NNetwork Strategies in Election Campaigns
Marco Alberto Javarone
Dept. of Mathematics and Computer Science, University of Cagliari, 09123 Cagliari,ItalyDUMAS - Dept. of Human and Social Sciences, University of Sassari, 07100 Sassari,Italy
Abstract.
This study considers a simple variation of the voter model with twocompeting parties. In particular, we represent the case of political elections, wherepeople can choose to support one of the two candidates or to remain neutral. Peopleoperate within a social network and their opinions depend on those of the peoplewith whom they interact. Therefore, they may change their opinions over time, whichmay mean supporting one particular candidate or none. Candidates attempt to gainpeople’s support by interacting with them, whether they are in the same social circle(i.e. neighbors) or not. In particular, candidates follow a strategy of interacting for atime with people they do not know (that is, people who are not their neighbors). Ouranalysis of the proposed model sought to establish which network strategies are themost effective for candidates to gain popular support. We found that the most suitablestrategy depends on the topology of the social network. Finally, we investigated therole of charisma in these dynamics. Charisma is relevant in several social contexts,since charismatic people usually exercise a strong influence over others. Our resultsshowed that candidates’ charisma is an important contributory factor to a successfulnetwork strategy in election campaigns.PACS numbers: 89.75.-k, 89.65.-s, 89.75.Fb a r X i v : . [ phy s i c s . s o c - ph ] M a y etwork Strategies in Election Campaigns
1. Introduction
In the last years, opinion dynamics [1] has attracted the attention of many scientistsand several models, to study the formation and the spreading of opinions, have beendeveloped (e.g., [2, 3, 4, 5, 6, 7]). In these dynamics, interactions among individualsand the topology of their network play a fundamental role [1, 8, 9]. One of the mostsimple models of opinion dynamics is the voter model [10, 3, 11, 12]. The latter describesa set of agents that change opinion over time by interacting among themselves. Thevoter model allows to represent the evolution of a population toward consensus in thepresence of different opinions. In general, from a physical perspective, by this model itis possible to identify phase transitions in the system, as from a disordered state to anordered one [13, 14]; although as shown in [15] also non-linear dynamics, that entailsthe system reaches a final phase characterized by the coexistence of different opinions,can be introduced. Moreover, the voter model can be implemented in several ways,with the aim to catch a particular character or behavior of real systems, as politicalelections [16, 17, 18] and, more in general, competitions [19, 20, 21]. In this work, weintroduce a variant of the classical voter model, putting our attention on the case ofpolitical elections, in order to study the best strategies to gain the popular consensus.In the proposed model, there are two competitors (or candidates) that try to convincea community of agents. In turn, agents are neutral or they have a preference for onecompetitor. Therefore, we consider a system with three possible opinions [22]. Agentsare arranged in a network and they change opinion over time, by considering those oftheir neighbors. During the evolution of the system, competitors try to affect the opinionof agents by defining temporal connections with them. In particular, agents temporarilyconnected with competitors consider them as normal neighbors while compute their nextopinion. Therefore, each competitor considers very important to identify best agentsfor generating these temporal connections. In this context, best agents are those thatallow to increase the candidate’s consensus as fast as possible in the whole population.It is worth to note that the described dynamics is based on the structure of underlyingadaptive networks [23, 24, 25]. We perform a comparison among different networkstrategies, used to perform the selection of agents. Results of numerical simulations showa relation between the best strategy and the topology of the agent network. Moreover,we investigate whether the definition of network strategies should consider also thecompetitors’ charisma, as this quality is considered fundamental in social contexts. Theremainder of the paper is organized as follows: Section 2 introduces the proposed model,for studying network strategies to gain the popular consensus. Section 3 shows resultsof numerical simulations. Finally, Section 4 ends the paper.
