Small-world networks

Transcription

1 Small-world networks c A. J. Ganesh, University of Bristol, 2015 Popular folklore asserts that any two people in the world are linked through a chain of no more than six mutual acquaintances, as encapsulated in the phrase six degrees of separation. Some evidence in support of this claim arose from the experiments of an American sociologist, Stanley Milgram, in the 1960s. He gave letters to people in Nebraska and Kansas, addressed to a recipient in Boston about whom they were provided with some information. They had to ensure that the letter reached the recipient, but only by forwarding it to someone they knew personally, who would have to forward it likewise until it reached the target. Each intermediary added their contact information to the letter, so that Milgram could reconstruct the chain it had passed through on route to the target. In these experiments, the median length of chain among those letters which reached the target (which were a minority of the set of letters initially handed out) was around six. Milgram s experiments have been subsequently repeated on and Instant messenger networks, with qualitatively similar results. This leads to the question of whether we can develop mathematical models of networks that can reflect these features of real-world networks of friendship or acquaintance. It turns out that very simply mathematical models called random graphs can indeed exhibit very small diameters. In this model, the number of nodes and the mean node degree (or, possibly, the degree distribution) are fixed, but the edges are then assigned randomly between nodes. (We won t describe the model of randomness in detail here as that is not the primary focus of this chapter, but you can look up the Erdős- Renyi random graph, and the configuration model for further details of such random graph models and their properties.) But it turns out that random graphs are not good models of social networks because they show no social structure - they lack triangles and other small dense subnetworks, which are typical of real social networks. This led Strogatz and Watts to propose a small network which is constructed by a two-stage process. In the first stage, we start with a base network which has considerably locality of 1

2 connection; an example might be nodes placed on a ring or a lattice, with each node connected to its k nearest neighbours, for some fixed parameter k that is not too large. In the second stage, some of these nearest neighbour links are selected and rewired randomly, meaning that the link is broken, and re-made with some other node chosen at random, possibly far away in the base network. Strogatz and Watts then look at the relationship between the number (or proportion) of such rewired links and the diameter of the resulting graph. It turns out that once a sufficiently high proportion of links is rewired, the diameter becomes small (the rewired links have a random graph structure), but the links that haven t been rewired retain the locality that is more typical of a social network. Thus, a fairly simple mathematical model can exhibit the feature of small diameter observed in real-world networks. This feature has come to be known as the small-world property, and networks of the form described above have come to be called small-world networks. Jon Kleinberg came to Milgram s experiments from a slightly different perspective. He observed that not only did the network of social acquaintance have small diameter, it was also navigable. What the letter term means is that many of the recipients of Milgram s letters were able to find short paths to the target addressee. This would be straightforward if they knew the global structure of the network (the acquaintance relationships among all people in the world), but obviously they cannot know this! How then did they find short paths to the target? One could conjecture plausible explanations in terms of the information they might have used; perhaps they tried to move geographically closer to the target, while also matching some other feature such as type of employment, or education institution attended, or church membership. But Kleinberg wanted to construct mathematical network models that would have the feature of being navigable, and understand what structural properties give rise to this feature. We shall describe his work in this chapter. Kleinberg considered the following, quite stylised, network model, which is very similar to the small-world model of Strogatz and Watts. We start with a finite d-dimenionsal lattice on n d points. For concreteness, we shall work with d = 2, though all the results generalise to arbitary d (and were proved in that setting by Kleinberg). Thus, we have a lattice on n 2 points. Instead of breaking and re-wiring some of these links, we instead simply add some more random links. This has the advantage that the lattice maintains connectivity (random rewiring could break the network into disjoint pieces), 2

