Complexes of Tournaments, Directionality Filtrations and Persistent Homology
CCOMPLEXES OF TOURNAMENTS, DIRECTIONALITY FILTRATIONSAND PERSISTENT HOMOLOGY
DEJAN GOVC, RAN LEVI, AND JASON P. SMITH
Abstract.
Complete digraphs are referred to in the combinatorics literature as tourna-ments. We consider a family of semi-simplicial complexes, that we refer to as “tournaplexes”,whose simplices are tournaments. In particular, given a digraph G , we associate with it a“flag tournaplex” which is a tournaplex containing the directed flag complex of G , but alsothe geometric realisation of cliques that are not directed. We define several types of filtra-tions on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexesprovide finer means of distinguishing graph dynamics than the directed flag complex. Wethen demonstrate the power of these ideas by applying them to graph data arising from theBlue Brain Project’s digital reconstruction of a rat’s neocortex. The idea of this article arose from considering topological objects associated to directedgraphs, or digraphs . Digraphs have been playing an ever increasing role in the study ofnetworks where connections between nodes have a prescribed direction. A notable exampleof such networks are neural networks, where the use of digraphs is particularly prevalent[18]. The digraphs we consider in this article are finite and simple i.e., loop-free and any pairof distinct vertices may be connected by reciprocal edges, but not by more than one edge inthe same direction.An n -clique is a complete graph on n -vertices. Any undirected graph G gives rise to asimplicial complex called the clique complex, or flag complex, where n -simplices are the ( n + 1) -cliques in G . The clique complex is a very common construction that can be foundin both theoretical and applied works and is highly prevalent in topological data analysis.However, in many contexts one needs to deal with digraphs rather than undirected graphs,in which case forgetting edge orientation will imply loss of information. One solution for thisproblem was proposed in [17] with the introduction of the directed flag complex.By analogy to the undirected case, a directed n -clique is a digraph, whose underlying undi-rected graph is an n -clique, and such that the orientation of its edges defines a linear orderon its vertices. Given a digraph G , one associates with it the directed flag complex [17], thatis, the complex whose n -simplices are the directed ( n + 1) -cliques in G . The directed flagcomplex was used successfully in [17] as a way of gaining information about structural andfunctional properties of the Blue Brain Project reconstruction of the neocortical microcir-cuitry of a juvenile rat [10]. However, while more naturally fitting for use with digraphs,this construction still raises two natural problems. The first is that, while edge orientationdictates which cliques are used in the construction of the complex, once the construction isdone, it plays no further role in computations. A second problem is that the directed flagcomplex does not give a proper topological representation to cliques that are not directed,i.e. those cliques whose vertices are not linearly ordered via the orientation of their edges.This leads to the central idea of this article, namely to the construction of a complex builtof all possible cliques in the digraph, while preserving partial information that is determinedby edge orientation. a r X i v : . [ m a t h . A T ] J a n DEJAN GOVC, RAN LEVI, AND JASON P. SMITH
Finite digraphs whose underlying undirected graphs are cliques are referred to in thecombinatorics literature as tournaments . Tournaments have been studied since the 1940’sby many authors from an applicable point of view [1, 5, 12, 6], as well as theoretically [11].Tournaments naturally lend themselves to be considered as standard simplices, since aninduced subgraph of a tournament is clearly also a tournament. This lead us to the ideathat there is a natural family of complexes that are built out of tournaments. We refer tosuch complexes as tournaplexes . The directed flag complex of a digraph, and in fact anyordered simplicial complex is in particular a tournaplex. By analogy to the flag complex of agraph and the directed flag complex of a digraph, we introduce the flag tournaplex associatedto a digraph. The flag tournaplex of a digraph G contains the directed flag complex of G asa subcomplex, as well as other naturally occurring sub complexes.A key advantage of the flag tournaplex, over other combinatorial constructions one canassociate to a digraph, is that it invites the use of persistent homology [2], where the fil-tration parameters arise naturally from the digraph in question, without any extra weights.Furthermore, this can be done in a number of different ways, which makes the flag tour-naplex, at least in theory, a more powerful instrument by which one can study propertiesof digraphs and networks. The idea arose from considering a simple numerical invariant oftournaments that was introduced in [17] - the directionality of a tournament.In this paper we define three types of directionality, two that are local in nature and donot depend on an ambient tournaplex the tournament is a part of, and a third that doesdepend on the containing tournaplex. Local directionality invariants divide tournaments intoequivalence classes, and since they are always defined by means of the orientation of the edgesin a tournament, they could potentially provide an efficient dimension reduction method instudying dynamics on a digraph. Since the number of non-isomorphic n -tournaments growsvery rapidly with n , such a method is essential for any applicable purposes. However, inthis work we concentrate on the filtrations of tournaplexes that arise from directionalityinvariants, examine the way in which certain simple examples behave with respect to thosefiltrations, and apply the technique to some more involved data sets. Thus, we aim toconvince the reader that the flag tournaplex of a digraph, combined with its directionalityinvariants and persistent homology, are a powerful tool in the study of neuroscience, networktheory, and in general any subject that involves digraphs. Furthermore, we show that thesemethods have the potential to reveal more information about a digraph than some other,more standard, topological tools.After introducing the concept of a tournaplex in Section 1, we proceed in Section 2 bydefining several types of directionality. We mainly discuss three types: local directionality,3-cycle directionality and global directionality. We concentrate on the first two, which areclosely related to each other, as demonstrated in Proposition 2.3. We then discuss somebasic properties of local and 3-cycle directionality, and in particular derive in Proposition2.10 a probabilistic formula for the expectation of tournaments of a given order (number ofvertices) and local directionality in a random digraph.In Section 3 we show how directionality can be used to define filtrations on tournaplexes.In particular we produce simple examples that show the potential these filtrations have todistinguish tournaplexes with the use of persistent homology in cases where the homotopytypes of the corresponding geometric realisations are identical. We note that each isomor-phism type of a tournament