Random growth on a Ramanujan graph
RRANDOM GROWTH ON A RAMANUJAN GRAPH
JANKO BÖHM, MICHAEL JOSWIG, LARS KASTNER, AND ANDREW NEWMAN
Abstract.
The behavior of a certain random growth process is analyzed on arbitraryregular and non-regular graphs. Our argument is based on the Expander MixingLemma, which entails that the results are strongest for Ramanujan graphs, whichasymptotically maximize the spectral gap. Further, we consider Erdős–Rényi randomgraphs and compare our theoretical results with computational experiments on flipgraphs of point configurations. The latter is relevant for enumerating triangulations. Introduction
Many collections of mathematical objects can be equipped with a graph structure, andso their enumeration can be considered as visiting all nodes of such a graph. Standardmethods for visiting all nodes in a graph, say G = ( V, E ) , include the depth-first searchand breadth-first search algorithms. The common theme is that the set of visited nodesstrictly grows which each step, until it covers the entire node set V . This naturally leadsto the following vast generalization. We call any increasing sequence P ⊆ P ⊆ · · · ⊆ V of node sets a growth process on G , provided that (cid:83) i P i = V . Our main tool is acertain random growth process, Algorithm A, which maintains a queue in addition tothe randomly constructed set P t of nodes which have been processed at time t . Thiswill be employed to estimate the number of nodes of a finite graph, which is given onlyimplicitly. Similar in spirit is a random sampling algorithm for estimating the size of atree by Hall and Knuth [13, 21].Naturally, estimating the size of a graph via a random process can only yield mean-ingful results if that graph is somehow “well connected”. While there are several notionsmeasuring this, here we we settle for expansion expressed in terms of spectral propertiesof the adjacency matrix; cf. [3, 16]. For a d -regular graph the largest eigenvalue is d , andthe second largest (in absolute value), denoted λ , defines the spectral gap d − λ . Ourfirst main result, Theorem 5, is a lower bound on the expected size of the queue during arandomized breadth-first search through an arbitrary regular graph. That lower bound Mathematics Subject Classification.
Key words and phrases. graph expansion; spectral gap; flip-graphs of triangulations.Research by J. Böhm on this work has been supported by Project II.5 of SFB-TRR 195: “SymbolicTools in Mathematics and their Application” of Deutsche Forschungsgemeinschaft.Research by M. Joswig is supported by Deutsche Forschungsgemeinschaft (EXC 2046: “MATH + ”,SFB-TRR 109: “Discretization in Geometry and Dynamics”, SFB-TRR 195: “Symbolic Tools in Mathe-matics and their Application”, and GRK 2434: “Facets of Complexity”).Research by L. Kastner is supported by Deutsche Forschungsgemeinschaft (SFB-TRR 195: “SymbolicTools in Mathematics and their Application”).Research by A. Newman is supported by Deutsche Forschungsgemeinschaft Graduiertenkolleg “Facetsof Complexity" (GRK 2434). a r X i v : . [ m a t h . C O ] O c t JANKO BÖHM, MICHAEL JOSWIG, LARS KASTNER, AND ANDREW NEWMAN becomes tighter the larger the spectral gap is, and the asymptotic optimum is markedby the regular Ramanujan graphs. The proof of Theorem 5 is based on the ExpanderMixing Lemma. In Theorem 6 we use that result to also derive upper and lower boundsfor the number of nodes of a regular graph from data obtained through randomizedbreadth-first search. We experimentally check our estimates on a regular Ramanujangraph constructed by Lubotzky, Phillips and Sarnak [23].Our motivation stems from enumerating triangulations of point sets in Euclidean space.Standard methods are based on traversing the flip graph of the point configuration,which has a node for each triangulation, and the edges correspond to local modificationsknown as flips (or Pachner moves). To estimate the number of triangulations from theinput alone is next to impossible; see [10, §8.4] for what is known. We explore howour Theorems 5 and 6 can help. Again we do experiments, this time on the flip graphsof convex polygons, a topic intimately linked to the combinatorics of Catalan numbers.These particular flip graphs are regular, but the exact spectral expansion of these graphsseems to be unknown. An explicit computation reveals that the flip graph of a k -gon isa regular Ramanujan graph for k ≤ , whereas the spectral gap seems to vanish prettyquickly for higher values of k .Most flip graphs are neither regular, nor do they appear to exhibit good expansion.We address both issues separately. First, we need to generalize our results to the non-regular setting. We follow the standard approach to study the spectral expansion ofnon-regular graphs via the normalized adjacency matrix; cf. [9]. Our main contributionhere is a specific non-regular version of the Expander Mixing Lemma, which seems to benew. This allows to interpret Theorems 5 and 6 also in the non-regular setting. Second,we study the expected asymptotic behavior of Algorithm A on random graphs in theErdős–Rényi model, both theoretically and experimentally.The the paper is closed with yet one more experiment, on the quotient flip graph of theregular 4-cube. The “quotient” comes from looking at triangulations modulo the naturalsymmetry of the cube. The estimates resulting from Theorem 6 work surprisingly well,although the assumptions made in that result are not met by the quotient flip graphof the 4-cube. Finally, again for the quotient flip graph of the 4-cube, we compare ourestimates with those obtained from the Hall–Knuth sampling procedure [13, 21].2. Expansion of regular graphs and a random growth process
