Growth Dynamics of Value and Cost Trade-off in Temporal Networks
Sheida Hasani, Razieh Masoomi, Jamshid Ardalankia, Mohammadbashir Sedighi, Hamid Jafari
GGrowth Dynamics of Value and Cost Trade-off inCompetitive Temporal Networks
Sheida Hasani a , Razieh Masoomi b , Jamshid Ardalankia c , MohammadbashirSedighi a , Hamid Jafari b a Department of Management, Science and Technology, Amirkabir University of Technology,Tehran, Iran b Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran c Department of Financial Management, Shahid Beheshti University, G.C., Evin, Tehran,19839, Iran
Abstract
The question is: What does happen to the real-world networks which cause themnot to grow permanently? The idea here is that real-world networks have topay the cost of growth. We investigate the growth and trade-off between valueand cost in the networks with cost and preferential attachment together. Since,the preferential attachment in BA model does not consider any stop againstthe infinite growth of networks, we introduce a modified version of preferentialattachment of BA model. This idea makes sense because the growth of realnetworks may be finite. In the present study, by combining preferential attach-ment in the science of temporal networks (interval graphs), and, the first orderdifferential equations of value and cost of making links, future equilibrium of anevolving network is illustrated. During the process of achieving a winning posi-tion, the variables against growth such as the competition cost, besides with theinternally structural cost may emerge. At the end, by applying this modifiedmodel, we found the circumstances which a trade-off between value and costemerge.
Keywords:
Temporal Network, Cost and Value Trade-off
Preprint submitted to Physica A September 2, 2019 a r X i v : . [ q -f i n . M F ] A ug . Introduction In the real world, networks inevitably pay the cost of growth. That is whyany event corresponding to growth is not out of charge. Hence, real networksmay not have infinite growth. A phenomenon like friction against growth -say cost of making new attachments- emerges in the networks.In this study, we investigate the growth dynamics of networks which emergetrade-off between value and cost . Firstly, we consider a network which itsgrowth is affected by preferential attachment [1–5] based on node activity [6].The value and the cost of the aforementioned network are separately describedby first order coupled differential equations, so we obtain a modified preferentialattachment of Barabasi-Albert model. Applying this modified model to such anetwork helps to clarify and predict the circumstances which the network stopsgrowing or continues to grow, or, there may exist some trade-off. As a back-ground corresponding to present research, Jafari et al [6] considered two typesof drivers which play role in the network’s dynamics such as node activity andalso memory effects. Indeed, memory and cost play the role of friction againstthe growth of network size. They found a critical time-scale, so-called ‘charac-teristic time’, which the attachment regime alters. Besides this, they found thatthe high event-wise temporal density -which implies higher cumulative degree,or similarly, the cumulative sum of past participation of a node in making links-cause more distinct and distinguishable critical time.Other methods for creating links are popular. Some of these are selecting ade-quate connections to intensify collaboration among the weaker agents and alsoconnection based on networks’ preferences not just nodes’ preferences, directedor undirected [7], peripheral to peripheral nodes, peripheral to central nodes, central to peripheral nodes, central to central nodes in competing networks andincreasing centrality in an individual sub-network [8]. Also Zhang et al [9] con-sidered the influential nodes based on their participation in arising the globalefficiency of the complex network.Also, some other researchers studied on the networks with accelerating growth210–12]. Among them, Safdari et al [13] and Liu et al [10] investigated the ac-celerating growing networks (AGN) together with the aging effects by applyinga factor to nodes with older time stamps. In addition, Ikeda [12] focused oncircumstances which the rate of accelerating growth in the network contributesto scale-free networks and investigated some accelerating scenarios and showednonlinearity between the rate of adding new nodes and the growth of the net-work. Zhao et al [14] investigated the evolutionary of instability of financialmarkets.From a microscopic point of view, on the contrary to investigation on the evo-lution of eigen constituents for extracting information of dynamic behaviorsand trends reversal [15, 16]. We introduce differential equations for the drivers( value and cost ). Due to microscopic and internal dynamics ( value and cost )in a competitive atmosphere, each node interacts with the newcomer links. Incomparison with Iranzo et al [7], we evaluate the behaviors of possible wan-dered links of the new comer temporal nodes without collaboration with thehosted network. Accordingly, the networks here are considered permanentlygrowing, dead, or living in a trade-off among negative growth and permanentpositive growth. This idea makes sense because in a non-collaborative economicworld [17] constituents seek to maximize their utility. Furthermore, in the in-vestment world, cost -as a barrier against networks’ growth- can be interpretedas the attraction of parallel markets and opportunities. Hence, the investmentopportunities without considering other parallel markets does not make sense,and also, evaluating corporate strategies without considering customers’ earned value and cost aspects in a market is not precise.
2. Value and Cost Trade-off and Competing Communities
Some researchers [7, 17–19] investigated temporal network’s topology in fi-nancial and marketing atmosphere. To do this, Grauwin et al [17] and Iranzo et al [7] investigated networks containing sub-networks which include strong(high centrality) and weak (low centrality) networks in a game of achieving the3trength (more centrality) and resulted that under circumstances based on Nashequilibrium and eigenvector centrality, the strongest is exposed to get weakerin a temporal process. The trade-off for being the strongest besides bearingthe threats (such as a situation which weaker competitors cooperate with eachother to overcome a certain stronger network) leads to a characteristic time.So, in the fate of the competition, dominant community is in danger of somedisturbing dynamic costs such as:- Originated internally (like costly structures or conflict of inner interests dueto
Agancy Theory in some growth level of financial [20–23], and social and be-havioral systems [24–26]);- Or, originated externally (like competitors changing strategy and underesti-mating the collective profitability of the united weaker ones [7] in statue quo ).A latent factor of these dynamics is agents’ utility function which in some scaleswill finally form the future state of the networks [17, 27].In our study, we discuss on 3 main concepts such as;- growth failure at the beginning;- permanent growth;- trade-off between stop or continue to grow.
3. Phase Transitions and Percolation Theory
Potential dynamics of interests in such systems emerge due to collectiveand collaborative effects -or simultaneously- emerge due to individual nature oflinks and sub-networks. This spectrum [17] on one side, is supposed to maxi-mize its own utility in a selfish manner, and on the other side of the spectrum,is supposed to minimize the global energy of the system in a cooperative andefficient manner. Based on the arrangement of constituents’ interests among thespectrum of selfish-cooperative trade-off, transition phase between growth andbarriers against growth (like cost ) appears. Accordingly, while constituents tryto reach the highest satisfaction, they may no longer face with their maximumsatisfaction, as Schelling’s segregation model proves [17].4ctually, in such networks, pairwise interactions among agents should be re-placed by interaction networks to reveal integrated behaviors. To illustratephase transitions and the interactions’ strength of linked clusters in a randomgraph, percolation theory is fruitful. Also, by percolation theory in transitiontime of networks, the giant components may dominate the general behavior ofthe network against attacks and diffusion [28]. Hence, the transition and break-down in the networks depend on the giant components’ critical behavior, theintegrated behaviors of clusters, and the global efficiency in networks.Dorogovtsev and Mendes, sec. 11.3 [11], by applying percolation theory thresh-olds , investigated the effects of sudden intentional random attack and damage(preferential elimination of nodes with highest degree which means more expo-sure to be targeted) in an undirected network. In this way, a certain networkgrows and are suddenly exposed to permanent damage. Hence, this approach isthe giant component’s preferential damage against preferential attachment. Thecalculation of the random damage threshold (which leads to network’s break-down) in scale-free networks was initially evaluated by Cohen et al [29]. Theyanalytically showed that by applying a preferential removal of a fraction of nodesin scale-free networks, the percolation transition of breakdown will be dependenton the power-law and network size. Interestingly, in the percolation transition,the effects of different power-laws are more significant than network-size effects.Another approach for investigating breakdown’s predecessors is working on thephenomenon of removing links rather than removing influential nodes [9]. Ourapproaches toward the mentioned question is somehow different. On the con-trary to a discrete evaluation of just failure or just growing, we are supposed toevaluate the trade-off between values and costs of attachments.Indeed, a question arises here. Why do we seek value and cost of creatinglinks in a network? A description for this approach relates to investment be-havior and marketing strategies, and as a whole, a trade-off between possiblescenarios in multi-agent situations. When a firm applies a platform, by passingtime, managers may find it not cost effective and may leave it. What does the5erm “ cost effective ” mean? indeed, managers may find that the cost is moresignificant rather than the earned value . This consciousness has a great deal ofimportance. Although, as soon as the links get to know that the cost of pres-ence in the network is more than the value which is earned, they will no longerbe motivated for presence in the network and they are not likely to attach it.In other words, once the cost gets larger than value , the whole network stopsgrowing. So, with the help of knowledge about value and cost behaviors, onecan describe possible scenarios of network growth.
4. Methodology
The main concept behind Barabasi-Albert model is preferential attachmentwhich states that nodes with more links are more exposed to be joined by newlinks [3, 4]. Initially, there exist N nodes with the initial Node Activity of k .For creation of an evolving process, at each time step m nodes among N nodesare selected based on a uniform random distribution. Then, each of the selectednodes attaches to its destination node with the probability proportional to itsnode activity [6]. A certain node with higher node activity, is more probable toattract a new attachment. We will have: k i ( t ) = t (cid:88) t =0 L i ( t ); (1)where L i ( t ) means the number of links added to the node i due to adding newnodes during t to t + dt time interval. Considering the initial number of links inthe network as m , after t time-step, we will have m mt links. Hence, duringan ‘intermediary process’ (adding links between constant number of previouslyexisted nodes)[12], the evolving rate of node activity is as follows [6]: dk i ( t ) dt = m + mk i ( t ) (cid:80) Nj (cid:54) = i k j ( t ) . (2)Since each link possesses two ends, the attachment of one end is selected bya random uniform distribution. However, the attachment of the other end iscreated by preferential attachment based on higher activity [6], Eq. 1. In this6egard, at each time step, higher node activity of a certain node causes higherrate of attachment to it by new comers.Consequently, there would be some competitions among nodes for attractingmore links. From relations above, it is obvious that during these competitions,nodes with higher activity in the network, have probably more chance to createlinks rather than nodes with less activity.By applying an analytical solution to above discussions, following relation isprovided for value behavior: k v,i ( t ) = 2 mt + c √ t (3)Where c is constant and depends on the initial conditions as below: c = k − mt √ t (4)If t → ∞ in Eq.3, the behavior of the network is more dominant by 2 mt . How-ever, for t →
0, the effects of c √ t will be more significant. t → ∞ : k v ∼ tt → k v ∼ √ t (5)There is a cross-over time, t ∗ , which is simultaneous with the transformation of value behavior of the network [6]. This cross-over time step is a separatingpoint between two behavioral regimes in Eq.3 and it is calculated by Eq.6: t ∗ = c m . (6)The behavioral transformation of value indicates that by increasing m , whilethe network crosses its cross over, t ∗ , the growth of the network increases with agreater acceleration [6, 11]. Since, total number of links grows nonlinearly [12]faster rather than linearly passing time steps, this phenomenon is “cumulativegrowth”. 7oteworthy, each agent has its own value and cost equations (Eq.7 andEq.8).Also, the fate of network will yield to the situations below: k v ( t ) − k c ( t ) > Possitive Growth (before characteristic time) k v ( t ) − k c ( t ) ≤ No Growth (after characteristic time), (7)where k v ( t ) and k c ( t ) refer to value and cost of making links, respectively. Forthe sake of negative growth, some researchers applied preferential damage[11].In this study we applied the cost of further growth.From now on, we consider that links not only grow due to value , but alsothey are exposed to costs of attachments which also change by time. For now,we consider a model which indicates changes of degrees in the existence of cost ,and, α is phase space. We will have [6]: k ν ( t ) = 2 mt + c √ tk c ( t ) = ( α + m ) t (8) Characteristic time is a temporal moment which cost curve intersects value curve. Consequently, at this moment value and cost are equal, k v = k c . Hence,in the intersection we have: 2 mt + c √ t = ( α + m ) tt characteristic = ( cα − m ) . (9)
5. Results
Fig.1 presents cross-over time, t ∗ , corresponding to different k and m pa-rameters. As illustrated, when initial degree, k , is smaller, the rise of links m ineach time step will shorten the cross-over time, t ∗ [6]. On the other hand, largerinitial degree in the network needs more rate of link creation, m , to shortencross-over time, t ∗ . This finding is crucial. This implies that the initial degree, k , acts like inertia against the cumulative growth and accordingly it takes moretime-steps for the network to pass the cross-over time, t ∗ .8 igure 1: Cross-over time, t ∗ , for different initial degrees of the network, ‘ k ’, versus thenumber of links, ‘ m ’, which is created at each time step is demonstrated. Accordingly, thevariations of k and m for t ∗ = 600, 1200 and 2400 is depicted.Figure 2: Left) Contour plot for ln(Characteristic Time) considering the changes of mk versusthe changes αk . Right) Contour plot for ln(Characteristic Time) considering the changes of k m versus the changes αm . Again, as seen in the Fig.1, the rise of m causes the cumulative growth of thenetwork occurs sooner. This implies that lowering the cross-over time will causethe network to escape from initial failure (this fact will be further shown byFig.3, too).In Fig.2, characteristic time and its contours are illustrated for different ra-tios of m/k vs. α/k (left panel), and k /m vs. α/m (right panel). Since noexistence of characteristic time is the best scenario for the network’s growth,it is vital for the network’s topology to encounter the links creation factor m equal to α . When the initial degree k → m/k → ∞ and α/k → ∞ ), theleft and right panel together clarify the situation. As shown in the right panelof Fig.2, when k → k /m →