gives rise to a directionality invariant on tournaplexes. Thisis similar to the idea behind 3-cycle directionality, which is basically an invariant that as-sociates with a tournament the number of regular 3-tournaments it contains (See Remark OMPLEXES OF TOURNAMENTS 3 p and q . We show that applying the method to 200 such random graphswith a fixed value of p and four different values of q , we obtain an almost perfect separationof the graphs into four families using a technique as simple as k -means clustering. This isused as a test case, which shows that the technique we develop here is indeed sensitive tonetwork structure.A second example is taken from data that was generated by the Blue Brain Project forthe paper [17]. Here we have response signals of a digital reconstruction of neocortical tissueof a rat [10] to a collection of stimuli that depend on two parameters and appear in 5 seeds(repeats of the same experiment). This gives 45 spike train datasets, which we then use togenerate a family of 450 tournaplexes. Once more, k -means clustering is applied to separatethe signals almost perfectly with respect to one of the two parameters and across all seeds.In both cases we compare the performance of this technique to that of a similar approachusing only the Betti numbers of the corresponding directed flag complex (as was done, usinga much larger spiking dataset and without an attempt at classification, in [17]), and observethat using tournaplexes with local directionality filtration gives much better results. SeeSection 4.2 for more detail about the Blue Brain Project model, and the data used in thisexample.We emphasise that the methods proposed in this article are applicable to any systemor phenomenon that can be encoded on a digraph. In many such applications researchersconsider motifs , namely subgraphs on a small number of vertices, as a means of characterisingvarious graph regimes and graph dynamics. Tournaments are nothing but motifs whoseunderlying undirected graph is a clique, and can thus be considered as building blocksof any other motif. Directionality invariants give ways of controlling the distribution oftournaments in a digraph, without getting lost in the abundance of their isomorphism types,and the associated filtrations provide the means of studying persistent homology on a digraphwith intrinsically defined weights. These features are independent of any specific applicationone may wish to apply the methods to, rather than only to neuroscience.Most of our computations were carried out by purpose designed modification of the soft-ware package Flagser [8], designed for computation of persistent homology of directedflag complexes. The package which we named
Tournser is designed specifically for thecomputation and manipulation of flag tournaplexes associated to digraphs.
Flagser and
DEJAN GOVC, RAN LEVI, AND JASON P. SMITH
Tournser are freely available at [7] and [20], respectively, and interactive online versionsare available at [22] and [21].As a “spin-off” project, the first named author investigated the homotopy types thatmay arise as directed flag complexes associated to tournaments. He showed in particularthat this gives an invariant of tournaments that provides a finer partition within a givendirectionality class of tournaments [3]. The methods in this paper were instrumental infinding the example in Section 3 demonstrating that a multi-parameter persistence moduleof tournaplexes associated to two or more directionality invariants can be more informativethan the the corresponding 1-parameter modules associated to each of them separately.The authors acknowledge support from EPSRC, grant EP/P025072/ - “Topological Anal-ysis of Neural Systems”, and from École Polytechnique Fédérale de Lausanne via a collab-oration agreement with the second named author. We also thank J¯anis Lazovskis, HenriRiihimäki and Pedro Conceição for useful discussions.1.
Tournaments and Tournaplexes
We start by defining the basic objects of study and some of their properties. All digraphsconsidered in this article are assume to be finite and simple. By simple we mean loop-free ,namely edges of the form ( v, v ) are not allowed for any vertex v , and if v and w are twodistinct vertices then a digraph may contain the reciprocal edges ( v, w ) and ( w, v ) , but twoedges in the same orientation between v and w are not allowed.If G = ( V, E ) and G (cid:48) = ( V (cid:48) , E (cid:48) ) are digraphs, then a morphism f : G → G (cid:48) is a pair offunctions ( φ, ψ ) , where φ : V → V (cid:48) and ψ : E → E (cid:48) , such that if e = ( v, v (cid:48) ) is a directededge in G , then ψ ( e ) = ( φ ( v ) , φ ( v (cid:48) )) is a directed edge in G (cid:48) . If G is any digraph, then byits underlying undirected graph we mean the graph (cid:98) G on the same vertices and edges, whereedge orientation is ignored (i.e. any directed edge or pair of reciprocal edges is replaced bya single undirected edge). Definition 1.1.
For a non-negative integer n , an n -tournament is a digraph with no recip-rocal edges whose underlying undirected graph is an n -clique. We will generally denote tournaments by greek letters, σ, τ etc.
Definition 1.2. An n -tournament σ is said to be • transitive , if its edge orientation defines a total ordering on its vertex set, • semi-regular , if for each vertex v ∈ σ its in-degree and out-degree differ by at most 1, and • regular if for each vertex v ∈ σ its in-degree and out-degree are equal. Clearly a regular tournament must be odd, and a regular tournament is always semi-regular. A tournament containing a vertex whose out-degree and in-degree differ by exactlyone must be even.Let σ be an n -tournament. Then for any set of k vertices in σ , the induced subgraph is a k -tournament. Such sub-tournaments will be referred to as the faces of σ .We can now define complexes built out of tournaments in a way that is totally analogousto the definition of an ordinary abstract simplicial complex. Definition 1.3. A tournaplex is a collection X of tournaments, such that if σ ∈ X and σ (cid:48) is a face of σ , then σ (cid:48) ∈ X . OMPLEXES OF TOURNAMENTS 5 If X is a finite tournaplex, then the dimension of X is the largest integer n , such that X contains at least one ( n + 1) -tournament and no k -tournaments for k > n + 1 . Thus we willsometimes refer to ( n + 1) -tournaments as n -dimensional .1.1. Geometric realisation.
One may think of tournaplexes abstractly as in Definition1.3. It is also possible to associate a topological space, a chain complex and homology witha tournaplex. To do so, the simplest way is to think of tournaplexes as semi-simplicial sets.
Definition 1.4.
Let X be a tournaplex together with a fixed linear ordering on its set ofvertices (1-tournaments). Define a semi-simplicial set X • , whose n -simplices are the set X n of all ( n + 1) -tournaments in X . For every tournament σ ∈ X , the vertices of σ inherit atotal order v , v , . . . , v n . For all n > and ≤ i ≤ n , define face operators ∂ i : X n → X n − ,where ∂ i ( σ ) is the sub-tournament spanned by all v j except v i . The face operators ∂ i clearly satisfy the usual simplicial face relations, and hence X • is awell defined semi-simplicial set. Clearly, up to isomorphism of semi-simplicial sets, this doesnot depend on the choice of total ordering on X . Definition 1.5.
