A detailed investigation into near degenerate exponential random graphs
JJournal of Statistical Physics manuscript No. (will be inserted by the editor)
Mei Yin
A detailed investigation into neardegenerate exponential random graphs
Received: date / Accepted: date
Abstract
The exponential family of random graphs has been a topic of con-tinued research interest. Despite the relative simplicity, these models capturea variety of interesting features displayed by large-scale networks and allowus to better understand how phases transition between one another as tuningparameters vary. As the parameters cross certain lines, the model asymptot-ically transitions from a very sparse graph to a very dense graph, completelyskipping all intermediate structures. We delve deeper into this near degener-ate tendency and give an explicit characterization of the asymptotic graphstructure as a function of the parameters.
Keywords
Exponential random graphs · Phase transitions · Near degener-acy
Mathematics Subject Classification (2000) · Exponential random graphs represent an important and challenging classof models, displaying both diverse and novel phase transition phenomena.These rather general models are exponential families of probability distribu-tions over graphs, in which dependence between the random edges is definedthrough certain finite subgraphs, in imitation of the use of potential energy toprovide dependence between particle states in a grand canonical ensemble ofstatistical physics. They are particularly useful when one wants to constructmodels that resemble observed networks as closely as possible, but without
Mei Yin’s research was partially supported by NSF grant DMS-1308333.Mei YinDepartment of Mathematics, University of Denver, Denver, CO 80208, USAE-mail: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y specifying an explicit network formation mechanism. Consider the set G n ofall simple graphs G n on n vertices (“simple” means undirected, with no loopsor multiple edges). By a k -parameter family of exponential random graphswe mean a family of probability measures P βn on G n defined by, for G n ∈ G n , P βn ( G n ) = exp (cid:0) n (cid:0) β t ( H , G n ) + · · · + β k t ( H k , G n ) − ψ βn (cid:1)(cid:1) , (1)where β = ( β , . . . , β k ) are k real parameters, H , . . . , H k are pre-chosenfinite simple graphs (and we take H to be a single edge), t ( H i , G n ) is thedensity of graph homomorphisms (the probability that a random vertex map V ( H i ) → V ( G n ) is edge-preserving), t ( H i , G n ) = | hom( H i , G n ) || V ( G n ) | | V ( H i ) | , (2)and ψ βn is the normalization constant, ψ βn = 1 n log (cid:88) G n ∈G n exp (cid:0) n ( β t ( H , G n ) + · · · + β k t ( H k , G n )) (cid:1) . (3)Intuitively, we can think of the k parameters β , . . . , β k as tuning parametersthat allow one to adjust the influence of different subgraphs H , . . . , H k of G n on the probability distribution and analyze the extent to which specific valuesof these subgraph densities “interfere” with one another. Since the real-worldnetworks the exponential models depict are often very large in size, our maininterest lies in exploring the structure of a typical graph drawn from themodel when n is large.This subject has attracted enormous attention in mathematics, as wellas in various applied disciplines. Many of the investigations employ the ele-gant theory of graph limits as developed by Lov´asz and coauthors (V.T. S´os,B. Szegedy, C. Borgs, J. Chayes, K. Vesztergombi, . . . ) [8] [9] [10] [17] [18].Building on earlier work of Aldous [1] and Hoover [15], the graph limit the-ory creates a new set of tools for representing and studying the asymptoticbehavior of graphs by connecting sequences of graphs to a unified graphonspace equipped with a cut metric. Though