In this chapter, we consider several graph-theoretic and probabilistic models

Transcription

1 THREE ONE GRAPH-THEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graph-theoretic and probabilistic odels for a social network, which we do under different assuptions related to two basic paraeters stated earlier in Chapter 1 naely, the nuber of vertices (n) and the nuber of arcs (). We will take a social network to be a digraph. Models analogous to soe of these can also be considered for graphs and weighted digraphs, although we will not discuss the in detail. At the outset, we ention that when we talk of odels in this chapter, we do not iply that any of the is a typical realistic representation of the situation in real life. We are not trying to build or present such odels. Rather, the odels we present can be used as a sort of null odel with which one can standardize soe of the paraeters or statistics in the underlying social networks. Each odel generally stipulates in soe way the set of all possible digraphs of which the observed digraph is an eleent. A statistical odel, oreover, assigns a probability distribution, depending on soe paraeters, over the class of all possible digraphs. For soe purposes, one ay use a probability odel that does not copletely specify the probability of each possible 53

2 54 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS digraph. By a graph-theoretic odel, we siply ean a odel that uses soe digraph paraeters and is not probabilistic. The general procedure we adopt for obtaining a standardized easure of any characteristic of a social network is as follows. We start with an initial siple real valued easure X, which is a function of the observed network, for the characteristic under consideration. We then choose an appropriate odel and standardize X as follows. If the odel is statistical, we take P(X x) to be the standardized easure, where x is the value of X for the particular network observed. This easure lies between 0 and 1 and can be converted to a percentage by ultiplying by 100. In case the distribution of X is not known, one can still use (x E(X))/σ (X) as a easure and can get soe idea about the tail probability fro Chebychev s inequality. If the odel is graphical, we find the iniu x in and the axiu x ax of the values X takes. Then we scale the observed value x in its range and take x x in (3.1) x ax x in as the standardized easure. The situation is soeties coplicated by the fact that not all (integer) values between x in and x ax are attained by X. Then, perhaps one can choose one of the following alternatives: (1) use the above easure regardless of whether X takes all values between x in and x ax and (2) use the probabilistic easure assuing that the distribution of X gives equal probabilities to each of the values taken by X. Thus, under each of the graphical odels, we want to find the range (or at least the iniu and the axiu values) of the crude easure X. Under each of the probabilistic odels, we want to find the distribution of X, either exactly or at least approxiately, and, if this is not possible, then the ean and the variance of X. We consider the following variables X, soe of which will be used in the next chapter to construct easures of various characteristics of a social network: the nuber of arcs; the out-degree and the in-degree of a vertex; the axiu out-degree and the axiu in-degree in the digraph; the nubers of sources, sinks, and isolates;

3 Graph-Theoretic and Statistical Models 55 the nuber of syetric pairs; the diaeter; the radius; the nuber of reachable pairs; the nuber of pairs reachable in k steps (k = 2 or 3); the nubers p and q of strong and weak coponents, p q; the nuber of arcs within strong coponents; the nuber h of arcs between strong coponents; the nuber of arcs uv such that d(v, u) 2; and the clique nuber. Under the probabilistic odels, we will also look at the following, where G denotes a rando digraph. P(G is syetric), P(G is asyetric), P(G is coplete asyetric), P(G is a tree), P(G is acyclic), P(G is strongly connected), and P(G is weakly connected). To avoid trivialities, we will assue that n 2 in all the odels considered below, although we allow to be 0. Also note that the probability distribution of the sources, sinks, and isolates has received special attention as these iediately reveal soe iportant features of social structures. This is discussed later in Chapter 5. It is perhaps worth entioning that, in applying these to actual social networks, one has to odify soe of the above variables X for various reasons. First, X ay not precisely easure the characteristic it is intended to easure (often the latter has different versions or nuances, all of which cannot be captured by a single variable). Then, the value of X itself ay be very difficult

4 56 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS to find even for the observed network, not to ention the distribution of X, especially if the nuber of vertices is large, although soeties, using various coputational and theoretical techniques, it ay be possible to copute the value for reasonably sized networks. For any of the variables, the exact range or even the values of x in and x ax and the distribution or even the ean and variance of X are not known exactly, and only occasionally soe theoretical approxiations or bounds are known. Soeties it ay be possible to use iterative ethods or siulation to get approxiations, and one has to be satisfied with the. Finally, any of the definitions or properties of graphtheoretic variables are too stringent to be of use in real-life situations, and one has to either odify the variable or use soe sort of cutoff to decide whether one can consider a network to have the particular property. It ay not be out of context to also ention that discussions on specific graph paraeters such as sinks, sources, and isolates ay be overephasized in the later parts of this chapter. These ters have natural sociological interpretations, as discussed in Chapter 5. The theoretical study of such paraeters requires probabilistic arguents, and we thought it proper to derive soe results and present the hereinafter along with the description of other ore useful paraeters such as out-degree, in-degree, and reciprocity. The basic assuptions underlying different odels that will be considered in this chapter are suarized in the following table. We have conceptualized four categories of odels (I IV), penetrating step-by-step according to four different levels of available inforation related to the foration of the social network. Minially, since the level of inforation available is that of only the size of a network (i.e., the nuber of actors (n)), we begin with n as given. Therefore, the possible digraphs are all digraphs with vertex set v 1,v 2,...,v n with no additional assuptions. Models I.1 to I.3 fall under this category. At the next level, the quantu of interaction aong the n actors (i.e., total nuber of ties of interaction or arcs ()) is also assued to be known. Models II.1 to II.2 fall in this category. Inforation available on the out-degrees of the actors that is, (d 1, d 2,...,d n ) gives us the third level of odeling, and III.1 to III.3 deal with this aspect. Models IV.1 to IV.2 deal with the situation when both the out-degree and in-degree sequences are known. For each of the four categories, we have considered graph-theoretic (deterinistic) and statistical (probabilistic) versions along with appropriate special cases, if any. The reaining odels (V VII) are known probabilistic odels that can be related to those stated in Categories II and III above.