The geometric realisation of a tournaplex X is defined to be the geometricrealisation of the corresponding semi-simplicial set X • . The chain complex of a tournaplex X and its homology are simply the usual chain complexand homology of the corresponding semi-simplicial set.We proceed with some naturally occurring examples. Example 1.6.
Let X be an abstract simplicial complex, and let X denote the collection ofits 1-simplices. Fix an arbitrary orientation on each simplex τ ∈ X . Thus every simplex σ ∈ X becomes a tournament, and turns X into a tournaplex. Notice that the homeomorphism type of the geometric realisation of the resulting tour-naplex in Example 1.6 is independent of the choice of orientations on the 1-simplices. It isin fact just that of the original simplicial complex.
Example 1.7.
Let X be an ordered simplicial complex, i.e. a collection of finite orderedsets that is closed under subsets. Then each ordered simplex σ ∈ X can be thought of as atransitive tournament. Hence ordered simplicial complexes are special cases of tournaplexesin which every tournament is transitive. Definition 1.8.
Let G be a digraph. The flag tournaplex associated to G is the tournaplex tFl( G ) , whose n -simplices are the subgraphs of G that are ( n + 1) -tournaments. Notice that not every tournaplex is a flag tournaplex. For instance the boundary of a 2-simplex, with any edge orientation is a tournaplex because it consists of three 2-tournamentsand three 1-tournaments, but it is not a flag tournaplex because it does not contain theinterior of the 2-simplex. However, just like the case of ordinary flag complexes, everytournaplex is contained as a sub-tournaplex in the flag tournaplex generated by its 1-skeleton.The directed flag complex associated to a digraph G is the ordered simplicial complex dFl( G ) whose simplices are the transitive tournaments in G . Thus dFl( G ) ⊆ tFl( G ) , where equalityholds if and only if every tournament in G is transitive. Moreover, if G has no reciprocaledges then tFl( G ) is isomorphic to the flag complex of the underlying undirected graph.We end this section with a comment on the number of tournaments up to digraph isomor-phism. In low dimensions the numbers are small: the number of non-isomorphic tournaments DEJAN GOVC, RAN LEVI, AND JASON P. SMITH on 2, 3, 4 and 5 vertices is 1, 2, 4, and 12, respectively. However the numbers grow rapidlywith dimension with nearly 200,000 non-isomorphic tournaments on 9 vertices and nearly 10million of them on 10 vertices. It is clear therefore that it may be useful to find invariantsthat can divide tournaments into a relatively small number of classes in a meaningful way.This is what we aim to do in Section 2.2.
Directionality invariants of tournaments
Given a digraph G = ( V, E ) , one may associate with each vertex v ∈ V its signed direc-tionality. This allows us to define certain integer valued functions on the set of simplicesof a tournaplex. This in turn gives natural ways of filtering tournaplexes, and as such theybecome natural candidates for analysis by means of techniques of persistent homology. Definition 2.1.
Let G = ( V, E ) be a digraph. For a vertex v ∈ V , define the signed degreeof v in G , by sd G ( v ) = indeg G ( v ) − outdeg G ( v ) . Let U ⊆ V be a subset of vertices.i) Define the signed degree of U relative to G by sd G ( U ) = (cid:88) v ∈ U sd G ( v ) . ii) Define the directionality of U relative to G by Dr G ( U ) = (cid:88) v ∈ U sd G ( v ) . Figure 1.
A digraph consisting of two 5-tournaments “glued” along a 1-face.The flag tournaplex (Definition 1.8) of this graph consists of two 4-simplicesglued along an edge, and as such is contractible. By contrast, the directed flagcomplex of the graph has the homotopy type of a wedge of two circles and a2-sphere. However, the topology of the directed flag complex that is embeddedin the flag tournaplex can be revealed using the 3-cycle filtration (see Sec-tion 3).
OMPLEXES OF TOURNAMENTS 7
Let U be a finite set, and let R U be the real vector space generated by U , and let R U ∗ bethe dual space of linear functionals R U → R . Equip R U ∗ with an inner product defined by ( α, β ) = (cid:88) v ∈ U α ( v ) β ( v ) for α, β ∈ R U ∗ . If U is the vertex set of a digraph G , then the functions indeg , outdeg and sd can be extended to functionals indeg G , outdeg G , sd G ∈ R U ∗ , and Dr G ( U ) = (sd G , sd G ) . Inthe norm on R U ∗ defined by the inner product, Dr G ( U ) is the square length of the functional sd G , or equivalently the square distance between the functionals indeg G ( U ) and outdeg G ( U ) .Clearly, if f : G → H is an isomorphism of digraphs, and W = f ( U ) , then sd G ( U ) = sd H ( W ) and Dr G ( U ) = Dr H ( W ) . We now use these constructions to define graph invariants ontournaments. Definition 2.2.
For any n -tournament σ in a graph G , let V σ denote the vertex set of σ ,and define:i) Local directionality : Dr( σ ) = Dr σ ( V σ ) .ii) : Let c ( σ ) denote the number of regular 3-sub-tournaments in σ .iii) Global directionality: Dr G ( σ ) = Dr G ( V σ ) . Notice that local directionality and the 3-cycle directionality are both invariants of asingle tournament, independent from the ambient graph containing it. Global directionalityon the other hand takes into account the general connectivity of the ambient graph. Next,we observe that the local directionality of a tournament is strongly related to its 3-cycledirectionality.
Proposition 2.3.
Let σ be an n -tournament. Then Dr( σ ) = 2 (cid:18) n + 13 (cid:19) − c ( σ ) . (1) Proof.
