Girth, magnitude homology, and phase transition of diagonality
GGirth, magnitude homology, and phase transition ofdiagonality
Yasuhiko
Asao ∗ Yasuaki
Hiraoka † Shu
Kanazawa ‡ February 10, 2021
Abstract
This paper studies the magnitude homology of graphs focusing mainly on the relationshipbetween its diagonality and the girth. Magnitude and magnitude homology are formulations ofthe Euler characteristic and the corresponding homology, respectively, for finite metric spaces,first introduced by Leinster and Hepworth–Willerton. Several authors study them restrictingto graphs with path metric, and some properties which are similar to the ordinary homologytheory have come to light. However, the whole picture of their behavior is still unrevealed,and it is expected that they catch some geometric properties of graphs. In this article, weshow that the girth of graphs partially determines magnitude homology, that is, the largergirth a graph has, the more homologies near the diagonal part vanish. Furthermore, applyingthis result to a typical random graph, we investigate how the diagonality of graphs variesstatistically as the edge density increases. In particular, we show that there exists a phasetransition phenomenon for the diagonality.
The magnitude of finite metric spaces was introduced by Leinster [12] as a formulation of Eulercharacteristic of finite metric spaces. Magnitude has several interesting properties such as multi-plicativity property and inclusion-exclusion principle, which seems parallel to the case of ordinaryEuler characteristic of topological spaces. However, whole picture of the behavior of magnitude isunrevealed, and that is attracting people in several areas of mathematics. In particular, magnitudeof finite graphs, which takes values in formal power series with Z -coefficients, is studied by severalauthors so far ([1], [3], [8], [9], [11]). Throughout this article, we call a finite, simple, and undirectedgraph without loops just a graph.The magnitude homology of graphs is a categorification of magnitude, first introduced byHepworth–Willerton [9] as an analogy of ordinary homology theory. It is a bigraded abeliangroup whose Euler characteristic coincides with the magnitude, and the multiplicativity propertyand the inclusion-exclusion principle are formulated as the Künneth and the Mayer–Vietoris theo-rems, respectively [9]. Their beautiful theory enables us to compute the magnitude and magnitudehomology of graphs. For example, Gu [8] showed a remarkable compatibility of magnitude ho-mology with algebraic Morse theory, and he computed magnitude homology of several types ofgraphs including well-known classical ones. Bottinelli–Kaiser [3] study the magnitude homology ofmedian graphs, using the retraction between homology groups. More or less, the remarkable prop-erty concerned in their works is the diagonality of graphs, first suggested in [9], which guaranteessimpleness of the magnitude homology in some sense.In this article, we show that the girth of graphs partially determines magnitude homology, thatis, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore,by using this result, we investigate how the diagonality of graphs varies statistically as the edgedensity (proportion of the number of edges to that of possible edges) increases. In particular,we show that there exists a phase transition phenomenon for the diagonality. As shown in [9], atree (or more generally, a forest) which has low edge density is diagonal. It is also known thata few graphs with high edge density are diagonal. This fact is shown in [9] for complete graph, ∗ Center for Advanced Intelligence Project, RIKEN. [email protected] † Kyoto University Institute for Advanced Study, WPI-ASHBi, Kyoto University. Center for Advanced IntelligenceProject, RIKEN. [email protected] ‡ Kyoto University Institute for Advanced Study, Kyoto University. [email protected] a r X i v : . [ m a t h . A T ] F e b nd in [8] for pawful graph (see Definition 2.8). However, graphs with intermediate edge densityare more likely to be non-diagonal. To describe this phenomenon statistically, we consider theErdős–Rényi graph model which is a typical random graph model extensively studied since the1960s ([5], [6], [7]). Given n ∈ N and p ∈ [0 , , an Erdős–Rényi graph G n,p with parameters n and p is a random graph with n vertices, where the edge between each pair of vertices is addedindependently with probability p .Now, we explain our results in the following. We first state a relationship between girth ofgraphs and magnitude homology. They will be proved in an algebraic and combinatorial way inSection 3. Let G be a graph and x ∈ V ( G ) be a vertex. We define the local girth of G at x by gir x ( G ) := inf { i ≥ | there exists a cycle of length i in G containing x } . We also define the girth of G by gir( G ) := min x gir x ( G ) . Note that the following statements are compatible with the computation of magnitude homologyfor trees and cycle graphs in [8] and [9], respectively. In particular, Corollary 1.4 is a generalizationof the computation of magnitude homology of trees in [9, Corollary 6.8]. Below, MH ∗ , ∗ ( G ) is themagnitude homology of G , and the superscript x of MH x ∗ , ∗ ( G ) indicates the restriction on thestarting point (see Section 2.2 for the definitions). Theorem 1.1.
Let (cid:96) ≥ . If gir x ( G ) ≥ , then MH x(cid:96),(cid:96) ( G ) ∼ = Z deg x , where deg x denotes the degree of the vertex x . The following is also obtained by Sazdanovic–Summers in [14, Thoerem 4.3].
Corollary 1.2.
Let (cid:96) ≥ . If gir( G ) ≥ , then MH (cid:96),(cid:96) ( G ) ∼ = Z E ( G ) , where E ( G ) denotes the number of edges of G . The following are extensions of the above.
Theorem 1.3.
Let (cid:96) ≥ and i ≥ . If gir x ( G ) ≥ i + 5 , then MH x(cid:96) − j,(cid:96) ( G ) ∼ = (cid:40) Z deg x , j = 0 , , ≤ j ≤ i. Corollary 1.4.
Let (cid:96) ≥ and i ≥ . If gir( G ) ≥ i + 5 , then MH (cid:96) − j,(cid:96) ( G ) ∼ = (cid:40) Z E ( G ) , j = 0 , , ≤ j ≤ i. The above results will be proved by using algebraic Morse theory. The following gives a criterionfor the diagonality of graphs. Let e ∈ E ( G ) be an edge. We define the local girth of G at e by gir e ( G ) := inf { i ≥ | there exists a cycle of length i in G containing e as its edge } . Note that we have gir( G ) = min e gir e ( G ) . Theorem 1.5.
Let G be a graph and e ∈ E ( G ) be an edge. If k := gir e ( G ) ∈ [5 , ∞ ) , then MH ,(cid:96) ( G ) (cid:54) = 0 for (cid:96) = (cid:98) k +12 (cid:99) . Corollary 1.6. If G is a diagonal graph, then gir( G ) = 3 , , or ∞ . By considering k = 2 i + 5 or i + 6 in Theorem 1.5, it turns out that the range ≤ j ≤ i guaranteeing the vanishing of magnitude homology groups in Corollary 1.4 is optimal.Next we state stochastic properties of magnitude homology with respect to the Erdős–Rényirandom graph model. They will be shown in Section 4. In the study of the Erdős–Rényi graph G n,p , one is usually concerned with the asymptotic behavior of G n,p as the number of vertices n tends to infinity, where p is typically regarded as a function of n . For a graph property P , we say2hat G n,p satisfies P asymptotically almost surely (a.a.s.) if lim n →∞ P ( G n,p satisfies P ) = 1 . Wealso use the Bachmann–Landau big- O /little- o notation with respect to the number of vertices n tending to infinity. Additionally, for non-negative functions f ( n ) and g ( n ) , f ( n ) = ω ( g ( n )) meanthat g ( n ) = o ( f ( n )) . One of the most classical themes is searching the threshold probability p ( n ) for various graph properties P . Here, we call the probability p ( n ) a threshold for P if p = o ( p ( n )) implies that G n,p satisfies P a.a.s. and p = ω ( p ( n )) implies that G n,p does not satisfy P a.a.s. Forexample, p ( n ) = n − is the threshold probability for the appearance of a cycle in G n,p .The first result exhibits a phase transition for the diagonality of Erdős–Rényi graphs. This iswhere the magnitude homology of Erdős–Rényi graph suddenly becomes non-diagonal. Theorem 1.7.
Let G n,p be an Erdős–Rényi graph with parameters n and p . Then, the following (1) , (2) , and (3) hold. (1) If p = o ( n − ) , then G n,p is diagonal a.a.s. (2) If p = cn − , then lim n →∞ P ( G n,p is non-diagonal ) = (cid:40) − √ − c exp( c/ c / c / c / , < c < , , c > . (3) If p = ω ( n − ) and p = o ( n − / ) , then G n,p is non-diagonal a.a.s. Figure 1: The limiting function of c appearing in Theorem 1.7 (2).As seen in Figure 1, the probability that G n,c/n is non-diagonal approaches an explicit constantbounded away from one whenever c < . Meanwhile, when c > , G n,c/n is non-diagonal a.a.s.A graph property P is said to be monotone increasing if whenever a graph G satisfies P and G is a subgraph of a graph G (cid:48) then G (cid:48) also satisfies P . Every monotone property has a thresholdprobability in Erdős–Rényi graphs [4]. However, since non-diagonality is not a monotone increasinggraph property, it is natural to seek what happens in the regime of p that Theorem 1.7 does notcover. The following theorem partially answers this question. Theorem 1.8.
Let ε > be fixed, and let G n,p be an Erdős–Rényi graph with parameters n and p . Then, p ≥ (cid:32) (3 + ε ) log nn (cid:33) / implies that G n,p is diagonal a.a.s. The behavior of the probability that G n,p is non-diagonal in the regime of p that both The-orems 1.7 and 1.8 do not cover should be studied as a further theme. At this moment, even theexistence of the threshold where G n,p again becomes diagonal is still unknown.Finally, we show the asymptotic behavior of each rank of magnitude homology around thethreshold probability. The following result can be regarded as a weak law of large numbers for therank of magnitude homology. Theorem 1.9.