5 Graph-Theoretic and Statistical Models 57 Model assuptions Model no. Assuptions I.1 None I.2 All digraphs are equally likely I.3 P(v i v j is an arc) = p for i = j, distinct pairs being independent II.1 II.2 III.1 III.2 III.3 IV.1 IV.2 V The nuber of edges is The nuber of edges is and all possible digraphs are equally likely d + (v i ) = d i for i = 1, 2,...,n d + (v i ) = d i for i = 1, 2,...,n and all possible digraphs are equally likely N(v i ) =d i, N(v i ) P i for i = 1, 2,...,n and all possible digraphs are equally likely d + (v i ) = d i and d (v i ) = e i for i = 1, 2,...,n d + (v i ) = d i and d (v i ) = e i for i = 1, 2,...,n and all possible digraphs are equally likely P(v j v i A v i v j A) = P(v j v i A) + τ P(v j v i / A) and P(v i v j A) = d i /(n 1) whenever i = j VI P((0, 0)) = exp(λ ij ); P((1, 0)) = exp(λ ij + θ + α i + β j ); P((0, 1)) = exp(λ ij + θ + α j + β i ); P((1, 1)) = exp(λ ij + 2θ + α i + β j + α j + β i + ρ); for an ordered pair of vertices (i, j) in a digraph G with the scores for dyadic oveents expressed by 2 2 = 4 cobinations of 0s and 1s; αi = 0 and β i = 0 VII P(G) = const. exp(θ + ρs + α i d i + β i e i ) for any digraph G; αi = 0 and β i = MODELS FIXING THE TOTAL NUMBER OF VERTICES Model I.1 This is the siplest graphical odel one can consider and takes the vertex set V ={v 1,v 2,...,v n } coprising n vertices as fixed and assues that all the 2 n(n 1) digraphs on V are actually possible. Clearly, the range of the nuber of arcs is {0, 1,...,n(n 1)}.

6 58 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS The range of the out-degree d i (as well as the in-degree e i ) of the ith vertex is {0, 1,...,n 1} for each i. The range of d ax, defined as ax(d 1, d 2,...,d n ), as well as that of e ax, defined as ax(e 1, e 2,...,e n ), is also {0, 1,...,n 1}. The range of the nuber of sources (as well as the nuber of sinks) is {0, 1,...,n}. However, the range of the nuber of isolates is {0, 1,..., n 2, n} (note that if n 1 of the vertices are isolates, the reaining vertex is autoatically an isolate). The range of the nuber s of syetric pairs is {0, 1,...,n(n 1)/2}. The range of the diaeter is {1, 2,...,n 1, }. (To get a digraph with diaeter k <, take a coplete syetric digraph on n k + 1 vertices and attach a syetric path on k vertices at soe vertex.) The range of the radius is {1, 2,...,n 1, }. (To get a digraph with radius k <, take a directed path on k + 1 vertices and join the first vertex to the reaining n k 1 vertices; then, the first vertex of the path is a center, and the radius is k.) Next we study R, the nuber of reachable pairs of distinct vertices. (Note that we are leaving out pairs of the type (u, u), even though u is reachable fro u in a trivial sense.) Clearly, the iniu and axiu values of R are 0 and n(n 1), but the range S n of R is not continuous. It can be shown by induction on n that S n T n, where T n ={0, 1,...,n 2 3n + 4} {n 2 2n + 1, n 2 n}. (3.2) We give S n below for n up to 15 for ready reference. S 1 ={0} T 1, S 2 ={0, 1, 2}=T 2, S 3 ={0, 1, 2, 3, 4, 6}=T 3, S 4 ={0, 1,...,9, 12}=T 4, S 5 ={0, 1,...,14, 16, 20}=T 5, S 6 ={0, 1,...,22, 25, 30}=T 6, S 7 ={0, 1,...,28, 30, 31, 32, 36, 42}=T 7 {29}, S 8 ={0, 1,...,44, 49, 56}=T 8, S 9 ={0, 1,...,52, 54, 56, 57, 58, 64, 72}=T 9 {53, 55}, S 10 ={0, 1,...,67, 69, 72, 73, 74, 81, 90}=T 10 {68, 70, 71}, S 11 ={0, 1,...,84, 86, 90, 91, 92, 100, 110}=T 11 {85, 87, 88, 89}, S 12 ={0, 1,...,97, 99,...,103, 105, 110, 111, 112, 121, 132}, S 13 ={0, 1,...,117, 120,...,124, 126, 132, 133, 134, 144, 156}, S 14 ={0, 1,...,139, 142,...,147, 149, 156, 157, 158, 169, 182}, S 15 ={0, 1,...,163, 166, 168,...,172, 174, 182, 183, 184, 196, 210}.

7 Graph-Theoretic and Statistical Models 59 Moreover, Rao (2002) has shown that for n 208, x S n if and only if x k(n 1) S n k for soe k with 1 k n 1, (3.3) using which S n can be deterined for n 208. He also showed that if f (n) is defined by {0, 1,..., f (n)} S n and f (n) + 1 / S n, then f (n) (n n 0.57 )(n 1) and that this bound is fairly good. It has also been found epirically that the nuber of eleents in S n is close to (n n 0.45 )(n 1) at least for n 208. Until now, we considered the nuber of pairs reachable in an arbitrary nuber of steps. Let R (k) (G) denote the nuber of pairs (u, v) of distinct vertices in G such that v is reachable in k or fewer steps fro u, and let S n (k) be the range of R (k) (G) as G varies over all networks on n vertices. Rao (2002) has shown that S n (2) = S n for n = 1, 2, 3 and S n (2) ={0, 1,..., n(n 1)} whenever n 4; S n (3) = S n for n = 1, 2, 3, 4 and S n (3) ={0, 1,...,n(n 1)} whenever n 5. He has also shown that for every k 2, S n (k) ={0, 1,...,n(n 1)} provided, n k +(k + 1) The range is {1, 2,...,n} for the nuber of strong coponents, the nuber of weak coponents, and the clique nuber. It is easy to see that the range of the difference p q between the nuber of strong coponents p and the nuber of weak coponents q is also {0, 1,...,n 1}. The range of the nuber h of arcs between strong coponents is {0, 1,..., n 2 }. This is because such arcs cannot be reciprocated. The range of the iniu nuber P of paths, fored with arcs joining different strong coponents of G and covering the vertex set, is clearly {1, 2,...,n}. Model I.2 Model I.2 is a probabilistic version of Model I.1, and takes the vertex set as fixed (e.g., V ={v 1,v 2,...,v n }) and assues that all the 2 n(n 1) possible digraphs on V are equally likely. Let X ij be the rando variable taking value 1 if v i v j is an arc and 0 otherwise. Then under the present odel, P(X ij = 1) = 1/2 since there are 2 n(n 1) 1 digraphs with v i v j as an arc. Also, it is easy to see that the n(n 1) rando variables X ij (1 i = j n) are utually independent. Thus, the odel is equivalent to v i v j, which is an arc with probability 1/2, and the ordered pairs are all utually independent. Hence, a rando digraph G under the odel can be generated by aking v i v j an arc with probability 1/2 for each