the theory itself is tailored todense graphs, serious attempts have been made at formulating parallel re-sults for sparse graphs [2] [4] [6] [7] [12] [19]. Applying the graph limit theoryto k -parameter exponential random graphs and utilizing a large deviationsresult for Erd˝os-R´enyi graphs established in Chatterjee and Varadhan [13],Chatterjee and Diaconis [11] showed that when n is large and β , . . . , β k arenon-negative, a typical graph drawn from the “attractive” exponential model(1) looks like an Erd˝os-R´enyi graph G ( n, u ∗ ) in the graphon sense, where theedge presence probability u ∗ ( β , . . . , β k ) is picked randomly from the set ofmaximizers of a variational problem for the limiting normalization constant ψ β ∞ = lim n →∞ ψ βn : ψ β ∞ = sup ≤ u ≤ (cid:18) β u e ( H ) + · · · + β k u e ( H k ) − u log u −
12 (1 − u ) log(1 − u ) (cid:19) , (4) where e ( H i ) is the number of edges in H i . They also noted that in the edge-(multiple)-star model where H j is a j -star for j = 1 , . . . , k , due to the uniquestructure of stars, maximizers of the variational problem for ψ β ∞ for all pa-rameter values satisfy (4) and a typical graph drawn from the model is al-ways Erd˝os-R´enyi. Since the limiting normalization constant is the generatingfunction for the limiting expectations of other random variables on the graphspace such as expectations and correlations of homomorphism densities, thesecrucial observations connect the occurrence of an asymptotic phase transitionin (1) with an abrupt change in the solution of (4) and have led to furtherexploration into exponential random graph models and their variations.Being exponential families with finite support, one might expect exponen-tial random graph models to enjoy a rather basic asymptotic form, thoughin fact, virtually all these models are highly nonstandard as the size of thenetwork increases. The 2-parameter exponential random graph models havetherefore generated continued research interest. These prototype models aresimpler than their k -parameter extensions but nevertheless exhibit a wealthof non-trivial attributes and capture a variety of interesting features dis-played by large networks. The relative simplicity furthermore helps us betterunderstand how phases transition between one another as tuning parametersvary and provides insight into the expressive power of the exponential con-struction. In the statistical physics literature, phase transition is often asso-ciated with loss of analyticity in the normalization constant, which gives riseto discontinuities in the observed statistics. For exponential random graphmodels, phase transition is characterized as a sharp, unambiguous partitionof parameter ranges separating those values in which changes in parameterslead to smooth changes in the homomorphism densities, from those specialparameter values where the response in the densities is singular.For the “attractive” 2-parameter edge-triangle model obtained by taking H = K (an edge), H = K (a triangle), and β ≥
0, Chatterjee and Di-aconis [11] gave the first rigorous proof of asymptotic singular behavior andidentified a curve β = q ( β ) across which the model transitions from verysparse to very dense, completely skipping all intermediate structures. In fur-ther works (see for example, Radin and Yin [24], Aristoff and Zhu [3]), thissingular behavior was discovered universally in generic 2-parameter modelswhere H is an edge and H is any finite simple graph, and the transitioncurve β = q ( β ) asymptotically approaches the straight line β = − β asthe parameters diverge. The double asymptotic framework of [11] was laterextended in [26], and the scenario in which the parameters diverge alonggeneral straight lines β = aβ , where a is a constant and β → ∞ , wasconsidered. Consistent with the near degeneracy predictions in [3] [11] [24],asymptotically for a ≤ −