Let k, (cid:96) ∈ N and p = cn − for some fixed c > . Let G n,p be an Erdős–Rényi graphwith parameters n and p . Then, lim n →∞ E [rk(MH k,(cid:96) ( G n,p ))] n = cδ k,(cid:96) , here δ k,(cid:96) is the Kronecker delta function. Moreover, for any ε > , lim n →∞ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) rk(MH k,(cid:96) ( G n,p )) n − cδ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) = 0 . Remark . Theorem 1.9 immediately implies that for any vertex x in G n,p , lim n →∞ E [rk(MH xk,(cid:96) ( G n,c/n ))] = cδ k,(cid:96) . Note that the value c appearing here coincides with the limit of the expected degree of x in G n,c/n . This means that E [rk(MH xk,(cid:96) ( G n,p ))] and E [(deg x ) δ k,(cid:96) ] are asymptotically equal. On theother hand, it is shown in [9] that rk(MH xk,(cid:96) ( T )) = (deg x ) δ k,(cid:96) for any tree T and its vertex x .Therefore, E [rk(MH xk,(cid:96) ( G n,p ))] and rk(MH xk,(cid:96) ( T )) depend only on the degree of x asymptotically.This property is compatible with the fact that G n,c/n has locally tree-like structure.The magnitude G ( q ) of a graph G , which takes value in the formal power seriese Z [[ q ]] , isdetermined by the magnitude homology of G (cf. [9, Theorem 2.8]): G ( q ) = ∞ (cid:88) (cid:96) =0 (cid:32) (cid:96) (cid:88) k =0 ( − k rk(MH k,(cid:96) ( G )) (cid:33) q (cid:96) . For (cid:96) ≥ , define χ (cid:96) ( G ) as the coefficient of q (cid:96) in the above equation. Then, the following corollaryof Theorem 1.9 immediately follows. Corollary 1.11.
Let (cid:96) ∈ N and p = cn − for some fixed c > . Let G n,p be an Erdős–Rényi graphwith parameters n and p . Then, lim n →∞ E [ χ (cid:96) ( G n,p )] n = ( − (cid:96) c. Moreover, for any ε > , lim n →∞ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) χ (cid:96) ( G n,p ) n − ( − (cid:96) c (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) = 0 . This article is organized as follows. In Section 2, we briefly review some basic definitions of themagnitude homology of graphs. In Section 3, we study the magnitude homology of graphs and itsdiagonality from a viewpoint of girth. We use algebraic Morse theory and combinatorial argumentson graphs. Finally, in Section 4, we study the magnitude homology of Erdős–Rényi graphs usingtheorems obtained in Section 3 together with classical results on random graphs.
Acknowledgement
The first author is supported by RIKEN Center for Advanced Intelligence Project (AIP). Thesecond author is supported by JST CREST Mathematics (15656429), JSPS Grant-in-Aid for Sci-entific Research (A) (20221963), and JSPS Grant-in-Aid for Challenging Research (Exploratory)(19091210). The third author is supported by JSPS KAKENHI Grant Number 19J11237.
In this section, we recall some definitions of the magnitude homology of graphs. A finite simple undirected graph without loops is a pair of a nonempty finite set V and a collection E of subsets in V of cardinality two. We regard V and E as a vertex set and an edge set, respectively.Throughout this article, we call a finite simple undirected graph without loops just a graph. Below,we describe some notation and terminology for a given graph G = ( V ( G ) , E ( G )) . Definition 2.1.
We say that x ∈ V ( G ) is adjacent to y ∈ V ( G ) if { x, y } ∈ E ( G ) , and denote x ∼ y . For x ∈ V ( G ) , the degree deg x indicates the number of vertices that are adjacent to x .4 efinition 2.2. A tuple ( x , x , . . . , x k ) ∈ V ( G ) k +1 is called a path between x, y ∈ V ( G ) if x = x , x k = y , and x i − ∼ x i for all i = 1 , , . . . , k . A graph G is said to be connected if for any twovertices x, y ∈ V ( G ) , there exists a path between x and y . Definition 2.3.
Let i ≥ . An i - cycle or cycle in a graph G is a tuple ( x , . . . , x i ) of vertices in G satisfying• { x k , x k +1 } ∈ E ( G ) for ≤ k ≤ i − ,• x = x i ,• x , . . . , x i − are all distinct. Definition 2.4. A tree is a connected graph that has no cycles, while a connected graph that hasexactly one cycle is called a unicyclic graph .For vertices x, y ∈ V ( G ) , an extended metric d ( x, y ) is defined as the length of shortest pathbetween x and y , and if there exist no such paths, we set d ( x, y ) = ∞ . Let G = ( V ( G ) , E ( G )) be a graph. For a tuple ( x , x , . . . , x k ) ∈ V ( G ) k +1 , we define L ( x , x , . . . , x k ) := k (cid:88) i =1 d ( x i − , x i ) . Let (cid:96) ∈ Z ≥ be fixed, and for any k ∈ Z ≥ , we define a free Z -module MC k,(cid:96) ( G ) generated by a set { ( x , x , . . . , x k ) ∈ V ( G ) k +1 | x (cid:54) = x (cid:54) = · · · (cid:54) = x k , L ( x , . . . , x k ) = (cid:96) } . We note from the definition that MC k,(cid:96) ( G ) = 0 for k > (cid:96) . We can decompose MC k,(cid:96) ( G ) intospatially localized versions as follows. For any k ∈ Z ≥ and x, y ∈ V ( G ) , we define free Z -modules MC xk,(cid:96) ( G ) and MC x,yk,(cid:96) ( G ) generated by sets { ( x , x , . . . , x k ) ∈ V ( G ) k +1 | x = x (cid:54) = x (cid:54) = · · · (cid:54) = x k , L ( x , . . . , x k ) = (cid:96) } , and { ( x , x , . . . , x k ) ∈ V ( G ) k +1 | x = x (cid:54) = x (cid:54) = · · · (cid:54) = x k = y, L ( x , . . . , x k ) = (cid:96) } , respectively. Then we have obvious decompositions MC k,(cid:96) ( G ) ∼ = (cid:77) x ∈ V ( G ) MC xk,(cid:96) ( G ) ∼ = (cid:77) x,y ∈ V ( G ) MC x,yk,(cid:96) ( G ) . (2.1) Definition 2.5.
Given ( x , . . . , x i , . . . , x k ) ∈ MC k,(cid:96) ( G ) , we say that x i is a smooth point of ( x , . . . , x i , . . . , x k ) if L ( x , . . . , x k ) = L ( x , . . . , ˆ x i , . . . , x k ) ,that is, d ( x i − , x i +1 ) = d ( x i − , x i ) + d ( x i , x i +1 ) . Here, the hat symbol over x i indicates that this vertex is deleted from ( x , . . . , x i , . . . , x k ) . We saythat x i is a singular point of ( x , . . . , x i , . . . , x k ) if it is not a smooth point of ( x , . . . , x i , . . . , x k ) .For k ≥ , the boundary map ∂ k,(cid:96) ( G ) : MC k,(cid:96) ( G ) → MC k − ,(cid:96) ( G ) is defined as the linearextension of ∂ k,(cid:96) ( G )( x , . . . , x k ) = k − (cid:88) i =1 ( − i { x i is smooth } ( x , . . . , ˆ x i , . . . , x k ) for ( x , . . . , x k ) ∈ MC k,(cid:96) ( G ) . By convention, we also define MC − ,l ( G ) = 0 and ∂ ,l ( G ) = 0 . Then,it holds that ∂ k,(cid:96) ( G ) ◦ ∂ k +1 ,(cid:96) ( G ) = 0 for k ≥ , that is, ker ∂ k,(cid:96) ( G ) ⊃ Im ∂ k +1 ,(cid:96) ( G ) . The magnitudehomology group MH k,(cid:96) ( G ) of length (cid:96) is defined by MH k,(cid:96) ( G ) := ker ∂ k,(cid:96) ( G ) / Im ∂ k,(cid:96) ( G ) .5bviously, the boundary maps are compatible with the decompositions (2.1). Hence it inducesthe decompositions MH k,(cid:96) ( G ) ∼ = (cid:77) x ∈ V ( G ) MH xk,(cid:96) ( G ) ∼ = (cid:77) x,y ∈ V ( G ) MH x,yk,(cid:96) ( G ) . (2.2)Note that, if x and y are adjacent, we have a tuple ( x, y, x, . . . ) which is a homology cycle in MH x(cid:96),(cid:96) ( G ) . Hence we have rk(MH x(cid:96),(cid:96) ( G )) ≥ deg x . In particular, rk(MH (cid:96),(cid:96) ( G )) ≥ E ( G ) holdsfrom Eq. (2.2). Example 2.6 ([9, Corollary 6.8]) . Let T be a tree, and x ∈ V ( T ) be fixed. Then we have MH xk,(cid:96) ( T ) (cid:39) Z , k = (cid:96) = 0 , Z deg x , k = (cid:96) ≥ , , k (cid:54) = (cid:96). This is verified by using Mayer–Vietoris Theorem in [9, Theorem 6.6] after checking that it iscompatible with the decompositions (2.2). Moreover, Eq. (2.2) yields MH k,(cid:96) ( T ) (cid:39) Z V ( T ) , k = (cid:96) = 0 , Z E ( T ) , k = (cid:96) ≥ , , k (cid:54) = (cid:96). Definition 2.7 ([9, Definition 7.1]) . A graph G is called diagonal if MH k,(cid:96) ( G ) = 0 for k (cid:54) = (cid:96) . Definition 2.8 ([8, Definition 4.2]) . A graph of diameter at most two is called pawful if anydistinct vertices x, y, z ∈ V ( G ) with d ( x, y ) = d ( y, z ) = 2 and d ( z, x ) = 1 have a common neighbor.Here, for S ⊂ V ( G ) , a vertex w ∈ V ( G ) is said to be a common neighbor of S if w is adjacent toall the vertices in S . Example 2.9.
Trees are diagonal, as seen in Example 2.6. Join graphs, in particular completegraphs, are also diagonal [9, Theorem 7.5]. Moreover, pawful graphs are diagonal [8, Theorem 4.4].