Here we describe a specific random growth process on arbitrary graphs which weultimately want to study in the context of enumerating triangulations. Yet we first setout to examine this random process in a much more idealized setting. That is, we wouldlike to understand its expected behavior on an expander graph .Expander graphs have been studied quite extensively, and we refer the reader to thesurvey of Hoory, Linial, and Wigderson [16] or to [3, Section 9.2] for an overview of manyof their definitions and properties. Roughly speaking, for our purposes, an expandergraph is a sparse graph on which a random walk mixes quickly. To make this precise inour setting, we will introduce Ramanujan graphs.Let G = ( V, E ) be an undirected graph with adjacency matrix A . The eigenvalues of G are defined to be the eigenvalues λ , . . . , λ n of A . Since A is a symmetric matrixits eigenvalues are real, so we may assume that λ ≥ λ ≥ · · · ≥ λ n . Moreover, if G ANDOM GROWTH ON A RAMANUJAN GRAPH 3 is d -regular then λ = d , and for all i we have | λ i | ≤ d . Let λ ( G ) = max {| λ | , | λ n |} =max {| λ i | | i = 2 , . . . , n } . A Ramanujan graph G is d -regular graph with λ ( G ) ≤ √ d − .The reader familiar with expander graphs will not find this topic of the spectrum ofa graph surprising at all. On the other hand, to those unfamiliar with expander graphsthis definition of Ramanujan graphs may seem strange in light of the rough description ofexpander graphs in terms of random walks. The well-known Expander Mixing Lemma,which likely first appeared in [2] provides one (of many) connections between these twonotions. Moreover, the proof of our results will make extensive use of this lemma.The results we state here for G a d -regular graph will be expressed in terms of λ , whichwe take as a shorthand for λ ( G ) . Our results will be nontrivial whenever λ < d . It isstandard that this occurs exactly when G is connected and not bipartite. However, ourresults yield the strongest consequences when d − λ is as large as possible. That is inthe case of Ramanujan graphs by a result of Alon [1, 26], where the threshold √ d − is explained as follows; see also Bilu and Linial [6]. Theorem 1 (Alon–Boppana Theorem [1]) . For every d ≥ and every (cid:15) > , there areonly finitely many d -regular graphs G with λ ≤ √ d − − (cid:15) . Before presenting our results, we give a precise statement of the Expander MixingLemma. We first define the notation e ( S, T ) for a graph G = ( V, E ) and S, T ⊆ V tobe the number of edges ( s, t ) in E where s is in S and t is in T . In particular, e ( S, T ) counts edges with both endpoints in S ∩ T twice. Theorem 2 (Expander Mixing Lemma) . Let G = ( V, E ) be a d -regular graph on n vertices with λ < d . Then for any subsets S and T of V we have (cid:12)(cid:12)(cid:12)(cid:12) e ( S, T ) − d | S | | T | n (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ · (cid:115) | S | | T | (cid:18) − | S | n (cid:19) (cid:18) − | T | n (cid:19) . Here we are interested in the expected behavior of a randomized breadth-first searchon a regular graph G . Toward stating and proving our results we recall the randomizedbreadth-first search algorithm and introduce the notation that we use.Our random growth process works on a connected graph G as its input, and throughoutthe algorithm we keep track of three sets of vertices which always partition the verticesof G . The sets are denoted by P , Q , and U and respectively refer to the queued vertices,processed vertices, and unvisited vertices. Algorithm A
Random Growth
Require:
Some vertex v of the connected graph G Ensure:
Visit all vertices in G ; i.e., the final state is P = V , Q = ∅ , U = ∅ P ← ∅ ; Q ← { v } ; U ← V \ { v } while Q (cid:54) = ∅ do pick a vertex v uniformly at random from QN ← { w : w neighbor of v in U } P ← P ∪ { v } ; Q ← ( Q \ { v } ) ∪ N ; U ← U \ N end while JANKO BÖHM, MICHAEL JOSWIG, LARS KASTNER, AND ANDREW NEWMAN
Occasionally we will use the variable t ∈ { , , ..., n } to refer to an arbitrary step ofthe process and when convenient, P t , Q t , and U t refer to the sets P , Q , and U , at step t ,with | P | = 0 , | Q | = 1 and | U | = n − . Note that | P t | = t for all t ∈ { , , ..., n } , andwe denote by v t the unique vertex in P t \ P t − . The following is immediate. Lemma 3.
For each t ≥ let p ( t ) be the minimal index in { , , . . . , t − } such that v t is a neighbor of v p ( t ) . Then the edges { , p (2) } , { , p (3) } , . . . , { n, p ( n ) } form a spanningtree of G with root vertex v . We seek bounds on the expected size of Q t as t varies from 0 to n . As our results holdfor large n , it makes sense to consider the densities of P , Q , and U rather than theirsizes. Let(1) π := | P | /n , κ := | Q | /n , υ := | U | /n . Thus, rather than fixing n and deriving upper and lower bound for Q = Q t as t variesfrom 0 to n , instead we derive upper and lower bounds for κ ∈ [0 , in terms of π ∈ [0 , .The first result is a structural lower bound on κ in terms of π that derives from theExpander Mixing Lemma and the fact that there are never any edges from P to U . Notethat this structural lower bound holds regardless of how we pick the next vertex in Q . Proposition 4.
Let G , d , n , and λ < d be as in the statement of the Expander MixingLemma. Further, let t = πn be the number of steps completed by any random growthprocess on G . Then the density κ of the queue Q at step t satisfies κ ≥ − π − λ (1 − π ) d π + λ (1 − π ) . Proof.
Let P = P t , Q = Q t , and U = U t , and their densities π , κ and υ be as describedabove with | P t | = t = πn . Due to π + κ + υ = 1 , an upper bound on υ implies a lowerbound on π + κ . Thus we will upper bound υ in terms of π .Observe that e ( P, U ) = 0 . Indeed, during the entire process a vertex is only movedfrom Q to P once all of its neighbors have been found and added to Q . On the otherhand by the Expander Mixing Lemma we obtain e ( P, U ) ≥ d π n υ nn − λ (cid:112) π n υ n (1 − π ) (1 − υ ) . Thus we have ≥ d π υ − λ (cid:112) π (1 − π ) υ (1 − υ ) . Since π, υ ∈ [0 , , it follows that υ is bounded as υ ≤ λ (1 − π ) d π + λ (1 − π ) . The claim follows since κ = 1 − π − υ . (cid:3) While Proposition 4 is a deterministic statement that does not require any assumptionon the order in which the vertices are processed, we obtain the following improved lowerbound by introducing randomness.
ANDOM GROWTH ON A RAMANUJAN GRAPH 5
Theorem 5.
Let G , d , n , and λ < d be as in the Expander Mixing Lemma. Further, let t = πn be the number of steps completed by the random growth on G . Then the expecteddensity κ of the queue Q at step t satisfies E ( κ ) ≥ − π − exp (cid:18) − ( d − λ ) (cid:18) d − (cid:19) π (cid:19) . Proof.
