Social Network Effects Bertil Hatt ([email protected]) France Telecom R&D∗, EconomiX†

the service is useless if considered by a single user, and increasingly relevant whenever widely adopted. The positive externalities for having several agents using the same service are usually labeled as“network effects”. Most models in economics have considered the global number of such users; some have split them in a finite number of classes, whether these are the type of the users, or whether the externalities across classes are distinct [8, 36]. We would like to introduce the idea that agents only consider the limited number of those with whom they share a relevant social relationship.

October 10, 2006

Previous valuations of network utility were arguably increasing too1 fast: • each user has a utility linear to the number of other users, according to Metcalfe [26]; hence a quadratic growth of total welfare, unreasonable2 with a large population; • all possible sub-sets according to Reed [35]; hence an exponential utility—even more difficult to justify at a global scale;

Abstract Economists have called “network effect” the (positive) externalities for having several agents using a service; most models have considered the global number of such users, with no actual distinction from “club effect”—and at most split the population in a finite number of classes, or into multi-sided markets. In many cases, agents tend to consider the users whom they know. This local lens induce us into the difficulties of formalizing interpersonal relations and social ties. We introduce recent findings on complex network parameters into a simple economic situation. Social networks appear homo- (or endo-)phile, clustered, “small” but focusstructured and assortative. We describe the direct consequences of these properties and their combination on the epidemic of a coordination service; we consider directions for an endogenous dynamic interaction model between social ties and communication services.

1

Introduction

To compensate for these excessive modeled growth rates, Odlyzko and Tilly [33] have introduced logarithmic scale for utility to valuation—arguing a Zipfian utility distribution between users, with no endogenous model of adoption, however. A local preference might offer an apparently simple, justified and more realistic answer—allowing a limited value increase for large and growing communication platforms. By controlling the relevancy of a network with a neighboring criterion we limit the added utility of a large communication network to: • inter-individual relations: coordination, conversational or intimate communications [14]; and (See the section 7): 1

Technologies need successive prototypes to be mastered; lobby groups try to grow their influence; communication services and social meetings depend on significant subscription. Those are examples among many networks of users: ∗ B004, † K104,

´ 38-40 av. Gal Leclerc, F-92794 Issy-les-Mx Cedex 9 TECH/LEI, Universit´e Paris-X, 200 av. R´epublique, F-92001 Nanterre Cedex

2

The metric nature of utility (and it link to financials) is an endless debate we would happily contribute to in a more theoretic paper—through the angle of the agent’s structured perception and her ability to associate with progress and to retain its benefits. Metcalfe used this rule of thumb to justify the existence of a critical mass of need for a communication technology (in an unreferenced 35mm presentation in 1980), arguing in favor of his then just developed Ethernet protocol. His formula is the simplest representation of increasing returns, proving the existence of a threshold of equipped devices beyond which his standard is worth implementing—and best fit for a limited number of close peers.

• an extension to publication-type communications for well-connected, influential individuals, for whom the frontier is not distinct3 ;

incompatible services: beyond a threshold, the dominating player would become a standard de facto. Since, Azerty keyboard layout is still commonly used in France, Wallonia and French-speaking Africa, in spite of a widespread Qwerty-like system; Apple MacintoshTM Operating System (MacOS) held against a Microsoft WindowsTM sustained domination on the personal computer (PC). Many explanations were provided [7]:

we are also looking forward to include, in a forthcoming paper using a similar framework: • global long-range relations, occasional encounters or specific needs— similar to the “blackboard” communications seen on Usenet and Web fora [15]; • anticipation of utility from new relatives: a more efficient system allows more contacts, a widely-used system demands less effort to introduce and a well-known poses less interpretation risk.

Individual Collective

• Apple’s commitment to quality, only valued high enough by the fringe who resisted the network effect: a skewed distribution of preference, sensitive enough towards high-quality not to decide on network effects; • high switching costs from MacOS to Windows system: a barrier-toentry4 ; • some decisive software, mostly for the creative user5 or a sign to belonging to a “different”6 group: those are explanations through typology; etc.

Prior to using the service, the agent knows his contacts; anticipate them. Interpersonal Match-making, communications expected new relations Asymmetric Blackboard, forum publishing

Though all these reasons might explain a part of the platform sustaining too small a share, we prefer to consider that MacOS users know similar users— should it be: • because they work with users having similar demands (homophily), or who developed mind-sets adapted to the platform conception (endogenous homophily); • because they have convinced they friends or relatives (exogenous social ties); or • because a user-community developed around the platform (endogenous ties).

Table 1: Four types of communication structure This variety of possible distance leads towards recent findings by Barab´asi, Albert [1, 3] and Newman [21, 29, 31, 38]: their models for complex network offer estimates for the number of contacts, the degree of an agent, depending on the criterion—or rather, the distribution of this degree among the population. We will now include those findings into classical network models in economics.

1.1

Are the market shares on the global level relevant when your colleagues, friends and those to whom you turn for help and counsel are in the fringe? As we will see, the constitution of a usage convention in a local social network is a chicken-and-egg problem, strengthening none-the-less a minority. Ostracism certainly played a role, on a similar fashion to the appeal for distinction [?]—but like social classes, these can only make sense if the relevant scope and result is shared by relatives.

Counter-examples to global network effects

According to Arthur [2]’s model of lock-in through increasing return, a market with network effects cannot sustain uneven shares between competing and 3

This idea an extension of Andersen’s Long tail, as we consider the “head” as the well-known writings and the “tail” as the inter-individual conservation, with a much limited public. Their nature might seem different, as the latter are not meant to be read by anyone else then the addressee, rather then no able of reaching a large audience; blogs however seem to be able to accommodate both with one technology, and several degree of publicity (Advertisement, featured by partners, referenced by search-engines, unreferenced but available to URL holders, pass-word protected, etc.).

4 5

6

2

This argument can be pooled with the previous one by considering quality as an experience good—to model an “eye-opening” rhetoric. Adobe PhotoShopTM has been much quoted for that matter, but this is hardly true anymore, though some software only accessible on MacOS are considered “better”, back to first argument on sensitivity to quality. A recent campaign used intensively this very argument.

Endogenous Exogenous

Endophily A community developed around the platform. Members of a group use the same platform.

Homophily The platform induced a relevant mind-set. Similar needs are addressed by the service.