2. The Model
We introduce a simple variant of a voter model with two competitors, e.g., twocandidates during an electoral campaign. Competitors aim to gain the popular consensus etwork Strategies in Election Campaigns t = 0, all agents are in a neutral state (i.e., 0), with theexception of the two competitors that are in the state 1 and 2, respectively. Then,at each time step, agents change their state (i.e., opinion) according to the followingtransition probabilities: p x → y = σ y p x = 1 − (cid:88) i =0 | i (cid:54) = x σ i (1)with p x → y transition probability to change from the x th state to the y th state and p x probability to remain in the same state. The value of σ y is computed as σ y = n y /n t ,with n y number of neighbors in the state y th and n t total number of neighbors (i.e., thedegree of the agent). Eventually, the summation to compute p x considers the densities σ i of neighbors having all the feasible states different from the x th state. In so doing,at each time step, the agents’ states are defined by using a weighted random selectionwith the transition probabilities (Eq. 1) used as weights. Therefore, the evolution of thesystem is described by the following equations: N ( t + 1) = N · N (cid:88) i =1 | o i (cid:54) =0 σ i ( t ) − N (cid:88) i =1 | o i =0 σ i ( t ) + σ i ( t ) + N ( t ) N ( t + 1) = N · N (cid:88) i =1 | o i (cid:54) =1 σ i ( t ) − N (cid:88) i =1 | o i =1 σ i ( t ) + σ i ( t ) + N ( t ) N ( t + 1) = N · N (cid:88) i =1 | o i (cid:54) =2 σ i ( t ) − N (cid:88) i =1 | o i =2 σ i ( t ) + σ i ( t ) + N ( t ) (2)with N x number of agents in the x th state and (cid:80) i =1 | o i = x that indicates the i th agenthaving a state o i equal (or different) to x . Competitors do not change their state andthey try to gain the consensus of the population. In particular, they generate temporalconnections with agents that are not their neighbors, with the aim to affect the valueof their transition probabilities. These temporal connections last only for one time stepand each competitor generates, every time, a number of temporal connections equal toits degree (i.e., the number of its neighbors). Therefore, agents temporarily connectedwith a competitor compute their transition probabilities as follows: p x → y = σ ty (3) etwork Strategies in Election Campaigns p x = 1 − p x → y (4)with σ ty temporal density of neighbors in the y th state (i.e., the state of the competitorthat contacted the agent), computed as: σ ty = n y + 1 n t + 1 (5)In so doing, the equations to describe the evolution of the system become: N ( t + 1) T = N ( t + 1) + k · k (cid:88) j =1 σ tA [ j ]1 ( t ) − k · k (cid:88) j =1 | o j =1 σ tA [ j ]2 ( t ) N ( t + 1) T = N ( t + 1) + k · k (cid:88) j =1 σ tA [ j ]2 ( t ) − k · k (cid:88) j =1 | o j =2 σ tA [ j ]1 ( t ) N ( t + 1) T = N − N ( t + 1) T − N ( t + 1) T (6)with N x ( t +1) T number of agents in the x th state, considering the temporal connections,and k , k degree of the competitor 1 and 2, respectively. The exponent of σ tx in Eq. 6,i.e., A x [ j ], represents the j th agent among those selected by the x th competitor, forgenerating temporal connections at time t . During the electoral campaign, at each timestep, competitors have to select the most useful agents to generate temporal connections.In order to perform this selection, competitors can use one of the following networkstrategies: • S0 . Random selection; • S1 . Random weighted selections, using the degree of agents as weights; • S2 . 2nd degree connections: agents at distance 2 (i.e., neighbors of their neighbors); • S3 . 3rd degree connections: agents at distance 3.Figure 1 shows an example where two competitors generate a temporal connection byusing the strategy S2 and the strategy S3 , respectively. Strategies S0 and S1 canbe defined as “global strategies”, as competitors consider the whole network to selectagents. Moreover, by using the strategy S1 , agents with high degree have a higherprobability to be selected. On the other hand, strategies S2 and S3 can be defined as“local strategies”, as competitors select agents by considering only the small portionof the network around them (i.e., friends of friends, and so on). In order to evaluatewhether a best network strategy can be identified, among those listed above, we analyzethe proposed model by using scale-free networks and small-world networks to connectthe agents.
3. Results
We performed many numerical simulations of the proposed model with the aim toidentify the best network strategy to gain the popular consensus. Agents have beenarranged in scale-free networks, generated by the Barabasi-Albert model (BA model etwork Strategies in Election Campaigns Figure 1.
