Felipe D Cunha

Is it Possible to Find Social Properties in Ve h i c u l a r N e t wo r k s ?

Everyday, vehicles transit in a city and along their trajectories, they encounter other vehicles. The frequencyo f these encounters is influenced by many factors, such as: vehicle speed, destinations, traffic conditions, and the period of the day. However, these factors are justified by the public roads limits and the driver's behavior. The people present daily routinesa n d similar behaviors that have a great impact in the daily traffic evolution. In this work, we present a numerical analysis of real and realistic data sets that describe the mobility of a set of vehicles. Social metrics are computed, and the results obtained are compared to random graphs in the direction to verify if vehicular network presents a social behavior. Finally, we discuss new social perspectives in vehicular networks.

I. INTRODUCTION

Ve h i c u l a r A d H o c N e t wo r k s ( VA N E T s ) a r e a s p e c i a l t y p e of Ad Hoc networks, formed by vehicles with processing and wireless communication ability, traveling on streets or highways. Commonly, the vehicles can communicate directly or by the use of a roadside unit (RSU). Through this infrastructure, vehicles can access network services and obtain data from other networks, such as the Internet. Due to this nature, VA N E T s c a n b e e s t a b l i s h e d i n d i f f e r e n t e n v i r o n m e n t s s u c h a s in urban centers and highways [START_REF] Boukerche | Vehicular ad hoc networks: A new challenge for localizationbased systems[END_REF], [START_REF] Faezipour | Progress and challenges in intelligent vehicle area networks[END_REF].

VA N E T s a r e p r o p o s e d t o a s s i s t t h e d r i v e r s i n t h e t r a f fi c , t o avoid collisions with sending alerts, to provide entertainment to passengers, and to contribute to create an intelligent traffic system. An important application of this network is to alert the drivers about road conditions and imminent crashes. Recent studies show that 60% of car crashes could be avoided if the drivers were warned just few seconds before the collision [START_REF] Faezipour | Progress and challenges in intelligent vehicle area networks[END_REF]. In this context, the interaction among vehicles can strongly contribute to broadcast an alert message reducing the crash ratio [START_REF] Schoch | Communication patterns in vanets[END_REF]. Another example of application of VANETs is the assistance to drivers to find the best route. In this scenario, it is important that VANETs monitor and predict the traffic conditions on streets, to avoid traffic jams decreasing the trip time [START_REF] Tostes | From data to knowledge: City-wide traffic flows analysis and prediction using bing maps[END_REF].

The communication in VANETs is highly influenced by the vehicles density and mobility patterns. Vehicles move and stop according to the speed limits, and directions imposed by the public roads. Furthermore, the period of the day can influence the vehicles mobility. In rush hours, the traffic can be slower and traffic jams can force the driver to stop the car, or to change the route. This is a scenario where the vehicles density can be higher. On the other hand, weekends and early mornings are considered quiet traffic scenarios, with low vehicles density. These density variations in different hours reinforce the characteristic of dynamic topology of VANETs, making the communication a challenging task in vehicular networks.

Also, the mobility is influenced by the behavior of the driver and its routine. During the weekends, destinations such as camping, malls, churches are chosen to leisure and entertainment, and are frequently visited. However, on weekdays people tend to repeat their paths at the same time and to the same destination, such as: school, work, university, restaurant, etc. During their trajectories, vehicles encounter others vehicles, pass in the same streets, and suffers the same traffic conditions. These presented features suggest the study of the vehicular mobility under a social perspective, in order to understand better its evolution, and to apply the social concepts improving the services and the connectivity in VANETs.

Social analysis can be used to monitoring the traffic evolution during the day aiming to the understanding of the human routines, the similar trajectories, and the rush times. In literature, many works present the mobility analysis of vehicular networks in order to finding social features and to characterize this mobility [START_REF] Fiore | The networking shape of vehicularm o b i l i t[END_REF], [START_REF] Loulloudes | The dynamics of vehicular networks in urban environments[END_REF], [START_REF] Liu | Exploring social properties in vehicular ad hoc networks[END_REF]. However, these works are based only in analytical mobility models, and the evaluation was performed in short periods. A initial work [START_REF] Cunha | How effective is to look at a vehicular network under a social perception?[END_REF] presents a social analysis of one realistic trace with duration of one day. In this work, we focus on the investigation of the following question: What are the characteristics that a vehicular network needs to present in direction of to prove the existence a social behavior? Following we summarize the contributions of this paper, which intents to answer this question:

• As o c i a la n a l y s i so ft w ot r a c e st h a td e s c r i b ef e a t u r e so f different group of vehicles.

• Ac o m p a r i s o no ft h er e s u l t sw i t hr a n d o mg r a p h s ,w i t h the objective to increase the validity of the analysis.

• One definition about the social metrics that can be used to evaluate the mobility behavior in VANETs.

• Adiscussionaboutthedif ferentperspecti vestostudythe social mobility in VANETs, and what the factors can influence this analysis.

The remainder of this paper is organized as follows. In Section II, we define the main statements used to build the temporal and random graphs, and the evaluated metrics. In Section III, we present the traces data set, and the quantitative results according to each calculated metric. In Section IV, we discuss about the factors that may affect the social mobility analysis in vehicular networks. Finally, in Section V, we present the final conclusions and future perspectives.

II. MODELING

This section details the modeling used to create the temporal graph, the characteristics of the random graphs used as reference in the results and the definition of the evaluated metrics used in the social analysis.

A. Temporal Graph

The model considers the encounter among vehicles, i.e., when two vehicles are within communication range of each other. In our evaluation, we consider communication range of 100 m among vehicles, according to the protocol 802.11p and the downtown environment. The traces describes the mobility in urban areas and highways, and we consider 25 km/h as the average speed.

Then, we map the vehicles mobility and encounters described in the trace into a temporal graph. We first divide the whole trace into discrete time slots of duration t =1h.S i n c e vehicle traces usually describe fast changes in the topology, short time slots are recommended to better capture traffic changes. Thus, with the data in each slot we build a temporal graph. The temporal graph at the time t is an undirected, and can be formally defined as a graph G(t)=( V, E),w h e r e V represents the set with all vehicles v i and E represents the set of edges e ij .I nG(t),a ne d g ee ij (t) exists between the vehicles v i and v j during time t,w i t hi̸ =j.E a c hm e t r i c is evaluated hourly, considering the temporal graph G(t) that takes into account all the encounters in the period t.W eaggregate the encounters that happen during the slot window, and build the graph using the Growing Time Window technique [START_REF] Hossmann | Know thy neighbor: Towards optimal mapping of contacts to social graphs for dtn routing[END_REF]. For each trace, 24 graphs G(t) are generated describing the vehicles encounters during the day.

B. Random Graphs

Ar a n d o mg r a p hG R is a graph obtained from a set of vertex and its respective degree distribution. The criteriat o add an edge between two vertex in the graph G R is made in a random way, based in probabilities models. In this work, the model used to add edges uses urn algorithm that create random structures [START_REF] Johnson | Urn Models and Their Applications: An Approach to Modern Discrete Probability Theory[END_REF]. In this algorithm, the first step is to insert d i balls in an urn, marked with the identifier i of the vertex. The amount of balls of each vertex is equal to the degree of the vertex i.T h et o t a la m o u n to fballs in the insertion process in the urn, matches with the degree distribution of the original graph G.S i n c et h i ss t e p ,t w oballs b i and b j are randomly selected and their identifiers are compared. If i̸ =j and there are no edge between these vertex, an edge e ij is added. The selection step is made until there is no ball inside the urn, or there is no possibility to add more edges in the graph, according to the insertion criteria. In this work, for each temporal graph G(t) we generated its random graph G R , where the metrics are computed.

C. Metrics Description

We use social metrics to characterize the vehicles mobility in metropolitan areas. We divide the social metrics in macroscopic and microscopic group, according to the classification proposed in [START_REF] Uppoor | Insights on metropolitan-scalev e h i c u l a r mobility from a networking perspective[END_REF]. The macroscopic metrics represent measures of the network global state. We use the Distance, the Diameter, the Density, and the Edge Persistence as macroscopic metrics. The microscopic metrics define individual values for the vehicle representing the behavior of a unique vehicle. We select the Node Degree, the Cluster Coefficient and the Closeness Centrality as microscopic metrics.