3 and we only need to think about the diameter of the resulting network. The extra links are added as follows. Each node in the lattice has up to four nearest neighbour links (four in the interior, three on the boundary and two at the corners). In addition, each node chooses one other node at random, not necessarily uniformly, and establishes a link to it. A node x chooses a node y with probability proportional to r(x, y) s, where r(x, y) is the lattice distance between x and y (r(x, y) = x y 1 = x 1 y 1 + x 2 y 2, where x 1 and x 2 denote the first and second co-ordinate of node x and similarly for y), and s 0 is a parameter. The node y chosen according to this probability distribution is called the long-range contact of x, and we say that x establishes a shortcut to y. In addition, other nodes may also establish shortcuts to x. The choices of long-range contacts by distinct nodes are mutually independent. While the establishing of shortcuts is most easily described in directed terms, the link should be thought of as bi-directional, and the whole graph as undirected. The role of the parameter s is that it captures the characteristic length of the shortcuts. If s is large, then choosing y for which r(x, y) is large is highly unlikely; long links are penalised, and shortcuts typically go to nodes which are fairly close by. At the other extreme, if s = 0, then there is no dependence on distance, and all nodes are equally likely to be chosen as the long-range contact of x. Hence, the mean length of a shortcut will be of order n in this case, whereas, for sufficiently large s, it will be of order 1. We shall henceforth refer to the above graph model as Kleinberg s model, noting that it is parametrised by a single parameter s, which is a non-negative integer. Kleinberg wanted to know for what values of s the resulting graph is navigable. He established the remarkable result that there is only one value of s for which it is navigable, namely s = 2! (In the more general case of d dimensions, it is only navigable if s = d.) Before formally stating and proving this result, we need a mathematically precise defintion of navigability. But first, it is worth mentioning that navigability is a different property from having a small diameter, which is what we first considered in the context of small-world networks. It turns out that the above models have small diameter for all s sufficiently small, and certainly s 2, but are not navigable for s < 2. In order to define navigability, we first need to introduce the concept of a decentralised routing algorithm. An algorithm for routing a message from a source node u to a destination node v will forward this message through 3

4 a series of intermediate nodes: in other words, it will come up with a route u = x 0, x 1, x 2,..., x k = v along which the messages passes from u to v. In this case, we say that the message is delivered in k steps. The route has to be a path in the network, i.e., (x i, x i+1 ) has to be an edge, either an original lattice edge or a shortcut. But subject to this constraint, how is the route determined? The distinction between centralised and decentralised algorithms has to do with the information available to the routing algorithm at any stage. In both cases, we assume that the co-ordinates of the source, destination and current node are known. In addition, a centralised algorithm knows the full network (i.e., all the shortcuts) and can use this information to find an optimal route. There are a number of well-known algorithms for finding the shortest path between a source and a destination in a given graph; look up Dijkstra s algorithm, for example. On the other hand, we will call an algorithm decentralised if, at some intermediate stage after visiting nodes x 0, x 1,..., x m, the only information available to the algorithm is the coordinates of these nodes, and of their lattice and long-range neighbours. Thus, it has to choose the next node to visit based on this information only. However, we impose no constraints on how it should use this information; it may make arbitrarily complicated decisions, and these may be deterministic or random functions of the available information up to time m. Roughly speaking, we will prove the following results. We will show that, if s = 2, then a specific decentralised algorithm, namely the greedy algorithm (defined below) can efficiently route a message from an arbitrary source to an arbitrary destination. On the other hand, if s 2, then there is no decentralised algorithm that can efficiently route messages between arbitrary source-destination pairs. The notion of efficiency we use is the following: a message delivery algorithm on the n n grid is efficient if, for any sourcedestination pair, the expected number of messages required to deliver it is bounded above by a polynomial in the logarithm of n. If it is not so bounded, then it is inefficient. The expectation is with respect to the randomness in the distribution of shortcuts, as well as in the routing decisions of the algorithm if these are random. We will prove an explicit upper bound on the message delivery time of the greedy algorithm when s = 2, and an explicit lower bound for any decentralised algorithm when s 2. In the remainder of this chapter, distance will always refer to lattice distance, i.e., r(x, y) = x y 1. 4