In this section, we study algebraically the magnitude homology of graphs. First in Section 3.1, webriefly review algebraic Morse theory, which is a crucial tool for the latter parts. In Sections 3.2and 3.3, we compute the ( (cid:96) − i, (cid:96) ) -part MH (cid:96) − i,(cid:96) ( G ) of magnitude homology for a general graph G and for some ≤ i ≤ (cid:96) − . In Section 3.4, we give a criterion for graphs to be diagonal. All themain results proved in this section, especially Theorems 1.3 and 1.5, will be key lemmas for theprobabilistic study of magnitude homology in Section 4. For our computation, we use algebraic Morse theory studied in [15]. The matching that we con-struct is quite similar to that of Gu’s ([8]), while he constructs matchings for several special graphsin [8]. In this subsection, we briefly review the algebraic Morse theory. It is almost the sameinstruction as in [8], and see [15] for the detail.Let C ∗ = ( C ∗ , ∂ ∗ ) be a chain complex of finite rank free Z -modules. We set C k = (cid:77) α ∈ I k C k,α ∼ = (cid:77) α ∈ I k Z for each k ≥ . We denote differentials restricted to each component as f βα : C k +1 ,α (cid:44) → C k +1 ∂ k +1 −−−→ C k (cid:16) C k,β . Let Γ C ∗ be a directed graph whose vertex set is (cid:96) k I k , and directed edges are { α → β | f βα (cid:54) = 0 } .Recall that a matching of a directed graph is a subset M of the edge set such that any two distinctedges in M have no common vertices. For a matching M of Γ C ∗ , we define a new directed graph Γ MC ∗ by inverting the direction of all edges in M .6 efinition 3.1. The matching M is called Morse matching if the directed graph Γ MC ∗ is acyclic,and all homomorphisms of the form f βα : C k +1 ,α (cid:44) → C k +1 ∂ k +1 −−−→ C k (cid:16) C k,β corresponding to the edges in M are isomorphisms.Here we remark that Γ MC ∗ is acyclic if and only if there are no closed paths in Γ MC ∗ of the form a −→ b −→ · · · −→ b p − −→ a p = a with a i ∈ C k +1 and b i ∈ C k for some k . Theorem 3.2 ([15]) . For a Morse matching M , the chain complex C ∗ is homotopy equivalent tothe chain complex ˚ C ∗ defined as follows : Let ˚ I k be a subset of I k which consists of vertices containedin no edges in M . We define ˚ C k = (cid:77) α ∈ ˚ I k C k,α for each k ≥ . For each α ∈ ˚ I k and β ∈ ˚ I k − , let Γ Mα,β be the set of paths in Γ MC ∗ connecting α and β in this order. For γ ∈ Γ Mα,β , we define ˚ ∂ k : C k,α → C k − ,β as ˚ ∂ γ = ( − i/ f βv i ◦ f − v i v i − ◦ · · · ◦ f − v v ◦ f v α , where γ = ( α → v → · · · → v i → β ) . Then the differential ˚ ∂ k restricted on C k,α for α ∈ ˚ I k isdefined as ˚ ∂ k | C k,α = (cid:88) β ∈ ˚ I k − ,γ ∈ Γ Mα,β ˚ ∂ γ . In particular, we have ˚ ∂ k = 0 if the original differential ∂ k vanishes on ˚ I k . In this subsection, we study the diagonal part ( ( (cid:96), (cid:96) ) -part) of magnitude homology. In the following,we assume that (cid:96) ≥ unless otherwise noted. We first recall the definition of the local girth of agraph at a fixed vertex, as seen in the introduction. Definition 3.3.
Let G be a graph and x ∈ V ( G ) be a vertex. We define the local girth of G at x by gir x ( G ) := inf { i ≥ | there exists an i -cycle in G containing x } . We also deine the girth of G by gir( G ) := min x gir x ( G ) Our subject in this subsection is to prove Theorem 1.1. We use the algebraic Morse theory forthe proof. Let us consider a truncated chain complex −→ MC x(cid:96),(cid:96) ( G ) −→ MC x(cid:96) − ,(cid:96) ( G ) −→ and denote it by C ∗ . It is easy to see that the first homology of C ∗ is isomorphic to MH x(cid:96),(cid:96) ( G ) . Forgraphs that have neither - nor -cycles containing x as their vertex, we give a Morse matching to C ∗ . In the following, we give a Morse matching to C ∗ with gir x ( G ) ≥ . Lemma 3.4.
Let (cid:96) ≥ and i ≥ . Let G be a graph with gir x ( G ) ≥ for a vertex x ∈ V ( G ) . Let ( x = x , . . . , x (cid:96) ) ∈ MC x(cid:96),(cid:96) ( G ) be a chain, and suppose that x j is its singular point for ≤ j ≤ i − . Then x i ∈ { x , x } . roof. We prove by induction on i . For i = 1 , the statement is trivially true. Suppose that x j issingular for ≤ j ≤ i − and x j ∈ { x , x } for ≤ j ≤ i − . Then we have { x i − , x i − } = { x , x } because x i − (cid:54) = x i − . Note here that we have d ( x k , x k +1 ) = 1 for ≤ k ≤ (cid:96) − by the definitionof MC (cid:96),(cid:96) ( G ) . Then by the assumption that x i − is a singular point, we have d ( x i − , x i ) ≤ . If wehave d ( x i − , x i ) = 1 , then these three points x i − , x i − , x i form a -cycle containing x because x i − or x i − coincides with x , which is not the case (see Figure 2). Hence we obtain that d ( x i − , x i ) = 0 ,which implies that x i = x i − ∈ { x , x } . x i − x i − x i Figure 2: An illustration of a -cycle containing x in the case that d ( x i − , x i ) = 1 .Let T (cid:96) be a subset of generators in MC x(cid:96),(cid:96) ( G ) defined as T (cid:96) = (cid:8) ( x , . . . , x (cid:96) ) ∈ MC x(cid:96),(cid:96) ( G ) | some x i ’s are smooth for ≤ i ≤ (cid:96) (cid:9) . Whenever T (cid:96) (cid:54) = ∅ , we define a map f (cid:96) : T (cid:96) −→ MC x(cid:96) − ,(cid:96) ( G ) by deleting the first smooth point, thatis, f (cid:96) ( x , . . . , x (cid:96) ) = ( x , . . . , ˆ x i , . . . , x (cid:96) ) , where x j is a singular point of ( x , . . . , x (cid:96) ) for ≤ j ≤ i − , and x i is its smooth point. Lemma 3.5. If gir x ( G ) ≥ , the above map f (cid:96) is injective.Proof. Suppose that f (cid:96) ( x , . . . , x (cid:96) ) = ( x , . . . , ˆ x i , . . . , x (cid:96) ) = ( y , . . . , ˆ y j , . . . , y (cid:96) ) = f (cid:96) ( y , . . . , y (cid:96) ) . Then we have d ( x i − , x i +1 ) = 2 and d ( y j − , y j +1 ) = 2 . Because the other pairs of adjacent pointsare apart from each other by distance , we obtain i = j . If i = j ≥ , then we have x k , y k ∈{ x , x } = { y , y } for ≤ k ≤ i = j by Lemma 3.4. Then we obtain x i = y i by the assumptionthat x i − = y i − , that is, ( x , . . . , x (cid:96) ) = ( y , . . . , y (cid:96) ) . Suppose that i = j = 1 and x (cid:54) = y . Thenwe have x = y , x = y , d ( x , x ) = 2 , and d ( x , x ) = d ( x , x ) = d ( x , y ) = d ( y , x ) = 1 (seeFigure 3). Hence these four points form a -cycle containing x , which is not the case. Thus weobtain x = y , that is, ( x , . . . , x (cid:96) ) = ( y , . . . , y (cid:96) ) . x = y x y x = y Figure 3: An illustration of a -cycle containing x .By Lemma 3.5, we can define a matching M f (cid:96) to C ∗ by the injective map f (cid:96) . When T (cid:96) is empty,we define the empty matching. Lemma 3.6. If gir x ( G ) ≥ , then the above matching M f (cid:96) is a Morse matching. roof. Let ( x , . . . , ˆ x i , . . . , x (cid:96) ) ∈ MC x(cid:96) − ,(cid:96) ( G ) , where x i is a smooth point of the tuple ( x , . . . , x (cid:96) ) ∈ MC x(cid:96),(cid:96) ( G ) , but not the first one. Note that i ≥ . We show that the tuple ( x , . . . , ˆ x i , . . . , x (cid:96) ) is not in theimage of f (cid:96) , which implies that the directed graph Γ M f(cid:96) C ∗ is acyclic. Suppose that f (cid:96) ( y , . . . , y (cid:96) ) =( x , . . . , ˆ x i , . . . , x (cid:96) ) , and let y j be the first smooth point of the tuple ( y , . . . , y (cid:96) ) . Then we have ( y , . . . , ˆ y j , . . . , y (cid:96) ) = ( x , . . . , ˆ x i , . . . , x (cid:96) ) , hence we have i = j by the same argument in the proof of Lemma 3.5. Then we also have x i (cid:54) = y j .Because y j is the first smooth point, we have { y , . . . , y j } = { y , y } by Lemma 3.4. By theassumption that x k = y k for ≤ k ≤ i − , we obtain that y i = y i − = x i − . Because y i isadjacent to y i +1 = x i +1 , we have d ( x i − , x i +1 ) = d ( x i +1 , x i ) = d ( x i , x i − ) = d ( x i − , x i − ) = 1 (seeFigure 4). Then there is a - or -cycle containing an edge { y i − , y i − } = { y , y } = { x , x } unlesswe have x i − = x i +1 . The former case contradicts that gir x ( G ) ≥ . The latter case contradictsthe fact that x i is a smooth point of ( x , . . . , x (cid:96) ) ∈ MC x(cid:96),(cid:96) ( G ) . x i − = y i − x i y i = y i − = x i − x i +1 = y i +1 Figure 4: An illustration of a - or -cycle containing x in the case that x i − (cid:54) = x i +1 . A -cycleappears when x i and y i are adjacent, otherwise a -cycle appears. Proof of Theorem 1.1.