Two competitors (i.e., red and green nodes) generate a temporal edge,indicated by a dotted line, following a strategy: the red node uses the strategy S2 (i.e.,it selects 2nd degree connections), whereas the green node uses the strategy S3 (i.e.,it selects 3rd degree connections). hereinafter) [27], and in small-world networks, generated by the Watts-Strogatz model(WS hereinafter) [28]. In particular, to achieve small-world networks, we start from a 2-dimensional regular lattice with 6 neighbors per node, then we rewire with probability β = 0 . N = 10 , provided with an average degree (cid:104) k (cid:105) = 6. Moreover, we performed further simulations in scale-free networks with N > , to observe the effects caused by the presence of a greater number of hubs–see Appendix. We recall that scale-free networks, generated by the BA model, havea degree distribution P ( k ) characterized by a scaling parameter γ ≈
3. In order tocompare network strategies, we consider the number of agents that have a preferencefor each competitor and the number of neutral agents. In particular, we analyze thevariation of the density ρ of agents, in these three states, over time. In Figures 2and 3, we report the comparison among four different simulations, performed on scale-free networks and small-world networks, respectively. A first information, achievedanalyzing the curves ( ρ, t ), representing agents in different states, is that the numberof neutral agents falls to zero after about 5 . · time steps in scale-free networks andafter about 1 . · time steps in small-world networks. After that, in both kinds ofnetwork, the system seems to reach a steady-state, characterized by small fluctuations ofdensities between the two states 1 and 2. Moreover, in the curves ( ρ, t ), we identify twoimportant points, called T and T . These points constitute the intersections betweenthe density of neutral agents and those of agents in the other states. The point T is theintersection between neutral agents and agents with the preference for the competitor1, whereas T is the intersection between neutral agents and agents with the preferencefor the other competitor (i.e., the 2). As discussed below, points T and T are usefulto compare the network strategies. etwork Strategies in Election Campaigns Figure 2.
Comparison among the density of agents in the state 0 (labeled as N in the legend, i.e., neutral agents), in the state 1 (labeled as C in the legend, i.e.,Competitor 1) and in the state 2 (labeled as C in the legend) over time, performed onscale-free networks. Results are averaged over 20 different realizations. a ) Competitor1 uses the strategy S0 and competitor 2 uses the strategy S1 . b ) Competitor 1 usesthe strategy S1 and competitor 2 uses the strategy S2 . c ) Competitor 1 uses thestrategy S1 and competitor 2 uses the strategy S3 . d ) Competitor 1 uses the strategy S2 and competitor 2 uses the strategy S3 . A useful parameter, to compare network strategies, is the difference of densities ∆ ρ between agents in the state 1 and agents in the state 2, over time –see Figure 4. Thetopology of the agents network seems to play a crucial role, as we observe by comparingresults shown in panels c and d of Figure 4, related to scale-free networks and small-world networks, respectively. In particular, the strategy S2 is better than the strategy S3 in scale-free networks, but just the opposite occurs in small-world networks (i.e., thestrategy S3 is the best one). Therefore, we computed the average value of δρ , comparingall strategies in both kinds of networks –see panel a of Figure 5. As discussed before,the points T and T of diagrams ( ρ, t ) can provide an information about the speed ofcompetitors in the earning of the global consensus. In particular, as shown in panel b etwork Strategies in Election Campaigns Figure 3.
Comparison among the density of agents in the state 0 (labeled as N in thelegend, i.e., neutral agents), in the state 1 (labeled as C in the legend, i.e., Competitor1) and in the state 2 (labeled as C in the legend) over time, performed on small-worldnetworks. Results are averaged over 20 different realizations. a ) Competitor 1 usesthe strategy S0 and competitor 2 uses the strategy S1 . b ) Competitor 1 uses thestrategy S1 and competitor 2 uses the strategy S2 . c ) Competitor 1 uses the strategy S1 and competitor 2 uses the strategy S3 . d ) Competitor 1 uses the strategy S2 andcompetitor 2 uses the strategy S3 . of Figure 5, we computed the difference T − T for each curve ( ρ, t ). Values of avg (∆ ρ )highlight that, in scale-free networks, local strategies are better than global ones. Inparticular, the best strategy is S2 . Instead, considering the global strategies, the S1 ismuch more better than S0 . On the other hand, in small world networks, we found thatthe best strategy is S3 , followed by the strategy S1 . Therefore, also in this case a localstrategy is more efficient than global ones. It is interesting to note that the strategy S2 yields optimal results in scale-free networks, but it is the worst strategy (amongthose analyzed) in small-world networks. Eventually, comparing the global strategies,we found that S1 is always better than S0 , in particular in scale-free networks, due tothe presence of hubs (i.e., nodes with a high degree). A further information is providedby the histogram ( T − T ) in panel b of Figure 5. In particular, we can evaluate whichare the faster strategies to gain the popular consensus. As discussed before, after that etwork Strategies in Election Campaigns Figure 4.