1, a typical graph sampled from the “attractive”2-parameter exponential model is sparse, while for a > −
1, a typical graphis nearly complete. Although much effort has been focused on 2-parametermodels, k -parameter models have also been examined. As shown in [25], neardegeneracy and universality are expected not only in generic 2-parametermodels but also in generic k -parameter models. Asymptotically, a typicalgraph drawn from the “attractive” k -parameter exponential model where β , . . . , β k ≥ (cid:80) ki =1 β i = 0 and nearly com- plete above it. For the edge-(multiple)-star model, the desirable star featurerelates to network expansiveness and has made predictions of similar asymp-totic phenomena possible in broader parameter regions. Related results maybe found in H¨aggstr¨om and Jonasson [14], Park and Newman [21], Bianconi[5], Lubetzky and Zhao [20], Radin and Sadun [22] [23], and Kenyon et al.[16].These theoretical findings have advanced our understanding of phase tran-sitions in exponential random graph models, yet some important questionsremain unanswered. Previous investigations identified near degenerate pa-rameter regions in which a typical sampled graph looks like an Erd˝os-R´enyigraph G ( n, u ∗ ), where the edge presence probability u ∗ → u ∗ towards these two degenerate states is not at all clear. When atypical graph is sparse ( u ∗ → u ∗ → u ∗ obtained in these theorems then makepossible a deeper exploration into the asymptotics of the limiting normaliza-tion constant of the exponential model in Theorem 3, which indicates thatthough a typical graph displays Erd˝os-R´enyi feature, the simplified Erd˝os-R´enyi graph and the real exponential graph are not exact asymptotic analogsin the usual statistical physics sense. In the sparse region, the Erd˝os-R´enyigraph does seem to reflect the asymptotic tendency of the exponential graphmore accurately, as the two interpretations of the limiting normalization con-stant coincide when the parameters diverge. Lastly, Theorems 4 and 5 furtherextend the near degenerate analysis from 2-parameter exponential randomgraph models to k -parameter exponential random graph models. This section explores the exact asymptotics of generic 2-parameter exponen-tial random graph models where β ≥ β < P βn ( G n ) = exp (cid:0) n (cid:0) β t ( H , G n ) + β t ( H , G n ) − ψ βn (cid:1)(cid:1) , (5)where H is an edge and H is a different finite simple graph, the 2-parametermodels are arguably simpler than their k -parameter generalizations. As illus-trated in Chatterjee and Diaconis [11], when n is large and β is non-negative,a typical graph drawn from the “attractive” 2-parameter exponential model(5) behaves like an Erd˝os-R´enyi graph G ( n, u ∗ ), where the edge presenceprobability u ∗ ( β , β ) is picked randomly from the set of maximizers of a β β u ∗ ( β , β ) e β − e − β +3 β ) − . . − . . . . . . Table 1
Asymptotic comparison for “attractive” edge-triangle model near degen-eracy. variational problem for the limiting normalization constant ψ β ∞ : ψ β ∞ = sup ≤ u ≤ (cid:18) β u + β u p − u log u −
12 (1 − u ) log(1 − u ) (cid:19) , (6)where p is the number of edges in H , and thus satisfies β + β p ( u ∗ ) p − = 12 log (cid:18) u ∗ − u ∗ (cid:19) . (7)This implicitly describes u ∗ as a function of the parameters β and β , buta closed-form solution is not obtainable except when β = 0, which gives u ∗ ( β ,
0) = e β / (1 + e β ). For β large negative, u ∗ ( β ,
0) asymptoticallybehaves like e β , while for β large positive, u ∗ ( β ,
0) asymptotically behaveslike 1 − e − β . We would like to know if these asymptotic results could begeneralized. By [26], taking β = aβ and β → ∞ , u ∗ → a ≤ − u ∗ → a > −
