Hence, if [MATH] equals [MATH] , then [MATH] lies in the convex hull of the characteristic vectors of losing coalitions, finishing the proof. |
Open Questions The question about asymptotic behaviour of the critical threshold value of complete simple games remains open. These are the games with a total order of players by "winning power". Freixas and Kurz |
conjectured that the critical threshold value of a complete simple game with [MATH] players equals [MATH] . Recently, in it was shown that the critical threshold value of such games is [MATH] Acknowledgements. We would like to thank Ahmad Abdi for helpful comments on the first version of this note. |
# Source: arxiv 1806.03213 # Title: Impact of End-User Behavior on User/Network Association in HetNets # Sections: all # Downloaded: 2026-03-02T09:22:58.970114+00:00 |
Impact of End-User Behavior on User/Network Association in HetNets Abstract We study the impact of end-user behavior on user/network association in a HetNet with multiple service providers (SPs). Specifically, we consider the uncertainty in the service guarantees offered by SPs in a HetNet, and use Prospect Theory (PT)... |
Index Terms: user association, HetNets, spectrum allocation, game theory, prospect theory, Stackelberg game. Introduction The emergence of HetNets for network densification in future wireless networks has lead to extensive user association studies in this context |
. The range of approaches include evolutionary game theory , auction based models , matching theory , Stackelberg, the competitive approach using Colonel Blotto game |
, and other gametheoretic models . However, most of these mechanisms essentially are borne out of expected utility theory (EUT) based approaches. When a service provider (SP) controls access to end-users via differentiated and hierarchical monetary pricing, then the performance of the network is directly subject to end... |
In this work, we use Prospect Theory , a Nobel prize winning theory that explains real-life decision-making and its deviations from EUT behavior, to study user decisions in wireless HetNets. To do so, we first formulate the user association problem in HetNets as a Stackelberg game between SPs and user, in which WiFi an... |
The rest of this paper is organized as follows. In section II , after describing the HetNet model, we introduce the model of interactions between SPs and users as a Stackelberg game, and formulate user association problem. In section III , we discuss the best response strategies for all players, and in section IV , we ... |
II System Model and Problem Formulation Network Model To study user association in HetNets, we developed a two-tier HetNet scenario which includes [MATH] wireless users that are randomly distributed within the coverage area of [MATH] base stations. As shown in Fig. , in our HetNet model there is one macrocell LTE BS lo... |
Stackelberg Game for User Association in HetNets In our HetNet model, each user receives [MATH] different offers from all [MATH] base stations, i.e., one offer from the cellular BS and [MATH] offers from the WiFi BSs. To enable multihoming, we assume each user can be simultaneously connected to both cellular and WiFi S... |
and not the focus of this paper. In the remainder of this paper, we focus on one of these games. Using the index [MATH] for user’s preferred WiFi SP and the index [MATH] for cellular SP, we denote the [MATH] of WiFi and cellular SPs with triples [MATH] , and [MATH] , respectively, in which the first term shows the adve... |
[EQUATION] where [MATH] is the users benefit function which is a logarithmic concave function of the aggregate data rate achieved, [MATH] , by the user, and [MATH] is the user’s cost function which shows the aggregate price that must be paid by user to the SPs for each [MATH] strategy. The aggregate data rate is define... |
Once the user chooses its best response strategy, [MATH] , the SPs will respond with their best response strategies to maximize their own utilities based on the received user decision. The utility of the WiFi SP, [MATH] is defined as |
[EQUATION] and the utility of cellular SP, [MATH] is defined as [EQUATION] where, the first term in both of these equations is the SPs’ expected payoff from the user, and the second term is their incurred service cost. The SPs’ payoff from the user is equal to the offered price in their bids if the user accepts their o... |
[EQUATION] III Best Response Strategies for players User Optimization Problem Upon receiving the offers from the SPs, the user will run an optimization problem to find its best strategy with regard to received offers. We assume the user’s payoff function from the received data is a concave function, as defined in Eq. ,... |
Max1 Problem: User’s Utility Maximization —————————————————————————- [EQUATION] As shown above, the user has two major constraints for bid selection. The first constraint, shown in Eq. , is the user’s data rate constraint which ensures the expected data rate for the user is higher than its minimum required data rate, [... |
SPs Optimization Problems When the SPs receive user’s decision with regard to their offers, they choose their best response strategy. The the best response strategy [MATH] for the WiFi BSs is obtained by solving the optimization problem below (denoted as [MATH] ): |
Max2 Problem: WiFi SP’s Utility Maximization —————————————————————————- [EQUATION] in which, [MATH] is the maximum amount of bandwidth that can be allocated to the user by the WiFi SP, and [MATH] is the maximum achievable data rate by the user from the WiFi SP considering [MATH] and the gain of the channel between the ... |