Difference between densities of agents ∆ ρ in the state 1 and 2, varyingthe strategies adopted by the two competitors. Results are averaged over 20 differentrealizations. a ) Results achieved in scale-free networks, when the competitors usestrategies S3 and S1 , respectively. b ) Results achieved in small-world networks, whenthe competitors use the strategies S3 and S1 , respectively. c ) Results achieved inscale-free networks, when the competitors use the strategies S3 and S2 , respectively. d ) Results achieved in small-world networks, when the competitors use the strategies S3 and S2 , respectively. the number of neutral agents falls to zero, the system reaches almost a steady-state,with small differences between the density of agents in states 1 and 2. Therefore, as thetime is an important variable in competitions as political elections [29], a good strategyallows also to obtain the consensus in a few time steps. As result of this analysis, wefound that best strategies, identified in the histogram avg (∆ ρ ), are also faster thanthe other ones. Furthermore, it is worth to highlight that, although S2 is weaker thanglobal strategies in small-world networks (according to values of avg (∆ ρ )), it yields afast increasing of global consensus. Hence, in the event the time variable is critical (i.e.,competitors have a few time to gain the consensus), local strategies are better thanglobal ones. etwork Strategies in Election Campaigns Figure 5.
Comparison between results achieved in scale-free networks (blue bars)and those achieved in small-world networks (red bars), varying the network strategy.Results are averaged over 20 different realizations. a ) avg (∆ ρ ), i.e., average differencebetween density of agents in the two states, 1 and 2. b ) Difference between points T and T , indicated in the inset (that shows an enlargement of a diagram ( ρ, t )). According to recent studies [30, 31], the politicians’ charisma plays an important rolein the achievement of the popular consensus. In general, the charisma is a qualitydeemed relevant in several social contexts as charismatic people are able to exercisea strong influence on other people. Therefore, here we investigate the proposedmodel considering charismatic competitors. In particular, we modify the transitionprobabilities of temporarily connected agents as follows: p x → y = x = 012 if x (cid:54) = 0 (7)whereas, p x , i.e., the probability that the temporarily connected agents remain in thesame state, is always computed by Eq. 4. In so doing, a charismatic competitor gainsalways the consensus of neutral agents, whereas it has the 50% of probabilities to gainthe consensus of agents that prefer its opponent. Figure 6 shows results achieved in bothkinds of network (i.e., scale-free and small-world) varying the network strategies playedby competitors. It is interesting to note that the presence of charismatic competitorsstrongly affects results. In particular, considering the histogram (∆ ρ ) (panel a ofFigure 6), global strategies are better than local ones in both kinds of network. Inscale-free networks the best strategy is S1 , whereas in small-world networks the bestone seems to be S0 . Notwithstanding, observing the histogram ( T − T ), we can see thatin scale-free networks there are small temporal differences between strategies. Therefore,from this point of view, all strategies are similar. Instead, in small-world networks wefound that best strategies are also the fastest ones. Finally, even if we consider thepresence of charismatic competitors, the topology of networks still affects results. etwork Strategies in Election Campaigns Figure 6.
Comparison between results achieved in scale-free networks (blue bars)and those achieved in small-world networks (red bars), varying the network strategyand considering charismatic competitors. Results are averaged over 20 differentrealizations. a ) avg (∆ ρ ), i.e., average difference between density of agents in thetwo states, 1 and 2. b ) Difference between points T and T .