1.2

Interpersonal knowledge needs an ad hoc definition

“Who are your friends?”—or rather: “How many friends do you have?” This innocuous question7 demands at least clarification: e. g. a friend can be someone one can remember having seen, was introduced to, whom he calls by her first name, to whom he has written at least once, to whom he has phoned during the last month, or encounters usually every week, etc. Sociology studies social ties, weighting, measuring, grouping or structuring them [43]. Recent findings —mostly by computer scientists and condensedmatter physicists, actually [39]— have been proving that whatever is the criterion, or the actual objet of study, many large complex networks tend to share significant properties—similarly to the way distributions of numerous independent additive measures converge to a gaussian.

Table 2: Four types of local effects

On the same market, but with a strategic approach: most software makers offer rebates for students, who are considered more price-sensitive, while they are building their future preference — a fine reasoning, especially as they are easy to identify. Yet: why would Apple target the academics with attractive offers for teachers? Is it also a type-discrimination in favor of a price-sensitive population (more than company-dwellers) or a way to target “social hubs”, referential influencers, whose computer is weekly seen by numerous students?

This paper organizes as follows: • before our economic suggestions, an introductory section 2 will resume the recent findings; • section 3 is an initial side-note, on a classical observation: the sigmoidal adoption rate of an innovation; a simple construction (based on word of mouth with uniformly random encounters) leads to a similar adoption curve: accelerating and then decelerating; this proves the silhouette has more due with the breakdown of users and non-users in the population, rather than a peculiar social structure, based on earlier-adopters or referential intermediaries; • after a first introduction to the general formal model is made in section 4, we shall consider the economic consequence of each characteristic of a social network for a service with positive local externalities:

Another (less explicitly economic) case is the use of certain languages: how could dialects spoken by the tiniest fraction of global population be relevant for a community of users connected to the world? Enough relations are internal to the group, so the local agreement is more relevant—should it be two dialects along a frontier, or depending on the context. Such examples are not to prove Arthur wrong on all cases of “network effect”: his model was developed with in mind the mastering of a specific technology —nuclear power plants with either heavy or light water circuits, actually— where the small number of actors corresponds to a fully connected community. Global and local hardly differ, and the total number of experimental plants is relevant to measure the maturity of the technology for any party involved. Actually, Arthur remarks himself that the European different standard proves how a similar case can face distinct externalities, when embed in another technical and industrial network.

– in section 5, we will consider clustering, and how it stabilizes services, allowing local lock-ins; we shall suggest a group-percolation approach to model the diffusion; then – in section 6, we will extend this model to non-uniform services, and will show how homophily allows niche services, contrarily to

Let us consider again the language case: geographic closeness is a relevant measure, but not the only one; human concentration in towns allows more complex phenomenon; translators and polyglots have structuring role, so have leaders and sub-groups—but the nature of the similarity can be different: neighboring habitat, ethic origin, similar craft or mind-set, etc. “Closeness” has been hardly studied in economics, probably because it does not abide a strict definition.

7

3

Lists of contacts are commonly developed on recent communication services, and managing them is an essential feature. Some social network service (SNS) offer to promote a defined number of relations, e. g. MySpace’s “Top 8”. This challenges an implicit social reciprocity rule and the debate might lead to a “ICT-native” generation to whom the resolution of these questions are part of social conventions.

small-world: given a relation criterion, one can trace an exhaustive graph— made of one large cluster if the criterion is broad enough; there is therefore, between any two agents, a shortest path; the average (or the maximum) length of such a path is called the diameter `˜ of the graph; this diameter is often surprisingly short [27]: the best-know estimation is the “six degrees of separation” on a calling-by-the-first-name based network8 . More importantly, the formal result is that this diameter increase extremely slowly with the number of members of the network [31]: es˜ ) = κ. ln(ln(N )), with κ a setting constant and N the timates that `(N cardinal of the total population9 .

Arthur’s conclusions on how the largest user base leads to a global lock-in; – afterwards, in section 7, we will consider asymmetric services, the economic model for providers of such services, and suggest insights on how assortivity favors their diffusion; and – finally, we shall consider in section 8 how focus-structure prevents from what could seems a possible small-world effect; we shall model how this favors the most simple solutions, so as the services without a closed dedicated interpretation; • in section 9 we will also consider cross-effect between those properties: are those complementary, strengthening or controlling each other? • lastly, we will advocate in section 10 for a dynamic approach i. e. how such properties and emerging technologies can endogenously change the aspect of social networks.

2 2.1

2.2

Complex networks are “small”, uneven and stable

Digital technology made it possible to collect data on larger networks, offering challenges to the small-world assumption [32], and statistical analysis has been able to isolate more properties:

Basic properties of complex, scale-free and social network

clustering: independently of explicit common characteristics, an agent’s relatives tend to know each other; however similar, this is different from endophily, and more similar to saying the relationships are generally transitive; this can be due to people introducing once friend to another, so as a consequence of endophily; it can also reinforce endophily, with strengthening mimetism; intermediated: short diameter and endophily could seem contradictory: how to reach anyone and everyone with a limited number of small changes? Endophily is however not that exhaustive and systematic; more precisely, the “shortest path” tend to go though a distinct, limited set of (often very

Complex networks are endophile, clustered

Structural sociology isolated the following properties in social networks: homophily: agents tend to prefer and befriend similar agents [25]; this can be due both to: • an exogenous mechanism, symmetrical to ostracism: similar people and more empathic, and lower one’s need to explain, of justify for himself; or simply • shared experiences: common or close path entice both common features and respective reckoning—this preference might also be designated as “endophily” (liking insiders);

8

this mechanism is dominant, even if multi-dimensional analysis tend to also isolate some forms of complementary: heterophily: no difference between agents prevent them from exchanging anything—though the distinct aspects make relationships interesting [37]; this can however be over-estimated, as a framework is needed to compare and easy to (assume and) forget when comparing;

9

4

In his seminal and much debated experiment, Stanley Milgram send in 1967 hundreds of stacks of postcards to random individuals in the USA (found in the white pages); he asked them to transmit the stack to a relative, in order to reach a person, only knowing her name and hometown. Few stacks arrived, but more than what Milgram expected; the average number of steps their took was close to six. E. g., this could mean that a similar structure (κ ' 1.93) could lead to: 9 ) ' 6.00; ˜ • a small diameter for world-sized population: `(6.10 ˜ • a not-so-small one for a far smaller sample: `(40, 000) ' 4.54; and • and a surprisingly large degree of separation for a group smaller than most address ˜ books: `(135) ' 3.06.

connected, but not necessarily) individuals: intermediaries—e. g. at the international level: diplomats, migrants or flight attendants; power-law distribution: the number of very connected individuals might explain most of network efficiency10 , resilience11 , and self-structuring capability12 ; looking at the density of degree Barab´asi and Albert [1], have found that it often follows a power-law13 of factor γ, i. e.

f (d) =

1 −γ d ζ(γ)

and F (d) = 1 −

1 d−(γ−1) (γ − 1)ζ(γ)

1 F (n)

(1)

f (n) n

γ being generally estimated between 2 and 3 [4];R ζ(·) is Riemann’s zeta function, (used for fitting R f = lim+∞ F = 1). Barab´asi and Albert’s interpretation is that the network follows a fractallike structure, with clusters at any scale of grouping: for this reason, they call such a structure a scale-free network.

