Exactly Solvable Random Graph Ensemble with Extensively Many ...

Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles Fabi´ an Aguirre L´ opez,1, 2 Paolo Barucca,3, 4 Mathilde Fekom,5 and Anthony CC Coolen1, 2

arXiv:1705.03743v1 [cond-mat.dis-nn] 10 May 2017

1

Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, United Kingdom 2 Institute for Mathematical and Molecular Biomedicine, King’s College London, Hodgkin Building, London SE1 1UL, United Kingdom 3 London Institute for Mathematical Sciences, 35a South St, Mayfair, London W1K 2XF, United Kingdom 4 University of Z¨ urich, Department of Banking and Finance, Z¨ urich, ZH, Switzerland 5 UFR de Physique, Université Paris Diderot (Paris 7), 5 Rue Thomas Mann, 75013 Paris, France (Dated: May 11, 2017) We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles’ control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

I.

INTRODUCTION

Models with pairwise interacting elements are ubiquitous in physics and are sufficient to capture the phenomenology of many systems, ranging from condensed matter via biology to the social sciences and informatics. The properties of the network of interactions strongly affects the properties of the system under study, and hence the analysis of networks is central in modern physics. Testing statistically whether a specific network property influences the dynamics of a system requires sampling networks where such property is controllable, and for which the probability measure is known. This is why random graph ensembles, especially maximum entropy ones from which it is possible to sample networks systematically with controlled properties, have gained increasing popularity. They range from degree configuration models [1–6], i.e. where the number of connections or nodes are fixed, to more complicated models where a block structure is given to the nodes in the network, as in stochastic block models [7], or other ensembles where clustering, i.e. the tendency of nodes with common neighbours to be connected, is enhanced [8–15]. However, controlling analytically and numerically second or higher-order properties of networks, i.e. node properties that not only depend on first neighbours, such as the density of cycles, is still a great mathematical and analytical challenge, whose range of applications continues to grow [10, 16–22]. So far, nearly all analytical results obtained for random graph ensembles rely on the assumption of the absence of short cycles, the tree-like approximation, and we have analytical solutions only for random graphs were clustering is absent or too weak (or improbable) to be relevant. One of the first random graph ensembles in literature to include short cycles was [23], where a term depending on the number of 3-cycles of the graph was included as a modification to the well known Erd¨ os-Rényi model (ER). This was done in order to encourage this connec-

tion transitivity in the graph. However, as was found in simulations [23] and in a more rigorous way in [8, 11], unless the graph is particularly small, this approach does not allow for a tuneable number of triangles. Depending on the values and the scaling of the parameters, the model of [23] either stays in a phase very close to the ER model, with a very slight increase in triangles, or it collapses to a condensed phase, where the complete clique has probability one. In this paper we introduce an exactly solvable ensemble of 2-regular random graphs, with an exponential measure that controls the presence of short cycles up to any finite length. The imposition of 2-regularity removes the possibility of a complete clique forming, and forces the graph instead to be partitioned into a set of disconnected cycles of different lengths. This makes the ensemble analytically solvable and perfectly tuneable. The model displays a second-order transition, from a phase dominated by extensively long cycles, to a phase where only (extensively many) cycles of short lengths are present. In section 2 we introduce and solve the model, in its canonical formulation. In section 3 we describe analytically the phases of the ensemble and the critical hypersurface in the space of parameters; from this result we also compute analytically the spectral density of the ensemble. In section 4 we demonstrate the equivalence of the canonical and grand canonical formulations of the model, and in section 5 we show the agreement of the analytical predictions with numerical experiments. In a final discussion section we summarize the results and delineate the future directions for this research, which are twofold. The first is to relax the 2-regularity constraint of the ensemble, in order to make it more directly comparable to realistic networks. The second is to understand better recent analytical approaches to random graph ensembles that involve constraints on the number of closed paths of all lengths, which is equivalent to constraining random graphs via their spectra [24].

2 II.

DEFINITIONS

We define a random graph ensemble over the set of undirected simple regular graphs of degree 2, which we denote by GN . Any graph in GN is necessarily a set of disjoint cycles. The probability assigned to each graph A ∈ GN is chosen proportional to the exponential of a weighted sum of the number of triangles, squares, pentagons, . . . , K-cycles present in A. We refer to this as biasing with respect of the number of short cycles. Thus ! K X 1 exp p(A) = ℓαℓ nℓ (A) , (1) ZN (α) ℓ=3

Here nℓ (A) denotes the number of length-ℓ cycles, i.e. closed paths of length ℓ without backtracking and without over-counting, and α = (α3 , . . . , αK ) ∈ IRK−2 is a vector of control parameters. Note that isolated nodes (ℓ = 1) and dimers (ℓ = 2) cannot occur due to the degree constraint. The factors ℓ in (1) are included for later convenience. The partition function ZN (α) is given by ! K X X ZN (α) = exp ℓαℓ nℓ (A) . (2) A∈GN

ℓ=3

Expression (1) defines a maximum entropy random graph ensemble with respect to the K − 2 observables nℓ (A), whose ensemble averages are controlled by varying the parameters α. The average fraction of the N nodes that will be found in an ℓ-cycle is given by ℓ hnℓ (A)i. (3) N P where hf (A)i = A p(A)f (A). Following the statistical mechanics route, we define a generating function φN (α): mℓ =

φN (α) = N −1 log[ZN (α)/N !]

(4)

with n = (n3 , . . . , nN ) ∈ INN −2 . This decomposition reflects the fact that, in the particular case of GN , we are fortunate that each graph has to be a collection of cycles, and can therefore be identified fully by a sequence n = (n3 , . . . , nN ) that specifies the number of cycles of each possible length up to N , and a labelling of the nodes. The sum over graphs is then performed by summing over all possible sequences n, keeping track of the multiplicity of each sequence via an associated density of states D(n): D(n) =

ℓ=K+1

III. A.

=

N! 2π

Z

nℓ ≥0

π

dω exp iωN +

−π

ANALYTICAL SOLUTION

+

(8)

From this, in combination with (4), we infer that Z π 1 dω N fN (ω,α) φN (α) = log e N −π 2π

(9)

with fN (ω, α) = iω +

n

2ℓ !

N X e−iωℓ 2ℓ

ℓ=K+1

Summation over graphs

To evaluate the partition function (2) we need to perform a sum over graphs. Such sums are usually not analytically tractable, especially when the ensemble definition involves cycles, as is the case in (1). Here we are able to perform the summation by rewriting it as X PK ZN (α) = D(n)e ℓ=3 ℓαℓ nℓ , (6)

K X e(αℓ −iω)ℓ ℓ=3

(5)

The generator φN (α) is minus the free energy density, apart from a factor N ! which reflects (topologically irrelevant) node label permutations. Including this factor will ensure that the limit φ(α) = limN →∞ φN (α) exists.

(7)

PN Apart from the condition N = ℓ=3 ℓnℓ , this density is proportional to N ! but corrected for over-counting due to the indistinguishability of different length-ℓ cycles, giving a divisor nℓ !, and due to the different ways one can number the nodes in each ℓ-cycle without altering the graph (ℓ cyclic permutations, plus ℓ anti-cyclic permutations), giving a further divisor (2ℓ)nRℓ . Using the integral form of π the Kronecker delta δnm = −π (dω/2π)eiω(n−m) , we can thus write the partition function as ! K X Y N! ℓαℓ nℓ ZN (α) = e QN nℓ n ℓ=3 [(2ℓ) nℓ !] ℓ=3 Z π dω iω(N −PN ℓ=3 ℓnℓ ) e × −π 2π   Z K Y X e(αℓ −iω)ℓnℓ N! π   = dω eiωN 2π −π (2ℓ)nℓ nℓ ! ℓ=3 nℓ ≥0   N Y X e−iωℓnℓ   × (2ℓ)nℓ nℓ !