As in the proof of Proposition 4, instead of κ we will consider the density υ ofunvisited vertices at time t . With P t , Q t , U t , π , κ , υ as before we derive upper and lowerbounds on υ as a function of π . At any step t we have by the Expander Mixing Lemmathat the number of edges between Q t ∪ P t and U t satisfies e ( P t ∪ Q t , U t ) ≥ d | P t ∪ Q t | | U t | n − λ · (cid:115) | P t ∪ Q t | | U t | (cid:18) − | P t ∪ Q t | n (cid:19) (cid:18) − | U t | n (cid:19) . As there are no edges between P t and U t , this also serves as a lower bound for e ( Q t , U t ) .At step t , the expected number of vertices added to the queue at the next step equals(2) e ( Q t , U t ) / | Q t | . This expectation has the following lower bound e ( Q t , U t ) | Q t | ≥ d | P t ∪ Q t | | U t || Q t | n − λn | Q t | (cid:112) | P t ∪ Q t | | U t | ( n − | P t ∪ Q t | ) ( n − | U t | )= d − λn | U t | (cid:18) | P t || Q t | (cid:19) . Now | Q t | clearly satisfies the upper bound | Q t | ≤ ( d − t = ( d − | P t | , thus we have e ( Q t , U t ) | Q t | ≥ d − λn | U t | (cid:18) d − (cid:19) . Therefore, the expected number of vertices removed from U t at step t + 1 given the size of U t is at least d − λn | U t | (cid:16) d − (cid:17) . Thus, we may bound E ( | U t +1 | ) from above as follows: E ( | U t +1 | ) = n (cid:88) u =0 E ( | U t +1 | | | U t | = u ) Pr( | U t | = u ) ≤ (cid:18) − d − λn (cid:18) d − (cid:19)(cid:19) n (cid:88) u =0 u Pr( | U t | = u )= (cid:18) − d − λn (cid:18) d − (cid:19)(cid:19) E ( | U t | ) . Now as U is always the full graph minus a single vertex we have that E ( | U t | ) ≤ (cid:18) − d − λn (cid:18) d − (cid:19)(cid:19) t ( n − ≤ exp (cid:18) − ( d − λ ) (cid:18) d − (cid:19) tn (cid:19) ( n − . JANKO BÖHM, MICHAEL JOSWIG, LARS KASTNER, AND ANDREW NEWMAN . . . . . . . . . . . . . density π of processed nodes d e n s i t y κ o f q u e u e Figure 1.
Lower bound on the expected queue size from Theorem 5(blue) and a comparison to experimental data (red) for
LPS(13 , .Thus we get E ( κ ) ≥ − π − exp (cid:18) − ( d − λ ) (cid:18) d − (cid:19) π (cid:19) . (cid:3) If one wants to use our random growth process to estimate the size of a graph, theExpander Mixing Lemma may in some sense be reversed. We make this formal as follows.As in the Expander Mixing Lemma and in Theorem 5, the strongest results follow whenthe spectral gap d − λ is as large as possible, that is, for a Ramanujan graph. Theorem 6.
Let G , d , n , and λ < d be as in the Expander Mixing Lemma. For a givenstep t in the random growth process, let P denote the vertices that have been processed, Q denote the queue, W denote the visited vertices (that is, W = P (cid:116) Q ) and U denotethe unvisited vertices. Then n satisfies ( d − λ ) · | W | ( d − λ ) · | W | − e ( U, W ) ≥ n ≥ ( d + λ ) · | W | ( d + λ ) · | W | − e ( U, W ) . Proof.
Recall from (1) that υ is the density of U . The Expander Mixing Lemma directlyimplies the following upper bound and lower bounds on υ : ( d + λ ) · | U | · | W | n | Q | ≥ e ( U, W ) | Q | ≥ ( d − λ ) · | U | · | W | n | Q | . Therefore, e ( U, W )( d + λ ) · | W | ≤ υ ≤ e ( U, W )( d − λ ) · | W | . Now from the definition of υ , we have | W | = | P | + | Q | = n − υn . This yields n = | W | / (1 − υ ) , and the bounds in the statement follow from the bounds on υ . (cid:3) In our applications discussed in later sections often we will not know the value of λ and our results described above will serve as heuristics to model what we see in our ANDOM GROWTH ON A RAMANUJAN GRAPH 7 . . . . . . . . . . .
52 density π of processed nodes d e n s i t y o f t o t a l n o d e s Figure 2.