1. Equivalently, for ( β , β ) sufficiently far away from theorigin, u ∗ → β ≤ − β and u ∗ → β > − β . As regards thespeed of u ∗ towards these two degenerate states, simulation results suggestthat u ∗ is asymptotically e β in the former case and is asymptotically 1 − e − β + pβ ) in the latter case. See Tables 1 and 2 and Figure 1. Even for β with small magnitude, the asymptotic tendency of u ∗ is quite evident. Theorem 1
Consider an “attractive” -parameter exponential random graphmodel (5) where β ≥ . For large n and ( β , β ) sufficiently far away fromthe origin, a typical graph drawn from the model looks like an Erd˝os-R´enyigraph G ( n, u ∗ ) , where the edge presence probability u ∗ ( β , β ) satisfies: – u ∗ (cid:16) e β if β ≤ − β , – u ∗ (cid:16) − e − β + pβ ) if β > − β .Proof As explained in the previous paragraph, in the large n limit, the asymp-totic edge presence probability u ∗ ( β , β ) of a typical sampled graph is pre-scribed according to the maximization problem (7). For ( β , β ) whose mag-nitude is sufficiently big, u ∗ → β ≤ − β and u ∗ → β > − β .For β ≤ − β , we rewrite (7) in the following way: u ∗ e β +2 β p ( u ∗ ) p − = 1 − u ∗ . (8) =0 =1n=20 Fig. 1
A simulated realization of the exponential random graph model on 20 nodeswith edges and triangles as sufficient statistics, where β = 0 and β = 1. Thesimulated graph displays Erd˝os-R´enyi feature with edge density 0 . u ∗ ( β , β ) = 0 . − e − β +3 β ) = 0 . Take 0 < (cid:15) < /p , since u ∗ →
0, for ( β , β ) sufficiently far away from theorigin, ( u ∗ ) p − < (cid:15) . Using 1 − u ∗ ≤
1, we then have u ∗ ≤ e β +2 β p(cid:15) ≤ e β ( − p(cid:15) ) . (9)This implies that 2 β p ( u ∗ ) p − ≤ β pe β ( p − − p(cid:15) ) → β gets sufficiently large. Using 1 − u ∗ → u ∗ asymptotically behaves like e β .For β > − β , we rewrite (7) in the following way: v ∗ e − β − β p (1 − v ∗ ) p − = 1 − v ∗ , (11)where v ∗ = 1 − u ∗ →
0. Going one step further, we separate 1 − ( p − v ∗ from (1 − v ∗ ) p − : v ∗ e − β − β p +2 β p ( p − v ∗ = (1 − v ∗ ) e − β p (cid:80) p − s =2 ( p − s ) ( − v ∗ ) s ≤ − v ∗ , (12)as the dominating term in the exponent − β p (cid:0) p − (cid:1) ( − v ∗ ) carries a negativesign. Take 0 < (cid:15) < /p , since v ∗ →
0, for ( β , β ) sufficiently far away fromthe origin, v ∗ < (cid:15) . Using 1 − v ∗ ≤
1, we then have v ∗ ≤ e − β − β p +2 β p ( p − (cid:15) ≤ e β (1 − p + p ( p − (cid:15) ) = e β ( p − − p(cid:15) ) . (13) -3 -2 -1 0 1 2 30123 Region I: nearly complete graphsu * ~ 1-e -2( +3 ) ~ + u * ~ e ~ e /2Region II: sparse graphs Fig. 2
Asymptotic tendency in “attractive” edge-triangle model.
This implies that2 β p ( p − v ∗ ≤ β p ( p − e β ( p − − p(cid:15) ) → β gets sufficiently large, and since v ∗ →
0, also implies that the sum of allthe terms in the exponent − β p (cid:80) p − s =2 (cid:0) p − s (cid:1) ( − v ∗ ) s →
0. Using 1 − v ∗ → v ∗ asymptotically behaves like e − β + pβ ) , orequivalently, u ∗ asymptotically behaves like 1 − e − β + pβ ) .In the edge-(single)-star model where H is a star with p edges, due tothe unique structure of stars, maximizers of the variational problem for thelimiting normalization constant ψ β ∞ when the parameter β < n and ( β , β ) sufficiently far away from the origin, a typical graphdrawn from the “repulsive” edge-(single)-star model where β < G ( n, u ∗ ), where the edge presenceprobability u ∗ → β < u ∗ → β > − pβ . As regards thespeed of u ∗ towards these two degenerate states, simulation results suggestthat just as in the “attractive” situation, u ∗ is asymptotically e β in thesparse case and is asymptotically 1 − e − β + pβ ) in the nearly complete case.See Table 2. Even for β with small magnitude, the asymptotic tendency of u ∗ is quite evident. Theorem 2