5 Theorem 1 Consider Kleinberg s model with s = 2. Fix a source node u and a target node v, and consider the greedy routing algorithm where, at each step, the message is routed to whichever neighbour of the current node (lattice neighbour or long-range contact) is closest to the target; if there is more than one node at the same distance, the tie is broken uniformly at random. The expected number of steps that it takes for the message to reach its target is bounded above by c log 2 n, where c > 0 is a constant that does not depend on n, or on the choice of source and target nodes. Proof. We first outline the main steps in the proof. The proof will proceed by considering how long it takes before the distance to the target is halved. We will show that, at any step of the routing algorithm, there is at least a c/ log n chance that there is a shortcut from the current node that takes the message more than half-way towards the target. Hence, the number of steps before such a halving is stochastically dominated by a geometric random variable with mean c log n. We also note that, irrespective of where u and v are located, log 2 n = c log n halvings of the distance suffice to reach the target. Combining these yields the claim of the theorem. Throughout, we shall use c to denote a generic constant, not necessarily the same each time it is used. First, suppose that the message is currently at a node x, which is at distance D from the target v. Let U denote the set of all nodes within distance D/2 of the target. Let y be a node in U. Since r(x, v) = D and r(v, y) D/2, it follows from the triangle inequality that r(x, y) r(x, v) + r(v, y) 3D/2. This upper bound on the distance between x and y translates into a lower bound on the probability that x chooses y as its long-range contact, an event we denote by x y. We have P(x y) = r(x, y) 2 (3D/2) 2 r(x, z) 2 z x. (1) z x r(x, z) 2 The first equality is based on the model description, which says that P(x y) is proportional to r(x, y) 2 as s = 2; the denominator is the normalisation constant needed to make this a probability. We will now derive an upper bound on the denominator, in order to get a simpler lower bound on P(x y). The denominator involves a sum of r(x, z) 2 over all nodes z in the finite grid other than the node x itself. Let us rewrite this sum by grouping together all nodes which are at the same distance from x. It is fairly easy to see 5

6 that, for any d 1, the number of nodes at distance exactly d from x is at most 4d; in fact, it is exactly 4d in the infinite two-dimensional grid. Each of these nodes contributes a term d 2 to the sum. Moreover, the maximum possible distance between any two nodes in the finite grid is 2n, so it suffices to restrict the sum to d between 1 and 2n. Thus, we get r(x, z) 2 z x 2n d=1 4d d 2 4 ( 1 + 2n 0 x 1 dx ) = 4(1 + log(2n)). Substituting this in (1), we obtain that for any node y U, P(x y) 9 4D 2 (1 + log(2n)) c D 2 log n, (2) where c > 0 is some constant that does not depend on n. This lower bound holds for every node y U. Now, the node x chooses a single other node as its long-range contact; hence, for any two distinct nodes y and z, the events that x chooses y and that x chooses z are disjoint. Consequently, the probability that x chooses its long-range contact within the set U, which event we denote x U, is bounded below by P(x U) = y U P(x y) c D 2 U. (3) log n We now want a lower bound on U, the number of nodes that are within distance D/2 of the target node, v. First, we note that there are at least d nodes at distance exactly d, for any d n/2; this is because, if d n/2, then considering the square made up of 4d nodes at distance exactly d from v in the infinite two-dimensional lattice, at least one side of this square is completely within the finite grid made of n 2 points. (Drawing a picture might help you see this.) Hence, the number of nodes in U is bounded from below by U min(d/2,n/2) d=1 d min(d2, n 2 ) 8 D2 32, where the last inequality holds for all D 2n, which is the largest value of D we shall consider. Substituting this in (3), we get P(x U) c log n for some constant c that does not depend on n. (4) 6

7 On the event that {x U}, the greedy algorithm gets at least half-way to the target in the next routing step. Observe that the distance to the target decreases at each step under the greedy algorithm, because there is always a lattice neighbour of the current node which is one step closer to the target; hence, no node is visited twice by this algoithm. Since the longrange contacts of different nodes are chosen mutually independently, the probability of the event {x U}, conditional on the set of nodes visited so far, is the same as the unconditional probability. In other words, the lower bound in equation (4) holds for each node x, irrespective of history. This implies that the number of steps of the routing algorithm needed for this event occurs is stochastically dominated by a geometric random variable with success probability c/ log n. Consequently, the expected number of steps before the event occurs is bounded above by c log n for some (different) constant c. After this many steps, the algorithm halves the distance to the target. The rest of the proof is straightforward. Since log 2 n = log n/ log 2 = c log n halvings suffice to reach the target, and the expected number of steps between successive halvings is no more than c log n, the expected number of steps for the algorithm to deliver the message to the target is bounded above by c log 2 n, as claimed. Next, we consider the case s > 2 and show that, in this case, there is no decentralised algorithm that can achieve message delivery in a number of steps smaller than a (fractional) power of n, the number of nodes. In particular, no such algorithm can achieve message delivery in a polylogarithmic number of steps, which is our notion of efficiency. Theorem 2 Consider Kleinberg s model with s > 2. Fix a source node u and a target node v with r(u, v) > n/4. Then, for any decentralised routing algorithm, the expected number of steps for message delivery is at least cn (s 2)/(s 1), where c > 0 is a constant that does not depend on n, or on the choice of source and target nodes. The expectation is with respect to the random distribution of shortcuts, as well as any randomness in the routing algorithm. Remark. If the source and target are very close to each other, it may be possible for a routing algorithm to deliver the message in a small number of steps. The assumption that the distance between them is at least n/4 is not very stringent; it holds, for example, if the source and target are chosen 7

