Entropies of tailored random graph ensembles: bipartite graphs, generalised degrees, and node neighbourhoods
EEntropies of tailored random graph ensembles: bipartitegraphs, generalised degrees, and node neighbourhoods
ES Roberts †‡ and ACC Coolen †§ † Institute for Mathematical and Molecular Biomedicine, King’s College London, HodgkinBuilding, London SE1 1UL, United Kingdom ‡ Randall Division of Cell and Molecular Biophysics, King’s College London, New HuntsHouse, London SE1 1UL, United Kingdom § London Institute for Mathematical Sciences, 35a South St, Mayfair, London W1K 2XF,United KingdomPACS numbers: 89.70.Cf, 89.75.Fb, 64.60.aqE-mail: [email protected] [email protected]
Abstract.
We calculate explicit formulae for the Shannon entropies of several families oftailored random graph ensembles for which no such formulae were as yet available, in leadingorders in the system size. These include bipartite graph ensembles with imposed (and possiblydistinct) degree distributions for the two node sets, graph ensembles constrained by specifiednode neighbourhood distributions, and graph ensembles constrained by specified generaliseddegree distributions.
1. Introduction
Networks are powerful and popular tools for characterising large and complex interactingparticle systems. They have become extremely valuable in physics, biology, computerscience, economics, and the social sciences. One approach is to quantify the implications ofhaving topological patterns in networks and graphs, by viewing these patterns as constraintson a random graph ensemble. This provides a way to measure and compare topologicalfeatures from the rational point of view of whether they are present in a large or smallnumber of possible networks. Precise definitions of random graph ensembles with controlledtopological characteristics also allow us to generate systematically graphs and networks whichare tailored to have features in common with those observed in a given application domain,either for the purpose of statistical mechanical process modelling or to serve as ‘null models’against which to test the importance of observations in real-world networks.A previous paper [1] considered tailored random graph ensembles with controlled degreedistribution and degree-degree correlations; the more recent [2] covered the case of directednetworks. In each case, the strategy is to calculate the Shannon entropy, from which we candeduce the e ff ective number of graphs in the ensemble. Related quantities such as complexityof typical graphs from the ensemble and information-theoretic distances between graphsnaturally follow from the entropy, or can be calculated using similar methods.In this paper we calculate, in leading order, the Shannon entropies of three as yetunsolved families of random graph ensembles, constrained by three di ff erent conditions: abipartite constraint with imposed degree distributions in the two nodes sets, a neighbourhooddistribution (where the neighbourhood of a node is defined as its own degree, plus the degree a r X i v : . [ c ond - m a t . d i s - nn ] A p r values of the nodes connected to it), and an imposed generalised degree distribution. Theseare each interesting in their own right as stand-alone results, and turn out to be closely linked.The first two cases can be resolved exactly, and give practical analytical expressions. Thegeneralised degree case was already partially studied in [3], with only limited success, andhere we require a plausible but as yet unproven conjecture to find an explicit formula for theentropy.The generalised degrees concept appears in the literature in various forms. For example,the authors of [4], measured the number of direct neighbours s of a subset of t nodes. Theyderive conditions based on their definition of general degrees which can ensure that (for somegiven m and d ) there are at least m internally disjoint paths of length at most d . The diameterof the network is an obvious corollary - the smallest d corresponding to m ≥
1. Theseresults can be applied to questions of robustness of networks. The authors of [5] studied thespectral density of random graphs with hierarchically constrained topologies. This includesconsideration of generalised degrees, as well as more general community structures. Using thereplica method, in a similar way to [3], they achieve a form analogous to equation (40). Theyproceed numerically from that point, hence our approach to an analytical solution presentedin equation (49) is entirely novel.
2. Definitions and notation
We consider ensembles of directed and nondirected random graphs. Each graph is defined byits adjacency matrix c = { c i j } , with i , j ∈ { , . . . , N } and with c i j ∈ { , } for all ( i , j ). Twonodes i and j are connected by a directed link j → i if and only if c i j =
1. We put c ii = i . In nondirected graphs one has c i j = c ji for all ( i , j ), so c is symmetric. The degree of anode i in a nondirected graph is the number of its neighbours, k i = (cid:80) j c i j . In directed graphswe distinguish between in- and out-degrees, k in i = (cid:80) j c i j and k out i = (cid:80) j c ji . They count thenumber of in- and out-bound links at a node i . A bipartite graph is one where the nodes canbe divided into two disjoint sets, such that c i j = i and j that belong to the same set.We define the set of neighbours of a node i in a nondirected graph as ∂ i = { j | c i j = } .Hence k i = | ∂ i | . To characterise a graph’s topology near i in more detail we can define thegeneralised degree of i as the pair ( k i , m i ), where m i = (cid:80) j c i j k j counts the number of length-two paths starting in i . The concept of a generalised degree is discussed in [6]. Even moreinformation is contained in the local neighbourhoodn i = ( k i ; { ξ si } ) , (1)in which the ordered integers { ξ si } give the degrees of the k i neighbours j ∈ ∂ i . See alsoFig. 1. Since m i = (cid:80) s ≤ k i ξ si , the neighbourhood n i provides more granular information thatcomplements that in the generalised degree ( k i , m i ). We will use bold symbols when localtopological parameters are defined for every node in a network, e.g. k = ( k , . . . , k N ) and n = (( k ; { ξ s } ) , . . . , ( k N ; { ξ sN } )). Generalisation to directed graphs is straightforward. Here ∂ i = { j | c i j + c ji > } , and the local neighbourhood would be defined as n i = ( (cid:126) k i ; { (cid:126)ξ si } ) with the k i pairs (cid:126)ξ si = ( k s , in , k s , out ) now giving both the in- and out-degrees of the neighbours of i .Our tailored random graph ensembles will be of the following form, involving N built-inlocal (site specific) topological constraints of the type discussed above, which we will for nowwrite generically as X i ( c ), and with the usual abbreviation δ a , b = (cid:81) i δ a i , b i : p ( c ) = (cid:88) X p ( X ) p ( c | X ) p ( X ) = (cid:89) i p ( X i ) (2) • i k i = k i , m i ) = (4 , n i = ( k i ; { ξ si } ) = (4; 3 , , , Figure 1.
