Energies of Network Nastsemble

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1 Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada Costera 11, Treste, Italy Dpartmento d Economa Poltca, Unverstá degl Stud d Sena, Pazza San Francesco 7, Sena, Italy 3 Max Weber Programme, European Unversty Insttute, Va Delle Fontanelle 10, San Domenco d Fesole (FI), Italy I. EVALUATION OF Σ The entropy of a network ensemble of N nodes wth gven feature φ(g, q) s defned as Σ φ(g, q) = 1 N log {g G N : φ(g, q) = φ(g, q)}. (1) By usng methods of statstcal mechancs t s possble to defne a partton functon Z φ(g, q) countng the number of network n an ensemble such that Σ φ(g, q) = 1 N log Z φ(g, q). () For the dervaton of the equatons for gettng Σ φ(g, q) n a general ensemble we wll address the reader to the papers [1, ]. In the followng we wll show how to derve the form of Σ for the case of networks wth heterogeneous degree and gven communty structure and for the case of spatal networks wth heterogeneous degree. The features that we wll consder are, n the frst case, the degree sequence k together wth a communty structure A,.e. φ = { k, A(q, q }, and, for the second case, the degree sequence k and the number of exstng lnks B(d) between nodes at dstance d,.e. φ = { k, B(d)}. A. The entropy Σ of a network ensemble wth gven degree sequence and communty structure Suppose that we have a network together wth a assgnment of the nodes q = 1,..., Q wth Q N. The entropy Σ k,a(q,q ) measures the total number of possble networks wth the same degree dstrbuton k and the same communty structure A, that s wth A(q, q ) lnks between nodes and j wth q = q and q j = q, respectvely. If we ndcate wth g j the adjacency matrx of the consdered undrected network, A(q, q ) can be expressed as A(q, q ) = <j δ(mn(q, q j ) q)δ(max(q, q j ) q )g j. (3) The problem of evaluatng Σ k,a(q,q ) can be cast n a statstcal mechancs calculaton snce t s equvalent to calculate wth takes ndeed the form of a statstcal mechancs partton functon,.e. Z k,a(q,q ) Z k,a(q,q ) = ((a j )) δ(k j g j ) δ(a(q, q ) δ(mn(q, q j ) q)δ(max(q, q j ) q )g j )e <j hjgj (4) q q <j wth h j ndcatng auxlary felds. Expressng the delta s n an ntegral form, and performng the saddle pont approxmaton, up to the second order [1, ], we can evaluate Z k,a(q,q ). Fnally, usng () we get the entropy of the ensemble NΣ k,a(q,q ) k ω A(q, q )wq,q + ln [1 + e ω +ω j +w(mn(q,q j ),max(q,q j )) ] q q <j 1 ln(πα ) 1 ln(πα q,q ) (5) q<q

2 wth the Lagrangan multplers {ω }, {w q,q } satsfyng the saddle pont equatons k e ω +ω j +w(mn(q,qj),max(q,qj)) = 1 + e ω +ω j +w(mn(q,q j ),max(q,q j )) j e ω +ωj+w q,q A(q, q ) = δ(mn(q, q j ) q)δ(max(q, q j ) q ), (6) <j 1 + e ω +ω j +w q,q and wth α, α q,q defned as α α q,q = j e ω +ω j +w mn(q,q j ),max(q,q j ) ( ) (7) 1 + e ω +ω j +w mn(q,q j ),max(q,q j ) e ω +ω j +w q,q = δ(mn(q, q j ) q)δ(max(q, q j ) q ) ). (8) <j (1 + e ω +ω j +w q,q In the ensemble of networks wth feature φ = { k, A(q, q )}, the probablty of a lnk between node and j s equal to p j = Z k,a(q,q h ) j hj=0 (,j) = eω +ω j +w(mn(q,qj),max(q,qj)) 1 + e ω +ω j +w(mn(q,q j ),max(q,q j )). (9) therefore by defnng e ω = z and e w q,q = W (q, q ) we get for the probablty p j p j = z z j W (q, q j ) 1 + z z j W (q, q j ). (10) II. THE ENTROPY Σ d OF A NETWORK ENSEMBLE WITH GIVEN DEGREE SEQUENCE AND GIVEN DISTANCE STRUCTURE Suppose that we have to analyze an undrected spatal networks. The entropy Σ k,b(d) measures the total number of possble networks wth the same degree sequence k and the same dstance dstrbuton B(d) = b 1,... b D. Here, we have coarse graned dstances nto D ntervals (d = d 1,... d D ) and b l ndcates the number of lnks between nodes at dstance at a dstance d j [d 1, d ] (d 0 = 0). The bns d are chosen so that b l s large and of order N. In an undrected network, b l s gven by the followng expresson b l = <j χ l (d,j )g j. (11) where χ l s the characterstc functon ndcatng the dstance nterval I l = (d l 1, d l ). Followng the same steps as n the prevous cases we fnd that the entropy for such an ensemble n the large network lmt s gven by NΣ k,b(d) k ω b l u l + ( ) ln 1 + e ω +ω j + l χ l(d j) u l l <j 1 ln(πα ) 1 ln(πα l ) (1) wth the Lagrangan multplers {ω }, {u l } satsfyng the saddle pont equatons l k = j e ω +ω j + l χ l(d j )u l 1 + e ω +ω j + l χ l(d j) u l e ω +ω j+u l b l = χ l (d j ), (13) 1 + e ω +ω j +u l <j