Upper and lower estimates on the total number of vertices(normalized by the actual number of vertices) of
LPS(13 , .experiments. However, before moving on to such applications, we describe a particularcase where we do have a Ramanujan graph G .The celebrated 1986 paper of Lubotzky, Phillips, and Sarnak [23, §2] gave the first ex-plicit construction of Ramanujan graphs. For two distinct primes p and q both congruentto 1 mod 4, Lubotzky, Phillips, and Sarnak consider the Cayley graph of p + 1 speciallychosen generators of the projective linear groups PSL(2 , Z /q Z ) or PGL(2 , Z /q Z ) , de-pending whether p is a square modulo q or not. This yields a ( p + 1) -regular Ramanujangraph, which we will denote LPS( p, q ) . When p is a square modulo q , the Ramanujangraph LPS( p, q ) will be nonbipartite with exactly ( q − q ) / vertices.For comparison with our results we ran Algorithm A on the Lubotzky–Phillips–Sarnakconstruction with p = 13 and q = 61 . Our reasoning for these values of p and q is that wewant a Ramanujan graph which is of similar edge density to the flip graphs we considerlater. The resulting graph LPS(13 , is 14-regular and has , vertices. A MATLAB [24] computation shows that λ ≈ . , which is smaller than · √ ≈ . , and thisconfirms that LPS(13 , is a Ramanujan graph. In Figure 1 we show the comparisonbetween the density of the queue throughout the search and the lower bound on theexpected queue size described by Theorem 5.Next, we consider how well Theorem 6 does at predicting the size of the graph duringthe search process. For the experiment on G = LPS(13 , , we start at a single nodeand run the random growth process. At every 1000 steps in the process we output thesize of the queue and the number of processed nodes. Moreover, we want to estimate e ( U, W ) . This is done via a random sample. Observe that all edges from W to U , havetheir W endpoint contained Q . Therefore to estimate e ( U, W ) , we sample 100 verticesfrom Q to estimate the average number of edges a vertex in Q sends to U .For example, in the particular run we consider here, when there were 26,000 verticesprocessed, we had 84,102 vertices in the queue. We also know that the graph is 14-regular JANKO BÖHM, MICHAEL JOSWIG, LARS KASTNER, AND ANDREW NEWMAN so at this point we have that with certainty the number of vertices is at most (14 − λ )(26 ,
000 + 84 , (14 − λ )(26 ,
000 + 84 , − e ( U, W ) . To estimate bounds however we need estimates on λ and e ( U, W ) . Since we have aRamanujan graph λ is at most √ , and we take this as the estimate for λ . In thisparticular case a random sampling of size 100 from the vertices in the queue gave anestimate on e ( U, W ) of , . . Thus the estimated upper bound on the number ofvertices is(3) (14 − √ ,
000 + 84 , (14 − √ ,
000 + 84 , − , . ≈ , . . Similarly, one estimates a lower bound on the number of vertices of , . . So withonly about 23 percent of the vertices moved into the set P , we can give upper and lowerbounds on the total number of vertices that are both within 3 percent of the right answer.Figure 2 shows the curves giving the upper and lower estimates throughout the process.Note that Theorem 6 gives no finite upper bound early in the process, and the figurereflects this.Of course, as we mentioned above, in this particular case we do know a precise ap-proximation for λ . Replacing √ by . in (3) improves the estimated upper boundfor the number of nodes of LPS(13 , only slightly, to , . . However, in largerexamples we would not be able to expect to compute λ numerically, nor will that be thegoal, so we want to instead focus on the coarser estimate here.3. Flip graphs of point configurations
Let P ⊂ R (cid:96) a finite set of n points that affinely spans the entire space. A triangulation Σ of P is a simplicial complex which covers the convex hull conv P , such that the verticesof each simplex form a subset of the given points P ; cf. [10, §2.3.1]. The set of allsubdivisions of P is partially ordered by refinement, and the triangulations are preciselythe finest subdivisions. The triangulations form the nodes of a graph, where the edgesarise from local modifications known as flips ; cf. [10, Definition 2.4.7] and Figure 3. Thisis the flip graph of P .A certain class of triangulations is of particular interest, e.g., due to connections withalgebra [10, §1.3]. The triangulation Σ is regular if it is induced by a height function h : P → R in the sense that the lower convex hull of conv (cid:8) ( p, h ( p )) | p ∈ P (cid:9) ⊂ R (cid:96) +1 projects to Σ by omitting the last coordinate. The subgraph of the flip graph whose nodescorrespond to the regular triangulations is the flip graph of regular triangulations of P .In the literature this is often called the “regular flip graph”; however, as we also talk aboutregularity in the graph-theoretic sense we make effort to avoid ambiguity between the twonotions of regular. Structurally, it is essential that the flip graph of regular triangulationsis contained in the vertex-edge graph of a convex polytope, the secondary polytope of P ,which is defined up to normal equivalence [10, Theorem 5.3.1]. In particular, the flipgraph of regular triangulations is necessarily connected [10, Corollary 5.3.14]; in general,this does not hold for the flip graph of all triangulations [10, §7.3]. ANDOM GROWTH ON A RAMANUJAN GRAPH 9
For simplicity of the exposition we will now assume that the points in P are in convexposition, i.e., they form the vertices of their convex hull, conv P . In this way we can alsoafford some sloppiness by not distinguishing between a polytope and its set of vertices. Figure 3.
On the left: The flip-graph of a hexagon is the vertex-edgegraph of the -dimensional associahedron. On the right: The quotientflip graph for the cyclic group C of rotations. Example 7.
Let P ⊂ R be the set of vertices of a convex k -gon. This point configurationhas k − (cid:0) k − k − (cid:1) triangulations, which is a Catalan number. Each triangulation is regular,and it is determined by its k − diagonals, and each one gives rise to a flip. This entailsthat the flip graph of P is a regular graph of degree k − with k − (cid:0) k − k − (cid:1) nodes. The ( k − -dimensional associahedron, which is simple, is a secondary polytope. Table 1.
Data on the flip-graphs of k -gons, for k ≥ . Here n is thenumber of nodes, i.e., triangulations, m is the number of edges, i.e., flips,and d is the degree. The small cases k ∈ { , } are omitted since thoseflip graphs are bipartite (in fact, consisting of a single node for k = 3 , anda single edge for k = 4 ). k n m d λ · √ d − k
11 12 13 14 15 n m d λ · √ d − The spectral expansion of the flip graph of a polygon is unclear, although the weakercondition of low diameter is known; cf. [30] and [27]. Yet inspecting the values for λ and √ d − in Table 1 yields the following. Observation 8.
The flip graph of a k -gon is a regular Ramanujan graph for k ∈ { , , } . In Figure 4, we show the results of applying Theorem 6 on the random growth processapplied to the flip graph Φ of the 15-gon. We estimate e ( U, W ) / | Q | at every 1000 stepsby a random sampling of 100 vertices from Q . More importantly, rather than taking theactual value of λ (Φ) , we wish to see what Theorem 6 would tell us were Φ to have goodexpansion so we take λ = 2 · √ ≈ . to use Theorem 6 to give upper and lowerestimates on the number of vertices of Φ . This is compared with the true number ofvertices, 742,900. In Figure 4 the upper curve shows the ratio of the upper estimate ofTheorem 6 to the true number of vertices and the lower curve shows the same ratio forthe lower estimate. . . . . . . . . . . . density π of processed nodes d e n s i t y o f t o t a l n o d e s Figure 4.