0) the characteristic time will interestinglybe more sensitive to the changes of ratio of α/m (around αm = 1) rather than9arger k /m . Hence, when the initial degree k is extremely low, the trade-offbetween α and m is more crucial. On the other hand, when the initial degree k is extremely high, the aforementioned trade-off among α and m is less sharper.It leads us to the consciousness that the higher initial degree in the network’sbirth, lower the sensitivity of the network to future α parameter. As a typicalrule for the trade-off among α and m , the higher difference among them lowerthe characteristic time. Hence, the life of network will be shortened, as Fig.2shows. Conversely, in the case of α →
0, again, the t Characteristic → ∞ .As illustrated in Fig.2, due to activity functions of nodes for different m , thehigher m , the shorter cross-over time, t ∗ [6]. This implies that by increasingthe ability of whole nodes to find each other, the evolution of whole networkproceeds faster. Hence, network will reach the cross-over time sooner.However, more efficient communication and advertising tools will cause fasterprocesses in market research , awareness phase and popularity cycles of goodsand services in a market. Accordingly, businesses may experience faster growthphase , maturity phase , and possibly faster occurrence of death phase . Whenit comes to the real-world networks, faster information translation and linkscreation will contribute to higher frequency in business cycles. In such entangledmarkets, economic firms should lower their inertia internally and externallyand permanently plan to develop new features on their goods and services.This will lower their network’s cost toward growth and postpone occurrenceof characteristic time in the state of trade-off between growth of value and cost . According to issues raised, one may consider 3 scenarios for the network’sgrowth. Plus, the phase space relating to each aforementioned scenarios ispresented as demonstrated in Fig.3. Scenario of Failure ‘a’ . In the present scenario, the cost of making new linksamong temporal nodes is always more than value of that. Hence, characteristictime tends to zero. This phenomenon occurs for α >> m , as panel a.2 in Fig.3proves. 10pon this scenario, in the investment world the cost and expected return ofselecting a trading position, is higher than future value . Scenario of Ever-growing Conquer ‘b’ . As a matter of fact in this situ-ation, panel b.1 and b.2 in Fig.3, the network has successfully fulfilled all thecircumstances. In other words, the value behavior has accelerated enough andduring ‘all’ circumstances, cost is less than the value which is earned by mak-ing links. Accordingly, constituents totally are satisfied! Hence, the network‘continuously’ grows. In two areas along αm , the network has the chance of“Ever-growing Conquer”. These areas conclude αm →
1, as panel b of Fig.3 andEq.9 proves.In the stock markets this situation occurs when the future value of the stock inthe investors’ point of view will be continuously better. In this state, in-spiteof probable rise in cost (like fear, liquidity problems, bad news), the collectiveeffects of opinions toward value of the market as a whole overcome the cost . Inthe marketing, this scenario occurs when the firm has been launched properly,and has then passed the value behavior’s cross-over and the firm’s competitiveadvantage is stable during the investigated scale. Scenario of Trade-off . The trade-off scenario can be explained by two eventswhich both occur between failure and ever-growing. Accordingly, by equalizingthe characteristic time and the cross-over time , t ∗ , we have two answers: α = 3 mα = − m , (10)where second answer is ineligible. Hence, two states emerge:- Based on Fig.3, panel c , the configuration of network is formed. Yet, beforeacceleration of value behavior in the network, cost outpaces the value of makinglinks. Hence, before that the network has the chance of changing the value behavior, its growth stops.The phase space of this state is obtained as following: t characteristic < t ∗ → α > m. (11)11n the world of financial markets, this scenario occurs during speculation, orin the case of sudden (relating to investment horizon) bad news after trading,or trading on minor trends. Noteworthy, in marketing networks, it can implythat a platform has been properly launched but the environment of the industrychanges before the firm grows enough.