III. RESULTS

In this Section we present the trace evaluation results. In the Section III-A, we describe details about the two traces evaluated. The Section III-B presents the results to the macroscopic metrics, and the Section III-C the results to the microscopic metrics.

A. Traces Evaluated

In this work, we analyze two public data set traces that describe the vehicle mobility along a city. The first trace picked is the Zurich's Trace [START_REF] Naumov | An evaluation of intervehicle ad hoc networks based on realistic vehicular traces[END_REF], a data set realistic based in mathematical models, and generated in a mobility simulator that considers the census information about the Zurich's city. This trace represents the traffic evolution over the day including rush times and daybreak. The Zurich's trace describes the trajectory of 260.000 vehicles over the day, and the Figure 1-(a) represents the number of vehicles in each day time. It is possible to see two periods with the higher number of vehicles that we consider as the rush times: 6 -9am and 2 -5pm, and the two hours with the major number of vehicles: 173.470 vehicles at 7am and 132.633 vehicles at 3pm. The analysis of this trace takes into account the traffic in all the day, with focus on the rush times.

The second trace picked is the San Francisco's Trace. This is a real data set with GPS information of 551 taxis in San Francisco's city, during four weeks [START_REF] Piorkowski | Crawdad trace[END_REF]. This information is captured from GPS devices in each vehicle, at each minute, over its trajectory. In the Figure 1-(b), it is showed the traffic evolution over the day. It is possible to see that during the daybreak there is a decrease in the number of taxis in service. At 6am, this behavior changes and the value increases again, just in time which people start their workday. Thus, since this hour, 400 taxis in average keep in service during the day, and we note a constant behavior in the graphic.

B. Macroscopic Metrics

Distance: To analyze the distance between two vertices in the graph, it is important to remark that in this work we consider the distance as the number of edges, which belongs to the path connecting these vertices. In a vehicular network, this metric can represent the existence of common interest between two vehicles, i.e., if two vehicles cross the same routes or visitnear places, the distance between them tends to be short. Otherwise, vehicles that do not pass the same streets in the same period, or vehicles that transit in distinct regions tend to present long distances. The Figure 2 presents the hourly average for the distance in the two traces. To the Zurich's trace, Figure 2-(a), we attribute the distance 0 to the hours where there is no traffic. However, in hours with high traffic it is possible to see one variation in the distance value. The number of vehicles in transit increase and the encounters probability increase too. Despite this traffic variation, the distance does not suffer a huge increase. This happened because in rush hours, in downtown there are many traffic jams that increase the encounters probability contributing to keep a short average distance. Also, we note that the random graphs G R present a little difference. These are graphs generated in random way, where the edge between two vertices obeys ap r o b a b i l i t y ;d i f f e r e n t l yo ft h eZ u r i c h ' sg r a p h sw h e r et h e existence of an edge respects the direction limits of the public roads. Thus, in the random graphs the edge insertion can create shortcuts in the graph. In addiction, in the Figure 2-(b) we can see the average distance to San Francisco's trace. As expected, the distance the trace presents a small distance during all the day, and the curve perform a constant behavior. In the same mode, the random graph G R presents the same behavior; all the vehicles can reach one each other with just one hop. It is happening because this trace particularly describes taxis mobility, due to the mobility pattern and the high vehicles density, during one hour, many taxis encounter one each other and the graph is very dense. As the random graph is generated from the San Francisco's graph, the density is equal and the distance keeps close, just a little variation in the daybreak. Diameter: In agreement with the Distance, the Diameter presents the major distance between two vertices. With this metric, it is possible to better understand the topological structure of the graph and their evolution over the time. In VA N E T s , t h e d i a m e t e r c a n i n d i c a t e t h e m a j o r c o s t t o o n e vehicle reach, in terms of the hops quantity in the communication. In the Figure 3 we can see the diameter for the two traces. In the Zurich's trace, Figure 3-(a), the diameter presents a reduction in the rush times, because in this period the encounter frequency in the central regions increase. In the same direction, the San Francisco's trace presents a short diameter in periods with high traffic. Both of traces have larger diameters in the low traffic period, since these are periods where the vehicles perform distinct routes, without congruent points. In other words, larger diameter can indicate the nonexistence of common interests. Density: The Density is one metric that quantifies the number of encounters happen during one hour. The baseline of this metric is the complete graph, and we will compare the number of edges of the graph G(t) with it. In the Figure 4 we can see the daily Density evolution. The first observation when we compare the two traces confirms the mobility pattern of the two different data sets. Taxis transit all the day with varied targets, and the particular vehicles have defined routines, with schedule and destinations defined. In the Figure 4-(a), we can see the Density to Zurich's trace, and it is possible to verifya low density. This trace has many vehicles crossing a large area with distinct destinations, and because these characteristics it presents a small numbers of encounters. In another hand, the Density of the San Francisco's Trace showed in the Figure 4-(b), and we can observe a higher density. Due the small number of taxis and the continuous movements per hour, the density remains around 80% during the day, and 60% in daybreak in according to the traffic evolution; many number of taxis in transit, major is the encounter probability. Also, many of encounters of taxis can have happen in the moment to leave and to return to the parking, considering taxis that belong to the same enterprise. Edge Persistence: In this work, we compute the edge persistence as the number of times one edge repeat during the time, i.e., whether an encounter between two vehicles repeat. Thus, we evaluate the edged persistence in according to the format of each trace. To the Zurich's trace we calculate the persistence in the two rush times, where the concentration of vehicles is elevated. We compare the encounters in the start of the day (6 -9am) with the end of the day (2 -5pm). In the San Francisco's trace we calculate the persistence in each day hour during one week, and we capture the encounters repetition during the week. Giving the traffic and encounter distribution in Zurich we analyze 340.000 edges in each period, and the persistence for this trace is 43%,i . e . ,n e a r l y150.000 edges repeated, which indicate similar routines among this vehicles. In the same context, we analyze the San Francisco's trace and we found an edge persistence of 75%.I ns p i t eo f taxis did different trajectory; it is necessary to return to initial point to wait for new passengers, which increase the encounter probability. When we look to both results, it is interest to observe that this is a good metric to explain the presence of similar routines and behavior in the graph.

C. Microscopic Metrics

Node Degree: When we evaluate the node degree in a VA N E T, i t i s i m p o r t a n t t o r e m a r k t h a t t h e t r a f fi c d e n s i t y s u f f ers variation during the day, and the number of encounters that one vehicle can perform is influenced by the quantify of vehicle in transit, e.g., as more vehicles are circulating, higher ist h e probability of encounters. In the Figure 5 we can see the daily Node Degree evolution. It is possible to observe in the Figure 5-(a) the average, maximum and median value to the degree. During the rush hours, it is the moment that the degree reaches the high value (1.355 at 7am.), which can indicate that is the time where vehicles make long trips or cross high density regions. The same behavior can be observed at 3pm with high degree. Also, we can see the average and median value that presents the value to 200 encounters to the rush times. These are acceptable values, which with more traffic is expected that more encounters happen. Many of the social networks are considered scale free networks, because presents a power law distribution to the node degree. In this context, we evaluate the distribution for the two hours with major concentration of vehicles that we can see in the Figures 5-(b) and5-(c). In these hours, 7am and 3pm respectively, there are in transit approximately 100.000 vehicles and the graphics showed that the degree distribution follows a power law distribution. Thus, we can consider the Zurich's graphs as scale free: many vehicles with low degree and low vehicles with high degree.