For n = 3 the tournament σ is either transitive or regular. In the first case Dr( σ ) = 8 and in the other Dr( σ ) = 0 . Thus the claim follows. Proceed by induction on n . Assume (1)holds for all ( n − -tournaments and prove it holds for n -tournaments. Choose a vertex v ∈ σ , and let σ (cid:48) denote the face of σ corresponding to removing v . We split the vertices V σ (cid:48) of σ (cid:48) into two parts: V in consisting of the v ∈ V σ (cid:48) such that [ v → v ] ∈ σ and V out consistingof all vertices v ∈ V σ (cid:48) such that [ v → v ] ∈ σ .Define σ in to be the subgraph of σ (cid:48) induced by V in , and let σ out be the subgraph inducedby V out . Let σ in − out be the subgraph with vertex set V σ (cid:48) , and whose edges are incident toone vertex in V in and the other in V out .Now, write: (cid:88) v ∈ V σ sd σ ( v ) = (cid:88) v ∈ V in (sd σ (cid:48) ( v ) + 1) + (cid:88) v ∈ V out (sd σ (cid:48) ( v ) − + ( | V out | − | V in | ) = (cid:88) v ∈ V σ (cid:48) (sd σ (cid:48) ( v )) + 2 (cid:88) v ∈ V in sd σ (cid:48) ( v ) − (cid:88) v ∈ V out sd σ (cid:48) ( v ) + | V σ (cid:48) | + ( | V out | − | V in | ) . Note that (cid:80) v ∈ V X sd X ( v ) = 0 for any graph X with vertex set V X , since this is equivalent tosum of all in-degrees minus the sum of all out-degrees. So we can rewrite the second and DEJAN GOVC, RAN LEVI, AND JASON P. SMITH third term as follows: (cid:88) v ∈ V in sd σ (cid:48) ( v ) − (cid:88) v ∈ V out sd σ (cid:48) ( v ) = 2 (cid:88) v ∈ V σ (cid:48) sd σ (cid:48) ( v ) − (cid:88) v ∈ V out sd σ (cid:48) ( v )= 2 (cid:88) v ∈ V σ (cid:48) sd σ (cid:48) ( v ) − (cid:88) v ∈ V out sd σ out ( v ) − (cid:88) v ∈ V out sd σ in − out ( v )= − (cid:88) v ∈ V out sd σ in − out ( v )= 4 | V in || V out | − (cid:96), where | V in || V out | arises as the number of all edges in σ in − out and (cid:96) is the number of edges v → v with v ∈ V in and v ∈ V out . Therefore, Dr( σ ) = (cid:88) v ∈ V σ sd σ ( v ) = Dr( σ (cid:48) ) − (cid:96) + | V σ (cid:48) | + ( | V out | − | V in | ) + 4 | V in || V out | = Dr( σ (cid:48) ) − (cid:96) + | V σ (cid:48) | + | V σ (cid:48) | . Using the inductive hypothesis, this simplifies to
Dr( σ ) = 2 (cid:18) n (cid:19) + n ( n − − c ( σ (cid:48) ) + (cid:96) ) = 2 (cid:18) n + 13 (cid:19) − c ( σ (cid:48) ) + (cid:96) ) . Finally, observe that (cid:96) is precisely the number of regular 3-tournaments in σ that are notalready present in σ (cid:48) , and the proof is complete. (cid:3) Note that Proposition 2.3 can be derived from the first corollary to [11, Theorem 4].However, we include the above proof as we feel it gives a simpler combinatorial understandingof the link between local directionality and the -cycle directionality. Corollary 2.4 ([17, Supp. Meth. 2.1, Proposition 1]) . Let σ be an n -tournament. Then Dr( σ ) ≤ (cid:18) n + 13 (cid:19) , with equality obtained if and only if σ is transitive. Corollary 2.5.
Let σ be an n -tournament, and let σ (cid:48) be a face of codimension 1. Then c ( σ ) − c ( σ (cid:48) ) ≤ (cid:18) n − (cid:19) . Proof.
Let σ (cid:48) be an ( n − -face of σ , and let v ∈ V σ be the vertex not present in V σ (cid:48) .Partition V σ (cid:48) into two disjoint subsets V in and V out , where V in contains the vertices v ∈ V σ (cid:48) such that the edge between v and v is oriented towards v , and V out = V σ (cid:48) \ V in . Set (cid:96) = | V in | ,so | V out | = n − − (cid:96) . Then the largest value of c ( σ ) − c ( σ (cid:48) ) , that is, the number of directed -cycles in σ that are not in σ (cid:48) , is obtained if every edge between a vertex v (cid:48) ∈ V in and avertex v (cid:48)(cid:48) ∈ V out is oriented from v (cid:48) to v (cid:48)(cid:48) , and that number is (cid:96) ( n − − (cid:96) ) = − (cid:96) + ( n − (cid:96) .This quadratic function of (cid:96) obtains its maximum exactly when (cid:96) = n − , and so the maximalnumber of directed -cycles that can appear in σ and are not present in σ (cid:48) is (cid:0) n − (cid:1) . (cid:3) Example 2.6. If σ is an n -tournament and τ is a k -face of σ , then it is not true in generalthat Dr( τ ) ≤ Dr( σ ) . For instance a regular 3-tournament has local directionality 0 buteach of its 2-faces has local directionality 2. Similarly a semi-regular 4-tournament has localdirectionality 4 but has two 3-faces with local directionality 8. OMPLEXES OF TOURNAMENTS 9
One natural question in view of Proposition 2.3 is whether every number that can the-oretically appear as the local directionality of a tournament, is indeed realisable as such.The answer to this question follows from three theorems by Kendall and Babington-Smithin their 1940 paper [5, 8.(1)-(3)]. Using our terminology these results read as follows:
Theorem 2.7 ([5]) . The following statements hold for an n -tournament σ : i) c ( σ ) ≤ (cid:40) n − n n odd n − n n even . Furthermore, these bounds are sharp in the sense that thereexists an n -tournament with this number of 3-cycles. ii) For any integer k between 0 and the upper bounds in i), there exists at least one n -tournament σ such that c ( σ ) = k . Corollary 2.8.
For each pair of non-negative integers n and k , where k is at most aslarge as the bounds in Theorem 2.7, there exists at least one n -tournament σ such that Dr( σ ) = 2 (cid:0) n +13 (cid:1) − k . Corollary 2.9.