The main quantities of interest (3) for our graph ensemble (1) can be computed from (4) via mℓ = ∂φN (α)/∂αℓ .

N ! δN,PN ℓnℓ ℓ=3 . δnℓ ,nℓ (A) = QN nℓ n !] [(2ℓ) ℓ ℓ=3 A∈GN ℓ=3 N X Y

K X e(αℓ −iω)ℓ ℓ=3

2ℓN

+

N X e−iωℓ . (10) 2ℓN

ℓ=K+1

The limit N → ∞ of (9) can now be obtained by evaluating the integral over ω in (8) via steepest descent: φ(α) = lim extrω fN (ω, α). N →∞

(11)

The extremum is found by solving ∂f (ω, α)/∂ω = 0.

3 B.

Scaling with N of control parameters

We observe that for finite {αℓ } our model cannot exhibit nonzero cycle densities mℓ in the infinite size limit, since the α-dependent term in (10) vanishes for N → ∞. We are therefore led to redefining the parameters α with a size dependent shift, αℓ = α ˜ℓ +

1 log(N ), ℓ

(12)

K

where α ˜ℓ = O(1). We denote the vector of shifted O(1) ˜ = (˜ control parameters by α α3 , . . . , α ˜ K ), and we define ˜ This implies that for N → ∞ we will φN (α) = ϕN (α). ˜ have mℓ = ∂ϕ(α)/∂ α ˜ ℓ , in which now K N n X X e(α˜ ℓ −iω)ℓ e−iωℓ o ˜ = lim extrω iω + ϕ(α) . + N →∞ 2ℓ 2ℓN ℓ=3

fact that all nodes are typically in cycles of length K or less, so m∞ = 0. A second phase, the connected phase, is characterized by finding a finite fraction of the nodes in longer cycles, so here m∞ > 0. The transition separating the phases is marked by bifurcation of m∞ > 0 solutions. If limN →∞ x < 1, the second term of (17) vanishes for N → ∞, and we immediately obtain m∞ = 0. Hence we are in the disconnected phase, and here the asymptotic observables mℓ are simply found by solving

ℓ=K+1

(13)

m∞ = 0 : 1 =

1=

1 2

e(α˜ ℓ −iω)ℓ +

ℓ=3

1 2N

N X

e−iωℓ ,

(14)

ℓ=K+1

1 (α˜ ℓ −iω)ℓ e 2

(15)

This last identity, in combination with (14), prompts us P to introduce m∞ = 1 − ℓ≤K mℓ ∈ [0, 1], which gives the fraction of the nodes that are not in cycles of length K or less. It is for N → ∞ apparently given by 1 N →∞ 2N

m∞ = lim

N X

e−iωℓ

(16)

ℓ=K+1

It follows from (15), that the physical saddle point ω, after contour deformation, must be purely imaginary. We switch accordingly to the new variable x = e−iω ∈ IR+ 0, in terms of which our equations become: K

1=

1 X ℓ ℓα˜ ℓ 1 xe + 2 2N ℓ=3

mℓ = lim

N →∞

m∞ = lim

N →∞

C.

1 ℓ ℓα˜ ℓ xe 2 N 1 X 2N

N X

1 ℓ ℓα˜ ℓ x e . (20) 2

The condition x < 1 for this solution to exist will be met for large values of {α ˜ ℓ }. Upon reducing the control parameters {α ˜ ℓ }, the value of x will increase, and a transition to the connected phase occurs exactly when x = 1. This happens at the critical manifold in the K−2 dimensional parameter space, defined by validity of K X

eℓα˜ ℓ = 2.

(21)

ℓ=3

and that the asymptotic values of the observables mℓ are subsequently given by mℓ =

mℓ =

ℓ=3

Differentiation of this latter expression reveals that the value ω at the extremum is to be solved from K X

1 X ℓ ℓα˜ ℓ xe , 2

xℓ ,

(17)

ℓ=K+1

(18)

To confirm the equations of the connected phase, we need to investigate how the solution x of (17) scales with N as we approach x = 1. Substituting x = 1 − ξ/N , expanding (17) in N , and taking the limit N → ∞ gives 1 mℓ = eℓα˜ ℓ , 2

K

m∞

1 X ℓα˜ ℓ =1− e , 2

(22)

ℓ=3

and the link between ξ and m∞ is m∞ = (1−e−ξ )/2ξ. It turns out that all cycles of finite length L > K will always have vanishing density for N → ∞. This can be seen simply by replacing K → L in the previous analysis, but with αℓ = 0 for all K < ℓ ≤ L. The newly added control parameters with ℓ > K will give α ˜ ℓ = −ℓ−1 log N , and hence mℓ = limN →∞ 21 xℓ eℓα˜ ℓ = limN →∞ 12 xℓ /N = 0, in both phases. We knew that in the disconnected phase all nodes will typically be in the controlled short cycles of length K or less. We may now conclude that, in the connected phase, those nodes that are not in the controlled short cycles (the fraction m∞ > 0) will typically be found in cycles of diverging length. The ensemble’s Shannon entropy [25] is given by X SN = − p(A) log p(A) A∈GN

xℓ

(19)

ℓ=K+1

Phase phenomenology of the ensemble

We will now demonstrate that the solutions to the coupled equations (17,18) give rise to two phases of our graph ensemble. A disconnected phase is characterized by the

K h X ∂φN (α) i = log N ! + N φN (α) − αℓ ∂αℓ ℓ=3

K X mℓ = N log(N ) 1 − + O(N ) ℓ

(23)

ℓ=3

PK 1 PK 1 Since ℓ=3 (mℓ /ℓ) ≤ 3 ℓ=3 mℓ ≤ 3 , the leading order will for large N always scale as N log(N ), and be bounded according to SN ≥ 23 N log(N ) + O(N ), but

4 with a reduced prefactor if we increase the fraction of nodes in short cycles. The lower bound is achieved in the disconnected phase, when m3 = 1 and mℓ>3 = 0. D.

Spectral densities of adjacency matrices

A graph can be represented uniquely by its adjacency matrix {Aij }, where Aij ∈ {0, 1}, and Aij = 1 if and only if there is a link from j to i. The set GN contains only simple nondirected graphs, so our adjacency matrices are symmetric and with zero diagonal elements. The eigenvalue density of the adjacency matrix of a graph A, N 1 X δ[µ − µi (A)], ̺(µ|A) = N i=1

A∈GN

The adjacency matrix of a graph that consists of a single cycle of length ℓ has the Toeplitz form, and is therefore diagonalized trivially, leading to the density ̺ℓ (µ) =

ℓ−1 1X δ µ − 2 cos(2πr/ℓ) . ℓ r=0

(26)

If the cycle length ℓ diverges, this density becomes continuous (in a distributional sense), see e.g. [26], ̺∞ (µ) = lim ̺ℓ (µ) = ℓ→∞

1 θ(2 − |µ|) p . π 4 − µ2

(27)

Within the canonical approach one finds that, if N is sufficiently large, graphs generated randomly from (1) will all display the same values of the main intensive quantities, such as the fraction of ℓ-cycles (modulo finite size fluctuations). We expect a similar claim to hold if we sample randomly both the graphs and the number N of nodes, i.e. if we work with grand canonical graph ensembles. The grand partition function of our ensemble with weights wN = e−µN /N ! (where µ > 0) is given by Q(α) =

ℓ−1 N 2πr X 1 X ) nℓ (A) δ µ − 2 cos N ℓ r=0

=

ℓ=3

N

̺ℓ (µ).

K X e−µℓ ℓ=1

̺(µ) =

K X ℓ=3

mℓ ̺ℓ (µ) + m∞ ̺∞ (µ).

2ℓ

−

K X e(αℓ −µ)ℓ

2ℓ

ℓ=3

hN i =

+

1 log(1−e−µ ). (32) 2

=

K

K

ℓ=3

ℓ=1

1 X (αℓ −µ)ℓ 1 X −µℓ 1 e−µ + e − e 2 1−e−µ 2 2 −µ(K+1)

(29)

(30)

Its partial derivatives with respect to µ and α yield the average system size, via hN i = ∂Ω(α)/∂µ, and the average number of length-ℓ cycles (for ℓ = 3, . . . , K), via hnℓ (A)i = −ℓ−1 ∂Ω(α)/∂αℓ . We thereby find that

(28)

Upon averaging over the ensemble, using (3) and our earlier observation that for N → ∞ the fraction of nodes in cycles of finite length L > K vanishes, we immediately obtain the asymptotic ensemble-averaged spectrum corresponding to (1), expressed in terms of (26) and (27):

wN ZN (α),

with ZN (α) defined in (2). The divisor N ! in wN will simplify our calculation, without losing the benefits of the thermodynamic limit (since we will find that for µ → 0 the expected system size still diverges). Direct calculation of Q(α) now circumvents the integration over ω: ! ∞ K XY e−µℓnℓ Y ℓαℓ nℓ Q(α) = e (2ℓ)nℓ nℓ ! n ℓ=3 ℓ=3      K Y X e−µℓn Y X e(αℓ −µ)ℓn     = (2ℓ)n n! (2ℓ)n n! n≥0 ℓ>K ℓ=3 n≥0 ! K X e−µℓ X e(αℓ −µ)ℓ + = exp 2ℓ 2ℓ ℓ>K ℓ=3 ! K K X X e(αℓ −µ)ℓ e−µℓ 1 −µ = exp − log(1−e ) − 2ℓ 2 2ℓ ℓ=3 ℓ=1 (31) P where we used ℓ>0 xℓ /ℓ = − log(1−x). From Q(α) we obtain, in turn, the grand potential Ω(α) = − log Q(α):

ℓ=3

N X ℓnℓ (A)

∞ X

N =1

Ω(α) =

The set of eigenvalues for each A ∈ GN will just be the union of all the sets of eigenvalues of the disjoint cycles of which it is composed, taking multiplicities into account: ̺(µ|A) =

GRAND CANONICAL APPROACH

(24)

contains valuable information on the statistics of cycles in the graph. Here the sum runs over the set of (real) eigenvalues {µi (A)}i=1,...,N of A, taking into account multiplicities. For instance, the number of closed paths in A R is proportional to dµ ̺(µ|A)µℓ . Our main quantity of interest will be the expected density, averaged over the ensemble probabilities (1), in the infinite size limit, X (25) ̺(µ) = lim p(A)̺(µ|A). N →∞

IV.

K 1X

1e + 2 1−e−µ 2

e(αℓ −µ)ℓ

(33)

ℓ=3

and hnℓ (A)i =

1 (αℓ −µ)ℓ e 2ℓ

(34)

Clearly, hN i diverges for µ → 0, which gives our thermodynamic limit. In this limit we can then work out for

5 1.0

1.0

0.8

m3

0.8

0.6

m3

0.6

0.4 0.4

0.2

0.0

0.2 0e+00

2e+04

4e+04

6e+04

8e+04

1e+05

−0.2

0.0

0.2

FIG. 1. Examples of the evolution of the fraction m3 of nodes in triangles, measured during MCMC simulations, for K = 3. Time is defined as the number of accepted edge swap moves per link. The bottom two curves correspond to the connected phase of the ensemble, equilibrating to the values m3 = 0.125 for α ˜ 3 = 13 log(0.25), and to m3 = 0.45 for α ˜ 3 = 13 log(0.9). The top curve corresponds to the disconnected phase, here the MCMC process is equilibrating to the value m3 = 1.

ℓ ∈ {3, . . . , K} the ratios

µ↓0

ℓhnℓ i = lim µ↓0 hN i

e−µ(K+1) 1−e−µ

e(αℓ −µ)ℓ = 0 (35) PK + ℓ′ =3 e(αℓ′ −µ)ℓ′

Similar to the canonical case, any µ-independent α will asymptotically always yield a vanishing fraction of nodes in cycles of length ℓ ≤ K. Again a re-parametrization is required to obtain a non-trivial thermodynamic limit, which is here αℓ = α ˜ℓ + ℓ−1 loghN i. Upon following this prescription, we then reproduce the canonical result 1 ℓhnℓ i = e(α˜ ℓ −µ)ℓ , hN i 2

(36)

and our expression (33) for hN i now becomes hN i =

e−µ(K+1) . PK (1−e−µ ) 2 − ℓ=3 e(α˜ ℓ −µ)ℓ

0.6

0.8

FIG. 2. Values of m3 shown versus α ˜ 3 for ensembles with K = 3. Numerical results, measured upon equilibration of the MCMC processes, are shown as black dots with error bars for N = 1000, and as squares for N = 5000 (error bars for N = 5000 are not shown; their sizes are similar to or smaller than the squares). The solid line is the prediction of (38).

V.

lim

0.4

α ˜3

t

(37)

The re-parametrization of α now depends on α itself, via hN i, and has to be consistent with a nonnegative value P (α ˜ ℓ −µ)ℓ for (37), i.e. with 21 K ≤ 1. Expression (36) ℓ=3 e PK gives us the physical interpretation ℓ=3 ℓhnℓ i/hN i ≤ 1. PK In the limit µ ↓ 0 the condition becomes ℓ=3 eℓα˜ ℓ ≤ 2. PK In the case of inequality we have ℓ=3 ℓhnℓ i/hN i < 1, so we are in the connected phase. The case of equality reproduces our earlier phase transition condition (21) and we enter the disconnected phase; here the thermodynamic limit is reached already for nonzero µ, and we can again recover our canonical equations, with exp(−µ) now playing the role of the canonical order parameter x.

NUMERICAL SIMULATIONS

Calculating ZN (A) by numerical enumeration for nontrivial values of N is not a realistic option, since the size of the set GN grows super-exponentially with N . Instead, to test our theoretical predictions we have sampled graphs from the ensemble (1) using the Markov Chain Monte Carlo (MCMC) method described in e.g. [27] or [6]. Starting from an arbitrary 2-regular N -node graph, this stochastic process is based on executing repeated (degree-preserving) edge swap moves with appropriate nontrivial move acceptance probabilities, constructed such that the Markov chain’s equilibrium distribution is the target measure (1). In each simulation experiment, the MCMC process was first run for 105 to 106 accepted moves per link, and equilibration was confirmed by measuring the Hamming distance between the instantaneous and the initial state. After this randomization stage, the instantaneous state A arrived at by the chain was defined to be our graph sample. We have limited our simulations to ensembles with K = 3 and K = 4. The degree of equilibration achieved by the MCMC during a run of 105 accepted moves per link is illustrated in Figure 1, where we show typical evolution curves of the order parameter m3 during the stochastic process. For K = 3 we have just one control parameter α ˜3, and the order parameter is the fraction m3 of nodes in triangles. The theory claims that, for large N , the graphs from our ensemble will be collections of triangles and large rings. The key equations (20,21,22) reduce to the following predictions, with α ˜ c = 13 log(2) ≈ 0.23105 . . . : α ˜3 < α ˜c :

m3 = 21 e3α˜ 3 ,

connected phase

α ˜3 > α ˜c :

m3 = 1,

disconnected phase

(38)

Numerical simulations with sizes N = 1000 and N = 5000 show excellent agreement with these predictions, as

6 1.0

0.4

Disconnected 0.8

0.2

α ˜4

0.0

−0.2

−0.4

m4

* * Connected

−0.6

−0.8

−0.6

−0.4

0.6

0.4

0.2

*

0.0 −0.2

0.0

0.2

0.4

α ˜3

**

0.0

* 0.2

0.4

0.6

0.8

1.0

m3

FIG. 3. Left panel: the plane of control parameters for K = 4. The solid black line is the critical line e3α˜ 3 + e4α˜ 4 = 2 (here m∞ = 0). The dashed lines correspond to parameter combinations with constant m∞ , taking the values m∞ ∈ {0.75, 0.5, 0.25}, from bottom to top. The markers represent parameter combinations chosen for MCMC simulations. Right panel: the fractions (m3 , m4 ) associated with the control parameter combinations in the left panel. Here the markers represent the simulation results, which are indeed found on the respective lines predicted by the theory. Note that the theory predicts that all parameter combinations in the disconnected phase e3α˜ 3 + e4α˜ 4 ≥ 2, should be mapped to the line m1 + m2 = 1 in the right panel. Error bars were omitted, as they are as big as or smaller than the markers.

shown in Figure 2, both in terms of the values of m3 and in terms of the location of the transition. For K = 4 we have two control parameters, α ˜ 3 and α ˜4, and the theory claims that for large N the graphs from our ensemble will now be collections of triangles, squares and large rings. Here the key equations (20,21,22) predict that the transition line in parameter space is given by e3α˜ 3 + e4α˜ 4 = 2, and that the fractions m3 and m4 of nodes found in triangles and squares, respectively, are solved (together with the auxiliary order parameter x, in the disconnected phase) from: e3α˜ 3 + e4α˜ 4 < 2 :

m3 + m4 < 1,

connected phase

m3 = 21 e3α˜ 3 , m4 = 12 e4α˜ 4 e3α˜ 3 + e4α˜ 4 > 2 :

m3 + m4 = 1,

(39) disconnected phase

m3 = 21 x3 e3α˜ 3 , m4 = 12 x4 e4α˜ 4 Figure 3 (left panel) shows the resulting predicted phase diagram in the (˜ α3 , α ˜ 4 ) plane. The mapping (˜ α3 , α ˜ 4 ) 7→ (m3 , m4 ) will map the lower region of the phase diagram (the connected phase) to the interior of the triangle m3 + m4 < 1 in the right panel of Figure 3. The upper region of the phase diagram on the left (the disconnected phase), including the critical line, will be mapped to the line m3 + m4 = 1 in the right panel. To test also these predictions against numerical simulations, we have chosen multiple points (˜ α3 , α ˜ 4 ) in both regions of the phase diagram, grouped such that the predicted values of m∞ = 1−m3 −m4 were always in the set {0.25, 0.5, 0.75}. The prediction would therefore be that in the (m3 , m4 ) plane these groups of points should be found on the lines m3 + m4 = 1 − m∞ . Upon measuring the fractions m3 and m4 via MCMC in the corresponding graph ensembles, these predictions are once more validated convincingly. See Figure 3.

Graphs sampled from our ensemble with K = 4 do indeed typically consist of controlled numbers of triangles and squares, and a long ring. Figure 4 shows examples of such graphs, obtained via MCMC, together with the eigenvalue spectra of their adjacency matrices (obtained by numerical diagonalization). Also the observed spectra agree with the corresponding theoretical predictions (29).

VI.

CONCLUSIONS

In this paper we presented an analytical solution for an exponential random graph ensemble with a controllable density of short cycles. Whereas one would normally not expect such non-treelike graph ensembles to be solvable, here this is possible as a consequence of imposing a local degree constraint of strict 2-regularity. We found a second order phase transition, which separates a connected phase with large and small cycles from a disconnected phase where the graphs are typically formed only of extensively many short cycles. The short cycles appear in controlled proportions, for which we found analytical expressions in terms of the ensemble’s parameters. We also derived an analytical expression for the critical submanifold in the phase diagram, and for the expected eigenvalue spectrum of the graphs’ adjacency matrices. We analysed both the canonical and the grand canonical formulation of the ensemble. The canonical version was solved via steepest descent integration. In the grand canonical version one avoids steepest descent integration, but (as always) the chemical potential takes over the role of the steepest descent integration variable of the canonical version. In the thermodynamic limit, the canonical and grand canonical routes result in identical equations. These equations are found to give highly accurate predictions already for modest graph sizes, such as N = 1000,

7

135

410

interacts with

618

interacts with

635

interacts with

interacts with

262

interacts with

interacts with

798

539

interacts with

887

884

interacts with

interacts with

896

868

628

interacts with

interacts with

721

837

86

interacts with

interacts with

856

921

544

interacts with

interacts with

756

855

19

interacts with

interacts with

758

918

interacts with

interacts with

839

852

264

interacts with

interacts with

777

824

476

interacts with

interacts with

785

966

74

interacts with

interacts with

872

interacts with

690

995

668

925

730

813

405

interacts with interacts with

345 700 114

interacts with

interacts with

504

interacts with 820

interacts with

interacts with

785

94

interacts with

661 interacts with interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

529

289 733

571


interacts with 494 interacts with

interacts with

interacts with

158 178

interacts with 908 interacts with interacts with

918 interacts with

526 420

312

870

interacts with

970

916

861

768

766

976

724

894

700

697

703

659

6

766

interacts with

interacts with

interacts with

interacts with

472

interacts with

233 64

818 581

interacts with

249

interacts with

317 interacts with 387


730


964 interacts with

interacts with 213

223

588

169

323

333

36

71

291

520

227

265

384

61

772

506

286

interacts with

26

interacts with interacts with 986

interacts with 222

interacts with

interacts with

interacts with

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

192 interacts with

373 402

interacts with 200

301

189

672

116

117

537

432

67

92

355

interacts with

interacts with

78 291

interacts with

929

634

624

610

663 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

609

761

812

929

598

740

597

882

593

998

592

655

581

989

572

934

555

734

554

876

932

745

interacts with

interacts with

interacts with

interacts with

577

565 interacts with


784


764 509

719

interacts with

800

772

709

995

708

876

685

819

721

951

669

914

825

727

interacts with

535

interacts with

327

497 interacts with

475

interacts with

875

interacts with

2

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with interacts with interacts with

interacts with

268

949 356

interacts with

interacts with

210

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

241

interacts with

142

interacts with

321 252

882

interacts with

111


interacts with

interacts with

955

954

716

981

794

920

681

844

171

836 999

interacts with

338

686

986

739

907

602

975

990

821

898

922

621

857

670

interacts with

400 218

468

interacts with

interacts with

interacts with

interacts with 496 932

interacts with

418

interacts with

471

interacts with 977

interacts with

interacts with

28


145

231

interacts with

interacts with 988


323

362

929

interacts with 753

672


722 463

231

131

506

488

389

229

439

274

320

286

407

10

67

144

132

206

89

95

247

367

873

283


interacts with

interacts with

859 interacts with

interacts with

86 interacts with

971 946 933

634 347

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

386

interacts with

898

interacts with

interacts with

795

interacts with

interacts with

104

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

330 638

558

69

571

551

852

528

575

546

478

472

93

601

569

999

572

618

655

560

610

579

interacts with


365 959

interacts with

interacts with

interacts with

interacts with

interacts with

811

interacts with

interacts with

541

533

538

667

861

501

801

475

880

517

826

447

627

445

680

737

451

436

994

435

912

553

747

426

463

558

754

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

903

89

614 169

995

239

interacts with

interacts with

316

interacts with



interacts with

766

791

530

776

522

498

971

897

62


705 interacts with

36

623

interacts with

interacts with

717

interacts with

723

288

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

710 415

interacts with

interacts with

interacts with

480 879

634 477

interacts with 584

interacts with

interacts with

601

interacts with

interacts with

952

interacts with


25

interacts with

233

130

119

204

108

149

122

78

741

710

502

973

633

465

699

883

442

591

571

429

867

692 interacts with

interacts with 22

613

60

17

876 interacts with

interacts with


678

interacts with

interacts with

interacts with

interacts with

interacts with

507

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

954 760

833

68 187 interacts with

interacts with interacts with 16 705 interacts with

interacts with 311

interacts with

356

742

322

985

314

799

299

952

275

554

256

706

245

661

566

889

interacts with

799


33

interacts with

390 interacts with

422 27 838 interacts with

111

482

625

interacts with interacts with 483 488 interacts with interacts with

interacts with 574

538

interacts with

interacts with

964 interacts with

934

interacts with 164

interacts with

122

interacts with

interacts with

interacts with

interacts with 260

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 82

interacts with

924 667 74 583 interacts with interacts with

interacts with


273 727

164

58

158

130

332

275

385

52

330

37

311

9

369

interacts with

552

379 interacts with

953 431

interacts with

interacts with

interacts with

interacts with

interacts with

392

interacts with

509

255

328

495

724

796

743

409

603

86

interacts with


899 interacts with

103 295


interacts with

interacts with

455

interacts with

175

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

826

interacts with

interacts with

626 interacts with

89

290

interacts with interacts with 322 896 616

576

interacts with

interacts with

693

interacts with

424

420

627

415

992

414

683

405

612

858

528

675

604

805

981

395

930

383

853

382

993

374

523

370

543

478

818


990

interacts with

997

interacts with


26 191

622

interacts with

166

interacts with

10 interacts with

915


5

235

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

54

173

29

301

212

9

147

158

77


207

interacts with

105 682 920


771 459

741

interacts with 812

interacts with


interacts with

167 interacts with 173 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

157 412


51

372 484 729

interacts with

460

interacts with

751

541

803

862

582

939

419

994

416

978

415

899

878

562

959

674

404

400

399

601

947

470

600

844

476

474

612

521

80 interacts with

interacts with

interacts with 551 719 92

889

595 910

147 591

interacts with 610

341



interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


639

7

interacts with 439

interacts with

666

interacts with


interacts with

245

208

393

979 interacts with

570

240 interacts with

interacts with 274

434 736

451

interacts with

438

434

424

830

538

562

412

interacts with

611

720

389 interacts with

495

interacts with

interacts with

278

369 974 824

interacts with

interacts with 75


interacts with

interacts with

491

interacts with

221

128

124

204

225

308

22

172

251

282

146

299

25


24

interacts with

193

107 interacts with 170 interacts with interacts with


interacts with

164 865

interacts with

interacts with

1

28


6

940

58

42

23

76

84

40

44

740

890 592

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

543

interacts with

interacts with interacts with interacts with 702 interacts with 768 881

61

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

765 708

interacts with

117

interacts with interacts with interacts with interacts with interacts with

interacts with

48 interacts with

136

363

869

358

661

347

936

345

735

769

531

362

996

336

842

334

636

327

698

492

468

550

664

298

722

614

577 112

254

interacts with

183

169

349

394

821

156

508

144

605

115

972

570

366

221

466

956

608

interacts with

172

812


525


971 interacts with

258

interacts with

95 313

115 interacts with

631


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


770

605 485 154 911

interacts with

854

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 192

interacts with

160

754


interacts with

572

512

interacts with 758

380

788


251

310

163

953

510

339

113

100

interacts with 892

interacts with

110

269

interacts with

interacts with 755

interacts with

228 194


656

interacts with 413


624 774


957



426 30

334

510

361

620

931

851

341

337

904

695

534

304

302

432

590

interacts with


interacts with 156

interacts with

interacts with

interacts with

270

interacts with 618

403

746

324


464 interacts with

85

942

interacts with

interacts with

310

761 242

511 interacts with 724 106 9

589 interacts with

interacts with

interacts with 354

interacts with

206


191

32

14

936

interacts with



interacts with 578



279

118

interacts with

292

interacts with

interacts with

interacts with

interacts with


interacts with


interacts with

interacts with

526

interacts with

interacts with

interacts with

interacts with

interacts with

53 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

582

interacts with

interacts with

958

interacts with 492

413 252 530

124

931

460

457

943

809

497

85

856

466

746

352

384

636

13

177

interacts with

91

473

237

234

66

263

24

173

6

99

187

207

29

81


interacts with

interacts with 391

730

429

587

interacts with

779

652

221

580

181

interacts with

interacts with interacts with interacts with interacts with interacts with interacts with

383

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

502



45

70



437

interacts with



interacts with 568

interacts with

interacts with

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

63

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

843 interacts with

900



438


interacts with

170

interacts with

395

463

557

797

interacts with

226

625 414

interacts with 533

499

697

interacts with

944 interacts with

407

382

376

409

295

564

278

944

268

906

267

1000

249

513

233

290

217

547

216

952

213

795

212

630

211

594

191

647

845

396

633

696 201

950

641 interacts with

interacts with

interacts with



377

462 416

127

623

377

interacts with 349 interacts with interacts with interacts with

123


interacts with

interacts with

interacts with

196

155

206

interacts with

410

83

interacts with

513

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

292

interacts with

447

934

37

interacts with

14

16

34

851 interacts with


667 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

309 716

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

493 interacts with

11 interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

99

interacts with

160

649

interacts with

25 interacts with

223

220

216

545

671

225

941 981

499 658

645

662

503

821

679

861

interacts with

66 648 751

interacts with

781

interacts with

292

718

879

639

752

696

732

438

486

671

809

393

interacts with

interacts with

450

893 interacts with

62

61

857

interacts with

interacts with

921

763

interacts with

583 153

interacts with


335

742

interacts with

850


interacts with


898

344

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


124

22


interacts with 779

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with



interacts with 176

interacts with 962 318 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

557 23 interacts with 112

interacts with 417


interacts with

interacts with

interacts with

455

854

456

775

interacts with

interacts with

780

617

699

72

254

334

interacts with


interacts with

interacts with 945

interacts with 688

346

interacts with

20

998 817

31

64

102

17

114

140

123

46

73

80

76

14

57

519

828 364

interacts with

interacts with

688

948

446

interacts with

481

interacts with

90

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

333

66

143

interacts with

813

interacts with

510

interacts with

490 436

592

59

556

102


interacts with

interacts with

interacts with

interacts with

interacts with 655

319 interacts with

973

interacts with

713

642

141

interacts with

711 458

331

183

485

946

616

409

175

723

174

682

467

357

157

276

141

669

139

339

325

960

496

786

88

371

662

642

348

interacts with

interacts with


interacts with

829

615 interacts with

interacts with

154

940 114

343 interacts with

interacts with

interacts with

819 528

965

interacts with

interacts with

interacts with

489

interacts with

interacts with

interacts with

835

interacts with

878

interacts with

174

901

40

13

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

960

777

913

interacts with

interacts with 943

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


807

interacts with

interacts with

interacts with

interacts with

interacts with

269 843

3

1

870


866

interacts with

823

interacts with

interacts with

10

8

7

4

interacts with

interacts with

interacts with

interacts with

174

interacts with

978

interacts with

interacts with

interacts with

interacts with 203

609

interacts with

interacts with

121 365


interacts with 336

189

interacts with

205

178

177

820

578

171

567

159

899

117

94

800

interacts with


interacts with

834

698

203

718 712

interacts with

803 823 119

128

127

126

interacts with

165

interacts with

interacts with


500

729 784

300 interacts with

interacts with


900 interacts with

81


interacts with

120

683

671

interacts with


interacts with 140

interacts with

602 558

810 interacts with

interacts with 56


848

245

286

interacts with

250

792 906

interacts with 970

interacts with

804

565


interacts with

215

808

403

790

4 interacts with interacts with 210 interacts with

547

interacts with

interacts with


780 interacts with

interacts with

interacts with

922 225

interacts with

842

171 994

556

875

874

interacts with

910

interacts with

611

935


interacts with

interacts with

442 interacts with


881

interacts with 620

657 262 interacts with interacts with

84

interacts with 371

interacts with 271

82

84

41

13

4

5

interacts with 794

479

983

792

769


722

interacts with

682 interacts with

interacts with

interacts with 681

146

747

interacts with

interacts with

interacts with

interacts with

74

311

96

interacts with

interacts with

715

interacts with

interacts with

619

989


interacts with

interacts with

695

863

interacts with

interacts with

interacts with

883

782

interacts with 152

814

622

336

interacts with

277

270

296

656

141

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

38

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

300

637

676

829

549

interacts with 561

interacts with 632

532

529

interacts with

interacts with

516

interacts with

interacts with

interacts with

interacts with

257

253


interacts with

236

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 486

interacts with

interacts with

interacts with

interacts with


711


interacts with

interacts with

583

495

859

interacts with

87

624

interacts with

532

59

273

50

942

35

979

511

34

526

455

789

403

565

958

45 interacts with

interacts with

743

interacts with

interacts with


617

interacts with

interacts with

interacts with 750

87 631

1000

836

843

580

917

871

808

interacts with

980

928

783

interacts with

interacts with

125


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 822 interacts with 901

266

985

457

interacts with

822

interacts with

interacts with

interacts with

62 interacts with


874

548

539

534

interacts with


49

47

46

43

interacts with

460 125 279

interacts with

interacts with

interacts with

435

interacts with

993

588

905

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

281

interacts with

interacts with

interacts with

interacts with

12 interacts with

649

interacts with

interacts with

interacts with

interacts with


interacts with

interacts with

interacts with


781


interacts with 240

interacts with

239

236

231

interacts with

613 458

interacts with

909

691 interacts with

776

interacts with

550 227

interacts with 523

271 873 52

interacts with

interacts with

872

interacts with

892

776

interacts with

85

389

833

462

681

interacts with

728

997 927

982

411 interacts with

401 interacts with

186

398

394

390

interacts with

745

317

696

316

314

313

539 569

453 709

163 interacts with

interacts with


895

interacts with

633

580

433

828

600

467

408

525

813

482

477

interacts with

268

interacts with

783 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with



518

948

668

917

486

765

996

516

683

interacts with

interacts with

640


interacts with

interacts with 133

interacts with

937 interacts with

467 385

interacts with 660

interacts with

interacts with

272 234

804 502

interacts with

interacts with

interacts with 48

interacts with

674

interacts with



427

interacts with

interacts with

interacts with

763

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


811 interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

773 interacts with

641

interacts with

184 820

762

interacts with 57

interacts with

639 interacts with

interacts with

864 interacts with

370

807


937

interacts with 307 interacts with interacts with 694

857

161

interacts with

265 interacts with

interacts with

371

interacts with

781

8 interacts with

interacts with

645

924

622

613

545

689

530

546

865

672

910

969

497

584

726

707

792

665

673

727

453

643


interacts with

interacts with

264

3

259

258

404

314

401

395

390

386

378

373

interacts with

interacts with

337

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 912


interacts with

900

950

780

775

646

823

811

862

874

939

954

978

814

692

753

846

794

885

830

920

590

895

596

569



256 441 797 663

interacts with


890

364 673

977

570


interacts with interacts with interacts with 285

interacts with

interacts with interacts with 55 801


interacts with

679 interacts with

18

304 interacts with

504

interacts with

385

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

664

552

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

817 interacts with

interacts with interacts with 5 470

interacts with

983

573

interacts with

interacts with

315 interacts with

925

36

378

139

interacts with 424

interacts with

interacts with 498

interacts with 474

445 interacts with

533


871

interacts with

interacts with

491

interacts with

739

591

443

500

interacts with

118 144

707

interacts with


637

222

475

660

368

462

340

330

789 548

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with 108

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


975

961

853

485

787

interacts with 327

670 interacts with

549 987

745

690

723

740

392

524

522

507

506

504

498

493

480

479

473

472

458

457

444

440

431

423

419

418

411

406

824

interacts with 148

interacts with

interacts with

interacts with

interacts with

interacts with 676

858


199

555

445 150

519

587 499

interacts with 71

interacts with

98 849

110

362 interacts with

interacts with

interacts with

interacts with

786 interacts with

394

interacts with 382

515

713

351


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

927 410

interacts with

670

interacts with

interacts with

interacts with

interacts with

interacts with

219

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

694 939

773

interacts with

63


interacts with

interacts with

507

360

896 143

interacts with 662

46

interacts with

interacts with

18


193

interacts with 886

738


interacts with 838 575 interacts with 103

360

359

350

343

342

341

209

208

202

201

200

197

195

749

83

interacts with

interacts with 121


interacts with 646

interacts with

295

interacts with

interacts with


interacts with

120 interacts with

interacts with 31

interacts with 602

919

744

391

449

693

482

951

537

825

471

770

582

608

827

527

774

388

500

421

456

576

515

746

interacts with

interacts with

interacts with

962 645

915 643

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

223

interacts with

908

487

717

685

854

516

967

955

919

972

963

937

893

941

702

974

652

787

425

945

819

708

751


189

interacts with

702




938

interacts with 856

407

974



680

853 76

444

interacts with

942

589

interacts with

686

interacts with

interacts with

614 interacts with

388

353

941

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

304

887 109

interacts with

567

992

267 180

324

757

261

481

232

417

658

interacts with

744

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

110 interacts with

interacts with

interacts with

736

687

523

968

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

733

interacts with

969 842

589

938

989

847

503

interacts with

886

interacts with

interacts with

749

897 interacts with



interacts with 205

interacts with

interacts with

597

interacts with

559

interacts with

interacts with

793

interacts with 101

interacts with

interacts with 545

interacts with

361

interacts with

interacts with

interacts with

interacts with

interacts with

357 769

135

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with

42

interacts with

interacts with

interacts with

interacts with

387

386

381

380

378

376

373

372

366

364

359

353

350

349

346

344

342

338

335

329

326

322

461 761

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

916

866

248

interacts with

426

393

interacts with 451

interacts with

interacts with

60

826

17 907

905

interacts with

interacts with

interacts with


interacts with

406 interacts with

947

interacts with

interacts with 52


869

302 190

interacts with

188

185

182

179

647 151

interacts with 69

interacts with

63

60

57

56

55

54

933


777

interacts with


interacts with

834 563

interacts with

967

264 355

interacts with

interacts with

805 32


interacts with

449

interacts with

interacts with

interacts with

interacts with

775

interacts with

159 436

611

852

363 interacts with

interacts with 296

interacts with 790

553

interacts with 284

130

interacts with

984 930

interacts with

649

243

163

483

356

579

491

328

351

441

408

540

430

433

828

397

648

790

448

725

619

343

318

525

326

145

interacts with

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

845

997

831

529

984

514

736

505

733

561

586

660

518

948

877

799

935

658

764

850

657

573

810

interacts with

303

840

interacts with 560

137 interacts with


605 interacts with

interacts with

interacts with interacts with interacts with 404 interacts with

20

735

718

interacts with

interacts with

309

179

839 37


interacts with 253

interacts with

508

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

443

882

753

747

728

651

693

678

689

759

771

731

635

726

interacts with 721

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

593

185

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 976

750

interacts with

908

965

947

815

686

980

841

810

930

782

888

760

interacts with

758

interacts with

600 513

interacts with


interacts with

298

interacts with

687 815

interacts with

interacts with

635

interacts with


990

380


975


interacts with


interacts with

199

306

interacts with

689

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

478

860

505

312

230

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

309

307

306

297

294

293

288

285

266

261

259

258

255

254

253

252

250

246

244

242

241

interacts with


399

960 interacts with 452 211

interacts with

963

16

607 770

701

interacts with

interacts with 338

interacts with


interacts with

interacts with


interacts with

640

71 interacts with

748

717

704

654

648

647

632

621

606

599

596

590

584

interacts with


65

interacts with

959 652

interacts with

996

interacts with


270 980


interacts with



836

195

interacts with

352

19

353

interacts with


interacts with

interacts with

739



interacts with

697 34

814

650

interacts with


713

368

840

439

712

542

428

235

222

490

281

208

340

256

508

461

459

310

443

595

301

742

755


872

interacts with

549

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with 422

interacts with

interacts with

976

743 706

568

943

454

765

684

644

817

987

897

519

719

392

521

575

923

806

474

477

688

536

838

965

408

interacts with interacts with 841 interacts with 512 interacts with interacts with

520

880

357

593

903

946

732

459

307

367

767

734

402


68

interacts with

interacts with

interacts with

interacts with

945

731

interacts with

interacts with

902

interacts with

627

interacts with

675

interacts with

918

interacts with

991

interacts with

926

interacts with

879

interacts with

316

interacts with

597

interacts with

816

1000

542

923 interacts with

interacts with

interacts with

568

433


interacts with

800

interacts with 128

interacts with

944


951

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 196


220

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

131 276

interacts with

325 396 96

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

35

149

982

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with



659

692


interacts with

interacts with

interacts with

interacts with

interacts with

98 165

564

43


339 985

interacts with

726

535

interacts with

230

224

220

219

218

215

214

210

209

200

199

198

197

196

193

190

184

181

179

176

170

168

167

283

335

333

331

319

308

306

303

297

293

291

287

285

282

39 594

598 626

interacts with

501

interacts with

397

interacts with

interacts with

interacts with

interacts with

240

608 870

interacts with 788

interacts with

interacts with

741

188 interacts with

321


217


interacts with

585

567

interacts with

interacts with 139

interacts with

566

869

294 interacts with

58

855

346

388


444 448

543

interacts with


interacts with

interacts with

281 interacts with


923

928 464

963

435

594

638

579

615

288

501

329

162

531

427

198


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

166

interacts with

966

949

750

887

755

835

630

992

345

865

738

653

448

376 299 756

280

720

228

272

679

631

402

186

704

152

126

858

interacts with

205

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

922

15 interacts with

991

738

651

452

705

803

8

269

933

450

interacts with

interacts with

957

interacts with

450

interacts with

interacts with

interacts with

707

998 515

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

449

interacts with

964 544

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

259


411


350

interacts with

831

interacts with

interacts with

276

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

966

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

421 interacts with

374 615

888

176

175

172

168

167

165

159

151

150

148

140

138

137

interacts with

711

347 interacts with

198 interacts with

891

interacts with 913

interacts with

interacts with

interacts with

interacts with

703

442

interacts with

interacts with 880

162

155

154

153

151

150

149

147

145

142

138

interacts with

interacts with

238

469

425

630 interacts with

interacts with 358

522 187

interacts with


644 interacts with


interacts with

interacts with

805

interacts with

153 216

interacts with



932

931 914 598 757

interacts with

564

104

234

397

454

284

134

218

112

351

142

194

238

64

interacts with 456

484


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

969

621 interacts with

interacts with

interacts with

425

767

906

452

909

714

891

756

226

628

465

778

585

369

384 interacts with

interacts with 791

606

421 894

interacts with


interacts with

423


894

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

261

interacts with

290

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

612

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 737

interacts with

827

interacts with

91

129

340 468


400

interacts with

324 interacts with


342

607 41

38

35

33

31

30

24

21

20

19

15

12

11

interacts with

interacts with 885

609


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 381

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with


657

917 787

50


184


interacts with

interacts with

293

332 257


interacts with

interacts with


interacts with

interacts with

305 524

interacts with

542

673

588

680

212

267 224

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

698

864

907

interacts with

344 interacts with

interacts with 642

490

973


interacts with

73 interacts with

81 375 72


884

interacts with


75

664

interacts with

interacts with

interacts with

469 interacts with

603

294 interacts with

924

527

674

616

629

701

514

540

684

547

563

636

461

665

494

643

802

interacts with

interacts with

interacts with

interacts with

interacts with


interacts with

interacts with

interacts with

interacts with

950

interacts with

967

interacts with

936

interacts with

699

interacts with

873

interacts with

801

interacts with

831

interacts with

737

interacts with

744

interacts with

752

interacts with

637

interacts with

656

interacts with

725

interacts with

885

interacts with

928

895

840

interacts with

interacts with

181

116 97

interacts with


578

577

574

921

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 282

209

255


interacts with


interacts with

101

586

517

524

517

496

493

492

487

483

480

470

447

441

440

431

423

422

406

405

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with 830


315

471

398

interacts with

interacts with

interacts with

665

465

430

700

715

454



interacts with

interacts with


interacts with

562

331 interacts with

186

interacts with

interacts with

interacts with


332

795

619

313

325

418

296

511

559

768

358

375

283

263

428

576

381 interacts with

interacts with

interacts with

531 446 748 654

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

893

859

interacts with

912

850

479

620

837

676

916

848

846

867

536

988

798

644

839

604

interacts with


interacts with

interacts with

59 interacts with

50

432


280

278

277

302

interacts with 88

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

298

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

904 152



102 629

interacts with 197

interacts with

710

interacts with

847

interacts with

132

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

778 interacts with

interacts with

interacts with 249

247

246

244

239

237

236

235

230

228

227

224

217

215

214

213

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

211

interacts with

interacts with

263

interacts with

925


interacts with

interacts with

interacts with

573

interacts with

363

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

157

interacts with


interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

849

793

818

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

774

677

329

863

interacts with

interacts with

interacts with

interacts with

interacts with

192

595

399

437

550

868

806

260

370

354

266

323

420

544

289

320

646 interacts with

interacts with

interacts with

interacts with

806

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

355

845

986

691

827

970

904

666

911

379

586

659

581

785

860

387

735

interacts with

41

759

968

308

135

133

132

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

interacts with

476 725

79

829

interacts with

interacts with

interacts with

53



754

interacts with

214

interacts with

111

109

106

105

97

95

94

93

91

90

88

87

82

80

79

73

70

675

383

interacts with

interacts with

368

interacts with 825

interacts with

561


interacts with

303

interacts with 972 419 interacts with

interacts with

21

65 interacts with interacts with 884 553 interacts with interacts with

653

interacts with

684

interacts with

372

14

ρ(µ)

1.2

1.2

1.0

1.0

10

0.8

0.8

8

0.6

0.6

6

0.4

0.4

4

0.2

0.2

2

0.0

0.0 −2

−1

0

1

2

12

0 −2

−1

µ

0

µ

1

2

−2

−1

0

1

2

µ

FIG 4 Top row typ ca graphs samp ed numer ca y v a MCMC rom the canon ca ensemb e (1) o 2-regu ar nond rected s mp e graphs or N = 1000 Le t (m3 m4 ) = (0 0 0 06) and m∞ = 0 94 M dd e (m3 m4 ) = (0 25 0 56) and m∞ = 0 19 R ght (m3 m4 ) = (0 39 0 61) and m∞ = 0 The bottom row shows the e genva ue spectra o the correspond ng three ad acency matr ces computed by d rect numer ca d agona zat on The ocat ons o the peaks are seen to agree w th the theoret ca pred ct ons o (29) Note the d fferent sca e n the th rd spectrum graph to emphas ze the we ghts o the δ-peaks

as we confirmed n numer ca s mu at ons The parameter K represents the argest cyc e ength that s contro ed n our mode For K = 3 one contro s on y the number of tr ang es and our ensemb e becomes s m ar to that of Strauss 8 11 23 w th average degree two The rema n ng d fference s that n the Strauss mode the average degree s mposed mp c t y v a an overa soft constra nt wh e n the present mode a degree va ues are mposed as oca hard constra nts Due to th s d fference the degenerat on of the Strauss mode to a phase where the comp ete c que has probab ty one (so the number of tr ang es can no onger be tuned) s avo ded n the present ensemb e The comp ete c que s s mp y no onger an a owed configurat on and hence the number of tr ang es becomes fu y tuneab e f the mode parameters sca e appropr ate y w th the system s ze Our mode defin t on cou d obv ous y be genera zed to nc ude q-regu ar graphs w th q > 2 Numer ca exp orat ons for 3-regu ar vers ons w th K = 4 have been reported n 6 and show phenomeno ogy s m ar to that found here for q = 2 In part cu ar one aga n observes a d sconnected phase for arge va ues of α3 and α4 However ana yt ca so ut on for q > 2 a ong the nes fo owed here for q = 2 wou d not seem feas b e We cou d a so comb ne our present mode w th the Erd¨ os-Rèny ensemb e to produce connected random graphs w th a vary ng

number of short cyc es Aga n wh e the phenomeno ogy of such var at ons cou d be exp ored v a s mu at ons t s not c ear how one wou d be ab e to obta n ana yt ca so ut ons w thout the benefit of oca y tree- ke topo ogy In our v ew the ma n mer t of the present mode s that ts ana yt ca so ut on he ps us understand more comp cated oopy graph ensemb es It can serve as a benchmark mode aga nst wh ch more genera so ut on strateg es for non-tree ke random graphs can be tested such as 24 wh ch dea s w th spectra y constra ned max mum entropy graph ensemb es The moments of a graph s spectra dens ty are re ated to ts numbers of cyc es v a the traces of powers of the adjacency matr x In fact the present mode s a spec a case of the fam y of ensemb es stud ed n 24 from wh ch t can be obta ned by choos ng 2-regu ar degrees and an appropr ate po ynom a funct ona Lagrange parameter The ana yt ca and numer ca resu ts of th s paper suggest that to obta n phase trans t ons the funct ona Lagrange parameters n spectra y constra ned max mum entropy graph ensemb es 24 may need to have a spec fic sca ng w th the system s ze Acknowledgement FAL gratefu y acknow edges financ a support through a scho arsh p from Conacyt (Mex co)

8

[1] A. Békéssy, P. Bekessy, and J. Koml´ os, Stud. Sci. Math. Hungary 7, 343 (1972). [2] E. A. Bender and E. R. Canfield, J. Comb. Theory, Series A 24, 296 (1978). [3] M. Molloy and B. Reed, Random Structures & Algorithms 6, 161 (1995). [4] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, 026118 (2001). [5] M. Catanzaro, M. Bogu˜ n´ a, and R. Pastor-Satorras, Phys. Rev. E 71, 027103 (2005). [6] A. C. C. Coolen, A. Annibale, and E. Roberts, Generating Random Networks and Graphs (Oxford: University Press, 2017). [7] P. W. Holland, K. B. Laskey, and S. Leinhardt, Social Networks 5, 109 (1983). [8] J. Jonasson, J. Appl. Prob. 36, 852 (1999). [9] P. Holme and B. J. Kim, Phys. Rev. E 65, 026107 (2002). [10] J. Davidsen, H. Ebel, and S. Bornholdt, Phys. Rev. Lett. 88, 128701 (2002). [11] Z. Burda, J. Jurkiewicz, and A. Krzywicki, Phys. Rev. E 69, 026106 (2004). [12] P. L. Krapivsky and S. Redner, Phys. Rev. E 71, 036118 (2005). [13] M. E. J. Newman, Phys. Rev. Lett. 103, 058701 (2009).

[14] B. Bollob´ as, S. Janson, and O. Riordan, Random Structures & Algorithms 38, 269 (2011). [15] G. Bianconi, R. K. Darst, J. Iacovacci, and S. Fortunato, Phys. Rev. E 90, 042806 (2014). [16] M. S. Granovetter, Am. J. Sociol. 78, 1360 (1973). [17] R. V. Solé, R. Pastor-Satorras, E. Smith, and T. B. Kepler, Adv. Compl. Sys. 5, 43 (2002). [18] A. V´ azquez, Phys. Rev. E 67, 056104 (2003). [19] M. Marsili, F. Vega-Redondo, and F. Slanina, Proc. Nat. Acad. Sci. USA 101, 1439 (2004). [20] I. Ispolatov, P. Krapivsky, and A. Yuryev, Phys. Rev. E 71, 061911 (2005). [21] R. Toivonen, J.-P. Onnela, J. Saram¨ aki, J. Hyv¨ onen, and K. Kaski, Physica A 371, 851 (2006). [22] M. O. Jackson and B. W. Rogers, Am. Econ. Rev. 97, 890 (2007). [23] D. Strauss, SIAM Review 28, 513 (1986). [24] A. C. C. Coolen, J. Phys. C: Conference Series 699, 012022 (2016). [25] T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd Ed) (New York: Wiley, 2006). [26] B. D. McKay, Linear Algebra and its Applications 40, 203 (1981). [27] A. C. C. Coolen, A. De Martino, and A. Annibale, J. Stat. Phys. 136, 1035 (2009).

Exactly Solvable Random Graph Ensemble with Extensively Many ...

Exactly Solvable Random Graph Ensemble with Extensively Many ...

Suggest Documents

Exactly solvable potentials with finitely many discrete eigenvalues of ...

Exactly Solvable Pairing Models

Exactly Solvable Quantum Mechanics

Exactly Solvable SchrÃ¶dinger Equation with

New exactly solvable systems with Fock symmetry

Quasi-Exactly-Solvable Differential Equations

Exactly solvable models and ultracold Fermi gases

An Exactly Solvable Two-Way Traffic Model With Ordered Sequential

New Quasi-Exactly Solvable Sextic Polynomial Potentials

Exactly Solvable Potentials and Romanovski Polynomials ... - CiteSeerX

Exactly Solvable Almost Periodic Models

An Exactly Solvable Model of Interacting Bosons

Exactly solvable two-dimensional quantum spin models

Exactly Solvable Stochastic Processes for Traffic ...

An Exactly Solvable Asymmetric Neural Network

Madelung Representation and Exactly Solvable Schrodinger-Burgers

Quasi-exactly solvable models in nonlinear optics

EXACTLY SOLVABLE MODEL FOR NONLINEAR PULSE

Exactly-solvable problems for two-dimensional excitons

Exactly solvable model of uniaxial ferroelectrics

A unifying E2-quasi-exactly solvable model

Method of constructing exactly solvable chaos

Correlation properties of exactly solvable chaotic oscillators

Exactly Solvable Model for Two Dimensional Topological