8 60 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS ordered pair (i, j), with distinct ordered pairs being independent. Repeating this, one can generate any given nuber of rando digraphs and estiate the distribution of any statistic under the odel by siulation. It is easy to see that under the present odel, the nuber of arcs has the binoial distribution B(n(n 1), 1/2) since is the su of the n(n 1) independent Bernoulli rando variables X ij. The notation B(..) will be used for a binoial distribution without any further explanation. Since the out-degree d i of v i is j=i X ij, it follows that d i has the binoial distribution B(n 1, 1/2). Hence, E(d i ) = (n 1)/2 and V (d i ) = (n 1)/4. Since the d i s are independent, the distribution of d ax : = ax(d 1, d 2,...,d n ) can be coputed easily, although one cannot give a closed forula for it. Note that P(d ax k) = (P(d 1 k)) n. It is easy to see that the in-degree e i of v i also has the distribution B(n 1, 1/2), and d ax and e ax have the sae distribution. To give an exaple, let n = 3. Then each d i takes values 0, 1, and 2 with probabilities 1/4, 1/2, and 1/4. Hence, we have P(d ax = 0) = (1/4) 3 = 1/64. Also, P(d ax 1) = (3/4) 3 = 27/64, so P(d ax = 1) = 13/32 and P(d ax = 2) = 37/64. When n = 4, it can be checked that d ax takes values 0, 1, 2, and 3 with probabilities 1/4,096, 255/4,096, 2,145/4,096, and 1,695/4,096, respectively. The probability that v i is a source is 1/2 n 1. Also, different v i s being sources are independent events, so the nuber of sources has the distribution B(n, 1/2 n 1 ). It follows siilarly that the probability that v i is a sink is also 1/2 n 1, and the nuber of sinks has the distribution B(n, 1/2 n 1 ). The probability that v i is an isolated vertex is 1/2 2n 2. But the events in which different vertices are isolates are not independent. For exaple, if any n 1 vertices are isolates, it follows that the reaining vertex is also an isolate. So, to find the distribution of the nuber of isolates, we use the following forulae (see Feller, 1968). The probability that exactly k of the events A 1, A 2,...,A n occurs is k + 1 k + 2 S k k S k+1 + k S k+2 + +( 1) n k n k S n, (3.4) and the probability that at least k of A 1, A 2,...,A n occurs is k k + 1 n 1 S k S k+1 + S k+2 + +( 1) n k S n, (3.5) k 1 k 1 k 1

9 Graph-Theoretic and Statistical Models 61 where S k denotes P(A i1 A i2 A ik ), with the su being taken over all i 1, i 2,...,i k such that 1 i 1 < i 2 < < i k n. Clearly, now the probability that k given vertices are isolates is 1 2 k(k 1)+2k(n k) = 1 2 k(2n k 1). Hence, taking A i to be the event that the ith vertex is an isolate, we see that the probability that there are exactly k isolates is n 1 k + 1 n k 2 k(2n k 1) 1 k + 2 k k (k+1)(2n k 2) + k n 1 n ( 1)n k. (3.6) k (k+2)(2n k 3) k 2n(n 1) The probability that there are at least k isolates can be found by using (3.2). Taking the events that different vertices are isolates to be nearly independent, we see that the distribution of the nuber of isolates is approxiately B(n, 1/2 2n 2 ). However, this is not of uch iportance as the probability that there is no isolate is ore than for all n 10. By definition, the probability that G is any particular digraph (including the null digraph and the coplete syetric digraph) is 1/2 n(n 1). The probability that G is syetric is 1/2 n(n 1)/2 since, for G to be syetric, either none or both of v i v j and v j v i should be arcs for each unordered pair {i, j} with i = j. The probability that G is asyetric and the probability that G is coplete are both (3/4) n(n 1)/2 since, for G to be asyetric, at ost one of v i v j and v j v i should be an arc, and for G to be coplete, at least one of v i v j and v j v i should be an arc, for each unordered pair {i, j}, with i = j. The nuber of syetric pairs s(g) has the distribution B(n(n 1)/2, 1/4) since s(g) is the su of the n(n 1)/2 independent Bernoulli variables Y ij, where Y ij = X ij X ji for any unordered pair {i, j} with i = j. It is uch ore difficult to deal with probabilities of events depending on the distance between vertices because d(v i,v j ) depends not only on what happens at v i and v j but also on what happens in other parts of the digraph. For exaple, to find the probability that G is strongly connected, we have to find the nuber f (n) of strongly connected digraphs on the vertex set {v 1,v 2,...,v n }. A ethod for coputing this nuber f (n) for any n is known (see Harary, 1988). (The proble is usually referred to as enueration

10 62 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS Figure 3.1 G 1 :1 G 2 :6 G 3 :3 G 4 :6 G 5 :3 G 6 :3 G 7 :6 G 8 :6 G 9 :2 G 10 :6 G 11 :3 G 12 :6 G 13 :3 G 14 :3 G 15 :6 G 16 :1 of strongly connected labeled digraphs on n vertices, with the word labeled signifying that the vertex set is fixed and we are not counting nonisoorphic digraphs.) However, this involves generating functions, and no closed forula is known for f (n). Thus, even finding the probability that G is strongly connected is difficult. When n = 2, this probability is 1/4. If n = 3, out of the 64 possible digraphs, 18 are strongly connected, so the probability is 9/32. Here, the strongly connected digraphs consist of one digraph with six arcs; six digraphs with five arcs; three digraphs with four arcs foring two reciprocal pairs; six digraphs with four arcs, two of which for a reciprocal pair; and two digraphs with three arcs foring a circuit (see Figure 3.1). When n = 4, it can be checked that out of 4,096 possible digraphs, 1,606 are strongly connected, so the probability that G is strongly connected is 1,606/4,096 = When n = 2, the probability that G is weakly connected is 3/4. When n = 3, 54 out of the 64 possible digraphs are weakly connected, so the probability is 54/64 = When n = 4, it can be checked that out of 4,096 possible digraphs, 3,834 are weakly connected, so the probability is 3,834/4,096 = In Figure 3.2, we give the 16 nonisoorphic digraphs on three vertices. Along with each of these, we also give the nuber of digraphs on {v 1,v 2,v 3 } isoorphic to it. The probability that G has diaeter 1 is clearly 1/2 n(n 1). When n = 3, it is easy to check fro Figure 3.2 that the probability that G has diaeter 2 is 17/64, and the probability that G has diaeter is 23/32.

11 Graph-Theoretic and Statistical Models 63 Figure 3.2 v 2k v k+2 v 1. v 0. v k+1 v 2 v k v 2k v 2k+1. v 0. v k+2 v 1 v k+1 v 2 v k Let n = 3. Then P(r(G) = 1) = P(d ax = 2) = 37/64, where r(g) is the radius of G. It can be checked that 14 digraphs have radius 2 and 13 digraphs have radius, so P(r(G) = 2) = 7/32 and P(r(G)) = ) = 13/64. We ention that for general n and k, even expressions such as (3.3) are difficult to find for P(r(G) = k) and P(d(G) = k). However, it is easy to prove that P(d(G) = 2) 1 as n. To see this, let A i be the event that at least one of v 1 v i and v i v 2 is not an arc. Then P(A i ) = 3/4 for all i = 1, 2. Since P(v 1 v 2 is not an arc) = 1/2, it follows that P(d(v 1,v 2 )> 2) = (3/4) n 2 /2. Since P(d(v i,v j )>2) = P(d(v 1,v 2 )>2), whenever i = j, we have P(d(G)>2) = P(d(v i,v j )>2 for at least one pair (i, j)) n(n 1)(3/4) n 2 /2 0 as n. Since P(d(G) = 1) = 1/2 n(n 1) 0, we have P(d(G) = 2) 1 as n. Since P(d(G) = 2) 1, we also have P(r(G) 2) 1 as n. Now we show that P(r(G) = 1) 0 as n. For any fixed i, let E i be the event that v i v j is an arc for all j = i. Then P(E i ) = 1/2 n 1. Also, E i s are independent, so P(r(G) = 1) = P(E 1 E 2 E n ) = 1 P(Ē 1 Ē 2 Ē n ) = n n n = 2 n 1 ( 1) k 1 k 2 k(n 1) n 0 (3.7) 2n 1 k=1 as n. Here the inequality follows fro the fact that a k / a k+1 > 1 for all k, where a k denotes the kth ter in the su (it also follows fro the fact that P( E i ) P(E i )). Hence, P(r(G) = 1) 0, and P(r(G) = 2) 1 as n.

12 64 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS The distribution of the clique nuber ω(g) is difficult to find for general n, although it can be worked out for sall n using (3.2). It ay be noted that, for any n, P(ω(G) = 1) = P(G is asyetric) = (3/4) n(n 1)/2 and P(ω(G) = n) = 1/2 n(n 1). Hence, when n = 3, the clique nuber takes values 1, 2, and 3 with probabilities 27/64, 9/16, and 1/64. Next let n = 4. Then P(ω(G) = 1) = 729/4,096, so P(ω(G) 2) = 3,367/4,096. To find P(ω(G) 3), let A 1, A 2, A 3, A 4 be the events that {v 1,v 2,v 3 }, {v 1,v 2,v 4 }, {v 1,v 3,v 4 }, and {v 2,v 3,v 4 } induce coplete syetric digraphs. Then P(A i ) = 1/2 6, P(A i A j ) = 1/2 10 whenever i = j, P(A i A j A k ) = 1/2 12 whenever i, j, k are distinct, and P(A 1 A 2 A 3 A 4 ) = 1/2 12. So P(ω(G) 3) = 4/64 6/1, /4,096 1/4,096 = 235/4,096. Hence, P(ω(G) = 2) = 3,132/4,096, P(ω(G) = 3) = 234/4,096, and P(ω(G) = 4) = 1/4,096. However, if we try to find P(ω(G) 3) in the sae way when n = 5, we note that P(A i A j A k ) depends on what i, j and k are. Thus, the forulae becoe ore coplicated as n increases. Results of Siulation It should be evident that for soe of the statistics, the exact distributional properties are hard to derive. These are analytically intractable unless n is sall. For ready reference, therefore, we give the distribution of various statistics considered above, for soe values of n. These were obtained by siulation using 100,000 rando digraphs (except for sall values of n when the exact distribution can be coputed). The error in the estiate of any probability should not exceed and is expected to be uch less (less than when the probability is less than 0.01). We do not give the distributions of (G) and s(g) as these are binoial distributions. Note that a dash in an entry in the table eans that the probability is either 0 or is positive but less than Maxiu Out-Degree n

13 Graph-Theoretic and Statistical Models 65 Maxiu Out-Degree (Continuation) n = Maxiu Out-Degree (Continuation) n = Sources Isolates n Diaeter Radius n

14 66 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS Strong Coponents Weak Coponents n Clique Nuber n Model I.3 Under Model I.2, the expected nuber of arcs is n(n 1)/2. So when is very sall or very large (close to n(n 1)), Model I.2 is not appropriate, and we consider Model I.3. In this probabilistic odel, we again take the vertex set to be fixed and assue that all the 2 n(n 1) digraphs on V are possible. But we stipulate that v i v j is chosen as an arc with a fixed probability p (0 < p < 1) and that the ordered pairs are all utually independent. Note that Model I.2 is the special case of Model I.3 corresponding to p = 1/2. Let X ij be defined as before. Then, under the present odel, we have P(X ij = 1) = p, and X ij s are independent. So the nuber of arcs has the binoial distribution B(n(n 1), p), and the out-degree d i as well as the in-degree e i of v i has the sae binoial distribution B(n 1, p). Clearly, the probability of getting any particular digraph G 0 on V is p q n(n 1), where is the nuber of arcs in G 0 and q = 1 p. Hence, any