Upper and lower estimates on the total number of verticesof the flip graph of the 15-gon (normalized by the actual number of ver-tices)We compare this figure back to the idealized setting of Figure 2. Here there is morevolatility in the upper bound at the beginning and the two curves do not converge to eachother quite as quickly. Still, the two curves become quite close to one another long beforethe process has finished. Note that the flip graph of the 15-gon is 12-regular comparedto the 14-regular
LP S (13 , .Even when only 220,000 vertices have been processed in this particular run, there are476,113 vertices waiting in the queue, the data predicts that the total number of verticeswill be between 715,279 and 767,518. So with only 30 percent of the vertices processedand another 64 percent in the queue we already have the right number of vertices boundedbetween 96 percent and 103 percent of its true value.Next we describe other polytopes whose flip graphs are regular graphs. A split of P is a subdivision with exactly two maximal cells; it is necessarily regular [14, Lemma ANDOM GROWTH ON A RAMANUJAN GRAPH 11 P is called totally splittable . This is the case, e.g.,when P is the vertex set of a polygon: each diagonal defines a split; cf. Example 7 andFigure 3. Note that every subdivision of a totally split point configuration is regular.There is a full characterization of the totally split polytopes: Theorem 9 ([15, Theorem 9]) . A polytope P is totally splittable if and only if it has thesame oriented matroid as a simplex, a crosspolytope, a polygon, a prism over a simplex,or a (possibly multiple) join of these polytopes. This result has an immediate consequence on the associated flip graphs.
Corollary 10.
Let P be the set of k vertices of a totally splittable (cid:96) -polytope. Thenany secondary polytope is simple of dimension k − (cid:96) − . Consequently, the flip graph ofregular triangulations of P is ( k − (cid:96) − -regular.Proof. The dimension of the secondary fan modulo its lineality space equals k − (cid:96) − ; cf.[10, §5.1.3]. Now the claim follows from an inspection case by case. By [15, Remark 11]the secondary polytopes of totally splittable polytopes are (possibly multiple) productsof simplices, permutohedra, and associahedra. (cid:3) Non-regular graphs
Most flip graphs are not regular. Nonetheless, we aim for an analogue of Theorem 5 inthe absence of regularity. In the most classic setting expander graphs are defined in termsof regular graphs. However in the literature there is a more general notion of spectralexpansion for non-regular graphs in terms of the normalized adjacency matrix. For ourpurposes we will keep the assumption that G is a graph with no isolated vertices and let D = D ( G ) denote the degree matrix of G , which is diagonal and invertible. Then thenormalized adjacency matrix of G is defined as N := D − / AD − / , where A is the usual adjacency matrix of G . It is standard knowledge that the eigenvaluesof N fall in [ − , , with the multiplicity of the eigenvalue 1 corresponding to the numberof connected components of G ; see, e.g., [9]. We use µ i to denote the eigenvalues of N with µ > µ > · · · > µ n , and we write µ := µ ( G ) for the maximum of µ and − µ n .Before we justify comparing the flip graphs from our experiments to (non-regular)expander graphs, we present a non-regular version of the Expander Mixing Lemma. Thiswill allow us to be more precise when analyzing more experiments in Section 6. We firstneed to define the volume of a set of vertices in a graph. For a graph G and a subset U of the vertices of G , the volume of U is vol( U ) := (cid:88) v ∈ U deg( v ) . There are several versions of the Expander Mixing Lemma for non-regular graphs knownthat use this volume notion; see, e.g., [9]. However, none that we found gave preciselythe formulation that we need for our purposes, so we give our own formulation here. Ourargument essentially follows the proof of the d -regular analogue found in [2, Lemma 2.1]. Proposition 11.
Let G = ( V, E ) be a graph with µ = µ ( G ) . Then for any partition of V into two nonempty set S and T , one has (cid:12)(cid:12)(cid:12)(cid:12) e ( S, T ) − vol( S ) vol( T )vol( V ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ vol( S ) vol( T )vol( V ) . Proof.
Recall that N = D − / AD − / . Then f : V ( G ) → R defined by f ( v ) = (cid:112) deg( v ) is an eigenvector for the eigenvalue 1. For U and V given as in the statement let g : V ( G ) → R be defined by g ( v ) = − (cid:112) deg( v ) / vol( S ) if v ∈ S and g ( v ) = (cid:112) deg( v ) / vol( T ) if v / ∈ S . Now f is orthogonal to g since (cid:104) f, g (cid:105) = (cid:88) v ∈ S − deg( v )vol( S ) + (cid:88) v ∈ T deg( v )vol( T ) = − vol( S )vol( S ) + vol( T )vol( T ) = 0 . It follows from the fact that N is a symmetric matrix (and hence has an orthogonaleigenbasis) therefore that |(cid:104) N g, g (cid:105)| ≤ µ (cid:107) g (cid:107) . Also, (cid:107) g (cid:107) = S ) + T ) . It remains to compute (cid:104) N g, g (cid:105) . We have (cid:104)
N g, g (cid:105) = (cid:104) D − / AD − / g, g (cid:105) = (cid:104) AD − / g, D − / g (cid:105) . Let g (cid:48) = D − / g . Thus g (cid:48) ( v ) = − S ) if v ∈ S and g (cid:48) ( v ) = T ) if v ∈ T . It is easy tocheck that (cid:104) Ag (cid:48) , g (cid:48) (cid:105) = 2 (cid:88) ij ∈ E g (cid:48) ( i ) g (cid:48) ( j ) ; indeed this holds for any vector in R | V | . Now as an edge can either have both endpointsin S , both endpoints in T , or contribute to e ( S, T ) we have (cid:88) ij ∈ E g (cid:48) ( i ) g (cid:48) ( j ) = e ( S, S )vol ( S ) + e ( T, T )vol ( T ) − e ( S, T )vol( S ) vol( T )= vol( S ) − e ( S, T )vol ( S ) + vol( T ) − e ( S, T )vol ( T ) − e ( S, T )vol( S ) vol( T )= 1vol( S ) + 1vol( T ) − e ( S, T ) (cid:18) ( S ) + 2vol( S ) vol( T ) + 1vol ( T ) (cid:19) = 1vol( S ) + 1vol( T ) − e ( S, T ) (cid:18) S ) + 1vol( T ) (cid:19) = (cid:107) g (cid:107) (cid:18) − e ( S, T ) (cid:18) S ) + 1vol( T ) (cid:19)(cid:19) . Thus, (cid:12)(cid:12)(cid:12)(cid:12) − e ( S, T ) (cid:18) S ) + 1vol( T ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:104) N g, g (cid:105)(cid:107) g (cid:107) ≤ µ . Since S ) + 1vol( T ) = vol( V )vol( S ) vol( T ) , the claim follows. (cid:3) ANDOM GROWTH ON A RAMANUJAN GRAPH 13
As in the 15-gon example, we want to compare non-regular flip graphs to Ramanujangraphs. However, without regularity we need to specify what we mean by the termRamanujan graph. There does already exist a notion of non-regular Ramanujan graphs,posed by Lubotzky in [22] and discussed in [8] stated in terms of the spectral radiusof the universal covering tree of a graph. Here, we will not be so precise. Rather ourassumption will be that the degree sequence of the graphs under consideration have smallvariance. This will allow us to draw a comparison to a regular Ramanujan graph.In terms of the variance of the degree sequence we can obtain the following hybridversion of the Expander Mixing Lemma as a corollary to Proposition 11.