Consider a “repulsive” edge-(single)-star model obtained by tak-ing H a star with p edges and β < in (5). For large n and ( β , β ) β β u ∗ ( β , β ) e β − e − β +2 β ) − − . − − . . − . . − . . . . − . . Table 2
Asymptotic comparison for edge-2-star model near degeneracy. sufficiently far away from the origin, a typical graph drawn from the modellooks like an Erd˝os-R´enyi graph G ( n, u ∗ ) , where the edge presence probability u ∗ ( β , β ) satisfies: – u ∗ (cid:16) e β if β < , – u ∗ (cid:16) − e − β + pβ ) if β > − pβ .Proof For ( β , β ) whose magnitude is sufficiently big, we examine the max-imization problem (7) separately when β < β > − pβ .First for β <
0. Assume that β ≤ δ β for some fixed but arbitrary δ >
0. We rewrite (7) in the following way: u ∗ e β = (1 − u ∗ ) e β p ( u ∗ ) p − ≤ − u ∗ . (15)Using 1 − u ∗ ≤
1, we then have u ∗ ≤ e β ≤ e δ β . (16)This implies that 2 | β | p ( u ∗ ) p − ≤ | β | pe δ β ( p − → β gets sufficiently negative. Using 1 − u ∗ → u ∗ asymptotically behaves like e β .Next for β > − pβ . Assume that β ≥ − ( p + δ ) β for some fixed butarbitrary δ >
0. We rewrite (7) in the following way: v ∗ e − β − β p (1 − v ∗ ) p − = 1 − v ∗ , (18)where v ∗ = 1 − u ∗ →
0. Going one step further, we separate 1 from (1 − v ∗ ) p − : v ∗ e − β − β p = (1 − v ∗ ) e − β p (cid:80) p − s =1 ( p − s ) ( − v ∗ ) s ≤ − v ∗ , (19)as the dominating term in the exponent 2 β p ( p − v ∗ carries a negative sign.Using 1 − v ∗ ≤
1, we then have v ∗ ≤ e − β − β p ≤ e δ β . (20) This implies that2 | β | p ( p − v ∗ ≤ | β | p ( p − e δ β → β gets sufficiently negative, and since v ∗ →
0, also implies that the sum ofall the terms in the exponent − β p (cid:80) p − s =1 (cid:0) p − s (cid:1) ( − v ∗ ) s →
0. Using 1 − v ∗ → v ∗ asymptotically behaves like e − β + pβ ) , orequivalently, u ∗ asymptotically behaves like 1 − e − β + pβ ) .Though the 2-parameter exponential random graph G n looks like anErd˝os-R´enyi random graph G ( n, u ∗ ) in the large n limit, we also note somemarked dissimilarities. The limiting normalization constant ψ β ∞ for the 2-parameter exponential model (5) is given by (6), while the “equivalent”Erd˝os-R´enyi model yields that ψ β ∞ is − log(1 − u ∗ ) /
2. Since u ∗ is nonzerofor finite ( β , β ) [11], the two different interpretations of the limiting nor-malization constant indicate that the simplified Erd˝os-R´enyi graph and thereal exponential model are not exact asymptotic analogs in the usual statis-tical physics sense. When the relevant Erd˝os-R´enyi graph is near degenerate,Theorems 1 and 2 give the asymptotic speed of u ∗ as a function of β and β , allowing a deeper analysis of the asymptotics of ψ β ∞ in the following The-orem 3. The theorem is formulated in the context of the edge-(single)-starmodel, since the asymptotics of u ∗ are known in broader parameter regionsfor this model, but the statement for the “attractive” situation ( β ≥ Theorem 3
Consider an edge-(single)-star model obtained by taking H astar with p edges in (5). For ( β , β ) sufficiently far away from the origin,the limiting normalization constant ψ β ∞ satisfies: – ψ β ∞ (cid:16) e β / if β ≤ − β and β < , – ψ β ∞ (cid:16) β + β if β > − β and β > − pβ .Proof For ( β , β ) whose magnitude is sufficiently big, we examine the lim-iting normalization constant (6) separately in the sparse region and in thenearly complete region.In the sparse region ( β ≤ − β and β < ψ β ∞ = β u ∗ + β ( u ∗ ) p − u ∗ log u ∗ −
12 (1 − u ∗ ) log(1 − u ∗ ) (22)= β ( u ∗ ) p (1 − p ) −
12 log(1 − u ∗ ) . From Theorems 1 and 2, β ( u ∗ ) p − → − log(1 − u ∗ ) (cid:16) u ∗ (cid:16) e β . Thisshows that ψ β ∞ asymptotically behaves like e β / -3 -2 -1 0 1 2 3-3-2-10123 Region I: nearly complete graphsRegion III: non-trivial graphsu * ~ 1-e -2( +2 ) ~ + u * ~ e ~ e /2Region II: sparse graphs Fig. 3
Asymptotic tendency in edge-2-star model.