15 Graph-Theoretic and Statistical Models 67 two digraphs with the sae nuber of arcs are equally likely; in particular, P(G = G 0 ) = P(G = G c 0 ), where Gc 0 denotes the converse of G 0. Here, the probability that v i is a source is q n 1. Also, the nuber of sources has the distribution B(n, q n 1 ). Siilarly, the probability that v i is a sink is q n 1, and the nuber of sinks has the distribution B(n, q n 1 ). The probability that v i is an isolated vertex is q 2n 2. The probability that k given vertices are isolates is q k(2n k 1). Hence, the probability that there are exactly k isolates is n k k + 1 n q k(2n k 1) q (k+1)(2n k 2) k k + 1 k + 2 n n + q (k+2)(2n k 3) + +( 1) n k k k + 2 k q n(n 1). (3.8) Again, an expression, siilar to (3.2), for the probability that there are at least k isolates can be written down. The probability that G is a null digraph is q n(n 1), and the probability that G is the coplete syetric digraph is p n(n 1). The probability that G is syetric is (p 2 + q 2 ) n(n 1)/2, the probability that G is asyetric is (1 p 2 ) n(n 1)/2, and the probability that G is coplete is (1 q 2 ) n(n 1)/2. The nuber of syetric pairs s(g) has the distribution B( n 2, p 2 ). When n = 2, the probability that G is strongly connected is p 2. If n = 3, the probability is p 6 + 6p 5 q + 9p 4 q 2 + 2p 3 q 3. This can be seen easily fro Figure 3.1. Here also expressions for P(r(G) = k) and P(d(G) = k) cannot be found but, rather surprisingly, it is easy to prove that P(d(G) = 2) 1 and P(r(G) = 2) 1 as n, provided only that 0 < p < 1. To see this, let A i be the event that at least one of v 1 v i and v i v 2 is not an arc. Then P(A i ) = 1 p 2 for all i = 1,2. Since P(v 1 v 2 is not an arc) = 1 p, we have P(d(v 1,v 2 )>2) = (1 p)(1 p 2 ) n 2. It follows as before that P(d(G)>2) n(n 1)(1 p) (1 p 2 ) n 2 0 as n. Since P(d(G) = 1) = p n(n 1) 0, it follows that P(d(G) = 2) 1 as n. Hence, we also have P(r(G) 2) 1 as n. Now we show that P(r(G) = 1) 0 as n. For any fixed i, let E i be the event that v i v j is an arc for all j = i. Then P(E i ) = p n 1. Also, E i s are independent, so P(r(G) = 1) = P(E 1 E 2 E n ) = 1 P(Ē 1 Ē 2 Ē n ) = 1 1 p n 1 n. (3.9)

16 68 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS Moreover, P(E 1 E 2 E n ) P(E 1 ) + P(E 2 ) + +P(E n ) = np n 1 0. Hence, P(r(G) = 1) 0 and P(r(G) = 2) 1 as n. 3.3 MODELS FIXING THE TOTAL NUMBER OF VERTICES AND THE TOTAL OF ALL ARCS Model II.1 In Model I.1, we took only the nuber of vertices (n) as given and assued that all values of fro 0 to n(n 1) are actually possible. Often this is not realistic, particularly when n is large, so we introduce Model II.1. This graphical (nonprobabilistic) odel takes the vertex set V ={v 1,v 2,...,v n } and the nuber of arcs as fixed and assues that all the n(n 1) digraphs on V with arcs are actually possible. Note that 0 n(n 1), and refers to the total of all out-degrees of the n vertices, and this is also the sae as the total of all in-degrees. Under the present odel, the iniu value taken by the out-degree d i (as well as the in-degree e i ) of the ith vertex is ax(0, (n 2 2n + 1)). To see this, it is enough to notice that the nuber of arcs left in the coplete syetric digraph, when all the arcs leaving v i are dropped, is n 2 2n + 1. Trivially, the axiu value taken by the out-degree d i (as well as e i ) is in(n 1, ), and every integer value between the iniu and the axiu can actually be attained by d i (as well as by e i ). We now show that the range of d ax is { n, n +1,...,in (n 1, )}. Here a denotes the sallest integer greater than or equal to the nuber a. For exaple, 2=2 and 2.1=3. That the axiu value of d ax is in(n 1, ) is trivial to prove. That d ax in any digraph on n vertices with arcs is n follows fro ni=1 d i = since ax(d 1, d 2,...,d n ) d i /n. To construct a digraph G with d ax = n, let = nq r, where 0 r n 1. We consider the cases n odd and n even separately. First suppose n = 2k + 1. Then we show how to partition the set A of arcs of the coplete syetric digraph on V ={v 0,v 1,...,v 2k } into n 1 disjoint subsets of size n, each foring a single cycle (such a cycle is called a Hailtonian cycle). Arrange the vertices {v 1,v 2,...,v 2k } regularly on a circle with center v 0. The first Hailtonian cycle is [v 0,v 1,v 2,v 2k,v 3,v 2k 1,v 4,...,v k, v k+2,v k+1,v 0 ]. This is shown in the first diagra in Figure 3.2. For i = 2, 3,...,2k, the ith Hailtonian