Illustration of our definitions of local topological chacteristics in non-directedgraphs. At the minimal level one specifies for each node i (black vertex in the picture) onlythe degree k i = | ∂ i | = (cid:80) j c ij (the number of its neighbours). At the next level of detail oneprovides for each node the generalised degree ( k i , m i ), in which m i = (cid:80) j ∈ ∂ i k j = (cid:80) j c ij k j is thenumber of length-two paths starting in i . This is then generalised to include the actual degreesin the set ∂ i , by giving n i = ( k i ; { ξ si } ) (the ‘local neighbourhood’), in which the k i integers { ξ si } give the degrees of the nodes connected to i . To avoid ambiguities we adopt the rankingconvention ξ i ≤ ξ i ≤ . . . ≤ ξ k i i . Note that m i = (cid:80) j ∈ ∂ i k j = (cid:80) k i s = ξ si . p ( c | X ) = Z − ( X ) δ X , X ( c ) , Z ( X ) = (cid:88) c δ X , X ( c ) (3)The values X i for the local features are for each i drawn randomly and independentlyfrom p ( X ), after which one generates a graph c randomly and with uniform probabilitiesfrom the set of graphs that satisfy the N demands X i ( c ) = X i . The empirical distribution p ( X | c ) = N − (cid:80) i δ X , X i ( c ) of local features will be random, but the law of large numbers ensuresthat for N → ∞ it will converge to the chosen p ( X ) in (2) for any graph realisation, and theabove definitions guarantee that its ensemble average will be identical to p ( X ) for any N , (cid:88) c p ( c ) p ( X | c ) = N (cid:88) i (cid:88) X p ( X ) (cid:88) c δ X , X ( c ) Z ( X ) δ X , X i ( c ) = p ( X ) (4)If we aim to impose upon our graphs only a degree distribution we choose X i ( c ) = k i ( c ).Building in a distribution of generalised degrees corresponds to X i ( c ) = ( k i ( c ) , m i ( c )). If weseek to prescribe the distribution of all local neighbourhoods (1) we choose X i ( c ) = n i ( c ).A further quantity which will play a role in subsequent calculations is the joint degreedistribution of connected nodes. For nondirected graphs it is defined as W ( k , k (cid:48) | c ) = (cid:80) i j c i j δ k , k i δ k (cid:48) , k j (cid:80) i j c i j (5)and its average over the ensemble (2) is given by W ( k , k (cid:48) ) = (cid:88) X p ( X ) (cid:88) c W ( k , k (cid:48) | c ) δ X , X ( c ) Z ( X ) (6)In this paper we study the leading orders in the system size N of the Shannon entropyper node of the above tailored random graph ensembles (2), from which the e ff ective numberof graphs with the prescribed distribution p ( X ) of features follows as N = exp( NS ): S = − N (cid:88) c p ( c ) log p ( c ) = − N (cid:88) X (cid:81) i p ( X i ) Z ( X ) (cid:88) c δ X , X ( c ) log (cid:20) (cid:88) X (cid:48) (cid:81) j p ( X (cid:48) j ) Z ( X (cid:48) ) δ X (cid:48) , X ( c ) (cid:21) = − N (cid:88) X (cid:81) i p ( X i ) Z ( X ) (cid:88) c δ X , X ( c ) log (cid:20) (cid:81) j p ( X j ) Z ( X ) (cid:21) = (cid:88) X p ( X ) S ( X ) − (cid:88) X p ( X ) log p ( X ) (7)with S ( X ) = N log Z ( X ) = N log (cid:88) c δ X , X ( c ) (8)The core of the entropy calculation is determining the leading orders in N of S ( X ), whichis the Shannon entropy per node of the ensemble p ( c | X ) in which all node-specific values X = ( X , . . . , X N ) are constrained. For p ( X ) = p ( k ) this calculation has already been done in[1, 2]. For p ( X ) = p ( k , m ) it has only partly been done [3]. Here we investigate the relationbetween the entropies of the p ( k ) and p ( k , m ) ensembles and the entropy of the ensemble inwhich the distribution p ( n ) of local neighbourhoods (1) is imposed.
3. Building blocks of the entropy calculations
Since the generalised degrees ( k i , m i ) can be calculated from the local neighbourhoods (1)for any graph c , it is clear that the empirical distribution p ( k , m | c ) = N − (cid:80) i δ k , k i ( c ) δ m , m i ( c ) for any graph can be calculated from the empirical neighbourhood distribution p ( n | c ) = N − (cid:80) i δ n , n i ( c ) . If we denote with k ( n ) the central degree k in n = ( k ; { ξ s } ), we indeed obtain p ( k , m | c ) = N (cid:88) i δ k , k i ( c ) δ m , m i ( c ) (cid:88) n δ n , n i = (cid:88) n p ( n ) δ k , k ( n ) δ m , (cid:80) s ≤ k ( n ) ξ s (9)Less trivial is the statement that also the distribution W ( k , k (cid:48) | c ) of (5) can be written in termsof p ( n | c ). Using (cid:80) i j c i j = N ¯ k ( c ), with ¯ k ( c ) = N − (cid:80) i k i ( c ) we obtain W ( k , k (cid:48) | c ) = (cid:80) i δ k , k i ( c ) (cid:80) j ∈ ∂ i δ k (cid:48) , k j ( c ) N (cid:80) n p ( n | c ) k ( n ) = (cid:80) i (cid:80) n δ n , n i ( c ) δ k , k ( n ) (cid:80) s ≤ k ( n ) δ k (cid:48) ,ξ s N (cid:80) n p ( n | c ) k ( n ) = (cid:80) n p ( n | c ) δ k , k ( n ) (cid:80) s ≤ k ( n ) δ k (cid:48) ,ξ s (cid:80) n p ( n | c ) k ( n ) (10)Given the symmetry of W ( k , k (cid:48) | c ) under permutation of k and k (cid:48) we then also have W ( k , k (cid:48) | c ) = (cid:80) n p ( n | c ) δ k (cid:48) , k ( n ) (cid:80) s ≤ k ( n ) δ k ,ξ s (cid:80) n p ( n | c ) k ( n ) (11)The converse of the above statements is not true. One cannot calculate the neighbourhouddistribution p ( n | c ) from p ( k , m | c ) or from W ( k , k (cid:48) | c ) (or both). Note that by definition (andsince c is nondirected) we always have W ( k , k (cid:48) | c ) = W ( k (cid:48) , k | c ). Any nondirected graph c can always be decomposed uniquely into a collection of non-overlapping N -node subgraphs β kk (cid:48) , with k , k (cid:48) ∈ IN, which share the nodes { , . . . , N } of c but not all of the links. These subgraphs are defined for each ( k , k (cid:48) ) by the adjacency matrices β kk (cid:48) i j = c i j δ k , k i ( c ) δ k (cid:48) , k j ( c ) (12)Each graph β kk (cid:48) contains those links in c that go from a node with degree k (cid:48) to a node withdegree k . Clearly, all graphs β kk (cid:48) follow uniquely from c via (12). The converse uniquenessof c , given the matrices β kk (cid:48) , is a consequence of the simple identity c i j = c i j (cid:88) kk (cid:48) ≥ δ k , k i ( c ) δ k (cid:48) , k j ( c ) = (cid:88) kk (cid:48) ≥ δ k , k i ( c ) δ k (cid:48) , k j ( c ) c i j = (cid:88) kk (cid:48) ≥ β kk (cid:48) i j (13)The graph β kk (cid:48) is directed if k (cid:44) k (cid:48) , and nondirected if k = k (cid:48) . From the symmetry of c it followsmoreover that β kk (cid:48) ji = β k (cid:48) ki j for all ( i , j , k , k (cid:48) ), so β k (cid:48) k is specified in full by β kk (cid:48) . Although each β kk (cid:48) is an N -node graph, most of the nodes in β kk (cid:48) will be isolated: all nodes whose degrees inthe original graph c were neither k nor k (cid:48) will have degree zero in β kk (cid:48) .We now inspect the degree statistics of the decomposition graphs β kk (cid:48) , and their relationwith the structural features of c . If k (cid:44) k (cid:48) we find for the remaining degrees in β kk (cid:48) : k i ( c ) = k : k in i ( β kk (cid:48) ) = (cid:88) j ∈ ∂ i δ k (cid:48) , k j ( c ) , k out i ( β kk (cid:48) ) = k j ( c ) = k (cid:48) : k out j ( β kk (cid:48) ) = (cid:88) i ∈ ∂ j δ k , k i ( c ) , k in j ( β kk (cid:48) ) = β kk (cid:48) can be writen in terms of the empiricaldistribution of neighbourhoods of c , viz. p ( n | c ) = N − (cid:80) i δ n , n i ( c ) with n = ( k ; { ξ s } ): p kk (cid:48) ( q in , q out ) = N (cid:88) i δ q in , k in i ( β kk (cid:48) ) δ q out , k out i ( β kk (cid:48) ) = N (cid:88) i δ q in ,δ k , ki ( c ) (cid:80) j ∈ ∂ i δ k (cid:48) , kj ( c ) δ q out ,δ k (cid:48) , ki ( c ) (cid:80) j ∈ ∂ i δ k , kj ( c ) = N (cid:88) i (cid:20) δ k , k i ( c ) δ q in , (cid:80) j ∈ ∂ i δ k (cid:48) , kj ( c ) + (1 − δ k , k i ( c ) ) δ q in , (cid:21) × (cid:20) δ k (cid:48) , k i ( c ) δ q out , (cid:80) j ∈ ∂ i δ k , kj ( c ) + (1 − δ k (cid:48) , k i ( c ) ) δ q out , (cid:21) = (cid:88) n p ( n | c ) (cid:20) δ k , k ( n ) δ q in , (cid:80) s ≤ k ( n ) δ k (cid:48) ,ξ s ( n ) + (1 − δ k , k ( n ) ) δ q in , (cid:21) × (cid:20) δ k (cid:48) , k ( n ) δ q out , (cid:80) s ≤ k ( n ) δ k ,ξ s ( n ) + (1 − δ k (cid:48) , k ( n ) ) δ q out , (cid:21) (16)The two marginals of (16) are p kk (cid:48) in ( q ) = (cid:88) n p ( n | c ) (cid:20) δ k , k ( n ) δ q , (cid:80) s ≤ k ( n ) δ k (cid:48) ,ξ s ( n ) + (1 − δ k , k ( n ) ) δ q , (cid:21) (17) p kk (cid:48) out ( q ) = (cid:88) n p ( n | c ) (cid:20) δ k (cid:48) , k ( n ) δ q , (cid:80) s ≤ k ( n ) δ k ,ξ s ( n ) + (1 − δ k (cid:48) , k ( n ) ) δ q , (cid:21) (18)Hence p kk (cid:48) in ( q ) = p k (cid:48) k out ( q ), as expected. The average degree ¯ q kk (cid:48) = (cid:80) q in , q out q in p kk (cid:48) ( q in , q out ) = (cid:80) q in , q out q out p kk (cid:48) ( q in , q out ) of the graph β kk (cid:48) can be written, using identity (11) and the symmetryof W ( k , k (cid:48) | c ), as ¯ q kk (cid:48) = (cid:88) n p ( n | c ) δ k ( n ) , k (cid:88) s ≤ k ( n ) δ k (cid:48) ,ξ s ( n ) = ¯ k ( c ) W ( k , k (cid:48) | c ) (19)If k = k (cid:48) , the decomposition matrix β kk (cid:48) is symmetric. Here we find k i ( β kk ) = δ k , k i ( c ) (cid:88) j ∈ ∂ i δ k , k j ( c ) (20)Hence the degree distribution of β kk becomes p kk ( q ) = N (cid:88) i δ q ,δ k , ki ( c ) (cid:80) j ∈ ∂ i δ k , kj ( c ) = N (cid:88) i (cid:20) δ k , k i ( c ) δ q , (cid:80) j ∈ ∂ i δ k , kj ( c ) + (1 − δ k , k i ( c ) ) δ q , (cid:21) = (cid:88) n p ( n | c ) (cid:20) δ k , k ( n ) δ q , (cid:80) s ≤ k ( n ) δ k ,ξ s ( n ) + (1 − δ k , k ( n ) ) δ q , (cid:21) (21)The average degree in β kk is therefore¯ q kk = (cid:88) n p ( n | c ) δ k ( n ) , k (cid:88) s ≤ k ( n ) δ k ,ξ s ( n ) = ¯ k ( c ) W ( k , k | c ) (22)
4. Entropy of ensembles of bipartite graphs
Here we calculate the leading orders in N of the entropy per node (7) for ensembles of bipartitegrahs with prescribed (and possibly distinct) degree distributions in the two node sets. Thisis not only a novel result in itself, but will also form the seed of the entropy calculation forensembles with constrained neighboorhoods in a subsequent section.In a bipartite ensemble the N nodes can be divided into two disjoint sets A , B ⊆ { , . . . , N } such that c i j = i , j ∈ A or i , j ∈ B , leaving only links between A and B . Thisconstraint implies that there is a bijective mapping from the set of bipartite graphs on on { , . . . , N } to the set of directed graphs on { , . . . , N } , defined by assigning to each bipartitelink the direction of flow from A to B . This allows us to draw upon results on directed graphsderived in [2]. The directed graph c (cid:48) associated with the bipartite graph c would have j ∈ B or i ∈ A : c (cid:48) i j = j ∈ A and i ∈ B : c (cid:48) i j = c i j (24)and hence the in- and out-degree sequence (cid:126) k = (( k in1 , k out1 ) , . . . , k in N , k out N )) of c (cid:48) can be expressedin terms of the degree sequence k of c via i ∈ A : (cid:126) k i = ( k in i , k out i ) = (0 , k i ) (25) i ∈ B : (cid:126) k i = ( k in i , k out i ) = ( k i ,