The evolution for the San Francisco's Trace is presented in the Figure 6. The values to the average, median and maximum values can be seen in Figure 6-(a). It is noted that during 5 -7am the degree presents a reduction in according to reduction of the traffic density. Furthermore, accordance tothe density, the average of degree is high, with values nearby 450 encounters. With this behavior, we observe that considering the time-window for one hour, practically all the vehicles encounter the others vehicles. To verify the existence of the power law distribution, we select three diverse day moments: 6am, 12am and 6pm (Figures 6-(b), 6-(c) e 6-(d)). We can note that, with the day progresses, the encounter probability increases, mainly in the 6pm where the degree of all nodes is between 10 2 and 10 3 .B e s i d et h i s ,i ti sn op o s s i b l et of o u n d the power law in the pattern mobility. Then, we should not consider this trace as a scale free network. Cluster Coefficient: The cluster coefficient is a so important metric to analyze in the context of social network, because it indicates the presence of communities and users with common interests. In the context of vehicular network, a cluster can represent a group of vehicles that transit in the same direction, or that pass frequently in the same place, i.e., a group of vehicles that have common point in their trajectories. In the traditional paper [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], Watts and Strogatz present a discussion about the small worlds networks and comparison with random graphics. It is showed that de cluster coefficient of a social graph is one order of magnitude of a random graph G R .

We can see the cluster coefficient to the Zurich's trace and the respective comparison with random graphs in the Figures 7-(a) and7-(b). When we look to the maximum value, it is possible to note that there are vehicles in graph with 100% of their neighbors connected. We can attribute this result to vehicles that are in traffic jams areas, in downtown, mainly in areas that receives traffic flows from many adjacent roads. In these regions, there is a strong tendency to form vehicles cluster that share the same routes and traffic conditions. In the same graphic, we can observe that the cluster coefficient average is 50%,w h i c hi n d i c a t e st h a tah a l fo ft h ev e h i c l e s neighbors are connected in the rush hours. The comparison with G R can be seen in the Figure 7-(b), and we found a difference of one order of magnitude; the G R presents a low cluster coefficient compared with Zurich's graphs. This result shows that social behaviors can be found in Zurich's trace. In another way, we evaluate the cluster coefficient to the San Francisco's trace, and in the Figures 8-(a) and 8-(b) we show the results. It is possible to observe that the average, median and maximum values are near values, around 90%.Thismeans that only one cluster is present, where all the vertices are connected. This result indicates that there is no social behavior in the San Francisco's Trace. It is worth mentioning the impact of the parameters and assumptions used at the temporal graph generation (Section II-A). In particular, a time slot windows of 1h is used to infer vehicles encounters. In this context, any contact happened in this slot window generates an edge in the analyzed temporal graph. This could explain the high density and node degree found, and consequently the complete graph. When we compare the cluster coefficient with the random graphs, Figure 8-(b), it is possible to see the same behavior with the two graphs, with very similar values. Thereat, with the high value of the cluster coefficient, it is possible to verifyonly ac l i q u ei nt h eg r a p h ,i . e . ,au n i q u eg r o u pw i t hv e h i c l e st h a t have the same behavior. Taxis are vehicles that made random trips during all the day, i.e., they pick up clients in random locations in the city. In this sense, no routine is expected. Closeness Centrality: The evolution of the Closeness Centrality is significant to understand how close the vertices are in the graph. In VANETs, it is likely to evaluate this metric to appreciate the position of the vehicles in accordance to others and to know the mobility patterns of the vehicles. The Figures 9-(a) and 9-(b) show the results for the two trace analyzed. In the Figure 9-(a), we can verify that the Closeness modify according to the traffic evolution, with more traffic more is the number of vehicles to consider. However, it is possible to verify an increase in the value, indicating that the vehicles are more central. This happens because in the rush times there is a tendency to vehicles cross the downtown and encounter more vehicles. Also, more vehicles in transit contribute to generate more similar routes, which explain this small increase. Additionally, the random graphs present a similar behavior, with a little reduction with does not representative. Likewise, in the Figure 9-(b), we observe the values to closeness in the San Francisco's trace. Again, it is verified that the random graphs present a similar behavior to the values for closeness, and then a high value. According to the average Distance for this trace, around to one hop, it is possible to certify, with this metric, that the vertices are so close.

IV. VEHICULAR SOCIAL NETWORKS ANALYSIS

In this work, we have analyzed two vehicular traces under as o c i a lp e r s p e c t i v e ,a i m i n gt oc h a r a c t e r i z ei ta c c o r d i n gt o metrics that bring us to comprehend the vehicular mobility and the driver's routines. Therefore, to do this evaluation we chose some mobility parameters, and we do important assumptions to characterize the encounter among the vehicles (Section II-A). Through of these parameters, we create graphs and compute the metrics. However, when we look to the results we found social characteristics in the Zurich's trace, with strong directions to characterize this network as a Social Network. Except, for the San Francisco's trace we observed a high density and a huge number of encounters that collaborate to create a very dense graph, similar a complete graph. Thus, we look to the literature in order to understand the parameters that can affect the encounters improving this analysis.

Many of parameters can influence the existence of one encounter, such as: vehicle speed, communication range, time slot window, and minimal time to consider as one encounter. These are important parameters to define in the analysis when the focus is evaluate the encounter probability in the Vehicular Network. In the work [START_REF] Hossmann | Know thy neighbor: Towards optimal mapping of contacts to social graphs for dtn routing[END_REF], the authors present a discussion about how create the social graph with the contacts in a Delay Tolerant Network. They argue how the selection of the parameters and the contact aggregation can affect to the social graph. Beyond that the authors show a performance evaluation of routing protocols considering different techniques of contact aggregation to create the social graph. Another example of social analysis is presented in [START_REF] Liu | Exploring social properties in vehicular ad hoc networks[END_REF]. In this work the authors presented a study of the social interactions of the vehicles. They create graphs, and aggregate the encounters by the growing time window technique with a threshold to define one relationship between two vehicles. The authors show the presence of the social properties in two real traces.

Although, the correct definition of the parameters to find social features in VANETs, it is interest to take account the scenario of the vehicles transit: urban, rural and highways. According to the scenario, the vehicles speed can suffer many fluctuations, which impacts the encounter ratio. Another important feature to consider is the nature of vehicles. The taxis cross the whole city and perform random trajectories, without fixed time and trip duration. The buses transit the same routes and they obey the fixed schedule to perform the routes. In other hand, common people use their vehicles to perform predetermined trajectories according to their routines. Therefore, in order to characterize and to execute a good analysis, it is important take into account the vehicle nature, its mobility pattern in order to define the evaluated parameters according to the characteristics of this vehicle.

V. C ONCLUSION

In this work, we presented the social analysis of two vehicular traces describing the mobility of vehicles in the metropolitan area. We analyzed the Zurich's trace and San Francisco's trace that have different duration and routines. Considering the model that defines an encounter, the analyses of the Zurich showed the presence of the small world phenomenon.Also,inthistracethedegreedistributionfollowsthe power law, which describes this network as scale free network. Moreover, it presents features that indicate the presence of communities and people with similar interest. In the San Francisco's trace, we cannot find social properties. This is ag r a p hw i t ho n l yo n ec o m p o n e n ta n dh i g he d g e sd e n s i t y , which justifies the results (taxi mobility pattern and high encounters density). Besides all, according to results of the analysis we had shown that is possible to find social properties in VANETs, considering vehicles nature. As future works, we intend to extend the evaluation with the study of others social metrics and improve the analysis with variation of the parameters that define whether there is an encounter between two vehicles. Beyond that, we intend to perform a spatial and temporal metrics computation aiming to capture the difference of values in different points in the city. Also, we intend to design dissemination protocols aware of these social metrics evaluated.

Fig. 1 .

 1 Fig. 1. Daily traffic evolution.

Fig. 2 .

 2 Fig. 2. Daily Distance evolution.

Fig. 3 .

 3 Fig. 3. Daily Diameter evolution.

  of day(h) (a) Zurich.

  of day(h) (b) San Francisco.

Fig. 4 .

 4 Fig. 4. Daily Density evolution.

Fig. 7 .

 7 Fig. 7. Daily Cluster Coefficient evolution to the Zurich's Trace.

Fig. 5 .

 5 Fig. 5. Daily Distance evolution to the Zurich's Trace.

Fig. 6 .

 6 Fig. 6. Daily Distance evolution to the San Francisco's Trace.

  San Francisco × Random.

Fig. 8 .

 8 Fig. 8. Daily Cluster Coefficient evolution to the San Francisco's Trace.

Fig. 9 .