3 3 and α, α l are defned as α α l = j =,j e ω +ω j + l χ l(d j)u l ( 1 + e ω +ω j + l χ l(d j )u l e ω +ω j +u l ) χ l (d j )( ) 1 + e ω (14) +ω j +u l The probablty for a lnk between node and j s equal to p j = eω +ω j + l χ l(d j)u l. (15) 1 + e ω +ω j + l χ l(d j )u l Therefore performng the transformaton e ω = z and e u l = W (dl ) we get the expresson p j = z z j l χ l(d j )W (d l ) 1 + z z j l χ l(d j )W (d l ). (16)

4 4 III. MATLAB CODES FOR CALCULATING Σ The calculaton of Σ φ(g, q ), reported n the prevous secton, for networks wth gven communty structure and for networks wth gven spatal structure, requres the numercal soluton of the saddle pont equatons (6) and (13). These system of equatons can be solved recursvely. In the followng we report the matlab codes for the calculaton of Σ k,a(q,q ) and Σ k,b(d). Ths codes work well for networks of lmted sze, (up to,000 nodes). For networks of larger sze wrtng smlar codes n C or C++ languages s recommed. A. Matlab code for the calculaton of Σ k,a(q,q ) for networks wth gven degree sequence and communty structure % Ths program evaluates the entropy of networks wth gven class assgnment (type) % The nput of the program s a square adjacency matrx a of dmenson n % and a vector of dmenson n descrbng the assgnment type()=q % wth q nteger between 1 and Q dentfyng Q classes or s % The output of the program s the entropy Sgmac (Σ k,a(q,q )) of networks wth %gven degree sequence and communty structure nduced by the assgnment type(). functon [Sgmac,W]=Entropyclasses(a,types,n,Q) precson=10ˆ (-); p= undn=a; undn=undn>0; und=sum(undn) ; avgconn=sum(und)/n; % Compute entropy wth gven degree sequence and communty assgnment - A=zeros(Q,Q); for =1:Q for j=1:q A(,j)=sum(sum( tru( undn(type==,type==j) ) )) ; % compute exp(lagrangan multplers) z=rand(n,1); % z=exp(omega) W=rand(Q,Q); % W(,j)=exp( w(type(),type(j)) ) oldw=zeros(q,q); oldz=zeros(n,1); for kk=1:100 bgw=w(type,type); for k=1:10 U=ones(n,1)*z.* bgw; D=ones(n,n) + ( z*z.* bgw); U=max( U, 10ˆ (-15) ); D=max( D, 10ˆ (-15) ); z=und./ (sum( ( U./D - dag(dag(u./d)) ) ) ) ; z=max(z,10ˆ (-15)); z=mn(z,1); z=z.*(und>0)+(und==0)*0; for k=1:10