17 Graph-Theoretic and Statistical Models 69 cycle naely, [v 0,v i,v i+1,v i 1,v i+2,v i 2,v i+3,...,v k+i 1,v k+i,v 0 ] is obtained by rotating the first cycle clockwise around v 0 by i 1 steps. Now, we get G by taking the union of q of the Hailtonian cycles and dropping r arcs of one cycle if r > 0. Next let n = 2k + 2. Then we replace the arc bypassing the central vertex v 0 by a path of length 2 with v 2k+1 as the iddle vertex, as shown in the second diagra in Figure 3.2 (iagine v 2k+1 to be directly above v 0 in a different plane). As before, we get 2k Hailtonian cycles here also. Now we also have another set of n arcs: the 2k arcs in the earlier digraph, which were replaced by paths of length 2 and the two arcs v 0 v 2k+1 and v 2k+1 v 0. Note that every vertex has out-degree 1 w.r.t. the arcs in this last set also. Now, again, we get G by taking the union of q of the Hailtonian cycles (including the last set if necessary) and dropping r arcs fro one set if r > 0. To give the iniu and axiu values taken by the nuber of sources, let p denote the nuber of sources. Then in p = ax(0, n ). To see this, we note that p can be ade 0 whenever n by including a circuit on n vertices in the digraph. If n 1, the arcs can ake at ost vertices nonsources, so p n, and a digraph with p = n is obtained by taking a directed path on + 1 vertices together with n 1 isolated vertices. We now show that ax p = n /(n 1). Denote the right-hand side (RHS) by k. If S is the set of sources in any digraph on n vertices with arcs, then there is no arc entering any vertex in S, so p(n p) + (n p)(n p 1) = (n 1)(n p), which gives p k. A digraph G attaining p = k is obtained as follows: Let H be a coplete syetric digraph with vertex set V, where V =n, and let S be a subset of V with S =k. Reove all the arcs within S and all the arcs fro V S to S and a few ore if necessary (so that exactly arcs reain) to get G. Since changing the position of an arc can change the nuber of sources by at ost 1 and since any two digraphs on V with arcs each can be obtained fro each other by changing the position of one arc at a tie, it follows that the nuber p of sources actually takes all integer values between the iniu and the axiu. Finally, it is easy to see that the range of the nuber of sources is the sae as that of the nuber of sinks. To give the iniu and axiu values taken by the nuber of isolates, let q denote the nuber of isolates. Then it is easy to see that in q = ax (0, n 2) since the nuber of distinct vertices that are end (initial or terinal) vertices of the arcs is at ost 2, and the reaining vertices are isolated.we next show that ax q =, where = ax{k : (n k)

18 70 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS (n k 1)}. Clearly, we have (n q)(n q 1) and so q for every digraph. To get a digraph attaining, all one has to do is to put the arcs within n vertices. Again, it is easy to see that all values between the iniu and the axiu are attained. It is easy to see that the iniu and the axiu of the nuber s of syetric pairs are ax 0, n 2 and /2, respectively, and that every integral value between the iniu and the axiu is attained by s. If t is the nuber of arcs that are not reciprocated, we have (i) 2s + t = and (ii). Subtracting (ii) fro (i), we get s n s + t n 2 2. Since t 0, (i) gives s /2. The iniu and axiu values of the diaeter were obtained by Goldberg and Ghouila-Houri, respectively (see Berge, 1973). If a digraph G is strongly connected, the out-degree as well as the in-degree of every vertex in G is at least 1, so n. Moreover, if = n in G, then G is a circuit, and the diaeter is n 1. Hence, the iniu and the axiu values of the diaeter are if n 1. The iniu is n 1 and the axiu is if = n. Next we take n < n(n 1). Let n 1 = ( n + 1)q + r where 0 r < n + 1. Then the iniu value of the diaeter is 2q if r = 0 2q + 1 if r = 1 2q + 2 otherwise, (3.10) and the axiu value of the diaeter is n 1 if (n 2 + n 2)/2 n n 2 n otherwise. (3.11) It is easy to see that the iniu and axiu values of the radius are if < n 1. If n 1, the iniu radius is 1. For axiu radius, see Berge (1973). We now prove that the iniu nuber θ of strong coponents is n if = 0 θ = 1 if n. (3.12) n + 1 otherwise The result is trivial if 1. If n, then we can get a strongly connected digraph on n vertices with arcs by starting with a circuit on n vertices

19 Graph-Theoretic and Statistical Models 71 and adding n arcs arbitrarily, so θ = 1. So let 1 < < n. We first note that any strongly connected digraph on t 2 vertices has at least t arcs since the out-degree of each vertex ust be at least 1. Let G be any digraph on n vertices with arcs. Suppose exactly of the p strong coponents of G are singletons. If = n, then p = n n + 1. If <n, then n, so p + 1 n + 1. To get a digraph with exactly n + 1 strong coponents, consider a circuit on vertices together with n isolated vertices. We next prove that the axiu nuber of strong coponents is n n k + 1 k 0 = ax k : k n and, 2 2 which reduces to 1 2n n(n 1) + 1 if 2 n. (3.13) 2 1 (Note that 2 = 0 by definition.) For this, suppose G is an arbitrary digraph on n vertices with arcs. Let the strong coponents of G be C 1, C 2,...,C p, C i containing n i vertices. Then no arc between two C i s can be reciprocated, so n the nuber of arcs between C i s is at ost 2. Hence, n p ni n n p , i=1 where the second inequality follows on observing that p i=1 n i 2 is axiu when all but one of the n i s are 1 each. So p k 0, and it follows that k 0. To show equality, consider the digraph on the vertex set {v 1,v 2,...,v n } obtained as follows: If n 2, put arcs of the type vi v j with i < j. If > n 2, ake v i v j an arc whenever 1 i < j n and add n 2 arcs of the type v j v i with i < j within the first n k vertices. In this digraph, the last k 0 1 vertices for singleton strong coponents, so the digraph has at least k 0 and so exactly k 0 strong coponents. We now prove that the iniu nuber ξ of weak coponents is ax(1, n ). The result is trivial if = 0. If n 1, then we can get a weakly connected digraph on n vertices with arcs by starting with a path on n vertices and adding n + 1 arcs arbitrarily, so ξ = 1. Next let 0 < < n 1. We first note that any weakly connected digraph on t vertices has at least t 1 arcs (we oit the proof of this stateent). Now let G be any digraph on n