8 uniformly at random from among all nodes. The results of the theorem hold if the source-target separation is bigger than ɛn, for some ɛ > 0; the constant c in the statement of the theorem will then depend on ɛ as well. Proof. The outline of the proof is as follows. We shall show that the probability of encountering a long shortcut (where long will be precisely defined) at any given node is small. A decentralised algorithm has to make choices based on the nodes it has observed so far. It doesn t know if there is some other node nearby with a good shortcut. We will use an estimate of the probability of encountering a long shortcut, and the union bound, to show that the probability of encountering a long shortcut within the first cn (s 2)/(s 1) steps is smaller than a half. We will also show that, conditional on the event of not encountering long shortcuts, the routing algorithm could not have reached the target within this many steps. We now flesh out the details. First observe that for any two nodes x and y, the probability of x choosing y as its long-range contact is given by P(x y) = r(x, y) s z x r(x, z) s r(x, y) s, (5) since there is at least one node z at distance 1 from x, and so the denominator is bigger than 1. Now, fix a node x and let y be the random long-range contact chosen by it. Then, the probability that the distance between them is larger than k is bounded as follows: P(r(x, y) > k) = 2n j=k+1 z:r(x,z)=j) P(x z) 2n j=k+1 4j j s, (6) because there are at most 4j nodes at distance exactly j from x, and the probability of a shortcut to each of them is bounded above by j s by (5). Now, 2n j=k+1 4j j s 4 j=k+1 Substituting in (6), we find that j 1 s 4 P(r(x, y) > k) 4 s 2 k (s 2). k x 1 s dx = 4k2 s s 2. Exactly the same upper bound holds for the probability that some node y at distance bigger than k chooses x as its long-range contact. As the proof of 8

9 this follows along the same lines, we don t repeat it. Thus, the probability that any given node x has a long-range contact (that x chose, or that chose x) a distance further than k away is bounded by 8k (s 2) /(s 2). Let A x denote the event that node x has a shortcut of length bigger than n 1/(s 1). Then, as argued above, Next, let P(A x ) 8 s 2 s 2 n { s 2 n = min s 1. }n s 2 s 1, 16, 1 4 and, for a given decentralised routing algorithm, define A to be the event that one of the first n nodes visited by the routing algorithm has a shortcut of length bigger than n 1/(s 1). In other words, A = n i=1 A xi, where x i denotes the node visited at the i th step of the routing algorithm (or, more precisely, the i th distinct node visited by the algorithm, as it may backtrack). As the algorithm is decentralised, it has to choose nodes with no knowledge of their shortcuts. Hence, the bound on P(A x ) obtained above also applies to P(A xi ) for each i. Hence, by the union bound, P(A) P(A xi ) 8 s 2 s 2 n s 1 n 1 2. (7) n i=1 But on the event A c, the complement of the event A, no shortcuts of length bigger than n 1/(s 1) are encountered in the first n steps. Hence, by the triangle inequality, the total distance traversed in the first n steps is at most n n 1/(s 1). Substituting for n, we find that this distance is bounded above by 1 4 n s 2 1 n s 1 n s 1 = 4. But, by the assumption in the theorem, the source and target of the message are separated by a larger distance than this. Hence, on the event A c, the message is not delivered within the first n steps. Since the event A c has probability at least a half by (7), we conclude that the expected number of steps for message delivery is at least n /2. Looking at the expression for n, the theorem is proved. 9