In the nearly complete region ( β > − β and β > − pβ ), ψ β ∞ = β (1 − v ∗ ) + β (1 − v ∗ ) p − v ∗ log v ∗ −
12 (1 − v ∗ ) log(1 − v ∗ )(23)= β (1 − v ∗ ) p (1 − p ) −
12 log v ∗ . From Theorems 1 and 2, β v ∗ → v ∗ (cid:16) e − β + pβ ) . This shows that ψ β ∞ asymptotically behaves like β + β .We may also analyze the asymptotics of ψ β ∞ along the boundaries of thenear degenerate region. The boundary of the sparse region is given by β = 0and β <
0, and u ∗ satisfies β p ( u ∗ ) p − = 12 log (cid:18) u ∗ − u ∗ (cid:19) . (24)Though u ∗ → β in a complicated way, the asymptotic behaviorof ψ β ∞ can be characterized: ψ β ∞ = 1 − p p u ∗ log u ∗ − u ∗ −
12 log(1 − u ∗ ) . (25)Using log(1 − u ∗ ) (cid:16) − u ∗ , this shows that ψ β ∞ asymptotically behaves like(1 − p ) u ∗ log u ∗ / (2 p ). We recognize that the asymptotic behaviors of ψ β ∞ onthe boundary of and inside the sparse region are different: Inside, ψ β ∞ is asymptotically u ∗ / ψ β ∞ thoughalso converges to 0 is at a much slower rate.The boundary of the nearly complete region is given by β = − pβ and β <
0, and u ∗ satisfies − β p + β p (1 − v ∗ ) p − = −
12 log (cid:18) v ∗ − v ∗ (cid:19) . (26)Though v ∗ = 1 − u ∗ → β in a complicated way, the asymptoticbehavior of ψ β ∞ can be characterized: ψ β ∞ = β (1 − p )(1 − v ∗ ) − − p p (1 − v ∗ ) log (cid:18) v ∗ − v ∗ (cid:19) −
12 log v ∗ . (27)Since the dominating term on the left of (26) is − β p ( p − v ∗ , using log(1 − v ∗ ) →
0, we then have β p ( p − v ∗ (cid:16) log v ∗ /
2, which shows that | β | isasymptotically larger compared with | log v ∗ | and further shows that ψ β ∞ asymptotically behaves like β (1 − p ). We recognize that the asymptoticbehaviors of ψ β ∞ on the boundary of and inside the nearly complete regioncoincide. This section extends the investigation into near degeneracy from generic 2-parameter exponential random graph models to generic k -parameter expo-nential random graph models. For “attractive” models where β , . . . , β k ≥ β , . . . , β k < n is large and β , . . . , β k are non-negative, a typical graph drawn from the k -parameter exponentialmodel behaves like an Erd˝os-R´enyi graph G ( n, u ∗ ), where the edge presenceprobability u ∗ ( β , . . . , β k ) is picked randomly from the set of maximizers of(4), and thus satisfies β + β e ( H )( u ∗ ) e ( H ) − + · · · + β k e ( H k )( u ∗ ) e ( H k ) − = 12 log (cid:18) u ∗ − u ∗ (cid:19) , (28)where e ( H i ) is the number of edges in H i . We take H to be an edge andassume without loss of generality that 1 = e ( H ) ≤ · · · ≤ e ( H k ). Theorem 4
Consider an “attractive” k -parameter exponential random graphmodel (1) where β , . . . , β k ≥ . For large n and ( β , . . . , β k ) sufficiently faraway from the origin, a typical graph drawn from the model looks like anErd˝os-R´enyi graph G ( n, u ∗ ) , where the edge presence probability u ∗ ( β , . . . , β k ) satisfies: – u ∗ (cid:16) e β if (cid:80) ki =1 β i ≤ , – u ∗ (cid:16) − e (cid:80) ki =1 − β i e ( H i ) if (cid:80) ki =1 β i > . Proof