By Lemma 3.6, the chain complex C ∗ is homotopy equivalent to the chaincomplex generated by the unmatched generators of the Morse matching M f (cid:96) . The unmatchedgenerators in MC x(cid:96),(cid:96) ( G ) are exactly the tuples that have only singular points, and by Lemma 3.4,they are of the form ( x, y, x, y, . . . ) , where y is adjacent to x . Because the differential of MC x ∗ ,(cid:96) ( G ) vanishes on these generators, MH x(cid:96),(cid:96) ( G ) is isomorphic to a free module generated by the tuples ofthe form ( x, y, x, y, . . . ) . This completes the proof. We extend our matching constructed above to a larger part of magnitude chain complex. For atuple ( x , . . . , x n ) ∈ V ( G ) n +1 , we call ( x g , x g +1 ) a gap if d ( x g , x g +1 ) ≥ , and we call it the firstgap if additionally d ( x j , x j +1 ) = 1 for ≤ j ≤ g − . For ≤ i ≤ (cid:96) − , let T (cid:96) − i be a subset of MC x(cid:96) − i,(cid:96) ( G ) defined as T (cid:96) − i := (cid:26) ( x , . . . , x (cid:96) − i ) ∈ MC x(cid:96) − i,(cid:96) ( G ) (cid:12)(cid:12)(cid:12)(cid:12) some x j ’s are smooth point for ≤ j ≤ g − ,where ( x g , x g +1 ) is the first gap (cid:27) for i ≥ , and the subset T (cid:96) defined in the previous subsection for i = 0 . We simply say that x j is the first smooth point before the first gap of ( x , . . . , x (cid:96) − i ) if x j with ≤ j ≤ g − is a smoothpoint and x k ’s are singular points for ≤ k ≤ j − , where ( x g , x g +1 ) is the first gap. For i = 0 ,we mean just the first smooth point. Whenever T (cid:96) − i (cid:54) = ∅ , we define a map f (cid:96) − i : T (cid:96) − i −→ MC x(cid:96) − i − ,(cid:96) ( G ) by deleting the first smooth point before the first gap, that is, f (cid:96) − i ( x , . . . , x (cid:96) − i ) = ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) , where x j is the first smooth point of ( x , . . . , x (cid:96) − i ) before the first gap. Note that our definitionof T (cid:96) − i ’s and f (cid:96) − i ’s contain those of f (cid:96) and T (cid:96) defined in the previous subsection, respectively, byconsidering i = 0 . The image of the map f (cid:96) − i is disjoint from the subset T (cid:96) − i − for ≤ i ≤ (cid:96) − since the deletion of a point by f (cid:96) − i makes a new first gap before which there exists no smoothpoints. 9 emma 3.7. If gir x ( G ) ≥ , then f (cid:96) − i is injective for ≤ i ≤ (cid:96) − .Proof. As shown in Lemma 3.5, f (cid:96) is injective. Hence, we assume that i ≥ . Suppose that f (cid:96) − i ( x , . . . , x (cid:96) − i ) = f (cid:96) − i ( y , . . . , y (cid:96) − i ) . By the same argument in the proof of Lemma 3.5, thepositions of the first smooth point and the first gap of the both tuples are same. By looking at theparts before the first gap, the statement follows from the same argument in the proof of Lemma3.5.By Lemma 3.7, we can define a matching M f ∗ of MC x ∗ ,(cid:96) ( G ) by the injective maps f ∗ =( f (cid:96) − i ) ≤ i ≤ (cid:96) − . In the following, we assume i to be in the range ≤ i ≤ (cid:96) − unless otherwisementioned. Lemma 3.8. If gir x ( G ) ≥ , then the above matching M f ∗ is a Morse matching.Proof. Let ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) ∈ MC x(cid:96) − i − ,(cid:96) ( G ) , where x j is a smooth point of the tuple ( x , . . . , x (cid:96) − i ) ∈ MC x(cid:96) − i,(cid:96) ( G ) , but not the first smooth point before the first gap. The case for i = 0 has been already consideredin Lemma 3.6, hence we assume i ≥ . Let ( x g , x g +1 ) be the first gap of the tuple ( x , . . . , x (cid:96) − i ) .If j = g or g + 1 , then ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) is not in the image of f (cid:96) − i . It is because the firstgap ( x g − , x g +1 ) or ( x g , x g +2 ) of ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) must satisfy that d ( x g − , x g +1 ) ≥ or d ( x g , x g +2 ) ≥ respectively, while the first gap of an image of f (cid:96) − i must have distance 2. For thecase that j ≤ g − , we can show that the tuple ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) is not in the image of f (cid:96) − i by the same argument in the proof of Lemma 3.6. Hence the remained case is that j ≥ g + 2 . Inthis case, if we have ( x , . . . , ˆ x j , . . . , x (cid:96) − i ) = f (cid:96) − i ( y , . . . , y (cid:96) − i ) , the tuple ( y , . . . , y (cid:96) − i ) must be of the form ( x , . . . , x g , y new , x g +1 , . . . , x j − , x j +1 , . . . , x (cid:96) − i ) with d ( x g , x g +1 ) = 2 and y new is the first smooth point before the first gap by the definition of f (cid:96) − i . Then the first gap ( y g (cid:48) , y g (cid:48) +1 ) of ( y , . . . , y (cid:96) − i ) satisfies g (cid:48) ≥ g + 1 . Hence there cannot be acycle of the form a −→ b −→ · · · −→ a p −→ b p −→ a , in Γ M f ∗ MC x ∗ ,(cid:96) ( G ) with a k ∈ MC x(cid:96) − i,(cid:96) ( G ) , b k ∈ MC x(cid:96) − i − ,(cid:96) ( G ) because the position of the first gap of a k moves backward. This completes the proof.By Lemma 3.8, we obtain a chain complex ( ˚MC x ∗ ,(cid:96) ( G ) , ˚ ∂ ∗ ,(cid:96) ) consisting of unmatched generatorsby the Morse matching M f ∗ , which is homotopy equivalent to the original magnitude chain complex (MC x ∗ ,(cid:96) ( G ) , ∂ ∗ ,(cid:96) ) . The following lemma characterizes the generators of ( ˚MC x ∗ ,(cid:96) ( G ) , ˚ ∂ ∗ ,(cid:96) ) . Lemma 3.9.
Let gir x ( G ) ≥ . A tuple ( x , . . . , x (cid:96) − i ) ∈ MC x(cid:96) − i,(cid:96) ( G ) is unmatched by the matching M f ∗ if and only if it satisfies one of the following conditions : (i) It has no gaps and no smooth points,(ii) It has the first gap ( x g , x g +1 ) with g ≥ and d ( x g , x g +1 ) ≥ such that there is no smoothpoint before the first gap,(iii) It has the first gap ( x g , x g +1 ) with g ≥ and d ( x g , x g +1 ) = 2 such that there is no smoothpoint before the first gap. Furthermore, every vertex z adjacent to both of x g and x g +1 is thesecond smooth point of ( x , . . . , x g , z, x g +1 , . . . , x (cid:96) − i ) ,(iv) It has the first gap ( x , x ) with d ( x , x ) ≥ .Proof. Let ( x , . . . , x (cid:96) − i ) ∈ MC x(cid:96) − i,(cid:96) ( G ) satisfy none of the above conditions. We will show that ( x , . . . , x (cid:96) − i ) is matched. If there is a smooth point before the first gap, then it is in T (cid:96) − i , henceit is matched. Hence we can suppose that ( x , . . . , x (cid:96) − i ) has the first gap ( x g , x g +1 ) with g ≥ and d ( x g , x g +1 ) = 2 such that there is no smooth point before the first gap, and furthermore, there10s a vertex z adjacent to x g and x g +1 such that z is the first smooth point before the first gap of ( x , . . . , x g , z, x g +1 , . . . , x (cid:96) − i ) . Then we have f (cid:96) − i +1 ( x , . . . , x g , z, x g +1 , . . . , x (cid:96) − i ) = ( x , . . . , x (cid:96) − i ) , hence it is matched. Therefore the above conditions are necessary to be unmatched. The sufficiencyis straightforward.Now we look at the differential ˚ ∂ ∗ ,(cid:96) on ˚MC x ∗ ,(cid:96) ( G ) . Lemma 3.10.
Let gir x ( G ) ≥ . Let α be a tuple satisfying one of the conditions in Lemma 3.9.Then there are no paths of length ≥ in Γ M f ∗ MC x ∗ ,(cid:96) ( G ) that start from α .Proof. Note that there exist no directed edges α −→ β such that α ∈ MC x(cid:96) − i,(cid:96) ( G ) , β ∈ MC x(cid:96) − i +1 ,(cid:96) ( G ) by Lemma 3.9. Hence, let α −→ β be a directed edge in Γ M f ∗ MC x ∗ ,(cid:96) ( G ) with α ∈ MC x(cid:96) − i,(cid:96) ( G ) , β ∈ MC x(cid:96) − i − ,(cid:96) ( G ) . In order that this directed edge is extended to a path of length , β mustbe in the image of f (cid:96) − i . Note that, in order to be in the image of f (cid:96) − i , β must have the first gapwith distance exactly . Hence α and β must be of the form α = ( x , . . . , x g , x g +1 , . . . , x k , . . . , x (cid:96) − i ) ,β = ( x , . . . , x g , x g +1 , . . . , ˆ x k , . . . , x (cid:96) − i ) , where ( x g , x g +1 ) is the first gap of α and β with g ≥ , d ( x g , x g +1 ) = 2 , and g + 2 ≤ k ≤ (cid:96) − i − .Further, α must satisfy (iii) of Lemma 3.9 by the assumption. Hence every vertex y adjacent toboth of x g and x g +1 is the second smooth point of the tuple ( x , . . . , x g , y, x g +1 , . . . , ˆ x k , . . . , x (cid:96) − i ) , which implies that β cannot be in the image of f (cid:96) − i . Hence the statement follows.We obtain the following by Lemma 3.10 and Theorem 3.2. Lemma 3.11.
Let gir x ( G ) ≥ . The differentials on ˚MC x ∗ ,(cid:96) ( G ) are restrictions of those on MC x ∗ ,(cid:96) ( G ) . Now we further construct a Morse matching for ( ˚MC x ∗ ,(cid:96) ( G ) , ˚ ∂ ∗ ,(cid:96) ) . Before that, we study someproperties of the unmatched tuples of the matching M f ∗ by the following three lemmas. Lemma 3.12.
Suppose that gir x ( G ) ≥ . Let ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) ∈ ˚MC x(cid:96) − i,(cid:96) ( G ) , which satisfies the condition (ii) or (iii) in Lemma 3.9 with the first gap ( x g , x g +1 ) , g ≥ . If x g is its smooth point, then x g − is a singular point of the tuple ( x , . . . , x g − , ˆ x g , x g +1 , . . . , x (cid:96) − i ) .Proof. By Lemma 3.4, we have x m = x m +2 and x m +1 = x m +3 for ≤ m ≤ m + 3 ≤ g . Since x g is a smooth point, we have that d ( x g − , x g +1 ) = d ( x g − , x g ) + d ( x g , x g +1 ) . Then we have that d ( x g − = x g , x g − ) + d ( x g − , x g +1 ) = d ( x g , x g +1 ) + 2 d ( x g − , x g ) > d ( x g − = x g , x g +1 ) . Hence x g − is a singular point of the tuple ( x , . . . , x g − , x g − , ˆ x g , x g +1 , . . . , x (cid:96) − i ) . Lemma 3.13.