20 72 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS vertices with arcs. Suppose C 1, C 2,...,C q are the weak coponents of G, with C i having n i vertices. Then q i=1 (n i 1) = n q, so q n. To get a digraph with exactly n weak coponents, consider a path on + 1 vertices together with n 1 isolated vertices. We next prove that the axiu nuber of weak coponents is k 0 = ax{k : k n and (n k + 1)(n k)} = 12 2n (3.14) The proof is siilar to that for. Suppose G is an arbitrary digraph with weak coponents C 1, C 2,...,C q, with C i containing n i vertices. Then there are no arcs between C i s, so q n i (n i 1) (n q + 1)(n q), i=1 where the second inequality follows as before. Hence, q k 0 and k 0. To show equality, consider a digraph with all the arcs belonging to one weak coponent with n k + 1 vertices. We now consider the difference p q, where p and q denote the nuber of strong coponents and the nuber of weak coponents, respectively. We first note that p q. If 2, then a digraph with a circuit on in(, n) vertices has p = q. If = 1, then p = n and q = n 1. If = 0, then p = q = n. Hence, we have 1 if = 1 in p q = 0 otherwise. (3.15) We next show that if n 1 ax p q = 1 otherwise, (3.16) where is the axiu nuber of strong coponents. For this, it is enough to observe that there is a digraph attaining the axiu nuber of strong coponents and the iniu nuber of weak coponents siultaneously. We next consider the nuber h of arcs joining different strong coponents. The iniu value of h is 0 or 1 accordingly as 2 or = 1. To see this, all one has to do is to include a circuit on in(, n) vertices if 2. We

21 Graph-Theoretic and Statistical Models 73 next consider the axiu value of h. Clearly, ax h = if n 2. So let > n 2. Let i=1 n t = in i 2 : 1, ni 1 for i = 1, 2,...,, i=1 n i = n and n 2 + i=1 n i 2. Then we will show that ax h = n 2 t. Suppose G is a network on n vertices with arcs, of which h is between strong coponents. Suppose G has strong coponents C 1, C 2,...,C, with C i containing n i vertices. Then, for i = j, there cannot be arcs both fro C i to C j and fro C j to C i. Hence, h n 2 i=1 n i 2 n 2 t. To prove that the bound is attained, consider a digraph with v i v j an arc whenever i < j and with the sizes of the strong coponents equal to n 1, n 2,...,n (it is easy to see that such a digraph exists). We ention that t n 2, but the equality ay not always hold. For exaple, if n = 6 and = 20, then t = 6 attains when = 2 and n 1 = n 2 = 3, whereas n 2 = 5. Thus, ax h is 9 here. We finally coe to clique nuber. It is easy to see that the axiu value for the clique nuber of a digraph on n vertices with arcs is ax{k : k(k 1)}= (3.17) It can be shown that the iniu value of the clique nuber is ax k : n(n 1) k q 2 n rq where q = and r = n kq. k (3.18) We oit the proof of this result. We ention that the corresponding result for (undirected) graphs is known as Turan s theore. A digraph attaining the bound, which we denote by k, is obtained as follows: The vertex set is V 1 V 2... V k, where the V i s are pairwise disjoint, r of the V i s has size q + 1, and the rest have size q. The arcs are adjusted in such a way that there is no syetric pair within any V i. The clique nuber of this digraph is k because any clique in it can contain only one vertex fro each V i. Model II.2 This odel is the probabilistic version of Model II.1. It takes the vertex set V ={v 1,v 2,...,v n } and the nuber of arcs (0 n(n 1)) as fixed and assues that all the n(n 1) possible digraphs are equally likely. Note that a

22 74 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS rando digraph is now obtained by choosing of the n(n 1) ordered pairs v i v j, by siple rando sapling without replaceent, and aking the arcs. Since n(n 1) occurs repeatedly, we will write M for n(n 1) in what follows. Let X ij be defined as before. Then under the present odel, P(X ij = 1) = /M, but the X ij s are not independent. Since the out-degree d i of v i is the nuber of pairs chosen fro {v i v j 1 j n, j = i}, it follows that d i has the following hypergeoetric distribution: P(d i = k) = n 1 n 2 2n+1 k k M Hence, E(d i ) = /n and, ax(0, (n 2 2n + 1)) k in(n 1, ). V (d i ) = 1 n (3.19) 1 1 M n M 1. (3.20) It ay be noted that the distribution of d i is approxiately B(, 1/n) if n 1 and B(n 1, /M) if n 1. The distribution can also be approxiated by Poisson distribution with ean /n when << n(n 1) and a noral distribution with ean and variance given above when, M, and n are large subject to finite ean and variance in the liits. It is easy to see that the out-degree d i and the in-degree e i of v i have the sae distribution. Now, different d i s are not independent. For exaple, P(d 2 = k d 1 = ) = n 1 M 2(n 1) k k M (n 1) So the distribution of d ax is not easy to copute now. The probability that v i is a source is P(e i = 0) =, k = 0, 1,...,in(n 1, ). M (n 1) M. (3.21) Now, the events that different vertices are sources are not independent. However, the probability that k given vertices are sources is (n k)(n 1) M. (3.22) So an expression siilar to (3.1), for the probability that there are exactly k sources, can be written down. A sipler approxiation to the distribution of

23 Graph-Theoretic and Statistical Models 75 the expected nuber of sources can be obtained as follows when n is large: We assue that the events that different vertices are sources are independent. Then the nuber of sources has the distribution B(n, p), where p is given in (3.3). Now, n 2 p = i=0 1 1 M i n 1 exp M 2n/3 This approxiation sees to be good if < n. When n, p = 1 i=0 1 n 1 1 n 1 exp M i M 2/3 (n 1). M 2n/3 (n 1) M 2/3 sees to be better. In both cases, the ean of the true distribution is quite close to the ean of the binoial distribution, but the variance of the true distribution is soewhat saller than that of the binoial distribution. The probability that v i is a sink equals the probability that v i is a source, and the nuber of sinks and the nuber of sources have the sae distribution. The probability that v i is an isolated vertex is P(d i = 0 and e i = 0) = M 2(n 1) M, (3.23) which is approxiately exp( 2(n 1)/(M 4n/3)) if < 2n. When 2n, exp( 2(n 1)/(M 2/3)) is a better approxiation. The events that different vertices are isolates are not independent. The probability that k given vertices are isolates is (n k)(n k 1) M. (3.24) So an expression siilar to (3.1), for the probability that there are exactly k isolates, can be written down. However, again, the nuber of isolates is approxiately B(n, p), where p is given by (3.4). By definition, any two digraphs with arcs (in particular, any digraph and its converse) have the sae probability naely, 1/ M. The probability that G is syetric is M/2 /2 / M, provided that is even (and 0 otherwise). The probability that G is asyetric is M/2 2 / M. To find the probability that G is coplete, we note that given k distinct unordered pairs {i 1, j 1 }, {i 2, j 2 },...,{i k, j k }, the probability that none of