The proof follows a similar line of reasoning as in the proof of Theorem1. Expectedly though, the argument is more involved since we are workingwith k -parameter families rather than 2-parameter families.For (cid:80) ki =1 β i ≤
0, we rewrite (28) in the following way: u ∗ e β + (cid:80) ki =2 β i e ( H i )( u ∗ ) e ( Hi ) − = 1 − u ∗ . (29)Take 0 < (cid:15) < /e ( H k ), since u ∗ →
0, for ( β , . . . , β k ) sufficiently far awayfrom the origin, ( u ∗ ) e ( H i ) − < (cid:15) for 2 ≤ i ≤ k . Using 1 − u ∗ ≤
1, we thenhave u ∗ ≤ e β + (cid:80) ki =2 β i e ( H i ) (cid:15) ≤ e (cid:80) ki =2 β i ( − e ( H i ) (cid:15) ) . (30)This implies that2 β j e ( H j )( u ∗ ) e ( H j ) − ≤ β j e ( H j ) e ( e ( H j ) − (cid:80) ki =2 β i ( − e ( H i ) (cid:15) ) → ≤ j ≤ k as β , . . . , β k get sufficiently large. Using 1 − u ∗ → u ∗ asymptotically behaves like e β .For (cid:80) ki =1 β i >
0, we rewrite (28) in the following way: v ∗ e − β − (cid:80) ki =2 β i e ( H i )(1 − v ∗ ) e ( Hi ) − = 1 − v ∗ , (32)where v ∗ = 1 − u ∗ →
0. Going one step further, for 2 ≤ i ≤ k , we separate1 − ( e ( H i ) − v ∗ from (1 − v ∗ ) e ( H i ) − : v ∗ e (cid:80) ki =1 − β i e ( H i )+ (cid:80) ki =2 β i e ( H i )( e ( H i ) − v ∗ = (1 − v ∗ ) e − (cid:80) ki =2 β i e ( H i ) (cid:80) e ( Hi ) − s =2 ( e ( Hi ) − s ) ( − v ∗ ) s ≤ − v ∗ , (33)as the dominating term in the exponent − (cid:80) ki =2 β i e ( H i ) (cid:0) e ( H i ) − (cid:1) ( − v ∗ ) carries a negative sign. Take 0 < (cid:15) < /e ( H k ), since v ∗ →
0, for ( β , . . . , β k )sufficiently far away from the origin, v ∗ < (cid:15) . Using 1 − v ∗ ≤
1, we then have v ∗ ≤ e (cid:80) ki =1 − β i e ( H i )+ (cid:80) ki =2 β i e ( H i )( e ( H i ) − (cid:15) ≤ e (cid:80) ki =2 β i (1 − e ( H i )+ e ( H i )( e ( H i ) − (cid:15) ) = e (cid:80) ki =2 β i ( e ( H i ) − − e ( H i ) (cid:15) ) . (34)This implies that2 β j e ( H j )( e ( H j ) − v ∗ ≤ β j e ( H j )( e ( H j ) − e (cid:80) ki =2 β i ( e ( H i ) − − e ( H i ) (cid:15) ) → ≤ j ≤ k as β , . . . , β k get sufficiently large, and since v ∗ →
0, alsoimplies that the sum of all the terms in the exponent − (cid:80) ki =2 β i e ( H i ) (cid:80) e ( H i ) − s =2 (cid:0) e ( H i ) − s (cid:1) ( − v ∗ ) s →
0. Using 1 − v ∗ → v ∗ asymptotically behaves like e (cid:80) ki =1 − β i e ( H i ) , orequivalently, u ∗ asymptotically behaves like 1 − e (cid:80) ki =1 − β i e ( H i ) . Theorem 5