For any n ≥ there exists an n -tournament σ of minimal local directionalitythat is regular if Dr( σ ) = 0 ( n odd) and semi-regular if Dr( σ ) = n ( n even). Distribution of tournaments in digraphs by local directionality.
The motiva-tion for introducing tournaplexes is the idea that tournaments are basic building blocks ina geometric object that can be associated to a digraph. Rather than ignoring edge orien-tation and considering the ordinary flag complex, or neglecting cliques that are not linearlyordered and considering the directed flag complex, the flag tournaplex allows us to considerall tournaments as simplices, and as such forms a topological object that is richer in struc-ture, and better informing about properties of the digraph in question. See Figure 2 for asimple example of how these three constructions differ. However, the vast number of isomor-phism types of tournaments in higher dimensions makes using them as individual data unitsquite impractical. We have thus introduced directionality as one way of taking advantage ofgeneral tournaments, without the constraints imposed by the large number of isomorphismtypes.
Figure 2.
The flag complex of this digraph (ignoring orientation) is a 2-simplex. The directed flag complex is a 2-simplex with an additional arc onits bottom 1-face. The flag tournaplex is a cone formed of two 2-simplicesattached on two of their edges. Its homotopy type can be distinguished fromthat of the flag complex by using the directionality of its 3-tournaments.To demonstrate the potential of tournaments to inform on network structure, we examineda number of networks, and recorded the distribution of tournaments by local directionalityin Figure 3. In particular the probability of each possible value of directionality of a random tournament in a fixed dimension is known up to n = 10 , by calculations of Alway [1] andKendall and Babington-Smith [5]. More generally it was shown by Moran [12] that thedistribution of the number of 3-cycles in a random n -tournament tends to normal for n sufficiently large (see also [11, Theorem 6]). All these results are stated in terms of thenumber of 3-cycles in a tournament.Let T n,j be the number of all n -tournaments σ on a fixed vertex set, such that c ( σ ) = j .One can compute the distribution of local directionality in n -tournaments and T n,j recursively[1]. The following gives the said distributions for n ≤ .n 1 2 3 4 5 j Dr( σ ) T n,j T n,j we can express the expected number of tournaments of various -cycle direc-tionalities in Erdős-Rényi graphs. Recall that an Erdős-Rényi digraph with n vertices andconnection probability p is a random subgraph of the complete digraph with n vertices, whereeach of the n ( n − possible directed edges is included with probability p independently fromevery other directed edge. Proposition 2.10.
Let G be a directed Erdős-Rényi graph with n vertices and connectionprobability p . Let X k be the total number of k -tournaments that occur in G and let X k,j bethe total number of k -tournaments containing exactly j -cycles that occur in G . Then theexpected values of X k and X k,j are given by the formulas E ( X k ) = (cid:18) nk (cid:19) k ) p ( k ) and E ( X k,j ) = (cid:18) nk (cid:19) T k,j p ( k ) . Proof.
First note that the number of all possible k -tournaments on an n -vertex set is givenby (cid:18) nk (cid:19) k ) , as each one is obtained by choosing k out of n vertices and then orienting each of the (cid:0) k (cid:1) edges in one of two possible ways.The number of all possible k -tournaments on an n -vertex set containing exactly j -cyclesis given by (cid:18) nk (cid:19) T k,j , as we have to choose k vertices and then by definition T k,j is the number of tournaments onthat vertex set containing exactly j -cycles.Now, any specific k -tournament σ will occur in G with probability P ( σ ⊆ G ) = p ( k ) , as it has exactly (cid:0) k (cid:1) edges, each of which occurs independently with probability p . Let Y σ be the random variable which takes value if σ occurs in G and otherwise. It follows that E ( Y σ ) = p ( k ) . Note that X k is just the sum of Y σ over all possible k -tournaments σ . Similarly, X k,j is thesum of Y σ over all possible k -tournaments σ containing exactly j -cycles. Therefore, theresult follows by linearity of expectation. (cid:3) OMPLEXES OF TOURNAMENTS 11
We can consider T k,j as the theoretical distribution of k -tournaments σ with c ( σ ) = j .In Figure 3 we show the distribution of tournaments in all values of local directionality in anumber of sample networks. Notice the rather accurate estimates in the case of Erdős-Rényigraphs. It is also worth noticing how close the distribution of local directionality valuesis between the connectivity graph of C. Elegans [23] and that of the Blue Brain Projectmicrocircuit simulating a section of the neocortex of a rat [16]. The bottom three datasetscan be found in [9, 4, 14]. The bottom five datasets are available in Tournser formatat [13].
Figure 3.
Distribution of tournaments by local directionality in various net-works. The rows correspond to the graphs indicated on the left. The columnsare ordered left to right by the number n of vertices in a tournament for n ≥ .Each histogram shows the distribution of n -tournaments by local directional-ity. The numbers on top refer to the number c of 3-cycles in the tournamentsrepresented by the relevant column of the histogram. From this number localdirectionality can be computed as (cid:0) n +13 (cid:1) − c . The asterisk indicates theexistence of further distributions to the right.3. Filtrations
In this section we discuss three different ways to define a filtration on a tournaplex usingthe idea of directionality.Let K be a tournaplex with vertex set V . An increasing filtration on K is an increasingsequence of sub-tournaplexes ∅ = F ( K ) ⊆ F ( K ) ⊆ F ( K ) ⊆ · · · ⊆ F n ( K ) = K. An increasing filtering weight function on K is a function W : K → R with the propertythat if σ is a tournament in K and σ (cid:48) ⊆ σ is a face, then W ( σ (cid:48) ) ≤ W ( σ ) . Similarly onedefines decreasing filtrations and decreasing filtering weight functions.An increasing filtering weight function W on a tournaplex K naturally gives rise to in-creasing filtrations on K as follows: Fix an increasing sequence of real numbers r < r < · · · < r n − . Set F ( K ) = ∅ , F n ( K ) = K , and F i ( K ) = { σ ∈ K | W ( σ ) ≤ r i } , for ≤ i ≤ n − . Similarly a decreasing filtering weight function gives rise to a decreasingfiltration on K . If the weight function on K is a step function, the one has an obvious choicefor the sequence defining the filtration, namely the sequence of all possible values of thefunction in increasing order. We now use local, 3-cycle and global directionality as ways ofdefining filtrations on any tournaplex.We start with local directionality. As pointed out in Example 2.6, it is not generally thecase that if σ is a tournament and τ is a face of σ , then Dr( τ ) ≤ Dr( σ ) . Thus one is led tomake the following definition. Definition 3.1.