Figure 1: Power-law distribution, density and degree illustration

2.3 10 11 12

13

Social networks are assortative and fuzzy-structured by focus

In addition to that, conscious14 social networks seem to present:

Among many other criteria, the structure observed tend to permit to individuals that might interact to be generally close, while having a limited number of connections to maintain. Newman [31] has modeled the possibility for a network to stay rather efficient in spite of both random and targeted attacks on links or on individuals. In Milgram’s experiment, the most striking is not the path lengths that he considers the potential shortest, but the fact that his subjects can find some of the shortest, without knowing much about their surroundings. Such ability—relying on representation and favoring potential intermediaries— have been measured by [39, 42]. Some consider the degree distribution is closer to a log-normal distribution [24, 22]. This considerable debate has three main arguments:

assortivity: agents with many contacts tend to have a larger share (therefore more) of highly connected relations; they are no decisive results yet on the actual average number of their less connected relatives; explanations for this include distinction mechanisms, the number and degree of contacts being a consequence and a signal for the social ability of an agent; this result is equivalent to a degree-homophily, with major structural consequence; a structure by focus: family, work, alumni, clubs, etc. are dimensions along witch most agents organize their relatives [13]; however universal, such axes can be partially incompatible: an employee’s wife is a acceptable subject of the frame on his desk but—however close (and interested)— she has no legitimate part a priori in deciding how he can use his phone and email for personal convenience. As social action need to be ac-

• the type of the relations considered —in particular whether there are symmetric— could justify different cases; one could imagine that each detailed type distribution follows a power law, and that the sum of them all, the general contact distribution, follows a log-normal distribution; • the distribution could be combination of several standard function, with thresholds —a refinement of the previous argument; most cases have been described as log-normal “bodies” (the central part near the mode), with a power-law tails (the few most connected are significantly more frequent than what a power-law would forecast); • an understanding of the actual mechanisms involved in the building of the relations would be more helpful that econometry: log-normal distribution appears with multiplicative combination of many random events, while power-law needs more skewed mechanisms [28].

14

5

Newman has compared assortivity coefficient of several networks[30]; two groups emerge, with respectively significantly positive and a negative coefficient, respectively; the main distinction between them appears to be that in the first case, agents are conscious and decide to establish and maintain a link, while in the other case, structure is “spontaneous”.

counted for [5], this can prevent an adoption outside of an accepted focus. Pervasive information and communication technologies (ICT) rendered the coexistence of such foci possible: e. g. employees working on their laptop at home, and answering personal calls on their cell-phone at work.

1 p(t)α=3 D1

p(t)α=1 D2

However insightful, those properties are not always needed to explain nonlinear social stylized-fact.

3

Adoption can be sigmoidal without local preference

D1

4.1

1 1 + e−α(t−D)

D∈R

N = [[1, N ]] the set of the agents in the population, arbitrarily numbered; {Xk }1≤k≤m all the foci and {RXk }1≤k≤m the corresponding social relations on N ; except in section 7, RX is reciprocal, i. e.:

(2)

(3)

i RX j ⇔ j RX i;

The interest of the local approach we are pleading for is therefore not so much to prove that there is an irregular adoption rate —accelerating then decreasing— or to pinpoint a reasonable function to model it but rather to isolate the different implicit aspects and to understand how important each is and why they evolve.

4

16

(4)

and generally not (completely) transitive; b = {j|∃i ∈ X, iRj} the set of relatives of all the members of a given set; X Granovetter [16, 17] introduced “tie strength”, the weight of each relation: k dX i→j is the intensity of the relation between i and j (if R is symmetric) in the context Xk —or the interest i has in j otherwise; this formulation is ego-centric, and can consider asymmetry17 of the focus-structure.

General model of our research

As our research associate two different fields (quantitative sociology and industrial economics) we need to adapt the existing notations. 15

Formal social model

To formalize social network, let us consider:

The solution15 to that is the family of logistic16 functions of factor α: p(t) =

t

Figure 2: Logistic function graph plot

Let us consider an appearing service that relies on word-of-mouth (an experience good, or has to be known and does no central marketing); let p(t) be the share of adopters in the total relevant population. Agents meet each other uniformly and randomly, and one adopts the service if he is a non-user (1 − p(t)) and comes across a user (p(t)).We sum up the adoption probability and encounter rate in a ratio α: p0 (t) = α p(t) (1 − p(t))

D2

17

See proof in the appendix. Note that α can change with time t, provided it is constant by pieces. The adoption p(·) is then “logistic by pieces”.

In the case of incompatible structures, we should have actually introduced an ego-dependent i —but we would then have needed to number of focus mi and user-specific foci X1i , . . . , Xm i represent all possible agreement on what the foci covered with equivalences: Xki = Xkj0 . This would have been too complex for this paper.

6

4.2

4.3

Formal utility model

For our notations to be coherent,

Let S ⊂ N be the set of subscribers to the service S: Our general approach is to consider the benefits for the user i to subscribe to a service S, in the contexts, or foci, X1 , . . . , Xm :   X X X k   uSi = aSi .di→j − cSi (5) 1≤k≤m

j RXk i

j∈Sb

k dX i→j 6= 0

Xk k dX i). i→j = 1(j R P P Xk k As dX i→j are dummies, we can simplify 1≤k≤m j∈Sb di→j to be naturals; therefore let:

• a closed service can only be reached by users of the same service: Sb = S—e. g. closed IM; and • asymmetric communication, or publication, is designed for the message to be widely available: Sb = N ; e. g. billboards for advertisers, an editing and publishing facility;

di = #{j|iRj} be the degree of an agent, her number of contacts; and dX i = #{j|iRj; j ∈ X} the degree of i within the set X. uSi = aSi .dSi − cSi

(6)

b

On a more general level, and according to this simplification, let

these are the two cases we’ll consider in this paper, but further research shall look into two other cases with several services: • a service can be an extension of an other service S 0 : Sb = S 0 , e. g. SMS over mobile telephony; and • a service can be compatible with other services S1 , . . . , Sp , e. g. S several providers abiding the same standards: Sb = S ∪ 1≤`≤p S` .