0) (26)The directed graph will thus have the joint degree distribution p ( q in , q out ) = | A | N δ q in , p A ( q out ) + (1 − | A | N ) p B ( q in ) δ q out , (27)with the degree distributions p A ( k ) = | A | − (cid:80) i ∈ A δ k , k i ( c ) and p B ( k ) = | B | − (cid:80) i ∈ B δ k , k i ( c ) in thesets A and B of the bipartite graph. Our bipartite ensemble is one in which we describe thedistributions p A ( k ) and p B ( k ), together with the probability f ∈ [0 ,
1] for a node to be insubset A , and we forbid links within the sets A or B . Conservation of links demands that thetwo distributions cannot be independent, but must obey ¯ q = (1 − f ) (cid:80) q qp B ( q ) = f (cid:80) q qp A ( q ),where ¯ q is the average degree. Our bijective mapping to directed graphs shows that the entropyof any bipartite ensemble can be calculated by application of (7,8) to an ensemble of directedgraphs, with X i = ( τ i , k i ). Here τ i ∈ { A , B } gives the subset assigment of a node. We then find S = (cid:88) τ , k (cid:20) (cid:89) i p ( τ i , k i ) (cid:21) S ( τ , k ) − f log f − (1 − f ) log(1 − f ) − f (cid:88) k p A ( k ) log p A ( k ) − (1 − f ) (cid:88) k p B ( k ) log p B ( k ) (28)with p ( τ, k ) = f δ τ, A p A ( k ) + (1 − f ) δ τ, B p B ( k ) (29) S ( τ , k ) = N log (cid:88) c (cid:18) (cid:89) i ,τ i = A δ (cid:126) k i , (0 , k i ) (cid:19)(cid:18) (cid:89) i ,τ i = B δ (cid:126) k i , ( k i , (cid:19) (30)The latter quantity follows from the calculation in [7], with the short-hand π ¯ q ( q ) = e − ¯ q ¯ q q / q !and modulo terms that vanish for N → ∞ : S ( τ , k ) = ¯ q [log( N / ¯ q ) + + (cid:88) q (cid:20) f δ q , + (1 − f ) p B ( q ) (cid:21) log π ¯ q ( q ) + (cid:88) q (cid:20) f p A ( q ) + (1 − f ) δ q , (cid:21) log π ¯ q ( q ) = ¯ q log( N / ¯ q ) + f (cid:88) q p A ( q ) log π ¯ q ( q ) + (1 − f ) (cid:88) q p B ( q ) log π ¯ q ( q ) (31)This then leads to our final result for the entropy per node of tailored bipartite graphensembles, with imposed bipartite degree distributions p A ( k ) and p B ( k ), average degree ¯ k ,and a fraction f of nodes in the set A (modulo vanishing orders in N ): S = ¯ k log( N / ¯ k ) − f log f − (1 − f ) log(1 − f ) − f (cid:88) k p A ( k ) log (cid:18) p A ( k ) π ¯ k ( k ) (cid:19) − (1 − f ) (cid:88) k p B ( k ) log (cid:18) p B ( k ) π ¯ k ( k ) (cid:19) (32)If the sets A and B were to be specified explicity (as opposed to only their relative sizes), thecontribution S f = − f log f − (1 − f ) log(1 − f ) would disappear from the above formula.
5. Entropy of ensembles with constrained neighbourhoods
We now turn to the Shannon entropy per node (7) of the ensemble (2) in which for theobservables X i ( c ) we choose the local neighbourhood n i ( c ) defined in (1). For this we needto calculate the leading orders of S ( n ) = N − log (cid:80) c δ n , n ( c ) . We now use the one-to-onerelationship between a graph c and its decomposition c = (cid:80) qq (cid:48) β qq (cid:48) , to write S ( n ) = N log (cid:88)(cid:110) β kk (cid:48) (cid:111) δ n , n ( c ) (33)The next argument is the key to our ability to evaluate the entropy. It involves translating theconstraint n = n ( c ) into constraints on the decomposition matrices β kk (cid:48) . Let us define the setsof nodes in c which have the same degree, viz. I k ( n ) = { i ≤ N | k i ( c ) = k } . The constraint n = n ( c ) in (33) prescribes:( i ) all the sets I k of nodes with a given degree( ii ) for each node i ∈ I k which sets I k (cid:48) this node is (possibly multiply) connected toHence the constraint n = n ( c ) specifies exactly the in- and out-degree sequences of alldecomposition matrices β kk (cid:48) of c , which we will denote as (cid:126) q kk (cid:48) = ( q in , kk (cid:48) , q out , kk (cid:48) ), and whosedistributions we have already calculated in (16,21). We thus see that (33) can be written as S ( n ) = N log (cid:88)(cid:110) β kk (cid:48) (cid:111) (cid:89) kk (cid:48) δ (cid:126) q kk (cid:48) n ,(cid:126) q ( β kk (cid:48) ) (34)in which (cid:126) q kk (cid:48) n are the in- and out-degree sequences that are imposed by the local environmentsequence n on the decomposition matrix β kk (cid:48) , and whose distributions are known to be (16,21).Using the symmetry ( β kk (cid:48) ) † = β k (cid:48) k we may now write S ( n ) = N log (cid:20)(cid:18) (cid:89) k < k (cid:48) (cid:88) β kk (cid:48) δ (cid:126) q kk (cid:48) n ,(cid:126) q ( β kk (cid:48) ) (cid:19)(cid:18) (cid:89) k (cid:88) β kk δ (cid:126) q kk n ,(cid:126) q ( β kk ) (cid:19)(cid:21) = (cid:88) k < k (cid:48) (cid:26) N log (cid:88) β kk (cid:48) δ (cid:126) q kk (cid:48) n ,(cid:126) q ( β kk (cid:48) ) (cid:27) + (cid:88) k (cid:26) N log (cid:88) β kk δ q kk n , q ( β kk ) (cid:27) (35)We see that the entropy S ( n ) can be written as the sum of the entropies of sub-ensembles,which are the decomposition matrices β kk (cid:48) with prescribed degree sequences. The second sumin (35) is over nondirected ensembles, the first over directed ones. The sub-entropies were allcalculated, respectively, in [1] and [2] ‡ . The entropy of an N -node nondirected random graphensemble with degree sequence q was found to be (modulo terms that vanish for N → ∞ ): S q = N log (cid:88) c δ q , q ( c ) =
12 ¯ q [log( N / ¯ q ) + + (cid:88) q p ( q ) log π ¯ q ( q ) (36)in which ¯ q = N − (cid:80) i q i and π ¯ q ( q ) is the Poisson distribution with average ¯ q . The entropy ofan N -node directed random graph ensemble with in- and out-degree sequence (cid:126) q was found tobe (modulo terms that vanish for N → ∞ ): S (cid:126) q = N log (cid:88) c δ (cid:126) q ,(cid:126) q ( c ) = ¯ q [log( N / ¯ q ) + + (cid:88) q in , q out p ( q in , q out ) log[ π ¯ q ( q in ) π ¯ q ( q out )] (37)The above entropies depend in leading orders only on the degree distributions (as opposed tothe degree sequences), and since these distributions were already calculated (16,21), we cansimply insert (36,37) into (35), with the correct distributions (16,21), and find an expressionthat depends only on the local environment distribution p ( n ) = N − (cid:80) i δ n , n i : S ( n ) = (cid:88) k < k (cid:48) (cid:26) ¯ q kk (cid:48) [log( N / ¯ q kk (cid:48) ) + + (cid:88) q in , q out p kk (cid:48) ( q in , q out ) log[ π ¯ q kk (cid:48) ( q in ) π ¯ q kk (cid:48) ( q out )] (cid:27) + (cid:88) k (cid:26)
12 ¯ q kk [log( N / ¯ q kk ) + + (cid:88) q p kk ( q ) log π ¯ q kk ( q ) (cid:27) = (cid:88) k (cid:44) k (cid:48) (cid:26) ¯ kW ( k , k (cid:48) )[log( N / ¯ kW ( k , k (cid:48) )) − ‡ In [1, 2] the entropies were carried out for ensembles with prescribed degree distributions, but it was shown that,in analogy with (7), this is simply the sum of the Shannon entropy of the degree distributions and the entropy of thecorresponding ensemble with prescribed sequences. + kW ( k , k (cid:48) ) log[¯ kW ( k , k (cid:48) )] − (cid:88) q [ p kk (cid:48) in ( q ) + p kk (cid:48) out ( q )] log q ! (cid:27) + (cid:88) k (cid:26) ¯ kW ( k , k )[log( N / ¯ kW ( k , k )) − + kW ( k , k ) log[¯ kW ( k , k (cid:48) )] − (cid:88) q p kk ( q ) log q ! (cid:27) =
12 ¯ k [log( N / ¯ k ) − +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) − (cid:88) q (cid:20) (cid:88) k (cid:44) k (cid:48) [ p kk (cid:48) in ( q ) + p kk (cid:48) out ( q )] + (cid:88) k p kk ( q ) (cid:21) log q ! =
12 ¯ k [log( N / ¯ k ) − +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) − (cid:88) q (cid:88) n p ( n ) (cid:88) k , k (cid:48) (cid:20) δ k , k ( n ) δ q , (cid:80) s ≤ k ( n ) δ k (cid:48) ,ξ s ( n ) + (1 − δ k (cid:48) , k ( n ) ) δ q , (cid:21) log q ! =
12 ¯ k [log( N / ¯ k ) − +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) − (cid:88) n p ( n ) (cid:88) k log (cid:20)(cid:18) (cid:88) s ≤ k ( n ) δ k ,ξ s ( n ) (cid:19) ! (cid:21) (38)Insertion of this result into the general formula (7) gives us an analytical expression for theShannon entropy of the random graph ensemble with prescribed distribution p ( n ) of localneighbourhoods, modulo terms that vanish for N → ∞ . This expression is fully explicit,since ¯ k and W ( k , k (cid:48) ) are both determined by the distribution p ( n ), via ¯ k = (cid:80) n p ( n ) k ( n ) and(11) respectively: S =
12 ¯ k [log( N / ¯ k ) − +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) − (cid:88) n p ( n ) log p ( n ) − (cid:88) n p ( n ) (cid:88) k log (cid:20)(cid:18) (cid:88) s ≤ k ( n ) δ k ,ξ s ( n ) (cid:19) ! (cid:21) (39)
6. Entropy of ensembles of networks with specified generalized degree distribution
In this section we consider an ensemble of nondirected networks with a specified generalizeddegree distribution p ( k , m ) = N − (cid:80) i δ k , k i ( c ) δ m , m i ( c ) , where k i ( c ) = (cid:80) j c i j and m i = (cid:80) jk c i j c jk .Previous work [3] began this calculation, and reached (in leading order) the intermediate formset out below: S =
12 ¯ k [log( N / ¯ k ) + − (cid:88) k , m p ( k , m ) log (cid:18) p ( k , m ) π ( k ) (cid:19) + (cid:88) k , m p ( k , m ) log (cid:18) (cid:88) ξ ,.,ξ k δ m , (cid:80) ks = ξ s k (cid:89) s = γ ( k , ξ s ) (cid:19) (40)¯ k indicates the average degree; π ¯ k ( k ) is the Poissonian distribution with average degree ¯ k .The sum inside the logarithm in the final term of (40) runs over all sets of k nonnegative0integers ξ . . . ξ k . The function γ ( ., . ) is defined as the non-negative solution to the followingself-consistency relation: γ ( k , k (cid:48) ) = (cid:88) m (cid:48) k (cid:48) ¯ k p ( k (cid:48) , m (cid:48) ) (cid:80) ξ ...ξ k (cid:48)− δ m (cid:48) − k , (cid:80) k (cid:48)− s = ξ s (cid:81) k (cid:48) − s = γ ( k (cid:48) , ξ s ) (cid:80) ξ ...