Let ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) ∈ ˚MC x(cid:96) − i,(cid:96) ( G ) , which satisfies the condition (iii) in Lemma 3.9 with the first gap ( x g , x g +1 ) . If gir x ( G ) > , then x g is a smooth point of ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) .Proof. Note that x ∈ { x g − , x g } by the same argument as that in Lemma 3.4. Assume that x g isa singular point of ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) . Let z be a vertex adjacent to x g and x g +1 . Thenwe have d ( x g − , z ) = d ( x g − , x g ) + d ( x g , z ) = 2 so that it satisfies (iii) of Lemma 3.9. Hence wehave x g − (cid:54) = z . Since x g is a singular point of ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) , we have d ( x g − , x g +1 ) < d ( x g − , x g ) + d ( x g , x g +1 ) = 3 . If d ( x g − , x g +1 ) = 2 , then there exists a -cycle containing x because the point adjacent to x g − and x g +1 do not coincide with x g or z . This contradicts the assumption. If d ( x g − , x g +1 ) = 1 , thenthere exists a -cycle containing x . Further, we have d ( x g − , x g +1 ) (cid:54) = 0 because d ( x g , x g − ) = 1 and d ( x g , x g +1 ) = 2 . Therefore, we conclude that x g can never be a singular point.11 emma 3.14. Let i ≥ . Let ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) ∈ ˚MC x(cid:96) − i,(cid:96) ( G ) , which satisfies the condition (ii) or (iv) in Lemma 3.9 with the first gap ( x g , x g +1 ) , g ≥ . Supposethat x g is a singular point of ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) . If gir x ( G ) ≥ i + 4 , then ( x , . . . , x g , y, x g +1 , . . . , x (cid:96) − i ) ∈ ˚MC x(cid:96) − i +1 ,(cid:96) ( G ) , where y is taken as x g − for g ≥ and as an arbitrary vertex adjacent to x that lies in a shortestpath connecting x and x for g = 0 .Proof. Let x g = p −→ · · · −→ p d ( x g ,x g +1 ) = x g +1 be a shortest path connecting x g and x g +1 . When g = 0 , we can take y = p so that y becomesa smooth point. If we have g ≥ and p = x g − , then we can take y = p = x g − so that d ( x g , x g +1 ) = d ( x g , x g − ) + d ( x g − , x g +1 ) . Hence we suppose that g ≥ and p (cid:54) = x g − . Since x g is a singular point of ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) , there exist a shortest path x g − = q −→ · · · −→ q N = x g +1 with N < d ( x g , x g +1 ) and q (cid:54) = x g . Let j be the minimum number such that q j coincides withsome p m . Then x g −→ x g − = q −→ · · · −→ q j = p m −→ p m − −→ p = x g is a cycle of length < d ( x g , x g +1 )+2 because ( j, m ) (cid:54) = (1 , , (0 , . Note that we have d ( x g , x g +1 ) ≤ i + 1 because L ( x , . . . , x g , x g +1 , . . . , x (cid:96) − i ) = (cid:96) . Hence the obtained cycle has length < i + 4 . Since x , . . . , x g − are all singular points, we have x g − = x or x g = x by Lemma 3.4. Therefore thiscycle contains x as its vertex, it contradicts that gir x ( G ) ≥ i + 4 . Finally, we show that obtainedtuple ( x , . . . , x g , y, x g +1 , . . . , x (cid:96) − i ) is unmatched by the matching M f ∗ .• If d ( x g , x g +1 ) ≥ , then we have d ( y, x g +1 ) ≥ , hence it satisfies (ii) of Lemma 3.9.• If d ( x g , x g +1 ) = 3 and g = 0 , then we have d ( y, x ) = 2 . Let z be a vertex adjacent to bothof y and x . Then we must have d ( x , z ) = d ( x , y ) + d ( y, z ) because x = x and there is no -cycle containing x . Hence the tuple ( x , y, x , . . . , x (cid:96) − i ) satisfies (iii) of Lemma 3.9.• If d ( x g , x g +1 ) = 3 and g ≥ , then we have d ( y, x g +1 ) = 2 with y = x g − . Let z be a vertexadjacent to both of y and x g +1 . Then we must have d ( x g , z ) = d ( x g , y ) + d ( y, z ) becauseeither of x g or y = x g − coincides with x , and there are no -cycles containing x . Hence thetuple ( x , . . . , x g , y, x g +1 , . . . , x (cid:96) − i ) satisfies (iii) of Lemma 3.9.Now we consider the following truncated chain complex for i ≥ : −→ ˚MC x(cid:96),(cid:96) ( G ) −→ ˚MC x(cid:96) − ,(cid:96) ( G ) −→ · · · −→ ˚MC x(cid:96) − i − ,(cid:96) ( G ) −→ . We denote this chain complex by D ∗ in the following. Let U (cid:96) − j be the subset of generators of ˚MC x(cid:96) − j,(cid:96) ( G ) which consists of all the tuples satisfying (ii) or (iii) in Lemma 3.9 with smooth point x g . We define maps h (cid:96) − j : U (cid:96) − j −→ ˚MC x(cid:96) − j − ,(cid:96) ( G ) for ≤ j ≤ i by h (cid:96) − j ( x , . . . , x g , x g +1 , . . . , x (cid:96) − j ) = ( x , . . . , ˆ x g , x g +1 , . . . , x (cid:96) − j ) , where ( x g , x g +1 ) is the first gap. By Lemma 3.12, the image of h (cid:96) − j is disjoint from U (cid:96) − j − . Lemma 3.15.
Let i ≥ . If gir x ( G ) ≥ i + 5 , then h (cid:96) − j is injective for ≤ j ≤ i .Proof. Suppose that h (cid:96) − j ( x , . . . , x (cid:96) − j ) = h (cid:96) − j ( y , . . . , y (cid:96) − j ) . We can verify that the position ofthe first gaps of ( x , . . . , x (cid:96) − j ) and ( y , . . . , y (cid:96) − j ) are identical in the same manner as in Lemma 3.5.Then we have x k = y k except for k = g , where ( x g , x g +1 ) and ( y g , y g +1 ) are the first gaps. Since x k and y k are singular points of ( x , . . . , x (cid:96) − j ) and ( y , . . . , y (cid:96) − j ) , respectively, for ≤ k ≤ g − , we12ave { x , . . . , x g } = { x , x } and { y , . . . , y g } = { y , y } by Lemma 3.4. Hence we obtain x g = y g if g ≥ . Suppose that g = 1 and x (cid:54) = y . Since x and y are smooth points of ( x , . . . , x (cid:96) − j ) and ( y , . . . , y (cid:96) − j ) , respectively, there exist shortest paths x = x −→ x −→ · · · −→ x = y and x = y −→ y −→ · · · −→ x = y of length d ( x , x ) = 1+ d ( y , y ) ≤ j +2 . Then there exists a cycle of length ≤ j +2) ≤ i +4 containing x as its vertex, which contradicts the assumption. Hence we obtain that ( x , . . . , x (cid:96) − j ) =( y , . . . , y (cid:96) − j ) .By Lemmas 3.12 and 3.15, we can define a matching M h ∗ of D ∗ by injective maps h ∗ =( h (cid:96) − j ) ≤ j ≤ i . Lemma 3.16.
Let i ≥ . If gir x ( G ) ≥ i + 5 , then the above matching M h ∗ is a Morse matching.Proof. By Lemma 3.11, any differentials corresponding to edges in M h ∗ are isomorphisms (cf.Definition 3.1). Let ( x , . . . , x g , x g +1 , . . . , x (cid:96) − j ) ∈ ˚MC x(cid:96) − j,(cid:96) ( G ) with the first gap ( x g , x g +1 ) , g ≥ . Let ( x , . . . , x g , x g +1 , . . . , x (cid:96) − j ) = a −→ b −→ a −→ b −→ · · · be a path in Γ M h ∗ D ∗ with a p ∈ ˚MC x(cid:96) − j,(cid:96) ( G ) and b p ∈ ˚MC x(cid:96) − j − ,(cid:96) ( G ) for p ∈ N . Here the directededge a p −→ b p corresponds to a directed edge in Γ M f ∗ C ∗ . Again by Lemma 3.11, b is obtained bydeleting some smooth point of a . Hence b must be of the form ( x , . . . , x g , x g +1 , . . . , ˆ x k , . . . , x (cid:96) − j ) with g + 1 ≤ k ≤ (cid:96) − j − , and x g must be its singular point to be in the image of h (cid:96) − j byLemma 3.12. It follows that a is of the form ( x , . . . , x g , y, x g +1 , . . . , ˆ x k , . . . , x (cid:96) − j ) , where ( y, x g +1 ) is the first gap. Inductively, we conclude that the first gap of a i moves backwardas i increases. Hence there cannot be any cycle in Γ M h ∗ D ∗ . Proof of Theorem 1.3.
By Lemma 3.16, the chain complex D ∗ is homotopy equivalent to the chaincomplex consisting of all the unmatched tuples by M h ∗ . By Lemma 3.13, any tuples satisfying thecondition (iii) in Lemma 3.9 are matched. By Lennma 3.14, any tuples satisfying the condition (ii)or (iv) in Lemma 3.9 are matched. Hence it turns out that the unmatched tuples by M h ∗ are onlythose satisfying the condition (i) of Lemma 3.9 except for the tuples in MC x(cid:96) − i − ,(cid:96) ( G ) . Hence thestatement follows. We devote this subsection to proving Theorem 1.5 which gives a criterion of the diagonality ofgraphs. First we recall the definition of the local girth of a graph at a fixed edge, as seen in theintroduction.
Definition 3.17.
Let G be a graph and e ∈ E ( G ) be an edge. We define the local girth of G at e by gir e ( G ) := inf { i ≥ | there exists an i -cycle in G containing e as its edge } . Proof of Theorem 1.5.