Let K be a tournaplex. For each n -tournament σ ∈ K define the localdirectionality weight of σ to be W Dr ( σ ) def = Dr( σ ) + 2 (cid:18) n (cid:19) . Lemma 3.2.
Let K be a tournaplex and let W Dr : K → N be the local directionality weightfunction. Then W Dr defines an increasing filtration on K .Proof. It suffices to show that if σ is an n -tournament for n > and τ ⊆ σ is a face ofcodimension 1, then W Dr ( τ ) ≤ W Dr ( σ ) . By Proposition 2.3 and Corollary 2.5 W Dr ( σ ) − W Dr ( τ ) ≥ (cid:18)(cid:18) n + 13 (cid:19) − (cid:18) n − (cid:19)(cid:19) − n − = 0 . The claim follows. (cid:3)
Notice that it is possible for an n -tournament σ to have a face τ of codimension largerthan 1, whose directionality is larger than that of all codimension 1 faces. This justifiesadding the maximal possible directionality of a codimension 1 face to the local directionalityin order to create a filtration. See Figure 4 for such an example. Figure 4.
A regular 5-tournament (
Dr = 0 ) in which each 4-face is semi-regular (
Dr = 4 ). Hence each 4-face contains two 3-faces that are transi-tive (
Dr = 8 ). Definition 3.3.
Let K be a tournaplex. For each n -tournament σ ∈ K define the σ to be W c ( σ ) = c ( σ ) . OMPLEXES OF TOURNAMENTS 13
Clearly if σ is a tournament and τ is a face of σ , then c ( τ ) ≤ c ( σ ) , and hence this definesa filtration on any tournaplex. Remark 3.4.
Observe that tournaplexes may be naturally filtered in many other ways thatare similar to the 3-cycle filtration. For instance, if K is a tournaplex, define W T : K → R byletting W T ( σ ) be the number of transitive 3-tournaments in σ . Pushing this idea further, onecan consider any fixed (small) tournament τ , and filter K by the number of times τ appearsas a face in each σ ∈ K . This automatically yields a filtration, and relates nicely to wellknown approaches that regard the prevalence of certain motifs in networks as a determiningfactor in the emerging dynamics (see for instance [19] ). By Proposition 2.3, the 3-cycle weight function and the local directionality weight functionon an arbitrary n -tournament σ are related by the formula: W Dr ( σ ) = (cid:18) n (cid:19) n − − W c ( σ ) . (1)Thus, these filtrations are related to each other by a naturally occurring monotone transformthat is a function of simplex dimension. However, because of the intrinsic geometric datathat gives rise to the filtrations, the functions W Dr and W c are far from inducing similarfiltrations, in spite of this close relationship. Specifically, the local directionality filtration ofa tournaplex has its vertices in filtration 0, its edges and regular 3-tournaments in filtration 2and is, roughly speaking, a refinement of the dimension filtration. The relationship betweenthe 3-cycle filtration and dimension is more subtle. In filtration 0 one has the sub-tournaplexconsisting of all transitive tournaments (in any dimension). In higher stages of the 3-cyclefiltration the minimal possible dimension of an added simplex increases with the filtrationvalue (a 3-tournament cannot have a 3-cycle filtration larger than 1), but the dimension ofa simplex that is added on in any filtration value is not bounded above.In Figure 5 we have four digraphs whose flag tournaplexes realise the same homotopy type,but which are distinguished by the associated persistence diagrams with respect to localdirectionality filtration. Similarly it is easy to see that each of the four digraphs containsa different number of 3-cycles, and hence they are also distinguished by their persistencediagrams corresponding to their 3-cycle filtration. There are however examples of graphsthat cannot be distinguished by local directionality or 3-cycle filtration, as in Figure 6. Thus,a related question is whether there could be an advantage in using one of these filtrationsas opposed to another, or if there is a point in using a combination of both. In Table 3.1we show an example of the two non-isomorphic tournaments in Figure 7, where the 3-cyclefiltration is always contractible (see the bottom row), and where the local directionalityfiltration is much more interesting (see the right column). This gives a simple answer to thefirst question. The second will be discussed further below. Definition 3.5.
Let K be a tournaplex. For each tournament σ ∈ K , define the globaldirectionality weight of σ to be W K ( σ ) = Dr K ( V σ ) = (cid:80) v ∈ V σ sd K ( v ) , where K is the1-skeleton of K considered as a digraph. The global directionality weight function is clearly an increasing filtering function andas such induces an increasing filtration on a tournaplex. Its properties are quite differentfrom the local and the 3-cycle directionality functions in that any positive integer value canbe achieved as the global directionality of a tournament in some digraph. See Figure 6 foran example of some digraphs whose flag tournaplexes can be distinguished by the globaldirectionality filtration, but not by the local or the 3-cycle filtrations. On the other hand,the digraphs in Figure 5 cannot be distinguished by global directionality.
Figure 5.