FX be the distribution and fX the density of the number of contacts of members of X: FX (d) =

#{i ∈ X|di < d} , #X

fX (d) =

#{i ∈ X|di = d} ; #X

(7)

Y Y FX be the distribution and fX the density of the number of contacts in Y of members of X:

Each following section of this paper simplifies this equation to consider separately the impact that social network properties have over certain aspects of it.

Y FX (d) = 20

19

⇒

Because most studies defined a unique criterion of relation, few results have been conclusive about the intensity of the relation and the network structure emerging so far. Published findings with relation intensity [9, 10] concentrate on simulations, and are not modeled after the frame we use. Quantified conclusions on focus-structure are difficult to imagine. More intense relations are decisive in adoption20 but we chose here to simplify by only considering one level of relevant relations—therefore, our pak rameter dX i→j will mostly be a binary (“dummy”) variable in this paper:

aSi is the utility of the service: capabilities of the service (format, technology, usability, reliability, etc.), net of the marginal costs (financial and availability18 ); S ci are the costs for a user to subscribe19 to a service: efforts to subscribe and to understand the service, or even bundled-in advertising for free services, like certain instant messaging (IM) client; Sb is the frontier of S, the set of agents that can be reached by the service; we shall consider four cases:

18

Simplification with binary relations

The schedule to maintain relations is constrained, and hence time is a scarce social ressource— evenly distributed in the population. This intuition drives endogenous social networks [20, 19]; however, we will not consider this until a second paper. As we will note in section 7, not all these costs mean revenue for the provider.

7

#{i ∈ X|dYi < d} , #X

Y fX (d) =

#{i ∈ X|dYi = d} . #X

(8)

Haythornwaite [18] observed that closer relations meant more intense and diverse communication means, and more frequent and bold experimental attempts—risk-free, as these are covered by more empathy, confidence and other safe media: e. g. you probably checked that the fist e-mail that you sent arrived, by phoning or meeting with the recipient, who was a close relative.

5

Clustering and niche stability

a full-group can only be reached through an explicit shared belonging and commonly used information circuits. Let us consider a group B of n agents; this group can unanimously subscribe to a service S (that a member has knowledge of) if n ≥ δ + 1. A group B 0 can be connected enough to B to have interest23 in subscribing 0 (δ) = 0. Knowing so, they should subscribe too; together, the too: FBB∪B 0 two sets are stable subscribing base. B 0 , contrarily to B, can include less than n members: what is relevant is that each one has enough ties to B and B 0 .

Networks are clustered, i. e. organized by sets with more internal than external ties. Formally, if R is a social relation, all agents belong to at least21 one cluster:  i∈A (9) ∀i ∈ N , ∃A : FAA < FAAc

5.1

Local stability D

Let us consider a simple model with closed service S (Sb = S) and a uniform quality (aSi = a) and price (cSi = c); the minimal number of subscribed contacts for ego to have interest in subscribing is therefore δ = d ac e. If a set A is dense enough for the service, i. e. if FAA (δ) = 0, then ∀i ∈ A, A \ {i} ⊂ S ⇒ uSi = a.dSi − c ≥ a.δ − c ≥ 0

E A

G B C

22

H

We therefore can describe the diffusion of aSservice into a connected graph in Q steps, without risky anticipation, along ( 1≤i≤q B (i) )1≤q≤Q . This result encourage us to consider the group structure for costly products that need consideration and are worth discussing with relatives, e. g. telesurgical equipment used for specialists to co-operate, or going to a night-club. In general, expensive, closed, improvements on already possible communication depend on large enough fully-interconnected communities. By “group structure”, we imply to study groups as summit of a social meta-graph; they should be considered connected if they share members— or weakly connected if members know each other. The interpretation would

An active literature compares algorithmic methods to isolate those (from raw relation data). However, most of these programs assume an agent cannot belong to several clusters; Pissard [?] considered over-lapping. A following paper should try to link the cluster structure to foci. Dupuy [12] defines an information I as being of common knowledge (CK) among the population N = [[1, N ]] of agents (CKN (I)) if ∀(ij )1≤j≤k ∈ N k ,

C

F , G or H are not part of a group with more than 3 members. However, if the group {A, B, C, D, E} can adopt to a service with δ ≤ 4; F , G or H could then subscribe together.

An expending kernel

∀k ∈ N,

F

B

Figure 3: Two examples of a group of cardinal n = 5

Let us define a group as a fully-interconnected cluster: a group of n members therefore implies tr(n) = 12 n(n − 1) internal relations. We assume a group is self-aware, and information in the group is of common knowledge22 : this is easy to obtain for smaller sets, and our assumption for larger groups that 21

E A

(10)

and subscribing is a Nash-equilibrium for all agents in A. Hence, a service used de facto within a group A of the population can be locally relevant—even though it is a minority a priori representing too small a market share to be sustainable according to Arthur [2]; the intuition of the lock-in is exactly his, however. But how could such a service be launched? We consider two approches: a expending kernel of users; or anticipation mechanism.

5.2

D

Ki1 (Ki2 (. . . Kik (I)))

23

with Ki (I) signifying: “The agent i knows I is true.”

8

Once again, shared knowledge of the service (and of the intention to adopt) is needed to agree on a subscription equilibrium.

Note that any subset of a group is a group:

be more coherent, as each group should belong to one focus: two groups sharing a member could then be considered distant if their respective focus are not compatible, while two groups sharing a significant ratio of members, however based on different foci could have a de facto social legitimacy, e. g. a family making up half the staff of a company. However, the set B 0 in the graphic example in figure 3 is not a group, and such extension could be decisive in the diffusion of a service. In addition to groups of n members, we need to consider the sets in which any member as at least n connexions. The coordination within such sets could either rely on one or several subset groups —like the case we considered— or on a softer self-awareness. Such mechanisms claim a much more elaborate rationality model; the network statistics in three dimensions (size of the set, minimum internal degree, number of subset groups of the same) would suffice, and these imply difficult coordinations, with a priori little added value from the group structure. We would therefore favor:

∀G0 ⊂ G, ∀(i, j) ∈ G0

(i, j) ∈ G2 ⇒ i R j.