4. Discussion and Conclusions
In this work, we analyze network strategies to gain the popular consensus in the presenceof two competitors. We define a simple variation of the voter model with agents thatchange opinion according to transition probabilities, computed considering the opinionsof their neighbors. Moreover, we let competitors interact temporarily also with agentsthat are not their neighbors, with the aim to affect their opinion. Therefore, as observedbefore, the proposed model is based on an adaptive network. In particular, at each timestep, competitors select a number of agents, equal to their degree, to generate temporalconnections. This selection is performed by using a network strategy. Competitors canchoose between global strategies, i.e., random selection and weighted random selection(to select agents with a high degree), and local strategies, i.e., their 2nd connectionsdegree and 3rd connections degree. Simulations have been performed by arrangingagents in scale-free networks and in small-world networks. Results highlight that thetopology of networks strongly affects the outcomes of the model. In particular, in scale-free networks the best strategy to select agents is S2 , i.e., 2nd connections degree.On the other hand, in small-world networks is more efficient the strategy S3 , i.e., 3rdconnections degree. In general, we found that local strategies are more advantageousthan global ones in both kinds of network. In particular, although the strategy S2 seemsunfavourable in small-world networks, it is faster than both global strategies. We recallthat the term “fast”, in this context, is used to identify strategies that allow to increasethe global consensus in a few time steps. Furthermore, we performed simulationsconsidering “charismatic” competitors. We model the charisma of competitors byusing their probability to convince temporarily connected agents. In particular, thisprobability is 1 in the event they interact with neutral agents, whereas it is equal to etwork Strategies in Election Campaigns . ρ and T − T , show that S0 and S1 are better thanlocal strategies, and moreover, they yield similar results. On the other hand, in scale-free networks, global strategies are still better than local ones but, from a temporalperspective, there are small differences, i.e., all strategies allow to convince many agentsin a similar number of time steps. In order to conclude, results highlight that both thetopology of the agent network and the charisma of competitors should be considered toplan a successful strategy during electoral campaigns. Acknowledgments
The author wishes to thank Ginestra Bianconi for her helpful comments and suggestions.This work was supported by Fondazione Banco di Sardegna.
Appendix
In this section, we report results of the proposed model achieved by using scale-freenetworks with N = 5 · agents. In so doing, we can perform a further evaluation onthe effects of the hubs (i.e., nodes with a high degree) in these dynamics. As indicatedin Figure 7, on a quality level, results are similar to those achieved in smaller scale-freenetworks –see Figures 2 and 4. We observe that increasing N (i.e., the number of agents),the number of time steps to reduce neutral agents to zero increases. In particular, with N = 5 · , the density of neutral agents falls to zero after about 1 . · time steps,while with N = 10 it takes about 5 . · time steps. In Figure 8, we show resultsrelated to parameters avg (∆ ρ ) and T − T . Also in these diagrams, we found resultssimilar to those achieved in scale-free networks with N = 10 . Therefore, we can statethat also in the presence of more hubs in the agent network (considering the scale-freeconfiguration), on a quality level, the outcomes of the proposed model are similar tothose achieved in the main analysis. References [1] Castellano, C. and Fortunato, S. and Loreto, V., Statistical physics of social dynamics. Rev. Mod.Phys.
81 - 2 , (2009) 591–646[2] Martins, A.C.R., and Galam, S., Building up of individual inflexibility in opinion dynamics. Phys.Rev. E , (2013) 042807[3] Galam, S., SOCIOPHYSICS: a review of Galam models. International Journal of Modern PhysicsC , (2008) 409–440[4] Krapivsky, P. L. and Redner, S., Dynamics of Majority Rule in Two-State Interacting SpinSystems. Phys. Rev. Lett. , (2003) 238701[5] Sznajd-Weron, K. and Sznajd, J., Dynamics of Majority Rule in Two-State Interacting SpinSystems. International Journal of Modern Physics C , (2000) 1157[6] Holme, P. and Newman, M. E. J., Nonequilibrium phase transition in the coevolution of networksand opinions. Phys. Rev. E , (2006) 056108 etwork Strategies in Election Campaigns Figure 7.
On the top: comparison among the density of agents in the state 0 (labeledas N in the legend, i.e., neutral agents), in the state 1 (labeled as C in the legend, i.e.,Competitor 1) and in the state 2 (labeled as C in the legend) over time, performed onscale-free networks. a ) Competitor 1 uses the strategy S1 and competitor 2 uses thestrategy S3 . b ) Competitor 1 uses the strategy S2 and competitor 2 uses the strategy S3 . On the bottom: difference between densities of agents ∆ ρ in the state 1 and 2 inscale-free networks. c ) Results achieved when the competitors use the strategies S3 and S1 , respectively. d ) Results achievedwhen the competitors use the strategies S3 and S1 , respectively. Results are averaged over 20 different realizations.[7] Halu, A. and Zhao, K. and Baronchelli, A. and Bianconi, G., Connect and win: The role of socialnetworks in political elections. Europhysics Letters
102 - 1 , (2013) 16002[8] San Miguel, M. and Eguiluz, V.M. and Toral, R., Binary and Multivariate Stochastic Models ofConsensus Formation. Computing in Science and Engineering , (2005) 67 – 73[9] Kozma, Balazs and Barrat, Alain, Consensus formation on adaptive networks. Phys. Rev. E ,(2008) 016102[10] Sood, V. and Redner, S., Voter Model on Heterogeneous Graphs. Phys. Rev. Lett.