We first prove for the case that k is odd. We put k = 2 K + 1 . Let , , . . . , K + 1 be vertices of a (2 K + 1) -cycle with e = { , } . We suppose that each vertex i isadjacent to vertices i − and i + 1 , where we put K + 1 and K + 2 = 1 . Note that thedistance between each pair of vertices of this cycle in G is identical to that of the cycle graph itself.13f not, there will be cycles of length < K + 1 containing e , which contradicts the assumption. Inparticular, we have d (1 , K + 2) = d (2 , K + 2) = K . We show that the homology cycle [(1 , , K + 2)] ∈ MH ,K +22 ,K +1 ( G ) is non-trivial.Assume that we have [(1 , , K + 2)] = 0 , that is, there exist not necessarily distinct tuples α , . . . , α n ∈ MC ,K +23 ,K +1 ( G ) and a vertex a ∈ V ( G ) such that ∂ (cid:16) (1 , , a, K + 2) + ( − s α + · · · + ( − s n α n (cid:17) = (1 , , K + 2) . Here, s , . . . , s n ∈ { , } and we set s = 0 . Note that any tuples of the form (1 , a, , K + 2) do not appear in α i ’s, because L (1 , a, , K + 2) > K + 1 . We put α = (1 , , a, K + 2) and α i = (1 , x i , y i , K + 2) for i ∈ { , . . . , n } .Now we construct a graph A ( G ) with vertices { , a, x , y , . . . , x n , y n } . We span an edge between v, w if (1 , v, w, K +2) = α i or (1 , w, v, K +2) = α i for some i . Then we have the following lemma. Inthe following, we denote by (cid:104) v , . . . , v n (cid:105) a path in a graph consisting of edges { v , v } , . . . , { v n − , v n } in this order to make it easy to distinguish between paths and tuples. Lemma 3.18.
Let x be a vertex of A ( G ) which is connected to the vertex . Let (cid:104) , b , . . . , x (cid:105) bea shortest path in G connecting 1 and x . Then b = 2 .Proof. Let (cid:104) , a , a , . . . , x = a N (cid:105) be a path in A ( G ) connecting 2 and x . Note that a satisfiesthat d (1 ,
2) + d (2 , a ) + d ( a , K + 2) = K + 1 because (1 , , a , K + 2) = α m for some m . Let (cid:104) , b i , . . . , a i (cid:105) be a shortest path in G connecting 1 and a i . We show that b i = 2 by induction on i . If b (cid:54) = 2 , then a closed path obtained by concatenating three paths, (cid:104) , b , . . . , a (cid:105) , a shortestpath connecting a and 2, and the edge between 2 and 1 produces a cycle containing e . Note herethat the shortest path from to a does not pass through . If it goes through , then we have K +1 = d (1 , d (2 , a )+ d ( a , K +2) = 2+ d (1 , a )+ d ( a , K +2) ≥ d (1 , K +2) = K +2 . Because d (1 ,
2) + d (2 , a ) ≤ K , the obtained cycle is of length ≤ K , which contradicts the assumption.Hence we have b = 2 .Suppose b i = 2 and b i +11 (cid:54) = 2 . If (1 , a i , a i +1 , K + 2) = α m for some m , then a closed pathobtained by concatenating three paths, (cid:104) , b i , . . . , a i (cid:105) , a shortest path connecting a i and a i +1 ,and (cid:104) a i +1 , . . . , b i +11 , (cid:105) produces a cycle containing e . Note here that the shortest path from a i to a i +1 does not pass through in the same manner as discussed above. Because d (1 , a i ) + d ( a i , a i +1 ) ≤ K , the obtained cycle is of length ≤ K , which contradicts the assumption. Similarly,if (1 , a i +1 , a i , K + 2) = α m for some m , then a closed path obtained by concatenating three paths, (cid:104) , b i , . . . , a i (cid:105) , a shortest path connecting a i and a i +1 , and (cid:104) a i +1 , . . . , b i +11 , (cid:105) produces a cyclecontaining e . Because d (1 , a i +1 ) + d ( a i +1 , a i ) ≤ K , the obtained cycle is of length ≤ K , whichalso contradicts the assumption. Hence we have b i +11 = 2 .Now we divide the collection of tuples α = (1 , , a, K + 2) , α , . . . , α n into subcollections C , . . . , C M corresponding to the connected components of A ( G ) . Namely, two tuples α i and α j belong to thesame subcollection if the corresponding edges in A ( G ) are connected by some path. We supposethat (1 , , a, K + 2) ∈ C . Then we have ∂ (cid:16) (cid:88) i ≥ (cid:88) α j ∈ C i ( − s j α j (cid:17) = 0 . If not, there exists a tuple (1 , x, K + 2) (cid:54) = (1 , , K + 2) which appears in the left-hand side, andalso in ∂ (cid:16) (cid:80) α j ∈ C ( − s j α j (cid:17) with the opposite sign, because the total sum is (1 , , K + 2) . Thenit implies that the vertex x in A ( G ) belongs to two distinct connected components of A ( G ) , whichis a contradiction. Hence we have ∂ (cid:16) (cid:88) α j ∈ C ( − s j α j (cid:17) = (1 , , K + 2) , α m = (1 , x m , y m , K +2) ∈ C such that L (1 , x m , K +2) = K or L (1 , y m , K + 2) = K , because the right-hand side consists of odd terms. If L (1 , x m , K + 2) = K ,then a path in G obtained by concatenating a shortest path connecting and x m , and a shortestpath connecting x m and K + 2 is a shortest path connecting and K + 2 . Because a shortest pathconnecting and x m goes through by Lemma 3.18, we have d (2 , K + 2) = d (1 , K + 2) − d (1 ,
2) = K − , which is not true. We also have a contradiction from the same argument for the case that L (1 , y m , K + 2) = K . This completes a proof for the case that k is odd.Next we prove for the case that k is even. We put k = 2 K , and let , , . . . , K be vertices of K -cycle with e = { , } similarly to the odd case. Note that we have d (1 , K + 1) = K . We showthat the homology cycle [(1 , , K + 1) − (1 , K, K + 1)] ∈ MH ,K +12 ,K ( G ) is non-trivial. Assume thatwe have [(1 , , K + 1) − (1 , K, K + 1)] = 0 , that is, there exist tuples α , . . . , α n ∈ MC ,K +13 ,K ( G ) and vertices a, b ∈ V ( G ) such that ∂ (cid:16) (1 , , a, K + 1) + ( − s α + · · · + ( − s n α n − (1 , K, b, K + 1) (cid:17) = (1 , , K + 1) − (1 , K, K + 1) . Note that any tuples of the form (1 , a, , K + 1) and (1 , b, K, K + 1) do not appear in α i ’s, because L (1 , a, , K + 1) , L (1 , b, K, K + 1) > K . We put α = (1 , , a, K + 1) , α n +1 = (1 , K, b, K + 1) ,and α i = (1 , x i , y i , K + 2) for i ∈ { , . . . , n } . Similarly to the odd case, we construct a graph A ( G ) with vertices { , a, x , y , . . . , x n , y n , K, b } . Then the same statement in Lemma 3.18 holds. The proof is almost the same as that of Lemma3.18 as follows.
Proof of Lemma 3.18 for k = 2 K case. Let (cid:104) , a , a , . . . , x = a N (cid:105) be a path in A ( G ) connecting 2and x . Note that a satisfies that d (1 , d (2 , a )+ d ( a , K +1) = K because (1 , , a , K +1) = α m for some m . Let (cid:104) , b i , . . . , a i (cid:105) be a shortest path in G connecting 1 and a i . We show that b i = 2 byinduction on i . If b (cid:54) = 2 , then a closed path obtained by concatenating three paths, (cid:104) , b , . . . , a (cid:105) ,a shortest path connecting a and , and the edge between and produces a cycle containing e .Note here that the shortest path from to a does not pass through . If it goes through , thenwe have K = d (1 ,
2) + d (2 , a ) + d ( a , K + 1) = 2 + d (1 , a ) + d ( a , K + 1) ≥ d (1 , K + 1) = K + 2 .Because d (1 ,
2) + d (2 , a ) ≤ K − , the obtained cycle is of length ≤ K − , which contradicts theassumption. Hence we have b = 2 .Suppose b i = 2 and b i +11 (cid:54) = 2 . If (1 , a i , a i +1 , K + 1) = α m for some m , then a closed pathobtained by concatenating three paths, (cid:104) , b i , . . . , a i (cid:105) , a shortest path connecting a i and a i +1 , and (cid:104) a i +1 , . . . , b i +11 , (cid:105) produces a cycle containing e . Note here that the shortest path from a i to a i +1 does not pass through in the same manner as discussed above. Because d (1 , a i ) + d ( a i , a i +1 ) ≤ K − , the obtained cycle is of length ≤ K − , which contradicts the assumption. Similarly, if (1 , a i +1 , a i , K + 2) = α m for some m , then a closed path obtained by concatenating three paths, (cid:104) , b i , . . . , a i (cid:105) , a shortest path connecting a i and a i +1 , and (cid:104) a i +1 , . . . , b i +11 , (cid:105) produces a cyclecontaining e . Because d (1 , a i +1 ) + d ( a i +1 , a i ) ≤ K − , the obtained cycle is of length ≤ K − ,which also contradicts the assumption. Hence we have b i +11 = 2 .Now we can show that the vertices and b in A ( G ) belong to the same connected componentas follows. Divide the collection of tuples (1 , , a, K + 1) , α , . . . , α n , (1 , K, b, K + 1) into subcollections C , . . . , C M corresponding to the connected components of A ( G ) . Suppose that (1 , , a, K + 1) ∈ C and (1 , K, b, K + 1) ∈ C . By the same argument as that in the odd case, wehave ∂ (cid:16) (cid:88) i ≥ (cid:88) α j ∈ C i ( − s j α j (cid:17) = 0 . Because d (1 , K + 1) = K , every tuple α i has no singular points other than the end points. Hencetwo chains ∂ (cid:16) (cid:80) α j ∈ C ( − s j α j (cid:17) and ∂ (cid:16) (cid:80) α j ∈ C ( − s j α j (cid:17) must have a common term up to sign.It contradicts the disconnectedness assumption for C and C , hence the vertices and b in A ( G ) belong to the same connected component. Since the tuple (1 , K, b, K + 1) has no singular points,a path in G obtained by concatenating the edge between and K , and a shortest path connecting K and b is a shortest path connecting and b . This is a contradiction because every shortest pathin G connecting and b passes through at the first step by Lemma 3.18 for k = 2 K case.15 Stochastic properties of magnitude homology
In this subsection, we provide the proof of Theorem 1.7. We first prove Theorem 1.7 (1) whichfollows from the fact that a.a.s. G n,p has no cycles whenever p = o ( n − ) . In what follows, for i ≥ , we denote by C i the number of i -cycles in G n,p . Proof of Theorem 1.7 (1) . For i ≥ , a straightforward calculation yields E C i ≤ (cid:18) ni (cid:19) i !2 i p i ≤ ( np ) i i . Indeed, there are (cid:0) ni (cid:1) ways of selecting i vertices of an i -cycle from n vertices, and to each selection,there are i ! / (2 i ) ways of choosing the edges of the i -cycle. Lastly, the probability that the chosen i edges are included in G n,p is p i because of the mutual independence of edge appearance. As seenin Example 2.6, all trees, or more generally forests, are diagonal. Therefore, we have P ( G n,p is non-diagonal ) ≤ P (cid:32) ∞ (cid:88) i =3 C i ≥ (cid:33) ≤ ∞ (cid:88) k =3 E C i ≤ ∞ (cid:88) i =3 ( np ) i i . In the second inequality, we use Markov’s inequality. The right-hand side converges to zero as n → ∞ , which completes the proof.We now turn to proving Theorem 1.7 (2) (3). For their proofs, we divide the concerned regimeof p into two parts:(1) p = cn − for some < c < ,(2) lim inf n →∞ np > and p = o ( n − / ) .We then discuss the asymptotic behavior of P ( G n,p is non-diagonal ) in each part in different ways.For the estimate of P ( G n,p is non-diagonal ) in part (1), we use the following lemma which statesthat almost all vertices belong to tree components and that there exist no components containingmore than one cycle. Let T ( G n,p ) denote the number of vertices in G n,p belonging to some treecomponent. Lemma 4.1 (Theorem 5.7 (ii) and Corollary 5.8 in [2]) . Let p = cn − for some fixed < c < .Then, E [ T ( G n,p )] = n − O (1) . In addition, every component is either tree or unicyclic a.a.s. The following lemma is also useful.