Consider an “attractive” k -parameter exponential random graphmodel (1) where β , . . . , β k ≥ . For ( β , . . . , β k ) sufficiently far away fromthe origin, the limiting normalization constant ψ β ∞ satisfies: – ψ β ∞ (cid:16) e β / if (cid:80) ki =1 β i ≤ , – ψ β ∞ (cid:16) (cid:80) ki =1 β i if (cid:80) ki =1 β i > .Proof For ( β , . . . , β k ) whose magnitude is sufficiently big, we examine thelimiting normalization constant (4) separately in the sparse region and in thenearly complete region.In the sparse region ( (cid:80) ki =1 β i ≤ ψ β ∞ = k (cid:88) i =1 β i ( u ∗ ) e ( H i ) − u ∗ log u ∗ −
12 (1 − u ∗ ) log(1 − u ∗ ) (36)= k (cid:88) i =2 β i ( u ∗ ) e ( H i ) (1 − e ( H i )) −
12 log(1 − u ∗ ) . From Theorem 4, β i ( u ∗ ) e ( H i ) − → ≤ i ≤ k and − log(1 − u ∗ ) (cid:16) u ∗ (cid:16) e β . This shows that ψ β ∞ asymptotically behaves like e β / (cid:80) ki =1 β i > ψ β ∞ = k (cid:88) i =1 β i (1 − v ∗ ) e ( H i ) − v ∗ log v ∗ −
12 (1 − v ∗ ) log(1 − v ∗ ) (37)= k (cid:88) i =2 β i (1 − v ∗ ) e ( H i ) (1 − e ( H i )) −
12 log v ∗ . From Theorem 4, β i v ∗ → ≤ i ≤ k and v ∗ (cid:16) e (cid:80) ki =1 − β i e ( H i ) . Thisshows that ψ β ∞ asymptotically behaves like (cid:80) ki =1 β i .In the edge-(multiple)-star model, due to the unique structure of stars,maximizers of the variational problem for the limiting normalization constant ψ β ∞ satisfies (28) for any β , . . . , β k , and the near degeneracy predictionsmay be extended to the “repulsive” region. Using similar techniques as in[27], it may be shown that for ( β , . . . , β k ) sufficiently far away from theorigin and β , . . . , β k all negative, u ∗ → β < u ∗ → (cid:80) ki =1 β i e ( H i ) >
0. Then analogous conclusions as in Theorems 4 and 5 maybe drawn: – u ∗ (cid:16) e β and ψ β ∞ (cid:16) e β / (cid:80) ki =1 β i ≤ β < – u ∗ (cid:16) − e (cid:80) ki =1 − β i e ( H i ) and ψ β ∞ (cid:16) (cid:80) ki =1 β i if (cid:80) ki =1 β i > (cid:80) ki =1 β i e ( H i ) > Acknowledgements
The author is very grateful to the anonymous referees for the invaluablesuggestions that greatly improved the quality of this paper. She appreciatedthe opportunity to talk about this work in the Special Session on Topics inProbability at the 2016 AMS Western Spring Sectional Meeting, organizedby Tom Alberts and Arjun Krishnan.
References
1. Aldous, D.: Representations for partially exchangeable arrays of random vari-ables. J. Multivariate Anal. 11, 581-598 (1981)2. Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J.Probab. 12, 1454-1508 (2007)3. Aristoff, D., Zhu, L.: On the phase transition curve in a directed exponentialrandom graph model. arXiv: 1404.6514 (2014)4. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planargraphs. Electron. J. Probab. 6, 1-13 (2001)5. Bianconi, G.: Statistical mechanics of multiplex networks: Entropy and overlap.Phys. Rev. E 87, 062806 (2013)6. Borgs, C., Chayes, J., Cohn, H., Zhao, Y.: An L p theory of sparse graph con-vergence I. Limits, sparse random graph models, and power law distributions.arXiv: 1401.2906 (2014)7. Borgs, C., Chayes, J., Cohn, H., Zhao, Y.: An L pp