Corollary 12.
Let G = ( V, E ) be a graph with µ = µ ( G ) and so that G has averagedegree d , and the variance of its degree sequence is σ , then for any partition of V intotwo sets S and T one has the following lower bound on e ( S, T ) : e ( S, T ) ≥ (1 − µ ) ( d | S | − σ (cid:112) | S | n )( d | T | − σ (cid:112) | T | n ) dn . Proof.
The proof follows from Proposition 11 and the subsequent claim. (cid:3)
Claim.
For G as in the statement and S any subset of the vertex set of G one has | d | S | − vol( S ) | ≤ σ (cid:112) | S | n .Proof of claim. Let G and S be given as above. We have, (cid:0) d | S | − vol( S ) (cid:1) = (cid:32)(cid:88) v ∈ S ( d − deg( v )) (cid:33) = (cid:32)(cid:88) v ∈ G ( d − deg( v )) S ( v ) (cid:33) , where S denotes the indicator function of S . Thus by the Cauchy–Schwarz inequalitythe above quantity is at most (cid:88) v ∈ G ( d − deg( v )) (cid:88) v ∈ G ( S ( v )) = σ n | S | . And the claim follows. (cid:3)
It follows from the statement of Corollary 12 that for d much larger than σ and | S | and | T | both far enough away from 0 we essentially have the regular Expander MixingLemma with the degree of regularity replaced by the average degree of the graph. Therequirement that | S | and | T | both be far away from zero (where “far away" is determinedby how close σ is to zero) effectively creates two blind spots in a predicted lower bound.We don’t make a precise statement in terms of σ here, but rather point out that Corollary12 together with the proof of Theorem 5 gives us a heuristic explanation for applyingTheorem 5 with non-integer values of d . To be more precise we denote by β ( π, d, λ ) theexpression in the lower bound of Theorem 5, β ( π, d, λ ) = 1 − π − exp (cid:18) − ( d − λ ) (cid:18) d − (cid:19) π (cid:19) . In the next section, we make an experimental comparison between the behavior of thequeue of a non-regular flip graph with average degree d and the behavior of the queuesize modeled by β ( π, d, (cid:112) d − Recall that, by the Alon–Boppana Theorem, the threshold √ d − is essentially thesmallest possible value for λ ( G ) when G is a d -regular graph. Similarly, it follows froma result of Mohar [25] that λ ( G ) = 2 (cid:112) d − is the best one can hope for as far as thespectral gap of a non-regular graph is concerned.5. Random Graphs
To complete our theoretical analysis of the random growth process in Algorithm A wenow study its behavior on random graphs in the classical Erdős–Rényi model [11]. Thisis particularly interesting in view of a result of Friedman [12] which relates Ramanujangraphs with random graphs and because of which Ramanujan graphs are sometimes called“quasirandom”. Note, however, that Friedman’s random graphs are uniformly sampledfrom the class of all regular graphs. The Erdős–Rényi random graphs, which we denote G ( n, p ) , form an easier model to sample from and are more commonly studied. Therandom graph G ( n, p ) has n vertices and each of the (cid:0) n (cid:1) edges appearing independentlywith probability p .In the previous section we looked at non-regular graphs of average degree d . Thissuggests to compare with G ∼ G ( n, d/n ) as the expected degree of any fixed vertexunder this distribution is dn ( n − . Before we will do this comparison a few words aboutthe asymptotic behavior of G ∼ G ( n, d/n ) are in order. More background on Erdős–Rényi random graphs may be found in, e.g., [3], [7], or [17]. The first thing to mentionis that G ∼ G ( n, d/n ) is asymptotically almost surely disconnected if n is large. Theorem 13 (Erdős–Rényi [11]) . Let d be a fixed constant. If d < , then asymptoti-cally almost surely G ∼ G ( n, d/n ) is a disjoint union of components of order O (log n ) .However, if d > , then asymptotically almost surely G ∼ G ( n, d/n ) has a unique giantcomponent on (1 + o (1)) δ n vertices, where δ is the unique root of − x = e − dx in theopen interval (0 , . In the latter case the remaining components have order O (log n ) . This significant change in behavior at d = 1 is called the Erdős–Rényi phase transition and has been extensively studied; details may be found in [3, Chapter 9].Here we briefly want to look into the connection with Ramanujan graphs. The Ex-pander Mixing Lemma tells us that for G a d -regular graph and any subsets S and T ofthe vertices, e ( S, T ) is close to dn | S || T | . The term dn | S || T | is the expected number of edgesbetween S and T if G is taken to be an Erdős–Rényi graph distributed as G ( n, d/n ) .Thus a Ramanujan graph has roughly the number of edges between S and T for any S and T as would be expected in a random graph of the same edge density, where “roughly”is measured by the spectral gap. While Theorem 5 holds for a deterministic graph, herewe add a level of randomness in running our random growth process on a random graph.Now we investigate our random growth process on an Erdős–Rényi random graph. Weshould be a bit careful now that we have two levels of randomness, namely the graph andthe random selection of vertices from the queue. Note that we will only be interestedin the case where the giant component exists. We consider the following experiment forany n ∈ N and d > : ANDOM GROWTH ON A RAMANUJAN GRAPH 15
Algorithm B
Random Growth on a Random Graph Generate a graph G uniformly at random from G ( n, d/n ) . Pick a vertex v uniformly at random from G . Run Algorithm A starting at v , and return the tree described by Lemma 3.By Theorem 13, we know that, as n tends to infinity, the density of the final tree willeither be δ with probability δ , or it will be zero with probability − δ , because withprobability − δ we will pick a component on O (log n ) vertices. As we are interestedin searching the giant component, we consider Algorithm B conditioned on v being avertex in the giant component. Moreover, to simplify the exposition, we let this processrun to step n , with nothing happening after the queue first becomes empty. In this waywe do not have to concern ourselves with conditioning on the exact size of the giantcomponent. Even if we let Algorithm B run for n steps, we keep track of the first time t , where Q t = ∅ . Then P t is the component of v and U t comprises the vertices in thecomplement. That is, t = | P t | , and we define P r = P t , Q r = Q t , and U r = U t for all t < r ≤ n . With this notation we are prepared to prove the following theorem about thegrowth of the queue in a random search of the giant component of a random graph. Theorem 14.