Four non-isomorphic digraphs and their local directionality filtra-tions reflected in their persistence diagrams. The signed degree of each vertexin each digraph is 0. Hence the global directionality filtration cannot distin-guish these graphs. H , H and H are denoted in the diagrams by blue, orangeand red dots, respectively, and the numbers next to some dots correspond torank. All four tournaplexes have the homotopy type of a 2-sphere. Thesegraphs can however be distinguished by the homotopy type of their directedflag complexes.The multitude of directionality filtrations on tournaplexes suggest that two or more of themcan be used in conjunction to create a combined filtration or a multi-parameter persistencemodule. We examined this idea with respect to local directionality and -cycle filtrations.Let G and G be the 8-tournaments depicted in Figure 7. The flag tournaplexes tFl( G ) and tFl( G ) have identical 1-parameter persistence modules with respect to the -cycle direction-ality filtration and with respect to the local directionality filtration. The 3-cycle filtrationstages are all contractible (see the bottom row of Table 3.1). The local directionality filtra-tion is more interesting and is summarised in (2) (cf. also the right column in Table 3.1)below. H i ( X ) = [0 , ⊕ [0 , ∞ ); i = 0 , [2 , ; i = 1 , [10 , ⊕ [10 , ⊕ [10 , ; i = 2 , [20 , ⊕ [28 , ⊕ [28 , ⊕ [28 , ⊕ [28 , ; i = 3 , [44 , ⊕ [52 , ⊕ [52 , ⊕ [60 , ⊕ [60 , ⊕ [60 , i = 4 , [78 , ⊕ [94 , ⊕ [102 , ⊕ [102 , i = 5 , [150 , i = 6 , (2)where X is tFl( G j ) with local directionality filtration, for j = 1 , .However taking the two filtrations together yields two distinct 2-parameter persistencemodules. Considering the bifiltration of the geometric realisation of each tournaplex, andapplying the homotopy theoretic techniques used in [3], we were able to compute the homo-topy types of each bifiltration pair which are displayed in Figure 3.1. The difference between OMPLEXES OF TOURNAMENTS 15
Figure 6.
Four non-isomorphic digraphs and their global directionality filtra-tions. The numbers on the vertices denote the corresponding squared signeddirectionality, and the chart below each digraph is the persistence diagramcorresponding to the global directionality filtration on the corresponding tour-naplex. H , H and H are denoted in the diagrams by blue, orange and reddots, respectively. The numbers next to some dots correspond to rank. Each3-tournament in all four tournaplexes is transitive. Hence these graphs cannotbe distinguished by local directionality filtration or by 3-cycle filtration. Figure 7.
The two tournaments G (left) and G (right), whose 2D-persistence module is given in Figure 3.1. With their common edges shadedgrey, and differing edges in black. tFl( G ) and tFl( G ) reveals itself in bifiltration (44 , . In both cases we did not compute thefull persistence module structure (namely the maps between bifiltration stages). The pointof this calculation was to show that there are examples that are not distinguishable by localdirectionality or by 3-cycle filtration alone, but a combination of the two does separate them.Similarly one can construct a 1-parameter filtration by taking a function of two variablescombining the two filtration functions. For example, in our case, consider f ( σ ) def = max { W Dr ( σ ) , W c ( σ ) } . (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) W Dr W c ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
10 2
12 2
20 2 ∨ ∨ ∨ ∨ ∨ ∨
28 3
36 3
44 3 ∨ ∨ ∨ ∨ ∨
52 3 ∨ ∨ ∨ ∨ ∨ ∨ ∗ ∗ ∗ ∗ ∨ ∨ ∗ ∨ ∨ ∗ ∨ ∨ ∨ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Table 3.1.
The homotopy types of all bifiltration stages of tFl( G ) and tFl( G ) . Here ∗ n represents the disjoint union of n points, whereas m n repre-sents the wedge of n copies of the m -sphere. The homotopy types of tFl( G ) and tFl( G ) are in red and blue, respectively, and in black when they are thesame.Then f ( σ ) ≤ if and only if W Dr ( σ ) ≤ and W c ( σ ) ≤ . It is then easy to check,given the data in Table 3.1, that the filtration function f will distinguish the tournamentsin Figure 7 by 1-parameter persistence.4. Applications
The flag complex associated to a graph reveals higher order properties of the graph asthey are encoded in its topology. Similarly, the directed flag complex does the same fordigraphs. However, as already pointed out, the directed flag complex “sees” tournamentsthat are not transitive in the graph only by neglecting them, and as such they may or maynot be expressed as homology classes in the resulting complex. We introduced the flagtournaplex, motivated exactly by the idea that it will contain more information about thedigraph in question than the directed flag complex. This point is clear even without anyempirical evidence since in any tournaplex X , the subcomplex of transitive tournamentsappears as the -th stage in the -cycle filtration. Thus if X is a flag tournaplex, then anyhigher filtration can only add on the information present in the directed flag complex. Inthis section we demonstrate by several examples that this is indeed the case. OMPLEXES OF TOURNAMENTS 17
We consider the flag tournaplexes arising from two data sets. The first is a set of directedErdős-Rényi type digraphs, and the second is a collection of activity simulation data from theBlue Brain Project that was used in [17]. In both cases we consider a set of digraphs dividedinto subsets by some known parameters, and examine the capability of the topological metricsone can associate to flag tournaplexes with the local directionality filtration to cluster thedata into distinct classes, and compare the performance to that of the metrics associated tothe directed flag complex.Given a collection G of digraphs we wish to partition these graphs into groups of graphswith similar properties. In order to do this we must first map each graph G ∈ G to a featurevector V ( G ) of length k for a suitable positive integer k . We shall produce such a vectorby two methods. The first is based on a computation of the Betti numbers of directed flagcomplexes of the graphs in question. The second uses the persistent homology of the flagtournaplex tFl( G ) with the local directionality filtration. Each method produces a matrixwith as many rows as the number of graphs to be clustered and k columns. The rows of thesematrices are fed into a k -means clustering algorithm to produce our results, as explainedbelow. Definition 4.1.
For a digraph G , represent the persistent homology of the flag tournaplex tFl( G ) with respect to the local directionality filtration as a list of triples ( m, b, d ) , correspond-ing to ( dimension, birth, death ) . Let T (cid:96) ( G ) denote the resulting data set. Set (cid:98) T (cid:96) ( G ) as theset of the unique triples in T (cid:96) ( G ) and N G ( m, b, d ) as the number of times the triple ( m, b, d ) appears in T (cid:96) ( G ) . Erdős-Rényi type digraphs with varying local directionality distributions.