 9 Fig. 9. Daily Closeness Centrality evolution.


Andrew G Livingstone
email: livingstone@stir.ac.uk

Russell Spears

Antony S R Manstead

Martin Bruder

Strengthening identity in minority groups 1 Running head: STRENGTHENING IDENTITY IN MINORITY GROUPS The more, the merrier? Numerical strength versus subgroup distinctiveness in minority groups

Strengthening identity in minority groups 3

The more, the merrier? Numerical strength versus subgroup distinctiveness in minority groups

A C C E P T E D

M A N U S C R I P T

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Strengthening identity in minority groups 4

The more, the merrier?

Numerical strength versus subgroup distinctiveness in minority groups "The National Assembly of Québec recognizes that Quebecers wish to see the quality and influence of the French language assured, and is resolved therefore to make French the language of Government and the Law, as well as the normal and everyday language of work, instruction, communication, commerce and business." (Charter of the French language, S.Q., c. 5, 1977).

"Plaid Cymru the Party of Wales wants to see a fully bilingual nation, with each and every child given the opportunity to learn Welsh" (Plaid Cymru the Party of Wales manifesto, 2005).

Because minority groups are often low in power, their very distinctiveness may become a key source of meaning and value for their identity. For groups defined around an attribute that can be acquired, such as language, more people can in principle attain that attribute over time. However, we propose that increasing the number of those who have the prototypical attribute carries with it a threat to the distinctiveness of those who have had that attribute all along. In the present paper we test the proposition that prototypical members of minority groups may not always see increasing numerical strength as unequivocally positive, especially when their own position within the ingroup is made salient.

Consider the most salient example on the North American continent: French Canadians. Many French Canadians value their language very highly and feel threatened by the Anglophone culture that surrounds them [START_REF] Bourhis | Cross-cultural communication in Montreal: Two field studies since Bill 101[END_REF]; see also [START_REF] Bourhis | Vitality perceptions and language attitudes: Some Canadian data[END_REF]. If more Canadians spoke French, this would increase the numerical strength of French Canadians as a group and also help them to differentiate themselves from their more

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Strengthening identity in minority groups 5 powerful English-speaking neighbors south of the border. However, within Canada this could also undermine the group distinctiveness of Francophones. We investigate such tensions and argue that, even in cases where a language is seen as identity-defining for a whole country (as is true of Welsh for Welsh speakers), large increases in how widely it is spoken might paradoxically be aversive, at least under certain conditions.

The low-power position of many minority groups means that their members can perceive threats to the distinctiveness or even existence of their ingroup identity [START_REF] Jetten | Intergroup distinctiveness and differentiation: A meta-analytic integration[END_REF]. Such a threat often relates to the influence of a dominant, majority outgroup on the minority ingroup's distinctive characteristics, through pressure to assimilate with a majority group (Hornsey & Hogg, 2000) or the majority group's adoption of the minority group's cultural symbols. Threats to distinctiveness are particularly acute when group members define their ingroup in terms of specific cultural attributes, such as language. Here, the hegemony of a dominant outgroup's culture means that minority group cultures face a perpetual struggle for recognition. There are many examples; as well as

Francophones in Quebec, this applies to Basque and Catalan speakers in Spain, the Basques and Bretons in France, Welsh speakers in the UK, Native American languages in the USA, and Aboriginal and Maori cultures in Australia and New Zealand.

An important aspect of minority-majority relations in these contexts is that there is often variation in the extent to which category members have the attributes or characteristics that positively differentiate the minority group from the majority group. This is especially so in the case of large-scale categories such as gender, ethnicity, and nationality, in which particular characteristics are often promoted as category-defining by an active minority within the minority category (cf. [START_REF] Moscovici | Social influence and social change[END_REF] as part of a wider social and political agenda to influence minority-majority relations [START_REF] Simon | Politicized collective identity: A social psychological analysis[END_REF]. Note that our definition of a 'minority' here follows that offered by scholars such as [START_REF] Morris | Ethnic Groups[END_REF],

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Strengthening identity in minority groups 6 [START_REF] Tajfel | The social psychology of minorities[END_REF]Wagely and[START_REF] Wagley | Minorities in the New World: Six case studies[END_REF]. This definition emphasizes the importance of criteria such as power and status relations with a majority (i.e., dominant) outgroup rather than relative numerical size per se [START_REF] Simon | The social psychology of minoritymajority relations[END_REF]) -a definition that also maps onto group members' own conceptions of what defines majority and minority groups [START_REF] Seyranian | Dimensions of majority and minority groups[END_REF]. Having said this, numerical minorities also often tend to be less powerful than numerical majorities.

This raises interesting questions about how members of minority groups might react to changes in the prevalence of a potentially defining attribute or characteristic. A large and diverse body of research has demonstrated that when the distinctiveness of an ingroup category is questioned or threatened, ingroup members seek to re-establish ingroup distinctiveness by differentiating between the ingroup and outgroup on available dimensions of comparison (e.g., Brown, 1984a[START_REF] Brown | The effects of intergroup similarity and cooperative versus competitive orientation on intergroup discrimination[END_REF]Hornsey & Hogg, 2000;[START_REF] Jetten | Defining dimensions of distinctiveness: Group variability makes a difference to differentiation[END_REF], 2001;[START_REF] Jetten | Intergroup distinctiveness and differentiation: A meta-analytic integration[END_REF][START_REF] Mummendey | Social comparison, similarity and ingroup favouritism: A replication[END_REF]. This can include attempts by active factions within the minority category to spread and strengthen 'distinctive' ingroup characteristics, such as attempts by members of minority ethnolinguistic groups to strengthen the ingroup language (i.e., give it vitality: [START_REF] Bourhis | The language of intergroup distinctiveness[END_REF]Giles, Bourhis, & Taylor, 1977;Giles & Johnson, 1981). Indeed, efforts to protect ingroup identity and language cover the full spectrum of collective action, from promotional campaigns, through political action and demonstrations, to occasionally illegal and even violent forms of direct action.

On the basis of the above evidence it seems reasonable to expect that any increase in the strength or prevalence of a 'distinctive' ingroup attribute (e.g., increased vitality of the ingroup language) would be valued by minority group members -and that the larger this increase, the more favorably it would be regarded. However, we suggest that for members of minority groups who already have that attribute, things might not be this straightforward.

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Strengthening identity in minority groups 7

Although the prospect of a 'distinctive' attribute becoming more widespread within the ingroup would clearly strengthen the ingroup's position vis-à-vis the outgroup, it could also mean that the position of those who already have that attribute -and who may therefore have a distinct and prototypical position within the ingroup -is undermined. For example, as use of an ingroup language becomes more widespread, being able to speak it becomes less 'special,' perhaps diluting the identity claims of those who can speak the language. This is likely to be a particularly potent process in cases where an attribute is elevated in importance precisely because it is rare and difficult to attain, bolstering the 'special' claim over ingroup identity among those who have it [START_REF] Coupland | Home truths: Globalistion and the iconising of Welsh in a Welsh-American newspaper[END_REF].

This prediction is consistent with two influential theories of inter-group relations, social identity theory [START_REF] Tajfel | An integrative theory of intergroup conflict[END_REF] and optimal distinctiveness theory [START_REF] Brewer | The social self: On being the same and different at the same time[END_REF]. Social identity theorists propose that groups try to differentiate themselves from others that are too similar to their own, thereby threatening their group's distinctiveness [START_REF] Jetten | Intergroup distinctiveness and differentiation: A meta-analytic integration[END_REF]. In optimal distinctiveness theory distinctiveness is more explicitly conceptualized in terms of (relative) group size; it is argued that minority groups simultaneously satisfy a need for assimilation and inclusion, and a contrasting need to differentiate self from others (see also [START_REF] Hornsey | The individual within the group: Balancing the need to belong with the need to be different[END_REF]. Although the distinctiveness principles in both approaches are consistent with our predictions, the specific prediction we make here has not to our knowledge been made before. We propose that the distinctiveness motive may be sufficiently strong that it can lead those who have a 'distinctive' ingroup attribute to react negatively to the pervasive spread of that attribute within the ingroup category, even when that category is consensually defined by that attribute. This is because the spread of the attribute has the potential to undermine the distinctive group position of those within the broader ingroup category who already have the attribute.