ξ k (cid:48) δ m (cid:48) , (cid:80) k (cid:48) s = ξ s (cid:81) k (cid:48) s = γ ( k (cid:48) , ξ s ) (41)This equation does not yield to a straightforward solution, and can only be evaluatednumerically or in certain special cases. Without a physical interpretation of γ ( k , k (cid:48) ), thisintermediate answer is limited in how much insight it can provide. We will now show howthe entropy can be expressed in terms of measurable quantities.Our strategy is to derive an expression for the (observable) degree-degree correlations W ( k , k (cid:48) ), and show that these can be expressed it terms of the order parameter γ ( k , k (cid:48) ) thatappears in equation (40). We calculate the average of this quantity in our tailored ensemblesof the form (2), where we now define topological characteristics by specifying a generaliseddegree distribution p ( k , m ). We follow closely the steps taken in [3], and write for theensemble a specified generalised degree sequence ( k , m ): W ( k , k (cid:48) ) = Nk (cid:88) c p ( c | k , m ) (cid:88) rs c rs δ k , (cid:80) (cid:96) c r (cid:96) δ k (cid:48) , (cid:80) (cid:96) c s (cid:96) = N (cid:88) rs δ k , k r δ k (cid:48) , k s (cid:82) ππ d θ d φ e i( θ · k + φ · m ) − i( θ r + θ s + φ r k s + φ s k r ) (cid:81) i < j (cid:20) + kN (cid:18) e − i( θ i + θ j + φ i k j + φ j k i ) − (cid:19)(cid:21)(cid:82) ππ d θ d φ e i( θ · k + φ · m ) (cid:81) i < j (cid:20) + kN (cid:18) e − i ( θ i + θ j + φ i k j + φ j k i ) − (cid:19)(cid:21) + O ( 1 N ) = N (cid:88) rs δ k , k r δ k (cid:48) , k s (cid:82) ππ d θ d φ e i( θ · k + φ · m ) − i( θ r + θ s + φ r k (cid:48) + φ s k ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + ... (cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + ... + O ( 1 N ) = (cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) (cid:18) N (cid:80) rs δ k , k r δ k (cid:48) , k s e − i( θ r + θ s + φ r k (cid:48) + φ s k ) (cid:19)(cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + O ( 1 N ) = (cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) (cid:18) N (cid:80) r δ k , k r e − i( θ r + φ r k (cid:48) ) (cid:19)(cid:18) N (cid:80) s δ k (cid:48) , k s e − i( θ s + φ s k ) (cid:19)(cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + O ( 1 N ) = (cid:82) { d P d ˆ P } e N Ψ [ P , ˆ P ] (cid:18) (cid:82) d θ d φ P ( θ, φ, k )e − i θ − i φ k (cid:48) (cid:19)(cid:18) (cid:82) d θ d φ P ( θ, φ, k (cid:48) )e − i θ − i φ k (cid:19)(cid:82) { d P d ˆ P } e N Ψ [ P , ˆ P ] + O ( 1 N ) (42)Taking the limit N → ∞ therefore giveslim N →∞ W ( k , k (cid:48) ) = (cid:18) (cid:90) d θ d φ P ( θ, φ, k )e − i θ − i φ k (cid:48) (cid:19)(cid:18) (cid:90) d θ d φ P ( θ, φ, k (cid:48) )e − i θ − i φ k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) saddle − point { P , ˆ P } of Ψ (43)in which the function Ψ [ P , ˆ P ] is identical to that found in [3]. Using the the formulae in [3]that relate to the definition of the order parameter γ ( k , k (cid:48) ), we then obtain for N → ∞ theunexpected simple but welcome relation W ( k , k (cid:48) ) = γ ( k , k (cid:48) ) γ ( k (cid:48) , k ) (44)A similar, although slightly more involved, calculation leads to an expression for the jointdistribution W ( k , m ; k (cid:48) , m (cid:48) ); see the Appendix for details.1Our final aim is to use identity (44) to resolve equation (40) into observable quantities.Consider the nontrivial term in (40): Γ = (cid:88) k , m p ( k , m ) log (cid:18) (cid:88) ξ ,.,ξ k k (cid:89) s = γ ( k , ξ s ) δ m , (cid:80) ks = ξ s (cid:19) (45)At this point of the calculation, the e ff ect of factorising across nodes has been to break theexpression down into terms which, for every generalised degree ( k , m ), enumerate all thepossible ways of dividing m second neighbours between k first neighbours. The term insidethe logarithm sums for each k over all configurations { ξ . . . ξ k } which meet the condition (cid:80) ks = ξ s = m . To formalise this idea, we may re-aggregate the expression for any graphicallyrealisable distribution p ( k , m ) to write Γ = N log (cid:26) (cid:89) k , m (cid:18) (cid:88) ξ ,.,ξ k k (cid:89) s = γ ( k , ξ s ) δ m , (cid:80) ks = ξ s (cid:19) Np ( k , m ) = N log (cid:89) i (cid:18) (cid:88) ξ i ...ξ kii (cid:20) k i (cid:89) s = γ ( k i , ξ si ) (cid:21) δ m i , (cid:80) kis = ξ si (cid:19) = N log (cid:26) (cid:88) ξ ...ξ k . . . (cid:88) ξ N ...ξ kNN (cid:18) (cid:89) i δ m i , (cid:80) kis = ξ si (cid:19) (cid:89) i k i (cid:89) s = γ ( k i , ξ si ) (cid:27) (46)We can now see that the separate terms precisely enumerate all the permutations of degreesand neighbour-degrees for networks with a generalised degree sequence consistent with anypair ( k , m ) appearing N p ( k , m ) times. The Kronecker deltas δ m i , (cid:80) kis = ξ si tell us that each ξ si in anynonzero term is to be interpreted as the degree of a node j ∈ ∂ i , and must therefore appear alsoas the left argument in another factor of the type γ ( k j , . ). This insight allows the expressionto be substantially simplified, since we already know that γ ( k , k (cid:48) ) γ ( k (cid:48) , k ) = W ( k (cid:48) , k ) where W ( k (cid:48) , k ) is the correlation between degrees of connected nodes. Hence, any nonvanishingcontribution to the sum over all neighbourhoods inside the logarithm of (46) will be equal to arepeated product of factors W ( k , k (cid:48) ), with di ff erent ( k , k (cid:48) ). Since we also know that the numberof links between nodes with degree combination ( k , k (cid:48) ) equals N ¯ kW ( k , k (cid:48) ) in leading order in N , we conjecture that in leading order we may make the following replacement inside (46): (cid:89) i k i (cid:89) s = γ ( k i , ξ si ) → (cid:89) k , k (cid:48) W ( k , k (cid:48) ) ¯ kN W ( k , k (cid:48) ) (47)(where the factor in the exponent reflects the fact that two γ ( ., . ) factors combine to formeach factor W ( ., . )). With this conjecture we obtain, in leading order in N : Γ = N log (cid:26) (cid:88) ξ ...ξ k . . . (cid:88) ξ N ...ξ kNN (cid:18) (cid:89) i δ m i , (cid:80) kis = ξ si (cid:19)(cid:27) +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) = (cid:88) k , m p ( k , m ) log (cid:18) (cid:88) ξ ,.,ξ k δ m , (cid:80) ks = ξ s (cid:19) +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) ) (48)This implies that (40) simplifies to S =
12 ¯ k [log( N / ¯ k ) + +
12 ¯ k (cid:88) k , k (cid:48) W ( k , k (cid:48) ) log W ( k , k (cid:48) )2 − (cid:88) k , m p ( k , m ) log (cid:18) p ( k , m ) π ( k ) (cid:19) + (cid:88) k , m p ( k , m ) log (cid:18) (cid:88) ξ ,.,ξ k δ m , (cid:80) ks = ξ s (cid:19) (49)
7. Conclusion
Ensembles of tailored random graphs are extremely useful constructions in the modelling ofcomplex interacting particle systems in biology, physics, computer science, economics andthe social sciences. They allow us to quantify topological features of such systems and reasonquantitatively about their complexity, as well as define and generate useful random proxiesfor realistic networks in statistical mechanical analyses of processes.In this paper we have derived, in leading two orders in N , explicit expressions for theShannon entropies of di ff erent types of tailored random graph ensembles, for which no suchexpressions had yet been obtained. This work builds on and extends the ideas and techniquesdeveloped in the three papers [1, 3, 2], which use path integral representations to achieve linkfactorisation in the various summations over graphs. We show in this paper how the newensemble entropies can often be calculated by e ffi cient use and combination of earlier results.The first class of graph ensembles we studied consists of bipartite nondirected graphswith prescribed (and possibly nonidentical) distributions of degrees for the two node subsets.This case could be handled by a bijective mapping from bipartite to directed graphs, for whichformulae are available. The second class consists of graphs with prescribed distributions oflocal neighbourhoods, where the neighbourhood of a node is defined as its own degree plusthe values of the degrees of its immediate neighbours. This problem was solved using adecomposition in terms of bipartite graphs, building on the previous result. The final class ofgraphs, for which the entropy had in the past only partially been calculated, consist of graphswith presecribed distributions of generalised degrees, i.e. of ordinary degrees plus the totalnumber of length-two paths starting in the specified nodes. Here we derive two novel andexact identities linking the order parameters to macroscopic observables, which lead to anexplicit entropy formula based on a plausible but not yet proven conjecture,Since completing this work, our attention has been drawn to a preprint [8] whichconsiders the question of the entropy of random graph ensembles constrained with a givendistribution of neighbourhoods by a probability theory route, via an adapted ConfigurationModel. In that case, the neighbourhoods were specified as graphlets of an arbitrary depth. [8]also retrieves the entropy of an ensemble constrained with a specified degree distribution, asoriginally derived by [1]. Acknowledgements
ESR gratefully acknowledges financial support from the Biotechnology and BiologicalSciences Research Council of the United Kingdom.
References [1] Annibale A, Coolen A C, Fernandes L P, Fraternali F and Kleinjung J 2009
Journal of Physics A: Mathematicaland Theoretical Journal of Physics A Mathematical General Physical Review E Discrete Applied Mathematics
Journal of Physics A: Mathematical and Theoretical Networks: an introduction (OUP Oxford) [7] Roberts E S and Coolen A C 2012 Physical Review E arXiv:1308.5725 Appendix A. Generalised degree correlation kernel for ensembles with prescribedgeneralised degrees
The generalised quantity W ( k , m ; k (cid:48) , m (cid:48) ) in the ensemble with presecribed generalised degreedistributions p ( k , m ) can be calculated along the same lines as the calculation of W ( k , k (cid:48) ) inthe main text. It is defined as W ( k , m ; k (cid:48) , m (cid:48) | c ) = N ¯ k (cid:88) i j c i j δ k , (cid:80) (cid:96) c i (cid:96) δ k (cid:48) , (cid:80) (cid:96) c j (cid:96) δ m , (cid:80) (cid:96) c i (cid:96) k (cid:96) δ m (cid:48) , (cid:80) (cid:96) c j (cid:96) k (cid:96) (A.1)and its ensemble average takes the form W ( k , m ; k (cid:48) , m (cid:48) ) = (cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) (cid:18) N (cid:80) rs δ k , k r δ k (cid:48) , k s δ m , m r δ m (cid:48) , m s e − i( θ r + θ s + φ r k (cid:48) + φ s k ) (cid:19)(cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + O ( 1 N ) = (cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) (cid:18) N (cid:80) r δ k , k r δ m , m r e − i( θ r + φ r k (cid:48) ) (cid:19)(cid:18) N (cid:80) s δ k (cid:48) , k s δ m (cid:48) , m s e − i( θ s + φ s k ) (cid:19)(cid:82) ππ d θ d φ e i( θ · k + φ · m ) + k N (cid:80) ij e − i( θ i + θ j + φ ikj + φ jki ) + O ( 1 N ) (A.2)Now we will want to introduce a generalised order parameter, namely P ( θ, φ, k , m ) = N (cid:88) r δ k , k r δ m , m r δ ( θ − θ r ) δ ( φ − φ r ) (A.3)The previous order parameter used in he calculation of W ( k , k (cid:48) ) is a marginal of this, via P ( θ, φ, k ) = (cid:80) m P ( θ, φ, k , m ). This definition will give us W ( k , m ; k (cid:48) , m (cid:48) ) = (cid:18) (cid:90) π − π d θ d φ P ( θ, φ, k , m )e − i θ − i φ k (cid:48) (cid:19)(cid:18) (cid:90) π − π d θ d φ P ( θ, φ, k (cid:48) , m (cid:48) )e − i θ − i φ k (cid:19) (A.4) W ( k , k (cid:48) ) = (cid:88) mm (cid:48) W ( k , m ; k (cid:48) , m (cid:48) ) (A.5)in which the new order parameter and its conjugate are to be solved by extremisation of thegeneralised surface Ψ [ P , ˆ P ] = i (cid:88) km (cid:90) π − π d θ d φ ˆ P ( θ, φ, k , m ) P θ, φ, k , m ) + (cid:88) km P ( k , m ) log (cid:90) π − π d θ d φ e i( θ k + φ m − ˆ P ( θ,φ, k , m )) +