Let us define a shared user metric: h(G, G0 ) = #(G ∩ G0 ); and assume that the probability that a service S used by G is thereby used adopted by G0 , is proportional to the number of common members: S

c

P(G → G0 |G ⊂ S) = pS h(G, G0 ) 1(G0 ∈ G a )

(13)

Common agent are treated as vector. We could introduce a dependence to #G0 more subtle than a binary test, though this would involve endogeneity issues, and dynamic discussions how whether: • a greater benefit triggers faster adoption; • larger groups have more coordination issue in spite of our assumed common knowledge.

• either to consider the present group-approach, with a possibly less demanding criterion to allow “weal ties” that are usually transitive relation [16]; handling the accointances of each of ego’s contact is a considerable cognitive task: most agents resolve it by figuring coherent groups, generally near to be self-acknowledging, and thereby almost in common knowledge; • or the forthcoming anticipation based adoption, with better anticipation thanks to endophily.

5.3

2

A model of group percolation Figure 4: Connected groups and possible service paths

A service with a δ = ac ≤ n threshold can spread among connected groups larger than n members. We assume our groups are coherent (of the same focus) and a we have a relevant connection criterion (sharing a common user or having at least a pair of acquainted members). Note that our groups can overlap. Let G n be the set of all groups of size n:  G n = G ⊂ N |#G = n, ∀(i, j) ∈ G2 , iRj (11) Let G n be the set of all groups of size at least n: [  G n = G ⊂ N |#G ≥ n, ∀(i, j) ∈ G2 , iRj = Gν

From this percolation model, we should want to obtain similar statistics on groups than what we have on individuals: is the meta-network random, clustered, scale-free? There is no published results from actual data, and simulation is not relevant. To obtain such a base, we need actual data, and to adapt the only algorithm we know of to isolate over-lapping groups [?]. Then, we will be able to apply percolation theory [29] to the characteristic of a service with its possible diffusion among the groups of users. The two newt subsections frame our future perspective, to soften our group model.

(12)

ν≥n

9

5.4

Note that A and DiS are not independent; as both rely on multiplicative mechanisms, their outcome could follow a log-normal distribution, and so could their product. As the adoption is not definite, we need to introduce a penalty26 −k for unsubscribing: e. g., i has to warn his relatives, certain e-mail providers do not facilitate the export of the archives and address book; i’s utility to subscribe is therefore the random variable Ui = max(−k, A.DiS − ci ). With this model, we can now consider risk-averse agents. Linking this to agents’ degree would be an interesting insight—though difficult to get, as large social network data are often based on ICT-server logs, and risk-aversion is delicate to get from phone bills or semantic analysis of MySpace profile. Let us alight that, especially for risk-averse agents, Ui is decreasing if k, the cost to unsubscribe or switch is high. This effect could be stronger than the direct barrier-effect; actually it operates before it, as this reasoning happens when considering adoption. Therefore, lowering barrier to switch could be rational when facing a credible competition to come: however apparently counter-productive, it could attract risk-averse users, and help building a larger user-base with them—a decisive asset on markets ruled by network effects.

Anticipation-based growth

If a service S(a, c) is well known and understood, and its adoption perspective are clear, an agent i subscribes if his expected utility is positive: ue Si = a.di .F e {i} c

c a

−c

(14)

with F e {i} c (·) an estimated distribution of his contacts’ degree. However, some products involve too little cost to be worth the effort of a common decision, or are experience goods that cannot be valued prior to usage; or, with legitimacy constraints, only the information derived from a try out justifies to mention them to one’s contacts (in order to then coordinate on a joint adoption): e. g. downloading the client application of an IM. This second frame is similar to a percolation model, with the degree as the main variable. It does not takes into account the numerous uncertainty associated with innovation adoption: as the best approach to standard diffusion is scenario-based24 , we would prefer to consider a model with random-variable. Let us consider a uniform service S with known costs cSi = c and random benefits per co-user A. This can be because the service is still developed (“Beta”) or because the supporting company face uncertain future; the service can be dependent on social conventions more complex than just adoption, e. g. is it rude to send an SMS in such occasion? An agent i anticipates that on the long run, S will have been tried and discussed enough to be adopted by his relatives if relevant to them, i. e. if they have enough contacts25 . For each contact, he considers her subscription probability, and sums the outcomes to built a random variable of the number of subscribed contacts:   c  DiS B di ; F e {i} . (15) c A 24

25

5.5

Single-homing and multi-homing competition

With this simple model, competition between two services, if subscription to one and only one is necessary, is similar to considering the adoption of a service with cost and benefit being the difference between the two previous ones. The economic question is the same variation of a coordination game with complex interdependence. Non necessary adoption and more than two services increase the coordination problem, but not the fundamental result, which is that local settings are stability, provided they follow cluster structure. If multiple subscriptions are possible —what two-sided markets literature calls “multi-homing”— then we need to distinguish two cases:

Many agents refuse to adopt a technology which they might not understand, even though their expected utility is positive; this could have something to do with avoiding any negative selfrepresentation (failing to understand), or with rational framing issue: estimating the service utility demands to expand one’s mindset to understand what it does—and any estimation prior to a try out is misguided. We should deal with such issues in a coming paper on rationality and innovation in ICT. This could be parameterized by i’s second order contact likelihood to adopt, as we’ll see in the homophily section.

multi-dimensional each channel has its purpose, and the benefits add-up; besides from a common budget constraint, this corresponds to resolving a multiple case in parallel; 26

10

We could have considered a random variable −K for penalty, or a socially increasing cost k(di ).

platform competition only the best communication service is used, and only the maximum benefit a between two users is to consider in their total utility; this model is relevant to consider an agent that has to coordinate between several local stable solutions: he might consider using both, in spite of the sunk costs; if so, he has to split his network between each standard (other multi-homers are excluded, as their can rely on another platform), and each subscription is maintained as long as his relatives are numerous enough to justify to use the service.

late age and degree, as some assortative models do, then homophily (as the ability to empathize with one’s relative) favors the rational adoption of most efficient spreaders, fastening innovation.