Fix d > and let H ⊆ G ∼ G ( n, d/n ) be a largest component in therandom graph in the Erdős–Rényi model. We run Algorithm A on H starting at a vertex v in H , with the modification to let it run for n steps. Then for any π ∈ (0 , δ ) theasymptotic expected density of Q t , where t = πn , equals − π − exp( − dπ ) . If, however, π ∈ ( δ , the asymptotic expected density of Q t equals . Before we enter the proof, let us explain how the analysis of Algorithm A from The-orem 14 applies to Algorithm B. Since initially all vertices look alike, we do not know if v is actually contained in the giant component. However, this is easy to fix by keepingtrack of the sizes of the sets P , Q and U during in the random growth process. Thisleads to a numeric process by defining u t and q t for t ∈ N , as(4) u t = u t − − X , where X ∼ Bin( u t − , d/n ) , u = nq t = n − t − u t . Of course, q t will with probability 1 eventually become negative. The important obser-vation is that from the start to the first time t ∈ (0 , n ) where q t = 0 , these sequencesmeasure the size of the sets in an instance of Algorithm B. We begin with the followingsimple lemma about this numeric process. Lemma 15.
In the numeric process (4) the asymptotic expected value of u t /n equals exp( − dυ ) for t = πn , where π ∈ (0 , is fixed, and n tends to infinity.Proof. From the definition of { u t } t ≥ , we have that u t is distributed as a binomial randomvariable with n trials and success probability (cid:0) − dn (cid:1) t . Indeed, we may sample u t bythe following experiment: Begin with a set of n vertices, and at each step in , . . . , t choose a random set by including each vertex with independently probability d/n . Thisdistribution induces the binomial distribution Bin( u t − , d/n ) at step t on the verticesthat have not yet been selected. Thus u t is given by the number of vertices that are never picked in this process. From this we have that E ( u t ) = (1 − d/n ) υn n . Thus theexpected value of u t /n is asymptotically exp( − dυ ) . (cid:3) Now we will prove Theorem 14, which is about Algorithm A, by analyzing Algorithm Band using Lemma 15.
Proof of Theorem 14.
From Lemma 15 it follows that the expected value of q t /n at t = πn is − π − exp( − dπ ) . Of course, this only says something about Algorithm B in the casethat q r ≥ for all r ≤ t . The key observation is that after Steps 1 and 2 the component H of the chosen vertex v is deterministic, and the graph H is connected. Thus for π ∈ (0 , , we have | Q r | > for all r ≤ πn if and only if v lies in a componentof G ∼ G ( n, p ) of order at least πn . By Theorem 13, the probability of this event,asymptotically in n for π fixed, is δ for π < δ and zero for π > δ . From this itfollows that, with probability δ , the value q s measures the size of | Q s | in the randomgrowth process. On the other hand, with probability − δ , Algorithm B begins in acomponent of order O (log n ) , and so the asymptotic density of the queue at time t = πn and π > is zero. Thus for π > δ the expected density of | Q πn | is zero, while forsmaller π , the expected density is asymptotically δ (1 − π − exp( − dπ )) . But of course,if we condition on choosing a vertex in the giant component then the expected densityof Q t is asymptotically (1 − π − exp( − dπ )) if π < δ . (cid:3) Example 16.
Theorem 14 is an asymptotic result. To see how well it holds in practicewe conduct an experiment on a large random graph. We expect that, because of thequasirandomness of Ramanujan graphs, a random graph G with the same number ofvertices and edge density as LPS(13 , should exhibit the same behavior for the sizeof the queue as appears in Figure 1. To that end we generated an Erdős–Rényi randomgraph G with 113,460 nodes and 794,220 edges from the uniform distribution of graphswith the given numbers of nodes and edges. The resulting graph had average degree14.00 and the spectral gap of the normalized adjacency matrix was about 0.4848. Weran Algorithm A on G and examined the behavior of the queue. Figure 5 shows the resultscompared against the results for the Ramanujan graph already shown in Figure 1. Weomit the curve given by Theorem 14 with p = 14 / , as it is visually indistinguishablefrom the curve modeling the behavior of the queue in the random graph experiment.It is worth mentioning that our choice of edge density and number of vertices meansthat our actual randomly-generated graph ended up being connected. This is not sur-prising from the view point of the connectivity threshold of Erdős–Rényi random graphs.Indeed this classic result of [11] establishes that for p > log nn asymptotically almost surely G ∼ G ( n, p ) will be connected. In our case p = 14 / and > log(113460) ≈ . .To have a randomly-generated graph with edge density /n which is not connected wewould expect to need more than 1.2 million vertices. Moreover by Theorem 13, such agraph would have more than 99.9999% of its vertices in the giant component.6. Enumerating triangulations in practice
Standard software packages for enumerating triangulations include
TOPCOM [29],
Gfan [18] and
MPTOPCOM [19]. Both
TOPCOM and
Gfan employ random growth through the(regular) flip graph, whereas
MPTOPCOM uses reverse search [4]. We would like to estimate
ANDOM GROWTH ON A RAMANUJAN GRAPH 17 . . . . . . . . . . . . .