[EQUATION] in which [MATH] is the gain of bandwidth allocation which is less than one due to the guard bands between channels for preventing interference, [MATH] is a binary variable showing the activity of the users which [MATH] if the user [MATH] is active and has data demand, otherwise [MATH] , and [MATH] is also a ... |
[EQUATION] where, [MATH] is the transmit power of the BS, [MATH] is the channel gain between the user [MATH] and the SP [MATH] , and [MATH] is the noise variance over the transmission channel. Similarly, the cellular SP runs [MATH] optimization problem, which is defined exactly similar to [MATH] , except the index [MAT... |
Theorem 1 The SPs best response strategies, derived from [MATH] and [MATH] problems, will always satisfy the minimum data rate constraint in the boundary of its feasibility region, i.e. we always have [MATH] , and [MATH] for the WiFi and the Cellular SPs, respectively. |
Proof: We prove this by contradiction for the WiFi SP. Assume [MATH] is the optimal solution for [MATH] problem, and [MATH] is not a marginal BW, i.e. [MATH] . Considering Eq. 13 , we can infer that [MATH] (1). In this case [MATH] such that [MATH] (2). From (1) and (2), and considering the direct relation between [MATH... |
IV NE Existence Analysis under EUT We derive all the potential Nash Equilibrium strategies for the user in the proposed Stackelberg game under EUT. To see the effects of the SPs heterogeneity on the existence of NE, we consider both symmetric and asymmetric SPs cases. |
Symmetric SPs under EUT Under the symmetric model, we assume both SPs offer the same data rate and use the same pricing and cost functions, i.e. [MATH] [MATH] [MATH] [MATH] . In this situation, both SPs offer [MATH] with the price of [MATH] , and this offer costs [MATH] for them. Table summarizes the NE strategies for ... |
Consequently, because the SPs know that their offers are not going to be accepted by the user in 50% of times, they will also choose a mixed strategy of [MATH] which means in half of the times they prefer to stay silent and not offer any data rate to prevent negative utility as a result of being rejected by the user. |
Asymmetric SPs under EUT Under the asymmetric model, we assume the SPs have different pricing and cost functions, as defined in subsection II LABEL:SUBSEC:SM . We denote the WiFi and the cellular SPs best response strategies under asymmetric model as [MATH] , and [MATH] , respectively. Here we assume due to better chan... |
However, when the user’s payoff coefficient is bigger than a threshold shown in Table , then the extra payoff of the user achieved from the cellular SP is higher than the price asked by the cellular SP, and it convinces the user to accept both offers even though the WiFi SP offer is enough to guarantee its minimum data... |
NE Existence Analysis under PT So far, our analysis on the existence of NE and derivation of the user’s best response strategies was based on EUT. In fact, when the user is making decisions about a system with some uncertainty in system parameters, like the service guarantees in our model, EUT fails to describe the use... |
. To do so, we assume the SPs still make the same offers to the user as they did under EUT, however, we assume the user makes decisions about the received offers based on PT. In this work, we just focus on the probability weighting effect (PWE) of PT to see its effects on the NEs of our Stackelberg game. We use the Pre... |
to model the PWE under PT: [EQUATION] This function is a regressive and s-shaped function which is concave in [MATH] region and convex in [MATH] region, and [MATH] in the former domain while [MATH] in the later. Considering this function to model PWE of PT, we can infer that the user overestimates the service guarantee... |
Symmetric SPs under PT In this case, we assume both SPs use the same pricing and cost functions as described in subsection IV LABEL:SYMMETRICEUT for the EUT case. However, the user makes decision about the received offers based on PT, and thus user’s perception of the service guarantee will be affected by PWE of PT. In... |
Asymmetric SPs under PT In this case as described in subsection IV LABEL:ASYMMETRICEUT , we assume the SPs use different cost and pricing functions. As mentioned before, we assume the user will underestimate the service guarantees of the SP offers under PT, due to PWE. Hence, user’s estimation of the expected data rate... |
VI Proposed Bidding Strategy for SPs under PT In previous section, we inferred that under PT if the user underestimates the advertised service guarantees, it is more likely for the user to reject the SPs offers by choosing [MATH] strategy. This can reduce the utility of SPs if they do not redesign their bidding strateg... |
Bandwidth Expansion Under PT As mentioned before, under PT user replaces its subjected service guarantee, for example [MATH] for the WiFi SP, with the service guarantee advertised by the SPs, [MATH] for the WiFi. And if the advertised service guarantee is higher than [MATH] , the user will underestimate it, which means... |
[EQUATION] where, [MATH] is the WiFi SP bid under EUT. In fact the WiFi SP must expand its offered bandwidth so as to offer [MATH] data rate with the service guarantee of [MATH] , where [MATH] . This way, the extra bandwidth offered by the WiFi SP compensates the under estimation of the service guarantee by user under ... |
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