7 7.1

Asymmetric service and assortivity A degree-as-type model 0

6

We now consider an open service S 0 with a constant cost cSi = c on an asymmetric focus X 0 . As Sb0 = N , each agent who has knowledge of the service subscribe if he has enough contacts, independently of their subscription:

Homophile ties favor specific services

We have considered uniform services so far; most services are actually aimed at a given user-stereotype and adapted: the interpretation costs are lower and the utility per co-user higher for users similar to the intended target. All our findings can accomodate such a context, especially as homophily makes most clusters rather coherent with a given valuation (ci , ai ): for nonuniform service, the ratio of cost to benefit-per-known-user inside a group generally follows a concentrated distribution. If a group is not so unanimous,
we can applyc our findings by considering the largest ratio in the group: c i a G = maxi∈G ai , and compare it to group size; subgroups of users with higher valuation or lower cost might then be more interested in the service than the largest group itself. Homophily on a service valuation strengthens cluster convention; does it improve stability, though? For a service adapted to a target, adoption threshold are lowered, and the diffusion of the service is locally faster: better or more adapted services can overcome a previous version more easily because of homophily. A service that can be personalized (modeled as a large feature area, and cSi being the distance between i’s preference and S offering) is best to take advantage of that easier adoption. Hidden or unexpected personalization can make he limited heterophily a decisive barrier, while it generally constitutes a rare path, but a precious bridge to new users pools—highlighting the importance of a legible and open offering. Homophily is more about being able to understand and interpretation each other [37]: once that necessary step acquired, heterophily is richer. This ability to represent one’s relatives favors a realistic F e {i} c —and experience make agents more accurate and confident in their anticipation frames. If we corre-

uSi

0

0

= a.dX (16) i −c lcm 0 0 ⇒ uSi > 0. (17) therefore dX ≥ δ0 = i a With public knowledge of the service, this model is equivalent to a typebased representation, with δ the adoption threshold degree. We assume the provider retain all the subscription costs: this includes subscription fees and advertisement revenue (cost then being the hassle of advertisement, marginally equal to this revenue on an efficient market for ad-space) but it excludes cognitive efforts—reasonably as we mainly consider “powerusers”, with many potential readers. The provider’s profits with one exogenous technology a and with connection costs ξ per user is:   c  Πexo (c) = 1 − F (c − ξ) (18) Op a which is maximal for c verifying:   c  1  c  1−F − f (c − ξ) = 0 (19) a a a that is when the price impact equals the increase of subscribers: c c − ξ c 1−F = f (20) a a a and with a change of variable, d = ac :   ξ F (d) + d − f (d) = 1. (21) a 11

f (n) = k.n−2.5

This equilibrium is a classic contract theory result; however, with a powerlaw density (see equation (1)) f (d) =

1 −γ d ζ(γ)

and F (d) = 1 −

1 d−(γ−1) (γ − 1)ζ(γ)

  1 ξ 1 −γ d−(γ−1) − d − d (γ − 1)ζ(γ) a ζ(γ) γ−2 − d−(γ−1) (γ − 1)ζ(γ) γ−2 d (γ − 1) Therefore d∗ = ξ

=

If γ = 2.5, 1 +

=3

0 u = a.n − c∗

ξ 1 −γ = − d a ζ(γ) ξ = a

c∗ = 3ξ

ξ

1γ−1 and aγ−2

d

Figure 5: Power-law distribution and benefit 

1 c =ξ 1+ γ−2 ∗

 (22)

The optimal price is independent of a, the quality of the service, but not the profits: the better the service, the more subscribers.

Hence, in a focus with a rather regular network (γ < 3) the optimal price of the service can be several times the costs; for a much more centralized network (less, more important, experts), the provider of a publication service has interest in offering a more competitive price: the elite interest is clear, but it represents few subscribers, while cheaper price entice more to consider the service even more rapidly. Πexo Op |c=c∗

1 γ−2

γ = 2.5

 ∗  c 1−F (c∗ − ξ) a  ∗ 1 c f (c∗ − ξ)2 . a a

 = =

Πexo Op(c) u = a.n − c∗

with a power-law distribution, Πexo Op (a)|c=c∗

= = =

1 1 a ζ(γ)



c∗ a

−γ 

ξ γ−2 −γ 

 1 ξ(γ − 1) aζ(γ) a(γ − 2) aγ+1 (γ − 2)γ−2 γ−2 ξ ζ(γ) (γ − 1)γ

2 ξ γ−2

ξ

c∗ = 3ξ

d

Figure 6: Provider profits from subscription cost

2

Let us consider an endogenous quality of service: a company needs to spend the efforts ρ(a) to reach technology a; therefore its profits are worth: 12

Πendo Op (a)|c∗ =

aγ+1 (γ − 2)γ−2 − ρ(a) ξ γ−2 ζ(γ)(γ − 1)γ

φ(γ)

(23)

Therefore, we can establish a profit maximizing quality a∗ : d endo ∗ Π (c , a)|a∗ = 0 da Op a∗ γ (γ + 1) (γ − 2)γ−2 − ρ0 (a∗ ) = 0 ξ γ−2 ζ(γ) (γ − 1)γ a∗ γ ξ γ−2 ζ(γ) (γ − 1)γ = ρ0 (a∗ ) (γ + 1) (γ − 2)γ−2

ξ = 0.9

2

ξ = 0.7

1

ξ = 0.5 ξ = 0.3 1

2

3

4

5

6

γ

Figure 8: Monopoly quality, function of R&D factor λ

Let us consider exponential costs of innovation, i. e. decreasing marginal R&D output: ρ(a) = λaλ with 1 ≤ λ (24)

to innovate λ:

da∗ da∗ >0 >0 dξ dλ The effect of the network skewness γ is not clear:

1

a = cinv. λ 2

ξ = 1.1

(26)

λ = 2.2 a∗ λ = 2.8

1 da dc

1

2

= cinv. −

3

4

1.5 ξ ξ ξ ξ

λ−1 λ

1

ρ(a) = cinv.

= 1.1 = 0.9 = 0.7 = 0.5

ξ = 0.3 Figure 7: Quality as an exponential function of R&D investment γ−2

2

γ

(γ−1) For simplicity’s sake, let us define φ(γ) = ξ(γ+1)ζ(γ) (γ−2)γ−2 ∗γ a Following , ∗ λ−1 = φ(γ), therefore: a p a∗ = γ−λ+1 φ(γ)

3

4

5

6 γ

Figure 9: Monopoly quality, function of network skewness γ Note that limγ→+∞ a∗ = ξ: a completely central network is priced at its actual costs, with no R&D effort. There is a maximum in γ, before witch the optimal quality is increasing; after that (for an increasingly skewed network), it decreases towards ξ.