- Again, based on Fig.3, panel d , another state of the trade-off can be de-scribed as follows. The network may pass value behavior’s cross-over and con-sequently, it is able to rise acceleration of the value behavior. Yet, after passingthe value behavior’s cross-over, the cost of making newcomer temporal linksboosts insofar as the cost outpaces the value . Eventually, the network’s growthstops. Noteworthy, this phenomenon may not occur soon enough to avoid thestate of ever-growing. It actually depends on the targeted period.To obtain phase space in this state we have: t characteristic > t ∗ → α < m. (12)In the stock market, this is the situation that the investors do not prefer tostay in the trading position. In the marketing, the business cycle is started todownturn and will be no longer a boosting industry as it was in the past andnewer competitors may refuse to enter the industry.It is notable that the scaling features are significant in the trade-off scenario.Needless to say, to some scales, the network’s growth can be a cumulative pos-itive amplifier for its growth, and on the other scales it may be as a barrier togrowth anymore.To avoid the network to be vulnerable to “failure”, it can be kept safely in therange of 0 ≤ αm ≤ cost ) againstgrowth of the network earned value leads the whole system to smaller sub-networks and more wanderer agents. Interestingly, the emergence of wandereragents, their memory, and the ability of sub-networks to absorb them, willdetermine the fate of the whole. For the sake of forecasting possible scenarios ofsuch systems, it is crucial to investigate the cost of growth of the sub-networks.12 igure 3: a) Scenario of Failure, b) Scenario of Ever-growing Conquer, c) Scenario of Trade-offin the 1st State, d) Scenario of Trade-off in the 2nd state.
6. Conclusion
We found that some networks may fail at the beginning phase since their cost overcomes their value from the early beginning. Some are successful atthe beginning phase, however, they stop growing before reaching the positiveacceleration of value behavior (cross-over phenomenon). The trade-off scenariohappens when networks successfully pass their beginning phase. However, theyare exposed to failure after the emergence of characteristic time , because theirnetwork’s cost of growth overcomes its value simultaneous to the characteristictime.Individual nodes’ or sub-networks’ interests may be latent and be against col-lective motivations [17] of sub-networks and communities. This phenomenon isunderstood -for example- as internal cost of a certain sub-network, like agencytheory in corporate finance. Another criterion of cost can be structural cost ofaged links corresponding to previous customers and after sales services. Also tosome extent, when a network grows, covering highly distant customers amplifiescustomer-company costs, which one or both of them need to cover it.When it comes to financial markets, growth can be interpreted as investors mo-13ivation to gain future returns. This may contribute to a collective interest inthe market to lead other stocks. This collective behavior illustrates leader ofthe market [30–32] which determines general market trends. Then, the fear ofpossible turning point in the market trend, causes investors to be careful aboutthe stability of trend. These are examples of what we call the cost of growth ofthe strongest eigenvector [7]. Hence, a characteristic time (like turnover time)for stopping current trend comes to mind.Our proposed model implies that coexistence of cumulative value in preferentialattachment versus cost of attachments may cause the network to be not ever-growing, and under some circumstances growth of the network stops. Hence,depending on the quality of changes, a trade-off scenario is possible. In somescales the network’s growth accelerates itself and in others, it may cause somebarrier against itself to grow more.
Growth of the network (as an Amplifier) . The cumulative value whichis earned from increase in number of links, can help in developing the network.Development of the network rises earned value of constituents and contributesto attracting newcomers as Barabasi-Albert preferential attachment proves.The increase in the number of links around certain nodes may cause theemergence of monopoly. On the other hand, the network may be more vulnera-ble to the cost of any further growth. In this atmosphere, the competitors maybe more focused on the network. Hence, the attraction of competitors can beconsidered as the cost of attachments to the network. In long term periods, theattraction of proposed network decreases. As a result, probable links of newertime-steps may diverge from preferential attachment and cost effects becomemore significant.Aforementioned issues due to certain businesses can be considered in equa-tions. 14 eferencesReferences [1] M. E. J. 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