Lemma 4.2 (Corollary 4.9 in [2]) . Let p = cn − for some fixed c > . Then, for any m ≥ , ( C , C , . . . , C m ) → ( Z , Z , . . . , Z m ) in distribution as n → ∞ , where { Z i } mi =3 are mutually independent random variables, and each Z i follows the Poisson distri-bution with parameter c i / (2 i ) . In other words, for any m ≥ and ( a , a , . . . , a m ) ∈ Z m − ≥ , lim n →∞ P (( C , C , . . . , C m ) = ( a , a , . . . , a m )) = m (cid:89) i =3 { c i / (2 i ) } a i a i ! exp (cid:18) − c i i (cid:19) . Combining Lemmas 4.1 and 4.2, we obtain the estimate of P ( G n,p is diagonal ) in part (1) asfollows. Proposition 4.3.
Let p = cn − for some fixed < c < . Then, lim n →∞ P ( G n,p is diagonal ) = √ − c exp( c/ c / c / c / . Proof.
Let F and F denote the events that G n,p is diagonal and that G n,p does not contain anycycles of length at least , respectively. We additionally define E as the event that every componentin G n,p is either tree or unicyclic. We can confirm that every unicyclic component that has a cycleof length at least is non-diagonal. This follows from the Mayer–Vietoris Theorem for magnitudehomology [9, Theorem 6.6] combining with the fact that any cycle graphs of length at least are16on-diagonal (cf. [8, Theorems 4.6 and 4.8]). Therefore, we have E ∩ F ⊂ E ∩ F . On the otherhand, it holds that E ∩ F ⊃ E ∩ F by using again the Mayer–Vietoris Theorem with the fact thattree graphs and - or -cycle graphs are diagonal (cf. [9, Examples 2.5 and 5.4]). Consequently, weobtain E ∩ F = E ∩ F . Thus, it reduces to prove that lim n →∞ P ( F ) = √ − c exp( c/ c / c / c / . (4.1)Indeed, | P ( F ) − P ( F ) | = | P ( F \ E ) − P ( F \ E ) | ≤ P ( E c ) = o (1) from the second conclusion ofLemma 4.1.Now, let m ≥ be fixed, and let D denote the event that every cyclic component has at most m vertices. Then, we have P ( C = C = · · · = C m = 0) ≥ P ( F ) ≥ P ( { C = C = · · · = C m = 0 } ∩ D ) ≥ P ( C = C = · · · = C m = 0) − P ( D c ) . (4.2)From the first conclusion of Lemma 4.1, we can take a constant K , depending only on c , such that n − E [ T ( G n,p )] ≤ K for all n . Since the number of cyclic components that have more than m vertices is bounded above by { n − T ( G n,p ) } /m , we obtain P ( D c ) ≤ n − E [ T ( G n,p )] m ≤ Km using Markov’s inequality in the first inequality. Furthermore, Lemma 4.2 yields lim n →∞ P ( C = C = · · · = C m = 0) = m (cid:89) i =5 exp (cid:18) − c i i (cid:19) = exp (cid:32) − m (cid:88) i =5 c i i (cid:33) . Combining the above estimates with Eq. (4.2), we obtain exp (cid:32) − m (cid:88) i =5 c i i (cid:33) ≥ lim sup n →∞ P ( F ) ≥ lim inf n →∞ P ( F ) ≥ exp (cid:32) − m (cid:88) i =5 c i i (cid:33) − Km .
Eq. (4.1) follows from the equation above by taking m → ∞ , noting that exp (cid:32) − ∞ (cid:88) i =5 c i i (cid:33) = √ − c exp( c/ c / c / c / . For the estimate of P ( G n,p is non-diagonal ) in part (2), we use the following lemma. For agraph G , let us denote the number of connected components of G by ξ ( G ) . Lemma 4.4 ([6, Section 6]) . Let p = cn − for some fixed constant c > . Then, for any ε > , lim n →∞ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ξ ( G n,p ) n − u ( c ) (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) = 0 , where u ( c ) = 1 c ∞ (cid:88) i =1 i i − i ! ( ce − c ) i . Figure 5: Description of u ( c ) in Lemma 4.4.For a graph G , the circuit rank r ( G ) indicates the minimum number of edges that must beremoved from G to contain no cycles. As a well-known fact, it holds that r ( G ) = | E ( G ) | −| V ( G ) | + ξ ( G ) . 17 emma 4.5. Let p = cn − for some fixed constant c > . Then, there exists a constant δ > such that r ( G n,p ) ≥ δn a.a.s.Proof. We can verify that u ( c ) > − c/ whenever c > (see also Figure 5). Therefore, Lemma 4.4implies that for c > , there exists a constant δ > such that ξ ( G n,p ) ≥ (1 − c/ δ ) n a.a.s.Furthermore, since E ( G n,p ) follows the binomial distribution with parameters (cid:0) n (cid:1) and cn − , adirect computation yields E (cid:20) E ( G n,p ) n (cid:21) = c (cid:18) − n (cid:19) −−−−→ n →∞ c and Var (cid:18) E ( G n,p ) n (cid:19) = c n (cid:18) − n (cid:19)(cid:18) − cn (cid:19) −−−−→ n →∞ . Therefore, using the Minkowski inequality, E (cid:34)(cid:18) E ( G n,p ) n − c (cid:19) (cid:35) / ≤ E (cid:34)(cid:18) E ( G n,p ) n − E (cid:20) E ( G n,p ) n (cid:21)(cid:19) (cid:35) / + (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) E ( G n,p ) n (cid:21) − c (cid:12)(cid:12)(cid:12)(cid:12) = (cid:115) Var (cid:18) E ( G n,p ) n (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) E ( G n,p ) n (cid:21) − c (cid:12)(cid:12)(cid:12)(cid:12) −−−−→ n →∞ . (4.3)Thus, from Markov’s inequality, we have E ( G n,p ) ≥ ( c/ − δ ) n a.a.s. Combining these estimatesabove, we obtain a.a.s. r ( G n,p ) = E ( G n,p ) − n + ξ ( G n,p ) ≥ ( c/ − δ ) n − n +(1 − c/ δ ) n = δn .We now provide the estimate of P ( G n,p is non-diagonal ) in part (2). Proposition 4.6.
Let lim inf n →∞ np > and p = o ( n − / ) . Then, G n,p is non-diagonal a.a.s.Proof. Let X denote the number of edges e ∈ E ( G n,p ) such that gir e ( G n,p ) ∈ [5 , ∞ ) . FromTheorem 1.5, it suffices to prove that X ≥ a.a.s. We define Y as the number of edges that arecontained in some cycle. Then, Y ≥ r ( G n,p ) because of the definition of the circuit rank. Thus,by applying Lemma 4.5 with some fixed constant < c < lim inf n →∞ np , there exists a constant δ > such that Y ≥ r ( G n,p ) ≥ δn a.a.s. For i ≥ , we additionally define Y i as the number ofedges that are contained in some i -cycle. Then, P (cid:18) Y i > δ n (cid:19) ≤ δn E Y i ≤ iδn E C i ≤ iδn ( np ) i i = 32 δ n i − p i . The first inequality follows from Markov’s inequality. In the second inequality, we use a crudeestimate Y i ≤ iC i . Since p = o ( n − / ) , for i = 3 , , the right-hand side of the above equationconverges to zero as n → ∞ . Therefore, Y , Y ≤ δn/ a.a.s. Combining the estimates for Y , Y ,and Y , P (cid:18) X ≥ δ n (cid:19) ≥ P (cid:18) Y − Y − Y ≥ δ n (cid:19) ≥ P (cid:18) Y ≥ δn and Y , Y ≤ δ n (cid:19) −−−−→ n →∞ , which completes the proof.Combining Propositions 4.3 and 4.6, we obtain the conclusion of Theorem 1.7.Lastly, we prove Theorem 1.8. The notion of pawful graphs, introduced by Gu [8], is a key forthe proof. Recall from Definition 2.8 that a pawful graph G is a graph of diameter at most twosatisfying the property that for any distinct vertices x, y, z ∈ V ( G ) with d ( x, y ) = d ( y, z ) = 2 and d ( z, x ) = 1 , they have a common neighbor. Since pawful graphs are diagonal, the conclusion ofTheorem 1.8 follows immediately from the following Theorem. Theorem 4.7 ([10, Theorem 3.2]) . Let m ∈ N and ε > . Then, p ≥ (cid:18) ( m + ε ) log nn (cid:19) /m implies that every m vertices in G n,p have a common neighbor a.a.s. .2 Weak law of large numbers for the rank of magnitude homology In this subsection, we prove Theorem 1.9 using Theorem 1.3. We first give a general upper boundof the rank of magnitude homology of a graph.