24 76 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS v i1 v j1,v j1 v i1,v i2 v j2,v j2 v i2,...,v ik v jk,v jk v ik is an arc is M 2k / M. Hence, the probability that G is not coplete is n 2 M 2 M n 2 2 It is not difficult to see that M 4 M n M 6 M (3.25) P(s(G) = k) = ( n 2) k ( n 2 ) k 2 2k 2k n(n 1). (3.26) Fro this it follows that P(s(G) = k + 1) P(s(G) = k) ( 2k)( 2k + 1) = 4(k + 1)( n. 2 + k + 1) This ratio is greater than 1 or less than 1 accordingly as k < x or k > x, where x = ( + 4)( 1) 2n(n 1). (3.27) 2n(n 1) + 6 Hence, it follows that the distribution of s(g) is uniodal with ode at x. (The axiu probability ay be attained at ost at two consecutive integers.) We can also find the ean and the variance since s(g) is the su of the n(n 1)/2 variables Y ij, where Y ij = X ij X ji for any unordered pair {i, j} with i = j. Now E(Y ij ) = P(Y ij = 1) = M 2 2 M = ( 1) M(M 1). Let us denote ( 1)/(M(M 1)) by p for convenience. Then E(s(G)) = Mp 2 = ( 1) 2n(n 1) 2. (3.28) Now, V (Y ij ) = p(1 p). If {i, j} and {k,} are distinct, then cov(y ij, Y k ) = E(Y ij Y k ) E(Y ij )E(Y k ) = ( 1)( 2)( 3) M(M 1)(M 2)(M 3) p2. Noting that n 2 = M/2, we get

25 Graph-Theoretic and Statistical Models 77 V (s(g)) = M 2 p(1 p) + M 2 = E(s(G)) 1 E(s(G)) + M ( 2)( 3) 2 1 p (M 2)(M 3) p ( 2)( 3). (3.29) 2M 6 If n is large (n > 10, say), E(s(G)) 2 /(2n 2 ) and V (s(g)) E(s(G)) (1 /M) 2. Writing α = V (s(g))/e(s(g)) and β = (1 /M) 2, we actually have (M )(3(M )(M 1) M) α β = M 2, (M 1)(M 3) so α β 3/M. Note that the range of s(g) is [ax(0, M/2), [/2]]. It is now even ore difficult than in Model I.2 to deal with probabilities of events depending on the distance between vertices. For exaple, to find the probability that G is strongly connected, we have to find the nuber g(n, ) of strongly connected digraphs on the vertex set {v 1,v 2,...,v n } with arcs. No ethod for coputing this nuber g(n, ) is known. Thus, even finding the probability that G is strongly connected is difficult. If n = 3 and 2, the probability is 0. If n = 3 and = 3, out of the 20 possible digraphs, only 2 are strongly connected, so the probability is 1/10. If n = 3 and = 4, out of the 15 possible digraphs, 9 are strongly connected, so the probability is 3/5. If n = 3 and 5, then the probability is 1. The probability that G has diaeter 1 is clearly 1 or 0 according to whether = n(n 1) or not. When n = 3 and = 5, G has diaeter 2 with probability 1. When n = 3 and = 4, G has diaeter 2 with probability 3/5 and diaeter with probability 2/5. When n = 3 and = 3, G has diaeter 2 with probability 1/10 and diaeter with probability 9/10. When n = 3 and 2, G has diaeter with probability 1. When n = 3 and 4, G has radius 1 with probability 1. When n = 3 and = 3, G has radius 1 with probability 3/10 and radius 2 with probability 7/10. When n = 3 and = 2, G has radius 1 with probability 1/5, radius 2 with probability 2/5, and radius with probability 2/5. When n = 3 and 1, G has radius with probability 1. Results of Siulation To give an idea of the distributions of various statistics when both n and are fixed, we give the for n = 10 and a few values of. These were obtained by siulation using 100,000 rando digraphs.

26 78 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS Maxiu Out-Degree (Continuation) Maxiu Out-Degree Sources Isolates

27 Graph-Theoretic and Statistical Models 79 Syetric Pairs Syetric Pairs (Continued) = Syetric Pairs (Continued) = Syetric Pairs (Continued) = Diaeter

28 80 MODELS FOR SOCIAL NETWORKS WITH STATISTICAL APPLICATIONS Radius Clique Nuber Strong Coponents

29 Graph-Theoretic and Statistical Models 81 Weak Coponents MODELS FIXING ALL OUT-DEGREES OF INDIVIDUAL VERTICES Model III.1 Model II.1 leaves the possibility that all the arcs are within a few vertices. This is often not realistic, and any of the vertices ay have positive outdegree. In such a situation, as well as others in which the respondent is asked a question to nae his or her three best friends, Model III.1 is ore appropriate. This odel takes the vertex set V ={v 1,v 2,...,v n } and the out-degree d i of v i as fixed for i = 1, 2,...,n and assues that all the n 1 d i digraphs on V with d + (v i ) = d i for i = 1, 2,...,n are actually possible. Note that the d i s have to satisfy the following condition: 0 d i n 1 for all i. For the sake of convenience, we will assue in the discussion of this odel that d 1 d 2 d n. Moreover, G will denote a digraph with vertex set {v 1,v 2,...,v n } and with d + (v i ) = d i for all i. Clearly, the present odel fixes since = n i=1 d i. The nuber of sinks is also fixed since v i is a sink if and only if d i = 0. We now show that the range of the in-degree e j of v j is [ i= j ax(0, d i n + 2), i= j in(1, d i)]. Clearly, e j i= j ax(0, d i n + 2) in every G since for all i = j such that d i > n 2, v i v j is an arc. It is easy to see that the bound is attained. Also, e j i= j in(1, d i) in every G since for all i = j, v i v j can be an arc only if d i 1. Again it is easy to see that the bound is attained. To give the iniu and axiu values taken by the nuber of sources, let p denote the nuber of sources. Then we show that in p = ax(0, n ). Suppose G has a source u and a vertex v with in-degree at least 2. We ay