94 - 17 , (2005)178701[11] Galam, S.: From 2000 BushGore to 2006 Italian elections: voting at fifty-fifty and the contrarianeffect.
Quality & Quantity etwork Strategies in Election Campaigns Figure 8.
Results achieved in scale-free networks with N = 5 · agents, varyingthe network strategy. On the left, avg (∆ ρ ), i.e., average difference between densitiesof agents in the two states, 1 and 2. On the right, difference between points T and T . Results are averaged over 20 different realizations.two-state spin system. Phys. Rev. E , (2003) 046106[14] Castellano, C., Marsili, M., and Vespignani, A., Nonequilibrium Phase Transition in a Model forSocial Influence. Phys. Rev. Lett. , (2003) 3536–3539[15] Schweitzer, F., and Behera, L., Nonlinear voter models: the transition from invasion to coexistence.The European Physical Journal B , (2009) 301–318[16] Zschaler, Gerd and B¨ohme, Gesa A. and Seißinger, Michael and Huepe, Cristi´an and Gross, T.,Early fragmentation in the adaptive voter model on directed networks. Phys. Rev. E ,(2012) 046107[17] Fernandez-Gracia, J. and Suchecki, K. and Ramasco, J.J. and San Miguel, M. and Eguiluz, V.M.,Is the Voter Model a model for voters?. http://arxiv.org/abs/1309.1131, (2013)[18] Mobilia, M., Petersen, A., and Redner, S., On the role of zealotry in the voter model. J. Stat.Mech. (2007) P08029[19] Galam, S., Chopard, B., Masselot, A., and Droz,M., Competing species dynamics: Qualitativeadvantage versus geography. The European Physical Journal B , (1998) 529–531[20] Clifford, P. and Sudbury, A., A model for spatial conflict. Biometrika , (1973) 581–588[21] Castello, X. and Eguiluz, V.M. and San Miguel, M., Ordering dynamics with two non-excludingoptions: bilingualism in language competition. New Journal of Physics , (2006)[22] Marvel, Seth A. and Hong, Hyunsuk and Papush, Anna and Strogatz, Steven H., EncouragingModeration: Clues from a Simple Model of Ideological Conflict. Phys. Rev. Lett. , (2012)118702[23] Gross, T. and Hiroki, S., Adaptive Networks: Theory, Models and Applications. Springer BerlinHeidelberg, (2009)[24] Gross, T. and Blausius, B., Adaptive coevolutionary networks: a review. Journal of the RoyalSociety Interface , (2008) 259–271[25] Zschaler, G., Adaptive-network models of collective dynamics. The European Physical JournalSpecial Topics , (2012) 1–101[26] Szab´o, Gy¨orgy and Szolnoki, Attila, Three-state cyclic voter model extended with Potts energy.Phys. Rev. E , (2002) 036115[27] Barabasi, A.L. and Albert, R., Emergence of scaling in random networks. Science
286 - 5439 ,(1999) 509–512[28] Watts, D. J. and Strogatz, S. H., Collective dynamics of “small-world” networks. Nature ,(1998) 440–442 etwork Strategies in Election Campaigns [29] Hillygus, D.S., Encouraging Moderation: Clues from a Simple Model of Ideological Conflict. PublicOpinion Quarterly , (2011) 962–981[30] Merolla, J.L. and Zechmeister, E.J., The Nature, Determinants, and Consequences of Chvez?sCharisma: Evidence From a Study of Venezuelan Public Opinion. Comparative Political Studies , (2011) 28–54[31] Muftuler-Bac, M. and Keyman, E.F., The Era of Dominant-Party Politics. Journal of Democracy23