An Alternative Perspective

Recall that the pair approximation to the square grid hawk-dove model in Chapter 3 was closed by approximating values for the error terms ; (ijjjk) - equations (3.8) and (3.9) - which were justied

by assumptions on the distribution of neighbours, based on theory and observation. Using these assumptions, we can return directly to equations (3.4) and (3.6) and derive estimates EP

(ijk)for the

absolute number of triples (ijk) in the network, which hold for all i,j and k, as a function of the pair and single variables when each site has m = 4 neighbours:

(ijk)EP

(ijk)= (m ;1)

m (ij)(jk)(j) (4.1)

This formula has three natural and desirable properties for such an estimate:

i. It is proportional to the abundance of each constituent pair, (i-j and j-k), and is therefore zero if any of these are absent from the network.

ii. The total number of triples in the whole network is correctly predicted when each site in the network is occupied by a single species: we expect 12N triples overall (each of the 4N pairs can be extended to a triple by the connection to one of the other three neighbours), which equals 3=4(4N)

2=N, from equation (4.1).

iii. If the pair correlations vanish, so (ij) = 4(i)(j)=N, the expected number of i-j contacts if everybody has 4 neighbours, then the estimate EP

(ijk)reduces to

12N (i)_N (j)_N (k)_N

which is the expected number of i-j-k triples if all sites are independent of their neighbours the mean-eld estimate.

(Note that property ii. is really a special case of property iii. because pairwise correlations are trivial when there is only one species present. It will, however, prove useful to consider ii. directly later.) A natural question to ask next is what can be said about other elemental totals? Consider the following identity for 4-paths:

(ijkl) = X (jk) h Q_x(i);ik ih Q_y(l);lj i ;<ijk>il = X (jk) h Q (ijjk);ik+ x(ijjk) ih Q (ljkj);lj+ y(ljkj) i ;<ijk>il = (jk)(Q (ijjk);ik)(Q (ljkj);lj) + (jk); (ijjkjl);<ijk>il = (ijk)(jkl)_{(jk) + (jk); (i}jjkjl);<ijk>il (4.2)

The rst sum counts all i-j-k-l paths by counting all j-k paths multiplied by the number of distinct i neighbours of the j site and the number of l neighbours of the k site. However, this also counts triangles in which the i and l are the same site, and these must be subtracted to give just (ijkl).

Use has also been made of equation (3.3) and table (4.1). Because <ijk> = 0 on a square grid, the pair approximation estimates for (ijk) in equation (4.1) combine to give

(ijkl) (m;1) 2 m2 (ij)(jk)(kl) (j)(k) (4.3)

providing we also justify the original assumption of ; (ijjkjl)0 from Chapter 3. This looks like

a natural extension of equation (4.1) however, there is a complication. On a square grid with N individuals, there are 433N = 36N 4-paths in total, but 8N of these also form closed squares,

where the two ends neighbour each other. There is no particular reason, except convenience, why this extra connection between i and l should be ignored in this latter case, as it eectively is when setting ; (ijjkjl) = 0 to give equation (4.3). In fact, a similar problem arises with triangles in

the estimation of 3-path totals when they exist in a network, for example on a regular hexagonal grid where each individual interacts with its closest six neighbours. Under such circumstances the multinomial assumption of Chapter 3 would be hard to justify as two neighbours of a site, being neighbours themselves in two-fths of all cases, could not reasonably be assumed to be independent in a pair approximation.

Focusing on the 4-path ; (ijjkjl), we attempt to tackle this problem. Because, as discussed in chap-

ter 3, it is dicult to provide an estimate for ; (ijjkjl) directly we must consider an alternative.

One choice is to attempt to estimate (ijkl) by splitting it into the contribution of open 4-paths, where the i and l are not neighbours, and closed 4-paths, or squares, where they are. The squares are the most dicult, and we start with these.

We need to estimate ijkl]. Any amount of formal analysis along the lines considered so far in this thesis seems to quickly fall at on its face. Instead, consider a heuristic approach based on the estimate for EP

(ijk)found above, which aims to have the same three properties. For any constant c,

c(ij)(jk)(kl)(li)_(i)(j)(k)(l)

satises the rst property. We can choose c so that the second property also holds by solving c(4N)4N;4 = 8N (there are 8N countable squares in total), which gives an estimate that also

satises property three:

EP

ijkl]= N32

(ij)(jk)(kl)(li)

(i)(j)(k)(l) (4.4)

If we follow the same heuristic procedure to get an estimate for just the number of open 4-paths (where the i and l are not neighbours), of which we expect 28N in total, we obtain

7

16(ij)(jk)(kl)(j)(k)

which is just 28=36 of equation (4.3). Note the dierence between this and the square estimate EP

ijkl] is, up to proportionality, just a factor (il)=(i)(l) which alone is just the probability that from

(i)(l) possible i-l pairs, but only (il) realised i-l pairs). Combining the open and closed 4-path estimates, we have: EP (ijkl)= (ij)(jk)(kl)(j)(k) " 7 16 + N32(i)(l)(li) # (4.5) Figure (4.3) shows typical predictions of the estimate for EP

ijkl] when compared to the mean-eld

estimate 8N;3(i)(j)(k)(l) and the observed values of ijkl] from a square grid simulation. The pair

estimate is clearly much better than the mean-eld, and this was true in all cases for simulations at many dierent parameter values.