(25)

Our first result is that the optimal quality of the service is increasing both with the base cost of the service ξ, and—paradoxically— with the difficulty 13

Let us assume that some gents would use S1 : δ1 < δ2 < δ1/2 . Any agent with more contacts then δ1/2 subscribe to S2 ; any agent i with δ1 < δi < δ1/2 subscribes to S2 ; any agent with less contacts than δ1 doesn’t subscribe to any service. Therefore the profit of a bi-service monopoly27 is:

Similarly, the quality should increase for more irregular networks—a better service encourage a larger user-base; when the network is very irregular, the impact of small amelioration on the user-base is greater.

7.2

Differentiation and competition

S

Differentiation with separating services should allow to target different classes of users, adapt the price accordingly, and attract more rent—similarly to most type-based models. A provider offers two levels of services S1 (a1 , c1 ) and S2 (a2 , c2 ) with a1 < a2 and c1 < c2 . Any agent with more contacts then δ1 = ac11 (resp. δ2 = ac22 ) would use S1 (resp. S2 ); however, any agent with more contacts then δ1/2 = c2 −c1 a2 −a1 would prefer S2 over S1 .

1;2 ΠOp (a1 ; a2 ; c1 ; c2 )

=





d S1;2 Π dc2 Op

=

F (δ1 ) − F

(1 − F (δ1 )) (c1 − ξ) + (F (δ1 ) − F (δ1;2 )) (c2 − ξ)

c2 − c1 a2 − a1

 −f

c2 − c1 a2 − a1

 (c2 − ξ)

With a given a base-service (c1 ; a1 ) and the quality of the better service a2 , S1;2 |c∗2 = 0, the optimal c∗2 maximize the profits: ΠOp   ∗   ∗ c2 − c1 c2 − c1 +f (c∗2 − ξ) F (δ1 ) = F a2 − a1 a2 − a1

ui

uSi 1 = a1 .di − c1

And, considering power-law distribution: uSi 2 = a2 .di − c2

ui = 0

1−

1 −(γ−1) δ (γ − 1)ζ(γ) 1

=

a2

a1

c2 −c1 a2 −c2

c2 a2

c1 a1

di

c1

ΠOp c2 .f



c2 −c1 a2 −a1

i. e.



−(γ−1) δ1

 ∗ −(γ−1) 1 c2 − c1 (γ − 1)ζ(γ) a2 − a1  ∗ −γ 1 c2 − c1 + (c∗2 − ξ) ζ(γ) a2 − a1 1−

 =

c∗2 − c1 a2 − a1

c1 .f



c1 a1

−(γ−1) 

− (γ − 1)

c2







27

Figure 10: User valuation of two exclusive services 14

c∗2 − c1 a2 − a1

−γ 

c∗2 − c1 a2 − a1

 c∗2 − c1 ∗ − (γ − 1)(c2 − ξ) a2 − a1

−γ

(c∗2 − ξ)

−(γ−1)

= δ1

We consider two services of different quality with a single cost: this is coherent with having a full and a degraded service.

We recognize the ratios of a linear, two-products differencing model, combined with a non-linear demand distribution.

one focus, and seem illegitimate in other cases; e. g.: why a fax machine was always used for professional needs, and hardly ever, for instance, for parents to send to the grandparents a copy of that precious latest drawings that their 6 year-old grandchild made for them? Terminals were expensive, and the investment overcame budgetconstrained families, but the better-off families did not considered this usage, neither did industrials (that could have lowered the terminal price relying mass-market sales) of network operators (by cross-subsidizing with communications). Our benchmark, the available budget for family-talk, was not low then, as it included often long-distance, hour-long expensive phone conversations. While the small-world rely on criteria of a relation that are both general (for “neutrality”) and broad (to have a single, large, cluster), the actual diffusion of adoption abides more constraints. A jump between focus can happen, particularly if the same service is used by relatives in different foci, the connotation can be challenged; a universal service then can follow a much more efficient small-world like network. Let us consider a formal model : an agent i has relatives he knows though a finite set of contexts or foci, i. e. of possible relationship type: family, work, alumni, fishing buddies, etc.: Ci = {RF , RW , RM , . . .}. Each of these relation type imply a (possibly different) network structure28 but, more importantly, each of these structure is reckoning itself, and is auto-exemplifying— this meaning that justifications29 and explanations are costly outside a focus. Let us consider three cases :

Competition without socio-dynamic aspect is therefore similar to any typebased model—except the distribution of users is not uniform. For further development, we consider to model an endogenous knowledge of the service by the users: subscribers warn they readers, listeners or viewers about the service. In this frame, assortivity fasten the adoption by the most connected, hence the general adoption rate. Our goal would be to reach a suitable model for the economics of hosting blogs—and to explain their spectacular adoption rate thanks to degree-assortivity.

8

Focus-organized world

There are no universal list of foci, though common distinctions can be implicit. Blurred lines allowed increasingly frequent clashes between dimensions, proving how incompatible they can be; more importantly, a debate can appear on the lines, and whether such and such aspects are distinct or not. E. g.: a husband can consider a mistress has nothing to do with family life, while his wife believes her position as an exclusive partner is challenged; they would probably argue about her existing, but they also might disagree on which dimensions of their life has been implied. What interest us most in the multiplicity of foci is how it resolves the striking small-world result with the observed slow adoption rate of any innovation: if it takes five minutes to start a blog and tell all your friends, or an hour to buy a cell phone and become addicted, why would these services take years to spread? It should have taken six degrees of separation times five minute or an hour—less than a day. Even by including time to call all one’s friends, scarce availability, the diffusion rate doesn’t match the surprising short diameter— while word-of-mouth can take up in a thousand-more group in hours. We believe this is due to transitivity: passing the word about a story involving such teacher or student, such colleague or cousin, is a legitimate action inside the school, company, family. However, someone will never ask all his contacts (parents, boss, tax advisor, etc.) to be his “MSN buddies”: only certain foci make sense there—and the relation network of these foci is not as “small”, as there is hardly any far better connected individual. A relationship service, associated with the relation, can often be tied to

• if an agent uses a service S only with contacts all belonging to the same focus X , he cannot imagine it working elsewhere; • if an agent uses a service S with two different contacts, each belonging to different focus X and X 0 , he can make an effort e to have it accepted in any focus where it would be efficient; 28

29

15

We are considering another paper on the specific question of structuring social communities by clusters, hierarchies and partition, and the isolation of foci. Hardly anyone has extremely numerous close family members —though those who do make an network clearly assortative by degree— while some people know hundreds though clubs (that are generally homophile); asymmetric work hierarchical relations draw a network with executives as major hubs, but the longest diameter if we consider numerous level of management. For a more detailed structure of justification process, see De la justification (only available in French) [5].