10 Finally, we consider the case s < 2 and show that, in this case as well, there is no decentralised algorithm that can achieve message delivery in a number of steps smaller than a (fractional) power of n, the number of nodes. In particular, no such algorithm can achieve message delivery in a polylogarithmic number of steps. Theorem 3 Consider Kleinberg s model with s < 2. Fix a source node u and a target node v with r(u, v) > n/4. Then, for any decentralised routing algorithm, the expected number of steps for message delivery is at least cn (2 s)/3, where c > 0 is a constant that does not depend on n, or on the choice of source and target nodes. The expectation is with respect to the random distribution of shortcuts, as well as any randomness in the routing algorithm. Proof. The idea is as follows. We define a suitable neighbourhood U of the target node v and show that, for any node x, the probability of having a shortcut into U is small. This holds even for nodes x within U. Consequently, we argue that once this neighbourhood is reached, subsequent progress by the routing algorithm has to rely on lattice links because, with high probability, suitable shortcuts aren t available. We use this to obtain a lower bound on the expected number of routing steps. Intuitively, the key insight is that, when s < 2, shortcuts may have long range, but they are rather spread out. Hence, the routing algorithm may initially make rapid progress towards the target, but then gets stuck and is not able to zero in. Recall that for any two nodes x and y, the probability that x chooses y as its long-range contact is given by P(x y) = r(x, y) s z x r(x, 1, (8) z x z) s r(x, z) s since r(x, y) 1. We refine the upper bound on P(x y) by deriving a lower bound on the denominator. We have r(x, z) s j j s z x n/2 j=1 n/2 1 x 1 s dx (n/2)2 s 2 s. Why does the first inequality hold? In order to evaluate the sum, z x r(x, z) s, let us group together all nodes z at the same distance j from x. How many 10

11 such nodes are there? In the infinite two-dimensional lattice, there are exactly 4j of them, forming a square. If j n/2, then at least one side of this square is completely contained within the finite lattice being considered (just about, if x is at the centre of this lattice). Therefore, there are at least j nodes at distance exactly j from x, within the finite lattice. Each of these nodes contributes r(x, z) s = j s to the sum. Adding up the contributions from all nodes at distances between 1 and n/2, and ignoring contributions from nodes further away, gives us a lower bound on z x r(x, z) s and hence an upper bound on its reciprocal. Thus, substituting the above inequality in (8), we get P(x y) cn (2 s), (9) for some constant c. Now, define U = } {z : r(v, z) n 2 s 3, to be the set of all nodes wtihin distance n (2 s)/3 of the target v. Since the number of nodes at distance exactly j from v is at most 4j, the number of nodes in the set U is bounded by n (2 s)/3 U 1 + 4j cn 4 2s 3, j=1 for some constant c. (The extra 1 in U is for node v itself.) Let A x denote the event that there is a shortcut from x to U. For each node y U, the probability that x chooses y as its long-range contact is bounded above as in (9). Since the event of x choosing distinct nodes y in U as its contact are mutually exclusive events, the probability that x chooses some node in U as its contact is given by P(x U) = y U P(x y) cn (2 s) U cn 2 s 3, for some constant c. For each node y U, the probability that y chooses x as its long-range contact is exactly the same as the probability that x chooses y. Hence, a similar bound holds for the probability that some node in U chooses x as its long-range contact. Hence, we conclude that P(A x ) cn 2 s 3, (10) 11

12 for all nodes x, and for a constant c that does not depend on n or x. Let x i denote the i th distinct node visited by a given decentralised routing algorithm. Since the node choices of any such algorithm have to be made without knowledge of the shortcuts of the new node to be visited, the above bound on shortcut probability applies to each node x i. Define A = cn (2 s)/3 i=1 A xi, to be the event that there is no shortcut to the set U from any of the nodes visited in the first cn (2 s)/3 steps of the routing algorithm. Here c is a constant to be specified, not the same as the constant c in any other expression! Now, by (??) and the union bound, we can find a constant c such that P(A) cn (2 s)/3 i=1 P(A xi ) 1 2. (The constant 1/2 is arbitary above; we could have used any other constant, but the corresponding c would have been a different constant.) On the event A c, the complement of the event A, there is no shortcut into U in the first cn (2 s)/3 steps of the decentralised routing algorithm. Hence, even if the first step takes us to the boundary of the set U, further progress towards the target v has to be made using lattice steps. By the definition of the set U, at least n (2 s)/3 lattice steps are needed to reach v from the boundary of U. Since the event A c has probability bigger than a half, we conclude that the expected number of steps needed to reach the target is at least 1 2 min{c, 1}n(2 s)/3. This completes the proof of the theorem. 12