12 ¯ k (cid:90) d θ d φ d θ (cid:48) d φ (cid:48) (cid:88) kk (cid:48) mm (cid:48) P ( θ, φ, k , m ) P ( θ (cid:48) , φ (cid:48) , k (cid:48) , m (cid:48) )e − i( θ + θ (cid:48) + φ k (cid:48) + φ (cid:48) k ) (A.6)Variation of Ψ gives the following saddle-point equationsˆ P ( θ, φ, k , m ) = i¯ k e − i θ (cid:90) d θ (cid:48) d φ (cid:48) (cid:88) k (cid:48) m (cid:48) P ( θ (cid:48) , φ (cid:48) , k (cid:48) , m (cid:48) )e − i( θ (cid:48) + φ k (cid:48) + φ (cid:48) k ) (A.7) P ( θ, φ, k , m ) = P ( k , m ) e i( θ k + φ m − ˆ P ( θ,φ, k , m )) (cid:82) π − π d θ (cid:48) d φ (cid:48) e i( θ (cid:48) k + φ (cid:48) m − ˆ P ( θ (cid:48) ,φ (cid:48) , k , m )) (A.8)4Clearly ˆ P ( θ, φ, k , m ) = ˆ P ( θ, φ, k ) (i.e. it is independent of m ). We may therefore substituteˆ P ( θ, φ, k ) = i¯ k e − i θ ˆ P ( φ, k ) and findˆ P ( φ, k ) = (cid:90) d θ (cid:48) d φ (cid:48) (cid:88) k (cid:48) m (cid:48) P ( θ (cid:48) , φ (cid:48) , k (cid:48) , m (cid:48) )e − i( θ (cid:48) + φ k (cid:48) + φ (cid:48) k ) (A.9) P ( θ, φ, k , m ) = P ( k , m ) e i( θ k + φ m ) + ¯ k e − i θ ˆ P ( φ, k )) (cid:82) π − π d θ (cid:48) d φ (cid:48) e i( θ (cid:48) k + φ (cid:48) m ) + ¯ k e − i θ (cid:48) ˆ P ( φ (cid:48) , k )) (A.10)We observe as before in [3] that (cid:90) π − π d θ P ( θ, φ, k )e − i θ = (cid:88) m P ( k , m ) (cid:82) π − π d θ e i( θ ( k − + φ m ) + ¯ k e − i θ ˆ P ( φ, k )) (cid:82) π − π d θ d φ (cid:48) e i( θ k + φ (cid:48) m ) + ¯ k e − i θ ˆ P ( φ (cid:48) , k )) = (cid:88) m P ( k , m ) (cid:80) (cid:96) ≥ k (cid:96) ˆ P (cid:96) ( φ, k ) (cid:96) ! (cid:82) π − π d θ e i( θ ( k − − (cid:96) ) + φ m ) (cid:80) (cid:96) ≥ k (cid:96) ˆ P (cid:96) ( φ (cid:48) , k ) (cid:96) ! (cid:82) π − π d θ d φ (cid:48) e i( θ k + φ (cid:48) m − (cid:96)θ ) = (cid:88) m P ( k , m ) ¯ k k − ˆ P k − ( φ, k )( k − e i φ m ¯ k k ˆ P k ( φ (cid:48) , k ) k ! (cid:82) π − π d φ (cid:48) e i φ (cid:48) m = (cid:88) m k ¯ k P ( k , m ) ˆ P k − ( φ, k )e i φ m (cid:82) π − π d φ (cid:48) ˆ P k ( φ (cid:48) , k )e i φ (cid:48) m (A.11)Hence ˆ P ( φ, k ) = (cid:88) k (cid:48) m (cid:48) k (cid:48) ¯ k P ( k (cid:48) , m (cid:48) )e − i φ k (cid:48) (cid:82) π − π d φ (cid:48) ˆ P k (cid:48) − ( φ (cid:48) , k (cid:48) )e i φ (cid:48) ( m (cid:48) − k ) (cid:82) π − π d φ (cid:48) ˆ P k (cid:48) ( φ (cid:48) , k (cid:48) )e i φ (cid:48) m (cid:48) (A.12)After writing ˆ P ( φ, k ) = (cid:80) k (cid:48) γ ( k , k (cid:48) )e − i φ k (cid:48) we recover our familiar equation γ ( k , k (cid:48) ) γ ( k (cid:48) , k ) = k (cid:48) ¯ k (cid:88) m P ( k (cid:48) , m ) (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m , (cid:80) n ≤ k (cid:48) k n δ kk n (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m , (cid:80) n ≤ k (cid:48) k n (A.13)But now we can also work out the generalised kernel: W ( k , m ; k (cid:48) , m (cid:48) ) = (cid:18) (cid:90) π − π d θ d φ P ( θ, φ, k , m )e − i θ − i φ k (cid:48) (cid:19)(cid:18) (cid:90) π − π d θ d φ P ( θ, φ, k (cid:48) , m (cid:48) )e − i θ − i φ k (cid:19) = kk (cid:48) ¯ k P ( k , m ) P ( k (cid:48) , m (cid:48) ) (cid:18) (cid:82) π − π d φ ˆ P k − ( φ, k )e i φ ( m − k (cid:48) ) (cid:82) π − π d φ ˆ P k ( φ, k )e i φ m (cid:19)(cid:18) (cid:82) π − π d φ ˆ P k (cid:48) − ( φ, k (cid:48) )e i φ ( m (cid:48) − k ) (cid:82) π − π d φ ˆ P k (cid:48) ( φ, k (cid:48) )e i φ m (cid:48) (cid:19) = kk (cid:48) ¯ k P ( k , m ) P ( k (cid:48) , m (cid:48) ) γ ( k , k (cid:48) ) γ ( k (cid:48) , k ) (cid:18) (cid:80) k ... k k (cid:20) (cid:81) kn = γ ( k , k n ) (cid:21) δ m , (cid:80) n ≤ k k n δ k k , k (cid:48) (cid:80) k ... k k (cid:20) (cid:81) kn = γ ( k , k n ) (cid:21) δ m , (cid:80) n ≤ k k n (cid:19) × (cid:18) (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m (cid:48) , (cid:80) n ≤ k (cid:48) k n δ k k (cid:48) , k (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m (cid:48) , (cid:80) n ≤ k (cid:48) k n (cid:19) (A.14)We know that W ( k , k (cid:48) ) = γ ( k , k (cid:48) ) γ ( k (cid:48) , k ), and that P ( k , m ) k / ¯ k = W ( k , m ), so this can besimplified to W ( k , m ; k (cid:48) , m (cid:48) ) = W ( k , m ) W ( k (cid:48) , m (cid:48) ) W ( k , k (cid:48) ) (cid:18) (cid:80) k ... k k (cid:20) (cid:81) kn = γ ( k , k n ) (cid:21) δ m , (cid:80) n ≤ k k n δ k k , k (cid:48) (cid:80) k ... k k (cid:20) (cid:81) kn = γ ( k , k n ) (cid:21) δ m , (cid:80) n ≤ k k n (cid:19) × (cid:18) (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m (cid:48) , (cid:80) n ≤ k (cid:48) k n δ k k (cid:48) , k (cid:80) k ... k k (cid:48) (cid:20) (cid:81) k (cid:48) n = γ ( k (cid:48) , k n ) (cid:21) δ m (cid:48) , (cid:80) n ≤ k (cid:48) k n (cid:19)(cid:19)