81 density π of processed nodes d e n s i t y κ o f q u e u e Figure 5.
The observed behavior of the queue in G ∼ G (113460 , / (blue) compared to the observed behavior ofthe queue in LPS(13 , (red).when these programs will terminate early in the enumeration process. To this end wetry our methods developed for regular and nonregular graphs with good expansion. Incontrast to the previous sections our observations here are based on experiments only.Many interesting point configurations are highly symmetric. If there is a group actingon P , its action naturally lifts to the set of triangulations of P . Hence, one considers the quotient flip graph , having orbits of triangulations as nodes. Two orbits are connectedby an edge if there is a pair of representatives from both orbits, which differ by a flip.It suffices to visit the nodes of the quotient flip graph; orbits can be expanded later ifneeded (e.g., in an embarrassingly parallel manner). All software systems mentionedsupport computing in quotient flip graphs.For instance, a regular k -gon admits the natural action of the dihedral group D k byrotations and reflections. Figure 3 shows the quotient flip graph of a hexagon by thecyclic group of order six, which is a (normal) subgroup in D of index two. It has fournodes, and the average degree is d = (1 + 2 · / .There is no reason to believe that a general flip graph is a good expander. Moreover,most flip graphs are not regular. The same holds true for any of its quotients. Yet,we will argue heuristically that the situation is maybe somewhat more benign, from apractical point of view. Since flip graphs tend to be very large, in practice we will mostlyencounter point configurations which are rather small, both in terms of dimension andnumber of points. In many cases this will mean that the point configuration in questionadmits a substantial number of splits, even if there are many other coarsest subdivisions.We believe that this should result in a rather low variance in the degree sequence of flipgraphs within computational reach. Moreover, the vast majority of the triangulationsof a point configurations exhibits no symmetry at all, which means that most orbitsof triangulations have the size of the entire group acting. We believe that this should force that also the quotient flip graphs within computational reach still have a moderatevariance in their degree sequence. . . . . . . . . . . . . . density π of processed nodes d e n s i t y κ o f q u e u e Figure 6.
Lower bound on expected queue size from Theorem 5 (blue)and a comparison to experimental data (red) for the flip graph of the4-cube.
Example 17.
To see how strong of a comparison we can make between a flip graph and anon-regular Ramanujan graph, as in the discussion in Section 4, we consider the quotientflip graph of all triangulations of the 4-cube, which is connected [28]. The -dimensionalcube, with 16 vertices, has , , triangulations, partitioned into , orbits withrespect to the natural action of the Coxeter group of type B , whose order is . Exactly , orbits have the maximal length ; this is more than . . The correspondingnumbers counting only regular triangulations can be found in [10, §6.3.5].The flip graph of all triangulation of the 4-cube has the , orbits of triangulationsas its vertices and it has , , edges. Thus it has average degree d ≈ . . InFigure 6 we show a comparison to the size of the queue as a function of the density ofprocessed vertices, as the random growth process runs on this flip graph, compared to thelower bound given by β ( υ, . , √ . − as υ , the density of processed nodes,varies from 0 to 1. Visually, our heuristic works well in this case as the two curves arequite similar.Like the flip graphs of k -gons for higher k in the regular setting, the quotient flipgraph of the 4-cube itself does not actually have good expansion. The second largesteignvalue of the normalized adjacenty matrix is .9846 from computation in MATLAB [24].Nonetheless, however, the two curves in Figure 6 match up well.We don’t have a rigorous explanation for why random growth on flip graphs in ourexamples is so closely modeled by the behavior in regular Ramanujan-graphs in Theorems5 and 6. One possible reason is that the assumption of large spectral gap is stronger thanwhat is needed to reach our conclusions. It is known that a d -regular Ramanujan graph ANDOM GROWTH ON A RAMANUJAN GRAPH 19 on n vertices has diameter roughly log n when n is large (this is discussed, e.g., in [16]).While Table 1 seems to suggest that flip graphs of k -gons do not have large spectralgap, in general, Sleator, Tarjan, Thurston [30], and Pournin [27] show they do havelow diameter (again logarithmic in the number of vertices). Perhaps low diameter isenough to imply that random growth works similarly to how it works in a similarly-sizedRamanujan graph.We close this paper by returning to the triangulations of the 15-gon. MPTOPCOM is basedon the reverse search method of Avis and Fukuda [4], which is memory efficient and easyto parallelize [5]. That algorithm implicitly picks a spanning tree of the (quotient) flipgraph, in a deterministic way. We applied Hall–Knuth random sampling [13, 21] to
MPTOPCOM ’s reverse search tree of the flip graph of the 15-gon to estimate the size of thegraph. Figure 7 shows how increasing the number of samples makes that estimate moreprecise. Comparing with Figure 4 is a bit difficult, since the x -axes are not the same. Yetit seems that our estimates based on random growth and spectral expansion are superior,at least in this case. . . . . . . . . . · . . number of probes d e n s i t y o f t o t a l n o d e s Figure 7.
Hall–Knuth estimates for the reverse search tree in theflip graph of the 15-gon. Every point stands for one estimate. The y -coordinate of the point is the estimate divided by the actual number ofnodes ( , ). The x -coordinate indicates how many random probesinto the tree were made. Acknowledgments
The authors thank Marek Kaluba for generating the examples of Lubotzky–Phillips–Sarnak Ramanujan graphs used in Section 2, see [20]. Additionally, the authors thankStefan Felsner, Christian Haase, Nati Linial, and Tibor Szabó for helpful discussions andcomments on an earlier draft of this paper.
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J. Amer. Math. Soc. issn : 0894-0347. doi : . url : https://doi.org/10.2307/1990951 . (J. Böhm) Technische Universität Kaiserslautern Gottlieb-Daimler-Straße, Geb. 48,67663 Kaiserslautern, Germany
E-mail address : [email protected] (M. Joswig, L. Kastner, and A. Newman) Institut für Mathematik, MA 6-2, TechnischeUniversität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany
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