9.2

• if an agent uses a the same service S with the same contact in different focus, he freely assumes it is legitimate in any focus.

“Intermediaries”, on whom the small-world result depends, and who allow a service spread between foci, are often well-connected: the same who can encourage an extension are most likely to have it fit in a bigger picture. Based on the same association (yet to be proven) we can assume services that are convenient for people with many contacts, or who need to manage completely different worlds are preferred by these decisive individuals—therefore favoring asymmetric, neutral and efficient communication means. Would such intermediaries be considered more aware of interpretation issues, they should then prefer symmetric communication means: asymmetric services would then preferably develop along a clear, single focus.

Some network structures are more efficient for spreading; some are more stable. A service that can accommodate an assortive network and succeed in expanding to a less efficient homophile network is likely to have a rapid adoption, with a sustained usage. Our key result is however that ambiguity between the focus is essential to help agents figure out that services are not necessarily better at what they were meant for. If we consider endogenous relationships, communication services that help managing more focus increase their awareness, and reduce the bridges. As we will argue in a coming paper, this might be the only slowing down mechanism for more services, faster adoption and better tailored personalization.

9 9.1

Diffusion among foci depends on intermediaries

10

A dynamic approach would understand longer-term stability

In this paper, we assumed social networks had a constant structure: current research are focusing on this aspect:

Cross-properties strengthen these effects

• the Pew Internet & American Life Project [34] has isolated some of the evolutions due to more pervasive, rich, efficient communication services, isolating several forms of connection; • Donath and boyd [11] describe new ways to introduce one-self on multiformat, but new and constraint media; • boyd et al. [6] looks into “Social network services”, websites inspired by some of the results that we used; they try to recreate networks by managing explicit relations; as ambiguity was an essential part of forming a relation, a new sociology should emerge from the generation30 who practiced making friends on those site first.

Clustering reinforces our local conclusions

Endophily and clustering are closely related; we have been able to make a theoretic distinction, however, they are often impossible to tell from actual data. This chicken-and-egg situation challenges most interpretation attempts [23]. Our findings show nevertheless that both effects strengthen their respective impact on local stability. As assortivity is a form of endophily, similarly to this reinforcement, our expected results on the rapidity of the deployment of a publication service (asymmetric, preferred by the most connected) should be strengthened: the better connected shall hear about it even faster, as soon as it reaches their clusters, and spread it more efficiently. However, the diversity of foci slows these mechanisms: clustering favors inconsistent interpretation of a service, and might prevent an adoption. For instance, journalists and thinkers were fast to adopt blogs as an almostpublishing service, but they could not “understand” its adoption as a strictly personal and intimate communication medium.

We expect — but are eager to be proven wrong: • a more liquid network, with easier connexions and easier handling of distant relations; 30

16

As both the extend (measured by the number of users in the user range, up to 90 % for Facebook [40]) and the novelty of the behaviors (leaving on-line and freely accessible details, intimate feelings and sometime offensive or promiscuous material [41]) are spectacular, we are probably facing a major change in communication behavior.

• faster, less demanding fame, and even faster loss of interest; • the development of niches, and of a global view—leading to more acute foci distinctions, debates and contradictions; • not necessarily a larger share of drop-outs, but a widening understanding gap; • more complex and structured institutions and conventions to regulate communication behavior.

[9] Fr´ed´eric Dero¨ıan, Morphogenesis of social network and coexistence of technologies, Grequam, May 2000. [10]

, Formation of social networks and diffusion of innovations, Research Policy 1331 (2001), no. 1, 1–12.

[11] Judith Donath and danah boyd, Public displays of connection, BT Technology Journal 22 (2004), no. 4, 71–82. [12] Jean-Pierre Dupuy (ed.), Self-deception and paradoxes of rationality,, stanford university ed., C.S.L.I. Publications, 1998.

These will demand more, better-tailored services, more efficient technologies, matching higher interpretation abilities. Endogeneity between social structure and communication services could lead to more complex evolution.

[13] Scott L. Feld, The focused organization of social ties, The American Journal of Sociology 86 (1981), no. 5, 1015–1035.

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[31] 2003.

Resolution of the equation 2

[32] Mark E. J. Newman, S. Forrest, and J. Balthrop, Email networks and the spread of computer viruses, Physical Review E 66 (2002), no. 3, 03510.

If p(t) = 18

1 1+e−α(t−D)

with D ∈ R, then

p0 (t) αp(t)(1 − p(t))

=

0 − (−α)e−α(t−D) (1 + eα(t−D) )2 1 + e−α(t−D) − 1 1 + e−α(t−D) 1 + e−α(t−D) αe−α(t−D) = p0 (t) (1 − e−α(t−D) )2

= α =

1

(27) (28) (29)

Therefore p(·) is a solution to the equation, so as the limits for D → −∞, p(t) = 0 (i. e. no adoption) and for D → +∞, p(t) = 1 (i. e. the service has always been permanent and universal). Furthermore, there are no other solutions (besides these two trivial constant limits) as the equation are separated31 , and p(R) =]0, 1[ About the author Bertil Hatt has been a PhD student for a year, under a joint research contract (Cifre) between: • EconomiX, the economic lab of the University Paris-X – Nanterre, under the ´ Brousseau; supervision of Professor Eric ´ at France Telecom R&D, under • the Laboratory for Innovation Economics (LEI) the responsibility of Jean-S´ebastien Bedo. Prior to that, he was a energy-market consultant. He holds a MSc in Statistics and ´ e) and an MPhil in Institutional Economics (U. Paris-X). This Microeconomics (Ensa´ is his first contribution to an international conference.

31

By “separated” we mean that the equation can be written as: p0 (t) = g(p(t)).

19