Unfortunately it is dicult to tell whether or not EP

ijkl]is the best possible pair estimate (in general)

because of its heuristic derivation. It is certainly very natural (by comparison to EP

(ijk)), and it also

does better than many other guesses, such as 1 2N

p

(ij)(jk)(kl)(li) (4.6)

which is another feasible pair approximation to ijkl] that satises the three criteria - see also gure (4.3). (Actually, this estimate is it not directly proportional to each constituent pair frequency, because of the square root. But, importantly, it does vanish if any constituent pair is not present. In fact, we do not have proportionality in the strict sense in equation (4.1) either, because the set of all pair (ij) and single (i) variables is not independent).

The extra nonlinearities introduced into the EP

ijkl] equation, in comparison to that for E

P

(ijk), have

interesting consequences: Whereas the sum of the estimates EP

(ijk)for all possible i,j and k is easily

shown to be a constant 12N, the sum of the estimates EP

ijkl] is not constant

e.g. at (ii) = 2N,

(ij) = N, (jj) = 0 the sum is 656N=818:1N, compared to the ideal 8N. At (ii) = (jj) = 2N,

(ij) = 0, the sum is 16N. One should note, however, that this last distribution of strongly correlated pairs is highly articial. No real square lattice could attain such values.

When both estimates (4.3) and (4.5) for (ijkl) are compared graphically (not shown), there appears to be very little to choose between them. For some parameter values and some (ijkl) the estimate incorporating closed squares is closer to the observed value than the ; (ijjkjl) = 0 estimate for

others it is worse. Put another way, it appears that ; (ijjkjl) is equally well approximated by zero

as by (ij)(kl) (j)(k) " N 32(l)(i)(li) ; 1 8 # (4.7) which is the expression derived by combining equation (4.2) with the estimates in (4.1) and (4.5). This is small when i-l correlations are close to neutral: (il) = 4(i)(l)=N. There could be many reasons for this observation: The contribution from closed squares may be too insignicant com- pared to those omitted higher order correlations (triples, etc.), or perhaps there is a better square

approximation than EP

0 10 20 30 40 50 0 200 400 600 800 1000 1200 1400 [dhdh] time number a 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10 4 [dddd] time number b 0 10 20 30 40 50 0 500 1000 1500 2000 2500 [dddh] time number c 0 10 20 30 40 50 0 200 400 600 800 1000 1200 1400 [ddhh] time number d 0 10 20 30 40 50 0 200 400 600 800 1000 1200 1400 1600 1800 [dhhh] time number e 0 10 20 30 40 50 0 1000 2000 3000 4000 5000 6000 7000 [hhhh] time number f

Figure 4.3: Comparison between estimates for ijkl] (black) as observed in the lattice hawk-dove simulation: the mean-eld (blue), pair estimate EP

ijkl] (red) and

alternative pair estimate - equation (4.6) (yellow). The simulationwas run at s = 0:5 and N = 2500 from a random initial comprising 99% doves.

In deriving equation (4.5), we have attempted to use our knowledge of the actual spatial structure (in this case an abstract square grid) to incorporate the eect of a number of known neighbourly contacts which were originally ignored - namely the closed square contacts for 4 paths. In principle, however, there is no reason to stop with direct neighbours. Even for a pair approximation, the inuence of more indirect connections can be considered: The estimate EP

(ijk)for triples on a square

grid, for example, could be modied to incorporate the eect of a possible common neighbour of the i and k, in the case when ijk and a fourth individual form a closed square. One would separately consider the contribution from the `straight line' i-j-k triples and from the `bent' triples contained within ijkl] squares for every possible species l in the system. The next step would be to consider twice removed common neighbours, then three times removed neighbours, and so on.

Undoubtedly, the diculty in deriving the equations each time would rapidly increase as more re- mote common connections were considered. We may also expect a diminishing payo in terms of the returned extra accuracy of each successive estimate. After all, such a chain of models are still based only on pair correlations (nothing higher) and there is no reason why they should asymptotically approach the behaviour of an explicit space parent system. More and more information about the spatial structure of the network is required with each successive model, and probably sooner rather than later it would be more productive to consider higher order models based on higher correlations than pairs alone.