Lemma 4.8.
Let G be a graph, and let x ∈ V ( G ) be fixed. Then, for any k, (cid:96) ∈ N , rk(MH xk,(cid:96) ( G )) ≤ (cid:18) (cid:96) − k − (cid:19)(cid:16) max y ∈ V ( G ) deg y (cid:17) (cid:96) . Proof.
Recall that the generator set of MC xk,(cid:96) ( G ) is (cid:40) ( x , x , . . . , x k ) ∈ V ( G ) k +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x (cid:54) = x (cid:54) = · · · (cid:54) = x k , k (cid:88) i =1 d ( x i − , x i ) = (cid:96) (cid:41) = (cid:71) ( (cid:96) ,(cid:96) ,...,(cid:96) k ) ∈ N k (cid:96) + (cid:96) + ··· + (cid:96) k = (cid:96) { ( x , x , . . . , x k ) ∈ V ( G ) k +1 | x = x, d ( x i − , x i ) = (cid:96) i for ≤ i ≤ k } . Noting that for any u ∈ V ( G ) and r ∈ N , { v ∈ V ( G ) | d ( u, v ) = r } ≤ (cid:16) max y ∈ V ( G ) deg y (cid:17) r , we have { ( x , x , . . . , x k ) ∈ V ( G ) k +1 | x = x, d ( x i − , x i ) = (cid:96) i for ≤ i ≤ k }≤ k (cid:89) i =1 (cid:16) max y ∈ V ( G ) deg y (cid:17) (cid:96) i = (cid:16) max y ∈ V ( G ) deg y (cid:17) (cid:96) for any ( (cid:96) , (cid:96) , . . . , (cid:96) k ) ∈ N k with (cid:96) + (cid:96) + · · · + (cid:96) k = (cid:96) . Furthermore, a simple combinatorialargument yields { ( (cid:96) , (cid:96) , . . . , (cid:96) k ) ∈ N k | (cid:96) + (cid:96) + · · · + (cid:96) k = (cid:96) } = (cid:18) (cid:96) − k − (cid:19) . Thus, we conclude that rk(MH xk,(cid:96) ( G )) ≤ rk(MC xk,(cid:96) ( G )) ≤ (cid:18) (cid:96) − k − (cid:19)(cid:16) max y ∈ V ( G ) deg y (cid:17) (cid:96) . The following lemma gives a useful upper bounds of the probability that a binomial distributedrandom variable is larger than expected.
Lemma 4.9 ([13, Lemma 1.1]) . Suppose N ∈ N , p ∈ (0 , , and < k < N . Let X be a binomialrandom variable with parameters N and p , and set µ := E X = N p . If k ≥ e µ , then P ( X > k ) ≤ exp (cid:18) − k (cid:18) kµ (cid:19)(cid:19) . In what follows, let the Erdős–Rényi graph G n,p be constructed on an n -vertex set V n , and let o ∈ V n be an arbitrarily fixed vertex. Lemma 4.10.
Let k, (cid:96) ∈ N be fixed. It holds that for sufficiently large n and any x ∈ V n , E [rk(MH xk,(cid:96) ( G n,p )) ] ≤ (cid:18) (cid:96) − k − (cid:19) (log n ) (cid:96) . Proof.
Let D be the event that the maximum degree of G n,p is at most (log n ) / . Then, P ( D c ) ≤ (cid:88) y ∈ V n P (cid:16) deg y > log n (cid:17) = n P (cid:16) deg o > log n (cid:17) . deg o follows the binomial distribution with parameters n − and cn − , and set µ := E [deg o ] = ( n − cn − . Applying Lemma 4.9 with N = n − , p = cn − , and k = (log n ) / , wehave P (cid:16) deg o > log n (cid:17) ≤ exp (cid:16) − log n (cid:16) log n µ (cid:17)(cid:17) ≤ exp (cid:16) −
15 log n log log n (cid:17) = n − (log log n ) / for sufficiently large n . Therefore, for sufficiently large n and any x ∈ V n , we obtain E (cid:2) rk(MH xk,(cid:96) ( G )) (cid:3) ≤ (cid:18) (cid:96) − k − (cid:19) E (cid:20)(cid:16) max y ∈ V n deg y (cid:17) (cid:96) (cid:21) ≤ (cid:18) (cid:96) − k − (cid:19) (cid:26) E (cid:20)(cid:16) max y ∈ V n deg y (cid:17) (cid:96) ; D (cid:21) + n (cid:96) P ( D c ) (cid:27) ≤ (cid:18) (cid:96) − k − (cid:19) (cid:26)(cid:16) log n (cid:17) (cid:96) + n (cid:96) +1 − (log log n ) / (cid:27) ≤ (cid:18) (cid:96) − k − (cid:19) (log n ) (cid:96) . In the first inequality, we use Lemma 4.8.We now trun to proving Theorem 1.9 using Theorem 1.3.
Proof of Theorem 1.9.
Since MH k,(cid:96) ( G n,p ) = 0 if (cid:96) < k , we assume that (cid:96) ≥ k . For i ≥ , define E xi as the event that G n,p has at least one i -cycle containing x , and set E x := (cid:96) − k )+4 (cid:91) i =3 E xi . Applying Theorem 1.3, we have rk(MH k,(cid:96) ( G n,p )) n = 1 n (cid:88) x ∈ V n rk(MH xk,(cid:96) ( G n,p )) ≤ n (cid:88) x ∈ V n (cid:8) (deg x ) δ k,(cid:96) + rk(MH xk,(cid:96) ( G n,p ))1 E x (cid:9) = 2 E ( G n,p ) n δ k,(cid:96) + 1 n (cid:88) x ∈ V n rk(MH xk,(cid:96) ( G n,p ))1 E x . On the other hand, since rk(MH (cid:96),(cid:96) ( G n,p )) ≥ E ( G n,p ) , we have rk(MH k,(cid:96) ( G n,p )) n ≥ E ( G n,p ) n δ k,(cid:96) . Combining these estimates, we obtain (cid:12)(cid:12)(cid:12)(cid:12) rk(MH k,(cid:96) ( G n,p )) n − E ( G n,p ) n δ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:88) x ∈ V n rk(MH xk,(cid:96) ( G n,p ))1 E x . Therefore, using the triangle inequality, E (cid:12)(cid:12)(cid:12)(cid:12) rk(MH k,(cid:96) ( G n,p )) n − cδ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:12)(cid:12)(cid:12)(cid:12) rk(MH k,(cid:96) ( G n,p )) n − E ( G n,p ) n δ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) + E (cid:12)(cid:12)(cid:12)(cid:12) E ( G n,p ) n δ k,(cid:96) − cδ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:88) x ∈ V n E [rk(MH xk,(cid:96) ( G n,p ))1 E x ] + E (cid:12)(cid:12)(cid:12)(cid:12) E ( G n,p ) n − c (cid:12)(cid:12)(cid:12)(cid:12) δ k,(cid:96) ≤ E [rk(MH ok,(cid:96) ( G n,p ))1 E o ] + E (cid:12)(cid:12)(cid:12)(cid:12) E ( G n,p ) n − c (cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:2) rk(MH ok,(cid:96) ( G n,p )) (cid:3) / P (cid:0) E o (cid:1) / + E (cid:34)(cid:18) E ( G n,p ) n − c (cid:19) (cid:35) / . (4.4)20n the last line, we use the Cauchy–Schwarz inequality. The second term of Eq. (4.4) converges tozero as n → ∞ , as seen in Eq. (4.3). For the estimate of the first term in Eq. (4.4), we define C oi as the number of i -cycles containing o . We then have P ( E oi ) = P ( C oi ≥ ≤ E C oi = ( n − n − · · · ( n − i + 1)2 (cid:16) cn (cid:17) i ≤ c i n from Markov’s inequality, which implies that P ( E o ) ≤ (cid:96) − k )+4 (cid:88) i =3 P ( E oi ) ≤ n (cid:96) − k )+4 (cid:88) i =3 c i . From the estimate above and Lemma 4.10, the first term of Eq. (4.4) converges to zero as n → ∞ .Consequently, we obtain lim n →∞ E (cid:12)(cid:12)(cid:12)(cid:12) rk(MH k,(cid:96) ( G n,p )) n − cδ k,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , which implies the first conclusion. Again from Markov’s inequality, the above equation also impliesthe second conclusion. References [1] Y. Asao and K. Izumihara,
Geometric approach to graph magnitude homology , to appear inHomology Homotopy Appl., 2020.[2] B. Bollobás,
Random Graphs , 2nd ed., Cambridge studies in advanced mathematics, Cam-bridge University Press, 2001.[3] R. Bottinelli and T. Kaiser,
Magnitude homology, diagonality, medianness, Künneth andMayer–Vietoris , arXiv:2003.09271, 2020.[4] B. Bollobás and A. Thomason,
Threshold functions , Combinatorica (1987), 35–38.[5] P. Erdős and A. Rényi, On random graphs , Publ. Math. Debrecen (1959), 290–297.[6] P. Erdős and A. Rényi, On the evolution of random graphs , Publ. Math. Inst. Hungarian Acad.Sci. (1960), 17–60.[7] E. N. Gilbert, Random graphs , Ann. Math. Statist. (1959), 1141–1144.[8] Y. Gu, Graph magnitude homology via algebraic Morse theory , arXiv:1809.07240, 2018.[9] R. Hepworth and S. Willerton,
Categorifying the magnitude of a graph , Homology HomotopyAppl. (2017), 31–60.[10] M. Kahle, Topology of random clique complexes , Discrete Math. (2009), 1658–1671.[11] T. Leinster,
The magnitude of a graph , Math. Proc. Cambridge Philos. Soc. (2019),247–264.[12] T. Leinster,
The magnitude of metric spaces , Doc. Math. (2013), 857–905.[13] M. Penrose, Random geometric graphs , volume 5 of Oxford Studies in Probability. OxfordUniversity Press, Oxford, 2003.[14] R. Sazdanovic and V. Summers,
Torsion in the Magnitude homology of graphs ,arXiv:1912.13483, 2019.[15] E. Sköldberg,
Morse theory from an algebraic viewpoint , Trans. Amer. Math. Soc.358