5 5 d=zeros(q,q); bgw=w(type,type); for =1:Q, for j=1:q a=(type==)+0; b=(type==j)+0; M=(a*b ).* (z*z )./ ( ones(n,n) + (z*z * W(,j))) ; d(,j)=( sum(sum( M-dag(dag(M)) )) ); f d(,j)>0 W(,j)=A(,j)/d(,j); else W(,j)=0; f max(max(abs(w-oldw)))<precson && max(abs(z-oldz))<precson break oldw=w; oldz=z; M=log( ones(n,n) + ( (z*z ).*W(type,type) ) ) ; M=((z*z ).*W(type,type))/((ones(n,n)+(z*z ).*W(type,type)).ˆ ); alpha=sum(m-dag(dag(m))); for =1:Q, hspace7mmfor j=+1:q, alphaq(,j)=sum(sum(m.*((type==)*(type==j) )-dag(dag(m)))); hspace*5mm W=max( W, 10ˆ (-15) ); z=z+(und==0); Sc=(1/n)*( - sum(log(z).*und) - sum(sum( tru(a.* log(w) ) )) + sum(sum( tru( M,1) )) ); Sc=Sc-(1/n)*(0.5*sum(log(*p*alpha))-0.5*sum(sum(tru(log(*p*alphaq))))); Sgmac=Sc;

6 6 B. Matlab code for the calculaton of Σ k,b(d) for spatal networks wth gven degree sequence % Ths program evaluates the entropy of networks wth gven poston assgnment % The nput of the program s a square adjacency matrx a of dmenson n % and a matrx of dmenson n descrbng the dstance d(,j)=d=d(1),..,d(d) % between the nodes of the network % The output are % I) The entropy Sgmad (Σ k,b(d) ) of the undrected networks wth the same dstrbuton % of lnks at a gven dstance structure d of the network a; % II)The vector W(d) that modulates the probablty that a node s % connected wth a node j at dstance d from functon [Sgmad,W,l,z]=Entropydstance(a,d,Nd); precson=10ˆ (-); loops=100; p= ; n=max(sze(a)); Dmax=max(max(d)); [I,J,V]=fnd(d); dmn=mn(v); l=logspace(log(dmn)/log(10),log(dmax)/log(10),nd); for =1:n, for j=1:n, h=hst(d(,j),l); I=fnd(h); class(,j)=i(1); undn=+(a>0); und=sum(undn) ; avgconn=sum(und)/n; % Compute dstance entropy B=zeros(Nd,1); for d=1:nd B(d)=sum(sum(undN.*(class==d)))/; % compute exp(lagrangan multplers) z=rand(n,1); % z=exp(omega) W=rand(Nd,1); % W(d)=exp( w(d)) oldw=zeros(nd,1); oldz=zeros(n,1); for kk=1:loops bgw=zeros(n,n); for d=1:nd, bgw=bgw+w(d)*(class==d); U=ones(n,1)*z.* bgw; D=ones(n,n) + ( z*z.* bgw); z=und./ (sum( ( U./D - dag(dag(u./d)) ) ) ) ; z=max(z,10ˆ (-15)); z=z.*(und>0)+(und==0)*0;

7 B=zeros(Nd,1); for d=1:nd M=(class==d).* (z*z )./ ( ones(n,n) + ((z*z ).* bgw)) ; B(d)=( sum(sum( (M)-dag(dag(M)) )) )/; f (B(d)*B(d))>0.0 W(d)=B(d)/(B(d)); W(d)=max(W(d),10ˆ (-15)); W(d)=mn(W(d),10ˆ (15)); else W(d)=0; f max(max(abs(w-oldw)))<precson && max(abs(z-oldz))<precson break oldw=w; oldz=z; bgw=zeros(n,n); for d=1:nd, bgw=bgw+w(d)*(class==d); M=log( ones(n,n) + ( (z*z ).*bgw ) ) ; W=max( W, 10ˆ (-15) ); M=( (z*z ).*bgw )./( ( ones(n,n)+( (z*z ).*bgw) ).ˆ ); alpha=sum(m); Q=0; for =1:Nd q=sum(sum(m.*(class==d)) )/; f q > 10ˆ (-10) Q=Q + log(*p*q); z=z+(und==0); Sc=(1/n)*( - sum(log(z).*und) - sum(b.* log(w)) + sum(sum( tru( M,1) )) - ( sum(log(*p*alpha)) )/ - Q/ ); Sgmad=Sc; 7 [1] G. Bancon, The entropy of randomzed network ensembles, Europhys. Lett. 81, 8005 (008). [] G. Bancon, The entropy of network ensembles, arxv: