--- title: Edge-Triangle Random Graph Model url: https://www.emergentmind.com/topics/edge-triangle-model type: topic --- # Edge-Triangle Random Graph Model Searching arXiv for the cited papers and closely related work on the edge–triangle model. The edge-triangle model is a dense random-graph ensemble in which the primary observables are edge density and triangle density. In its microcanonical form, one fixes target values for these two subgraph densities and studies the asymptotic entropy of labeled graphs realizing them; in the dense limit, the problem becomes a constrained variational problem over graphons, that is, symmetric measurable functions \(g:[0,1]^2\to[0,1]\). The model is a canonical setting for analyzing phase structure, multipodality, and symmetry breaking in constrained network ensembles, and it also serves as a useful foil for edge-independent generative models, which cannot sustain high triangle density under bounded overlap without effectively memorizing a graph [1701.04444]. ## 1. Variational definition in the dense limit For a simple graph \(G\) on \(n\) vertices, the edge density and triangle density are \[ e(G)=\frac{\#(\text{edges in }G)}{\binom n2},\qquad t(G)=\frac{\#(\text{triangles in }G)}{\binom n3}. \] The microcanonical ensemble consists of all labeled graphs with prescribed \(e(G)\) and \(t(G)\), up to the \(O(1/n)\) rounding inherent in finite \(n\). The associated entropy density is defined by a large-\(n\) limit of the normalized logarithm of the number of such graphs. One formulation is \[ B(e,t)= \lim_{\delta\to0}\;\lim_{n\to\infty}\; \frac1{n^2}\,\ln\#\{G\in G_n:\;e(G)\in[e\pm\delta],\;t(G)\in[t\pm\delta]\}. \] In the dense regime, Chatterjee–Varadhan and Radin–Sadun imply a graphon variational principle. Writing \[ (g)_1=\iint g(x,y)\,dx\,dy,\qquad (g)_3=\iiint g(x,y)\,g(y,z)\,g(z,x)\,dx\,dy\,dz, \] and \[ S(g)=\iint H(g(x,y))\,dx\,dy,\qquad H(u)= -[u\ln u + (1-u)\ln(1-u)], \] one has \[ B(e,t)= \sup\{\,S(g)\;:\;(g)_1=e,\;(g)_3=t\}. \] An equivalent formulation in the \((\epsilon,\tau)\) notation uses \[ s(\epsilon,\tau)=\sup_W\{S(W):\int W=\epsilon,\;\int WWW=\tau\}, \] with the graphon entropy written as \[ S(W)=-\iint [W\ln W + (1-W)\ln(1-W)]\,dx\,dy. \] Whenever the supremum is attained at a unique graphon \(g_{e,t}\), up to measure-preserving relabelings, almost all large graphs with densities \((e,t)\) look like \(g_{e,t}\) [2211.10498]. ## 2. Feasible region and multipodal phase organization The admissible region of edge and triangle densities is \[ \Gamma=\{(\epsilon,\tau): 0\le \epsilon\le 1,\;\tau_{\min}(\epsilon)\le \tau\le \epsilon^3\}. \] The upper boundary \(\tau=\epsilon^3\) is the Erdős–Rényi curve. The lower boundary \(\tau_{\min}(\epsilon)\) is Razborov’s piecewise-quadratic “scalloped triangle,” with cusps at \[ (\epsilon,\tau)=\left(\frac{n}{n+1},\frac{n(n-1)}{(n+1)^2}\right),\qquad n=1,2,3,\dots \] The phase diagram described in the large-deviation literature contains four infinite families of interior phases together with one additional \(F\)-phase [1701.04444]. | Phase family | Symmetry | Defining feature | |---|---|---| | \(A(n,0)\) | \(n\) equivalent podes | Two edge-probabilities; touches the lower boundary at the cusp \(\left(\frac{n}{n+1},\frac{n(n-1)}{(n+1)^2}\right)\) | | \(B(n,1)\) | One special pode plus \(n\) equivalent podes | Four block-probabilities plus a size parameter \(c\) | | \(C(n,2)\) | Two equivalent small podes plus \(n\) equivalent large podes | Five block-probabilities plus \(c\) | | \(F(1,1)\) | Bipodal but not in the \(B\)-family | Occurs in the region \(\tau>\epsilon^3\) | Here a “pode” is a block in a finite partition of \([0,1]\) on which the entropy-maximizing graphon is constant. The reported synthesis of sampling, local stability analysis, and variational arguments yields a conjectured phase diagram with fourteen distinct phases together with a classification of which neighboring-phase boundaries are continuous and which are discontinuous [1701.04444]. ## 3. Bipodal graphons and explicit formulas near the Erdős–Rényi curve With two constraints, the dominant structures are often bipodal. A bipodal graphon is specified by block size \(c\in(0,1)\) and three link-probabilities \(a,b,d\in[0,1]\): \[ g_{a,b,c,d}(x,y)= \begin{cases} a, & x,yc,\\ d, & \text{otherwise}. \end{cases} \] For this ansatz, \[ e = c^2a + 2c(1-c)d + (1-c)^2b, \] \[ t = c^3a^3 + 3c^2(1-c)ad^2 + 3c(1-c)^2bd^2 + (1-c)^3b^3, \] and \[ S = c^2S_0(a) + (1-c)^2S_0(b) + 2c(1-c)S_0(d), \qquad S_0(z)=-\tfrac12[z\ln z +(1-z)\ln(1-z)]. \] Hence, within the bipodal approximation, \[ s(e,t)=\max_{a,b,d,c:\,e,t\ \text{fixed}} S(a,b,d,c). \] A particularly important regime lies just below the Erdős–Rényi curve \(t=e^3\). In the symmetric bipodal phase, denoted phase II in the symmetry-breaking analysis, the maximizer has \[ c=\tfrac12,\qquad a=b, \] and the stationarity relation \[ \delta^3=e^3-t \] gives \[ a=e-\delta,\qquad d=e+\delta,\qquad b=a. \] Thus the graphon has lower within-block density \(e-\delta\) and higher across-block density \(e+\delta\). Below a critical curve \(\Sigma\), the maximizer remains bipodal but becomes asymmetric: \(c\neq \tfrac12\) and \(a\neq b\), with two mirror-image solutions related by exchanging the two blocks. The cited analysis refers to this phase III as “crystalline,” in contrast to the “fluid-like” symmetric phase II [1604.07929]. ## 4. Symmetry breaking, local stability, and transition morphology The symmetry-breaking transition is conveniently described by fixing \((e,t)\), maximizing over \((a,b,d)\) at fixed \(c\), and studying the one-dimensional function \[ S(c)=\max_{a,b,d:\,e,t\ \text{fixed}} S(a,b,d,c). \] By symmetry, \(S(c)\) is even about \(c=\tfrac12\), so one expands \[ S(c)=S\!\left(\tfrac12\right)+\tfrac12S''\!\left(\tfrac12\right)\left(c-\tfrac12\right)^2+\frac1{4!}S^{(4)}\!\left(\tfrac12\right)\left(c-\tfrac12\right)^4+\cdots. \] In the symmetric phase, \(S''(\tfrac12)<0\), so \(c=\tfrac12\) is locally maximizing. Along the critical curve \(\Sigma\), one has \(S''(\tfrac12)=0\), and the numerical analysis reported for the bipodal problem finds \(S^{(4)}(\tfrac12)<0\). This yields a pitchfork bifurcation: \[ \left|c-\tfrac12\right|\approx \sqrt{-\,6\,S''(\tfrac12)\,/\,S^{(4)}(\tfrac12)}. \] Near the point \((e,t)=(\tfrac12,\tfrac18)\), the critical curve has the cubic expansion \[ t=e^3-8\left(e-\tfrac12\right)^3 + O\!\left((e-\tfrac12)^4\right), \] so \(\Sigma\) departs from the Erdős–Rényi curve with a cubic cusp [1604.07929]. Beyond the bipodal transition, the global phase diagram exhibits both continuous and discontinuous neighboring-phase boundaries. The continuous cases identified in the large-network phase analysis are \(A(n,0)\leftrightarrow B(n-1,1)\) for \(n\ge 2\), together with \(A(2,0)\leftrightarrow F(1,1)\) and \(B(1,1)\leftrightarrow F(1,1)\) along \(\tau=\epsilon^3\). All other neighboring-phase boundaries shown in the cited phase diagram are discontinuous. The proof strategy combines Euler–Lagrange equations, divergence-of-multipliers arguments ruling out spurious continuity, Hessian tests for \(A(n,0)\) stability, and a Landau–Anderson power-series analysis of the pitchfork at \(A(n,0)\leftrightarrow B(n-1,1)\) [1701.04444]. ## 5. The symmetric bipodal phase and its breakdown A later existence result establishes a genuine symmetric bipodal phase in the edge-triangle model. There is an open region \(O\) in the \((e,t)\)-plane containing the line segment \(e=\tfrac12\), \(00\) sufficiently small, the symmetric bipodal graphon is not the entropy maximizer [2211.10498]. The threshold \(e_0\) is obtained from the stability condition \[ 3\,H'''(e)^2 - H''(e)\,H^{(4)}(e)=0, \] which reduces to \[ 6e^2-6e+1=0. \] The proof of the positive result near \(e=\tfrac12\) uses a spectral decomposition \[ g(x,y)=e+\sum_{j=1}^\infty \lambda_j v_j(x)v_j(y), \] showing that the mass is concentrated in the first eigenmode, together with the pointwise Euler–Lagrange condition \[ H'(g(x,y))=\Lambda_e+\Lambda_t\int_0^1 g(x,z)g(z,y)\,dz. \] A cost-benefit expansion in the parameters \[ \alpha^2=\sum_{j\ge2}\lambda_j^2,\qquad \beta^2=\int (v_1(x)^2-1)^2,\qquad \gamma=\int v_1(x)\,dx \] then proves uniqueness of the symmetric bipodal maximizer for small perturbations of \(e=\tfrac12\). For \(e0\) [2211.10498]. ## 6. Relation to edge-independent models and triangle-density limitations The edge-triangle model imposes triangle density as a constraint and thereby encodes explicit higher-order structure. By contrast, edge-independent graph models sample each edge independently from a probability matrix \[ P=(P_{ij})_{i,j=1}^n\in [0,1]^{n\times n}, \] with \[ \Pr_{G\sim \mathcal G(P)}[G] = \prod_{(i,j)\in E(G)}P_{ij} \prod_{(i,j)\notin E(G)}(1-P_{ij}). \] This framework includes Erdős–Rényi, stochastic block models, Chung–Lu, Kronecker graphs, variational graph autoencoders, and NetGAN/CELL-style models. To exclude trivial memorization, one introduces the expected volume \[ V(P)=\sum_{i