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Strengthening identity in minority groups 8

We test these predictions in two very different contexts. In Study 1, we establish a context in which female participants are assigned to a high emotional intelligence subgroup of women, and are led to believe that emotional intelligence is a characteristic that differentiates women as a whole from men (see Hacker, 1951, for a more in-depth consideration of women as a minority group). In Study 2, we examine a natural context by sampling from a prototypical subgroup (speakers of the Welsh language) within a minority category (the Welsh) in the UK. In both cases, we manipulate the prospective numerical strength of the 'distinctive' but rare attribute (high emotional intelligence in Study 1; Welsh language ability in Study 2) within the minority group, and gauge the affective reactions of members of the minority group who have this attribute. We present participants with one of three scenarios in which future prospects for the 'distinctive' attribute are described. These indicate that there will be little change in the prevalence of the 'distinctive' attribute in the future (it will be held by 25% of the ingroup category population), a moderate increase (50%), or a large increase (75%). In the little change condition, those who have the 'distinctive' attribute clearly remain in the negative position of being a minority within the ingroup, with low numerical strength. In the large increase condition, they face the prospect of greatly increased numerical strength, but also of becoming a clear majority within the ingroup, bringing the potential for their own subgroup distinctiveness to be undermined. In the moderate increase condition, they also stand to increase their numerical strength but crucially -in contrast to the large increase condition -without the prospect of becoming a (non-distinct) majority within the ingroup.

The role of comparative context

How should the concerns discussed above be reflected in affective reactions to these different prospects? In line with principles from self-categorization theory [START_REF] Turner | Rediscovering the social group: A self-categorization theory[END_REF], one factor that is likely to shape affective reactions is the comparative context in

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Strengthening identity in minority groups 9 which group members consider these prospects (David & Turner, 1999;Haslam & Turner, 1992;Haslam, Turner, Oakes, McGarty, & Hayes, 1992). Specifically, does this inter-group context pit the inclusive ingroup category against a common outgroup, or does it focus on the ingroup category alone, such that divisions between those who do and do not have the 'distinctive' attribute become more apparent? It is important to note that, in line with selfcategorization theory [START_REF] Turner | Rediscovering the social group: A self-categorization theory[END_REF], both of the salient contexts discussed here constitute intergroup contexts, albeit at different levels of inclusion. However, to distinguish the two contexts, we refer to them as the inter-category (or inter-group) context, and the intra-category (or intra-group) context, respectively.

Where the wider relationship between the ingroup and outgroup categories is salient (i.e., in an inter-category context), we predict that affective reactions are likely to become more positive as a function of the projected increase in the prevalence of the attribute. This is because in this context group members will focus on the needs of the ingroup as a whole in relation to the dominant outgroup, as opposed to their own subgroup's position within the ingroup category.

In contrast, where the relationship between those who do and those who do not have the attribute is made salient (i.e., in an intra-category context), participants should focus on their own (subgroup) position within the ingroup category. In this context, a large increase is likely to threaten the distinctiveness and exclusivity of their position within the ingroup as a whole. Consequently we expect that for a large increase, participants focused on the intracategory context will be considerably less positive towards this change than those for whom the inter-category context of ingroup versus outgroup is salient. In contrast, a moderate increase in the prevalence of the attribute will be regarded just as positively as in the intercategory context because, as discussed above, this condition offers the prospect of increased

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Strengthening identity in minority groups 10 numerical strength while retaining an optimally-distinct position (neither a numerical minority nor a non-distinct majority) within the ingroup category.

Study 1

Participants

Participants were 198 British women with a mean age of 24.61 years (SD = 16.28).

They were randomly allocated by the survey software to one of the six conditions, resulting in cell sizes of between 25 and 40.

Design

The study had a 3 (subgroup numerical strength: no increase, moderate increase, large increase) x 2 (context: intra-category vs. inter-category) between-subjects design.

Materials and Procedure

The study was presented as an investigation into emotional intelligence among women, and consisted of a web-based questionnaire. Participants indicated their informed consent by clicking on an 'agree' button on a survey landing page. On doing so, participants were directed to the first page of the questionnaire.

Emotional intelligence questionnaire. The first page of the questionnaire contained a bogus emotional intelligence questionnaire. This consisted of 15 items on which participants indicated their agreement or disagreement with statements such as "When I see someone I know, I can usually tell how they are feeling," and "When someone I care about is sad, I feel sad too;" and 10 word-association items on which participants were presented with a target word (e.g., house) and had to select one of four other words (e.g., street; flat; room; home) that they most strongly associated with the target word.

After completing the questionnaire and clicking on the 'next page' button, participants were presented with a page that purportedly contained the results of the

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Strengthening identity in minority groups 11 emotional intelligence test. In reality, all participants received feedback that they were high in emotional intelligence.

In order to position participants as part of a prototypical minority subgroup, the feedback also emphasized that emotional intelligence was a defining characteristic for women, but that women who were high in emotional intelligence were actually a minority of around 25% of women as a whole. Future prospects for numerical strength manipulation. The manipulation of future prospects for subgroup numerical strength consisted of an article and graphs that described likely future trends for emotional intelligence among women. The conclusion of these was that women high in emotional intelligence would account for 25% (no increase condition),

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Strengthening identity in minority groups 12 50% (moderate increase condition), or 75% (large increase condition) of women as a whole by the year 2031. In each condition, the article began with the following description of the current level of emotional intelligence among women:

As indicated before, emotional intelligence is for many people at the heart of what it means to be a Woman, although current estimates suggest that high EI women make up only a small minority of around 25% of women as a whole.

The value and importance of emotional intelligence has been receiving renewed attention recently.

This is because of the great increase in Women's participation in activities and access to resources (e.g., self-improvement books and courses, internet networking resources, lifestyle training) that have the potential to improve emotional intelligence. A key question is therefore whether emotional intelligence could become more widespread among Women as a whole.

The article then described the future prospects for numbers of high EI women, followed by an illustrative graph. In the no increase condition, the article read:

The graph below shows that the projected number of EI women over the next 20 years is likely to show hardly any increase from what it is today, with no more than 25% of women falling into the high EI category. Despite the increased esteem and importance of emotional intelligence, it is most unlikely that the next few decades will see an increase in EI amongst Women.

In the moderate increase condition, the article read:

The graph below shows that the projected number of EI women over the next 20 years is likely to show a marked increase from what it is today, with up to 50% of women falling into the high EI category. With the increased esteem and importance of emotional intelligence, the next few decades could see an increase in EI amongst Women.

In the large increase condition, the article read:

The graph below shows that the projected number of EI women over the next 20 years is likely to show a dramatic increase from what it is today, with anything up to 75% of women falling into the high EI category. With the increased esteem and importance of emotional intelligence, the next few decades could see an increase in EI amongst Women.

The future prospects manipulation was checked using two items. The first item asked whether, according to the article, the number of women high in emotional intelligence was

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Strengthening identity in minority groups 13 likely to increase, decrease, or stay the same. Participants responded on a 7-point scale ranging from -3 (decrease strongly) through 0 (stay the same) to 3 (increase strongly). The second item asked what the proportion of women high in emotional intelligence would be in the year 2031 according to the trends described in the article. Participants responded to this item by clicking one of three options: same as now; up to 50%; up to 75%.

Eight items measured emotional reactions to the future prospects for subgroup numerical strength (α = .963). Participants were asked to indicate on a 7-point scale ranging from -3 (completely disagree) to +3 (completely agree) whether these future prospects made them feel delighted; pleased; relieved; happy; content; proud; elated; positive.

After completing the questionnaire, participants were directed to a debriefing page in which they were fully debriefed and thanked for their participation.

Results

Manipulation check. A 3 (future prospects for numerical strength: no change; moderate increase; large increase) X 2 (comparative context: inter-category vs. intracategory) ANOVA on the future prospects for numerical strength manipulation check revealed only a main effect of the future prospects for numerical strength manipulation, F(2, 192) = 29.96, p < .001, η p 2 = .240 (other F's < 1.33). Pairwise comparisons confirmed that, compared to the no increase condition (M = 0.44, SD = 0.98), future prospects for emotional intelligence were seen as greater in the moderate increase condition (M = 1.96, SD = 1.24), p < .001, and in the large increase condition (M = 1.99, SD = 1.39), p < .001. The difference between the moderate increase and large increase conditions was not significant, F < 1.

The second future prospects manipulation check also confirmed that 85.5%, 91.4%, and 90.4% of participants selected the correct option in the no, moderate, and large increase conditions, respectively 1 . Together, results on these checks suggest that the manipulation was successful.

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Strengthening identity in minority groups 14

Positive affect

A similar ANOVA on the affect scale revealed a significant main effect of future prospects for numerical strength, F(2, 191) = 14.40, p < .001, η 2 p = .131 (F < 1 for comparative context main effect). This was qualified by the expected interaction between the two factors, F(2, 191) = 3.81, p = .024, η p 2 = .038. This interaction is illustrated in the top panel of Figure 1. Simple effects analyses confirmed that the prospect of a moderate increase resulted in more positive affect than the prospect of no increase in both the intra-category and inter-category conditions, F(1, 191) = 6.20, p = .014, η 2 p = .031, and F(1, 191) = 22.92, p < .001, η 2 p = .107, respectively. Consistent with our main hypothesis, the simple main effect of context was significant only in the large increase condition, F(1, 191) = 6.32, p = .013, η p 2 = .032 (p = .144 in the no increase, and .601 in the moderate increase conditions).

Specifically, positive affect was lower in the intra-category condition than in the intercategory condition.

Discussion

The results of this study provide good support for our hypotheses. They confirm that prospects of a moderate increase in the numerical strength of a 'distinctive' but rare ingroup attribute evoked significantly greater positive affect than the prospect of no increase, regardless of whether the context was intra-category or inter-category. However when a large increase was projected a quite different pattern emerged as a function of context.

Specifically, the prospect of a large increase was greeted with less positive affect in the intracategory context than in the inter-category context. This supports our contention that while activists within minority groups may frequently base campaigns around strengthening the 'distinctive' characteristics that differentiate the ingroup from a comparison outgroup, increases in the numerical strength of such attributes may not always been seen as Second, our hypothesis suggests that the equivocal reaction to the prospect of a large increase in the intra-category condition is in part driven by perceptions that this increase may undermine the identity or special position of the 'prototypical' subgroup that already has the 'distinctive' attribute. In Study 2, we therefore measured the appraisal that the identity of the prototypical subgroup would be undermined in the case where the numerical strength of the 'distinctive' attribute was projected to grow sharply, and examined the mediating role of this appraisal. Third, our hypothesis would benefit from being tested it in a natural context in order to supplement the relatively contrived context examined in Study 1. To this end, in Study 2 we tested our hypothesis by sampling Welsh language speakers in Wales, UK. This is a context in which struggles over the strength of a numerically weak but prototypical characteristic are very real and passionate, and are linked to political campaigns and even to support for radical and illegal forms of social action [START_REF] Livingstone | Illegitimacy and identity threat in (inter)action: Predicting intergroup orientations among minority group members[END_REF].

Study 2

The role of the Welsh language has historically been a controversial issue (Davis, 1994). Although only approximately 25% of the population of Wales can speak it, the Welsh language is widely accepted as a defining dimension of Welsh identity by Welsh-speaking and non-Welsh speaking Welsh people alike (Welsh Language Board, 2003). Welsh

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Strengthening identity in minority groups 16 speakers in Wales provide a critical test-case, because of the identity-defining nature of the Welsh language for Welsh identity (accepted by non-speakers as well as speakers; see [START_REF] Coupland | Affiliation, engagement, language use and vitality: Secondary school students' subjective Strengthening identity in minority groups 30[END_REF][START_REF] Bourhis | Language as a determinant of Welsh identity[END_REF]Giles, Taylor, & Bourhis, 1977;Livingstone, Spears, & Manstead., 2009a), and the efforts being made to defend and expand the language after centuries of discrimination by the English.

Building on the design of Study 1, we presented Welsh-speaking participants with one of three scenarios in which future prospects for Welsh were described. These indicated that there would be little change in the use of Welsh in the future (it would be spoken by 25% of the population), a moderate increase (50%), or a large increase (75%). We also manipulated the comparative context (intra-national vs. inter-national).

In terms of affective reactions, we expected to replicate the pattern observed in Study 1. In addition, we expected the equivocal response to a large increase in the intra-national context to be evident in specific appraisals of how such a change would undermine the position of Welsh speakers in Wales.

Method

Participants

Participants were 163 adults who defined themselves as Welsh speakers. They were randomly allocated by the survey software to one of the six conditions, resulting in cell sizes of between 10 and 33 2 . Gender and age information were not recorded.

Design

The study had a 3 (future prospects for ingroup language use: no increase, moderate increase, large increase) x 2 (context: intra-national vs. inter-national) between-subjects design.

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Strengthening identity in minority groups 17

Materials and Procedure

The study consisted of a web-based questionnaire. Participants indicated their informed consent by clicking on an 'agree' button on a survey landing page. On doing so, participants were directed to the first page of the questionnaire.

Context manipulation. The first page of the questionnaire contained the manipulation of context. In the international context condition, the page was headed by images of Welsh, English, and UK flags alongside each other. Explanatory text stated that "We are interested in what you, as a Welsh person, consider to be the distinguishing characteristics of Welsh people and English people." Participants were instructed to provide up to three distinguishing characteristics of Welsh people in a box headed 'Welsh people,' and three distinguishing characteristics of English people in a box headed 'English people.' In the intra-national context condition, the page was headed by an image of the Welsh flag only.

Explanatory text stated that "We are interested in what you, as a Welsh speaker, consider to be the distinguishing characteristics of Welsh people who speak Welsh and Welsh people who do not speak Welsh." Participants were instructed to provide up to three distinguishing characteristics of Welsh-speaking people in a box headed 'Welsh-speaking Welsh people,'

and three distinguishing characteristics of non-Welsh-speaking Welsh people in a box headed 'non-Welsh-speaking Welsh people.' In order to sustain the effect of the manipulation, each subsequent page of the questionnaire was headed by the images of the condition-appropriate national flags.

Future prospects for ingroup language use manipulation. The manipulation of future prospects for ingroup language use consisted of an article and graphs that described likely future trends for ingroup language use. The conclusion of these was that efforts to spread the use of the ingroup language would result in 25% (no increase condition), 50% (moderate increase condition) or 75% (large increase condition) of the ingroup population being able to

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Strengthening identity in minority groups 18 speak the ingroup language. In each condition, the article began with the following description of the current level of ingroup language use:

The Welsh language is at the heart of Welsh national identity for many people, and has been so for a very long time. This is despite the grave threats it has faced down the centuries. At various points, the language stood on the brink of extinction. However, since the 1950's the language began to regain some of its lost status. This has been reflected in -and helped by -legislation such as the Welsh Language Acts of 1967 and 1993 that has removed the official stigma formerly placed upon the language.

One major indicator of the vitality of the Welsh language is of course the number of Welsh speakers in Wales. After a long period of decline, the number of people in Wales who could speak Welsh has recently been low but steady, at around 20% of the population. A key question has therefore been whether Welsh use could become more widespread across Wales as a whole.

The article then described the future prospects for ingroup language use, followed by an illustrative graph. In the no increase condition, the article read:

The graph below shows that the projected number of Welsh speakers in Wales over the next 40 years is likely to show hardly any increase from what it is today, with no more than 25% of the Welsh population being able to speak Welsh. Despite the increased esteem and prestige of the language, it is most unlikely that the 21 st century will see an increase in Welsh speaking as dramatic as its decline in the 20 th century.

In the moderate increase condition, the article read:

The graph below shows that the projected number of Welsh speakers in Wales over the next 40 years is likely to show an increase from what it is today, with anything up to 50% of the Welsh population being able to speak Welsh. With the increased esteem and prestige of the language, the 21 st century should see an increase in Welsh speaking, although not as dramatic as its decline in the 20 th century.

In the large increase condition, the article read:

The graph below shows that the projected number of Welsh speakers in Wales over the next 40 years is likely to show a dramatic increase from what it is today, with anything up to 75% of the Welsh population being able to speak Welsh. With the increased esteem and prestige of the language, the 21 st

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Strengthening identity in minority groups 19 century could see an increase in Welsh speaking that is just as dramatic as its decline in the 20 th century.

The future prospects manipulation was checked using two items. The first item asked whether, according to the article, the number of Welsh speakers in Wales was likely to increase, decrease, or stay the same. Participants responded on a 7-point scale ranging from --3 (decrease strongly) through 0 (stay the same) to 3 (increase strongly). The second item asked what the number of Welsh speakers in Wales would be in 2041 according to the trends described in the article. Participants responded to this item by clicking one of three options:

same as now; up to 50%; up to 75%.

Fourteen items measured emotional reactions to the future language use information (α = .973). These included eight positive emotion items (delighted; pleased; relieved; happy; content; proud; elated; positive) and six negative emotion items (uneasy; angry, irritated; annoyed; bitter; negative). Responses were made on a 7-point scale ranging from -3 (completely disagree) to +3 (completely agree), and responses on the negative emotion items were reverse-scored before the scale mean was calculated 2 .

Next came a 4-item scale measuring the extent to which the future language use information was perceived as undermining the position of Welsh speakers in Wales (α = .671). The scale contained the items, 'Any special importance of Welsh speakers in Wales would be undermined;' 'Part of my identity would be undermined;' 'My identity would be enhanced' (reverse-scored); 'Any special importance of Welsh speakers in Wales would be enhanced' (reverse-scored). Participants responded on a 7-point scale ranging from -3 (completely disagree) to +3 (completely agree).

After completing the questionnaire, participants were directed to a debriefing page in which they were fully debriefed and thanked for their participation.

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Strengthening identity in minority groups 20

Results

Manipulation checks

To check the effectiveness of the future prospects for ingroup language use manipulation, we submitted the first manipulation check item to a 3 (future prospects for ingroup language use: no increase, moderate increase, large increase) x 2 (context: intranational vs. inter-national) factorial ANOVA. This revealed only the expected main effect of future prospects for ingroup language use, F(2, 157) = 23.71, p < .001, η 2 p = .232. Pairwise comparisons confirmed that, compared to the no increase condition (M = 0.51, SD = 1.14), prospects for increased ingroup language use were seen as greater in the moderate increase condition (M = 1.78, SD = 1.26), F(1, 157) = 29.82, p < .001, η 2 p = .160, and in the large increase condition (M = 2.12, SD = 1.40), F(1, 157) = 37.28, p < .001, η 2 p = .192. The difference between the moderate increase and large increase conditions was also marginally significant, F(1, 157) = 2.89, p = .091, η 2 p = .018.

The second ingroup language prospects manipulation check also confirmed that 75%, 90%, and 78% of participants selected the correct option in the no, moderate, and large increase conditions, respectively 3 . Together, results on these checks suggest that the manipulation was successful.

Affective reaction

A similar ANOVA on the affective reaction scale revealed significant main effects of future prospects for ingroup language use, F(2, 157) = 138.37, p < .001, η 2 p = .638, and of context, F(1, 157) = 16.13, p < .001, η 2 p = .093. These were qualified by the predicted interaction between the two factors, F(2, 157) = 9.65, p < .001, η 2 p = .109. This interaction is illustrated in Figure 2. Simple effects analyses confirmed that the prospect of a moderate increase resulted in more positive affect than the prospect of no increase in both the intranational and inter-national conditions, F(1, 157) = 124.08, p < .001, η 2 p = .441, and F(1,

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Strengthening identity in minority groups 21 157) = 142.33, p < .001, η 2 p = .475, respectively. Consistent with our main hypothesis, the simple main effect of context was significant only in the large increase condition, F(1, 157) = 26.70, p < .001, η 2 p = .145 (Fs < 1 in the no increase and moderate increase conditions).

Specifically, positive affect was lower in the intra-national condition than in the international condition.

Welsh speakers' position undermined

A similar ANOVA on the appraisal of whether the position of Welsh speakers would be undermined revealed a significant main effect of future prospects for ingroup language use, F(2, 157) = 11.59, p < .001, η 2 p = .129, qualified by an interaction between future prospects for ingroup language use and context, F(2, 157) = 3.96, p = .021, η 2 p = .021. This interaction is illustrated in the lower panel of Figure 2. Simple effects analysis confirmed that the main effect of context was significant in the large increase condition, F(1, 157) = 8.74, p = .004, η 2 p = .053 (F's < 1 in the no increase condition and moderate increase condition). Specifically, the position of Welsh speakers was felt to be undermined to a greater extent in the intra-national condition than in the inter-national condition.

Relation between appraisals and affect

In order to test whether the appraisal of subgroup identity being undermined mediated the effect of comparative context on affective reaction, we set up a multi-group path model using AMOS 7 in which the direct and indirect (through the 'identity undermined' appraisal) effects of context on affective reaction were tested. The paths from comparative context to appraisal and affective reaction were free to vary in strength across future prospects for numerical strength conditions, reflecting the interactions described above. The path from appraisal to affective reaction did not differ in strength across future prospects for numerical strength conditions (∆χ 2 2 = 2.98, p = .225), and so was constrained across these conditions.

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Strengthening identity in minority groups 22

The results of this path analysis are illustrated in Figure 3. Consistent with partial mediation, the path from comparative context to affective reaction in the large increase condition was reduced from β = .474, p < .001 to β = .426, p = .003 when the appraisal was entered into the model. Using 5000 bootstrap samples, the indirect effect of comparative context on affective reaction through appraisal was also found to be significant in the large increase condition, b = .213 se = .113; p = .029 (95% CI: .022, .456).

Discussion

Results of Study 2 replicate and extend those of Study 1. They confirm that prospects of a moderate increase produced significantly greater positive affect than the prospect of no increase in both intra-national and inter-national contexts. However when a large increase was projected a quite different pattern emerged as a function of context. Specifically, the prospect of a large increase was greeted with a less positive affective reaction in the intranational than in the inter-national context.

Correspondingly, the prospect of a large increase in Welsh language use increased the extent to which Welsh speakers' position would be undermined in the intra-national condition compared to the inter-national condition. Moreover, this specific appraisal in turn partially mediated the effect of comparative context on affective reactions in the large increase condition, indicating that affective responses to the future prospects for the language are at least in part explained by what participants thought that these prospects would mean for their own position within the ingroup. That this was partial rather than full mediation probably reflects the way in which affective reactions to prospects for an ingroup's future are determined by multiple appraisals. As other research has shown, these prospects for the ingroup's wider intergroup position are also likely to relate to perceptions of entitlement/legitimacy (Ellemers et al., 1993;[START_REF] Van Zomeren | Toward an integrative social identity model of collective action: A quantitative research synthesis of three sociopsychological perspectives[END_REF], efficacy [START_REF] Van Zomeren | Put your money where your mouth is! Explaining collective action tendencies through group-based anger and group efficacy[END_REF], and identity threat [START_REF] Branscombe | The context and content of social identity threat[END_REF]Livingstone et al., A C C E P T E D M A N U S C R I P T

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Strengthening identity in minority groups 23 2009b). Implications for the position of one's own subgroup within the ingroup is therefore one among several factors that influence one's reaction to future prospects for the numerical strength of a 'distinctive' attribute within in a minority ingroup. Crucially though -and unlike the other appraisals mentioned above -this tension between the inter-category and intra-category implications of changes in numerical strength has not to our knowledge been considered as a contributing factor to minority group members' reactions to future prospects for the minority category.

General Discussion

For members of minority groups, the cultural characteristics that define the ingroup can be both highly important and vulnerable. This is particularly so when only a minority within the ingroup have access to them. It might therefore seem reasonable to expect that the prospect of these characteristics becoming more widespread within the ingroup would be seen as a good thing by ingroup members. Accordingly, minority-majority relations are often characterized by struggles for cultural recognition and protection on the part of minority group members, including campaigns aimed at revitalizing or establishing 'distinct' cultural or group-defining attributes [START_REF] Bourhis | The language of intergroup distinctiveness[END_REF]Giles & Johnson, 1981).

However, the prospect of increases in the numerical strength of a 'distinctive' attribute may also be disadvantageous for those who possess it already. Although such increases may enhance ingroup identity in relation to an outgroup, it may undermine the relatively 'special' position occupied by those within the ingroup who already have that attribute. The latter should be particularly true when the attribute becomes so widespread that it is a majority attribute within the group and thereby offers little opportunity for positive distinctiveness within the ingroup (cf. [START_REF] Spears | Strength in numbers or less is more? A matter of opinion and a question of taste[END_REF].

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Strengthening identity in minority groups 24

Inter-category vs. intra-category distinctiveness: A dilemma for minority group members?

Together, the findings of the present studies support our hypothesis that strengthening ingroup identity may be something of a double-edged sword for minority group members who already have an attribute that is 'distinctive' for the minority category, but is not necessarily widespread within it. This is because it involves balancing two important psychological concerns; namely, the distinctiveness and strength of the minority group identity in an inter-category context, and their own distinctiveness in an intra-category context. As the present findings indicate, increasing the numerical strength of the minority group's defining attributes in this way is most likely to be viewed positively when there is a focus on the wider inter-category context. In contrast, a focus on intra-category relations is likely to mean that the spread of 'distinctive' characteristics will be viewed in a rather more ambivalent manner.

The contextual contingency evident here is consistent with self-categorization principles [START_REF] Turner | Rediscovering the social group: A self-categorization theory[END_REF], in which comparative context shifts the self-focus from group to subgroup. It is also consistent with optimal distinctiveness theory [START_REF] Brewer | The social self: On being the same and different at the same time[END_REF], insofar as minority groups satisfy both inclusion and differentiation needs. In the present case our context manipulation simply shifts which self-category of our participants forms the minority group (i.e., being Female or an emotionally intelligent Female in Study 1; being Welsh or being a Welsh speaker in Study 2). The present research nevertheless extends previous empirical research on minority groups, by showing that the presumed desire to strengthen group-defining properties such as a language can be equivocal, depending on social context. These findings also help to make a broader theoretical point that distinguishes our approach from others that emphasize how identifying with a minority subgroup can help to satisfy distinctiveness needs (e.g., [START_REF] Hornsey | The individual within the group: Balancing the need to belong with the need to be different[END_REF]. Specifically, we suggest that the value of being part of distinctive subgroup within a minority is not a fixed property of the

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Strengthening identity in minority groups 25 subgroup's relative size. Rather, the 'value' of being a distinctive subgroup -and of changes in that distinctiveness -depends on the shifting intergroup context, both in terms of the salient frame of reference [START_REF] Turner | Rediscovering the social group: A self-categorization theory[END_REF], and the wider power and cultural relationships within which the minority category is embedded.

The present findings also have clear relevance for a wide range of settings in which inter-group relations are characterized by struggles over the valued symbols and attributes of a minority group identity. This relevance is most direct in relation to campaigns by members of minority groups to protect and extend a vulnerable group attribute such as language or different cultural practices. These campaigns are often targeted as much at other members of the minority category as they are at a majority outgroup. The intention is to raise consciousness of 'distinctive' features of and practices within the minority category and ensure that they live on. These campaigns are, in turn, often an important feature of struggles to achieve social, economic and political equality [START_REF] Tajfel | The social psychology of minorities[END_REF]. Our findings speak to the psychological factors that can work for or against social and political campaigns to strengthen a minority group identity in this way. A key implication here is that policy makers and campaigners need to be sensitive to the intra-category concerns of minority group members, as well as the inter-category position of the minority category as a whole. If the potential impact of identity-based campaigns on intra-group relations is ignored, then such campaigns may lose support from the very group members who already embody ingroup identity. Conversely, our approach suggests that support for identity-based campaigns is most likely when there is a clear focus on the 'bigger picture' of the ingroup's relation to a majority outgroup. Here, the increased vitality of ingroup identity is likely to be understood less as an intra-group threat, and more as an inter-group opportunity for equality and recognition.

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Strengthening identity in minority groups 26

Conclusion

The present findings shed light on an important, but as yet un-researched aspect of minority-majority group relations. Although it may seem logical that members of minority groups who possess a group-defining attribute would view the prospect of strengthening ingroup identity in unequivocally positive terms, we have shown here that this is not always the case. Rather, the context in which such prospects are evaluated is crucial in determining whether strengthening ingroup identity is regarded as a valuable weapon in inter-group struggle, or as a double-edged sword that helps to divide and rule.

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Strengthening identity in minority groups 27 Footnotes 1 Removal of participants who failed this check did not change the outcome of any subsequent analyses. They were therefore retained in order to preserve randomness of allocation to condition. 2 The software used in this study unexpectedly allocated fewer participants to the international, large increase condition (N = 10) than to other conditions (next smallest N = 26).

The consistency of the findings across both studies, the adherence of the data to distributional assumptions, and the use of Type III sums of squares together led us to conclude that although not ideal, this anomaly was ultimately inconsequential.

3 Analysis of the negative affect items as a separate scale (α = .94) revealed effects that directly mirrored those on the positive affect items as a separate scale (α = .97). Moreover, the two scales were highly correlated r = .771, p < .001. Given that the effects on the separate scales were therefore unlikely to be independent, we chose to combine the positive and negative affect items into a single scale.

4 Removal of participants who failed this check did not change the outcome of subsequent analyses, with the exception of the interaction on the Welsh speakers' position undermined measure, which was no longer significant, F(2, 128) = 2.32, p = .102, η 2 p = .035.

Nevertheless, the pattern of this interaction was identical, as were the results of simple effects analyses. Moreover, the effect size for this interaction was actually greater than that of the analysis reported in the main text (η 2 p = .021), increasing our confidence that the effect of removing those who failed the check was simply due to a loss of power rather than a qualitative change in the results. All participants were therefore retained in order to preserve randomness of allocation to condition, and to avoid a potential Type II error.

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Strengthening identity in minority groups 32

  Context manipulation. The next page of the questionnaire contained the manipulation of comparative context. In the inter-category context condition, the page was headed by images of three female figures and three male figures. Explanatory text stated that "We are interested in what you, as a high EI woman, consider to be the distinguishing characteristics of women and men." Participants were instructed to provide up to three distinguishing characteristics of women in a box headed 'women,' and three distinguishing characteristics of men in a box headed 'men.' In the intra-category context condition, the page was headed by an image of three female figures only. Accompanying text explained that "We are interested in what you, as a high EI woman, consider to be the distinguishing characteristics of high EI women and low EI women." Participants were instructed to provide up to three distinguishing characteristics of high EI women in a box headed 'high EI women,' and three distinguishing characteristics of low EI women in a box headed 'low EI women.' In order to sustain the effect of the manipulation, each subsequent page of the questionnaire was headed by the images of the condition-appropriate female or female and male figures.

  not least of all from the perspective of those who already have those attributes.Our aim in Study 2 is to build upon these findings by implementing three specific refinements to the design of Study 1. First, affective reactions in Study 1 were measured solely in terms of positive emotions. This leaves open the possibility that affective reactions scores reflected affective arousal, rather than a specific and meaningful positive reaction. To address this possibility, we included negative as well as positive affect items in Study 2.

Figure 1 .

 1 Figure 1. Interaction between future prospects for numerical strength and comparative

Figure 2 .

 2 Figure 2. Interaction between future prospects for numerical strength and comparative

Figure 3 .

 3 Figure 3. Appraisal partially mediates the effect of comparative context on affective reaction

This research was made possible by support from the Leverhulme Trust (ECF/2007/0050) and the ESRC (RES-148-25-0014), and forms part of the ESRC Identities and Social Action Programme. We are grateful to Matt Leighfield and Neil Williams for their technical support, and to Nicole Tausch for helpful comments and suggestions

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Strengthening identity in minority groups 28 The relationship between frame of reference, self-categorization and accentuation.

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