In this section we describe a validation experiment we ran where we attempt to show thatusing local directionality filtration and persistent homology provides greater discriminatorypower than that of Betti numbers of the associated directed flag complex for a fixed artificialdataset. This is a data set of random graphs denoted G ( n, p, q ) where n is the numberof vertices, indexed , . . . , n , and p and q are probability parameters. An instantiation of G ( n, p, q ) is generated by adding a directed edge ( i, j ) with probability (cid:40) p, if i > jq, if i < j . By varying the probability for different orientations we can control the frequency of tourna-ments with different local directionality.We constructed 50 instantiations in 4 groups with n = 250 , p = 0 . and q = 0 , . , . and . , a total of 200 graphs, which we denote by G ER . We use small values for q sothat the number of transitive tournaments remains largely unchanged, thus causing minimaldifference to the directed flag complexes. To check that this procedure achieved differentdistributions of local directionality, we computed the average distribution by dimension anddirectionality. The result is summarised in Figure 8. We then applied Algorithm 1 to producethe matrix V T ( G ER ) . Remark 4.2.
The aim of Algorithm 1 is to use persistent homology of tournaplexes to extractthe key functional information from the spike trains and express said data as a feature vectorthat can be used for classification. We begin by applying persistent homology and expressingthe resulting information as a vector, this is only possible since we know the directionalityfiltration has a finite number of possible values. However, the dimension of the resultingvector is too high to apply clustering algorithms. So the remaining steps of Algorithm 1 are a form of feature selection designed to extract the most important entries of the vector. This isdone by using standard deviation and selecting the filtration value and time bin pairs whichvary the most, the intuition being that these entries are the most likely to vary between input,thus giving the best classification accuracy.
Figure 8.
Distribution of local directionalities as a function of the proba-bility parameter q , normalised by connection density. The x -axis is labeledby Frequency/Density. The different colours correspond to the number n ofvertices in a tournament, with blue corresponding to a single vertex and pinkto 7 vertices. The vertical axis is labeled by the possible local directionalityweights in each dimension. Algorithm 1
Computing local directionality feature matrix V d T ( G ) Input:
A set G = {G , . . . , G n } of digraphs, and an integer d > Procedure: Compute persistent homology T (cid:96) ( G i ) of tFl( G i ) with respect to W Dr for all G i ∈ G Set (cid:98) T = (cid:83) ni =1 (cid:98) T (cid:96) ( G i ) , and fix an arbitrary ordering t , . . . , t r on its elements Set M = ( m i,j ) as the n × r matrix with m i,j = N G i ( t j ) Compute the standard deviation of each column of M (considered as a set of integers) Let C , . . . , C d be the d columns with the largest standard deviation Return the n × d matrix V d T ( G ) , given by concatenating the columns C · · · C d Let V β ( G ER ) denote the matrix where the i -th row is the vector V β ( G i ) def = [ β , β , . . . , β ] ofBetti numbers of the directed flag complex of G i ∈ G ER . Next, we applied k -means clusteringto the rows of each of the matrices V T ( G ER ) and V β ( G ER ) , using Python package scikit-learn [15]. The results are displayed in Figure 9. Comparing the results to the distributionof directionalities in Figure 8, one notices that the distribution of transitive tournaments OMPLEXES OF TOURNAMENTS 19 remains roughly the same regardless of q . Hence, it is to be expected that the Betti numbersof the directed flag complex will perform poorly in clustering the four families, which is indeedthe case. By contrast, the associated tournaplexes, filtered by local directionality give almostperfect separation, which is particularly remarkable in the cases q = 0 . and q = 0 . that give very similar distributions in Figure 8. This demonstrates that the topology of thetournaplex holds more information about the orientation of the edges in a digraph, comparedto the directed flag complex. Remark 4.3.
We select d = 6 in this computation for two reasons. Firstly, the Betti numbersare always zero for dimension and above in the directed flag complex. Secondly, for k -means clustering to give reliable results we require that the size of the vectors to be clusteredis significantly smaller than the size of the data set. Also, we use k -means clustering as it isa simple well known technique, but similar results are obtained using decision tree learningor linear discriminant analysis, as verified by Henri Riihimäki. Tournaplex . . . . q Directed Flag Complex . . . . q Figure 9.
The cluster assignment of k -means clustering applied to thecolumns of the matrices V T ( G ) (top) and V β ( G ) (bottom), where G is a setof 200 directed Erdős-Rényi type graphs with parameters p = 0 . and vary-ing q . The four colours represent the different clusters, each row correspondsto a different value of q .4.2. Brain activity simulation digraphs.
The data in this example was taken from BlueBrain Project’s reconstruction of the neocortical column of a juvenile rat [10]. The re-construction is a digital model of brain tissue based on basic biological principles. Theparticular model we used contains approximately 31000 digitally simulated neurons withroughly 8 million synaptic connections. There are 55 morphological types of neurons simu-lated in the column, in correspondence with known biological classification, and the columnis built in 6 layers, again following known biological principles. With the possible exceptionof more recent developments by the Blue Brain Project team, this model is the most accurateapproximation of real brain tissue currently available. In particular it allows researchers toobtain full connectivity data of the neural microcircuitry, and can be stimulated by spiketrains that are either recorded in vivo or artificially generated.The data set we use here was used in [17] to demonstrate the capability of the homologyof the directed flag complex to express certain properties of the simulations. In this dataset nine different stimuli were applied to the Blue Brain Project microcircuit. These stimuli,which take the form of spike trains that are injected directly into the digital tissue, can bedistinguished by two properties, their class and their grouping , with three possible valuesfor each property. Each class, denoted , or , represents a different temporal input ofthe stimulus, and each grouping, denoted a , b or c , represents a different spatial input of thestimulus. See [17, Figure 4A] for further information about the stimuli. Rather than using Algorithm 2
Computing local directionality feature matrix (cid:98) V m T ( D ) for functional data Input:
Spike trains D = { D , . . . , D n } on a graph G and three positive integers { m, t , t } Procedure: for i ∈ { , . . . , n } do for j ∈ { , . . . , (cid:100) L/t (cid:101)} do Compute transmission-response graphs G ij def = G D i j with parameters { t , t } Compute persistent homology T (cid:96) ( G ij ) of tFl( G ij ) with respect to W Dr Set (cid:98) T = ∪ i,j (cid:98) T (cid:96) ( G ij ) and fix an arbitrary ordering t , . . . , t l on its elements Set M j = ( m ij,c ) to be the n × l matrix with m ij,c = N G ij ( t c ) Let M be the concatenation of the matrices M , M , . . . , M (cid:100) L/t (cid:101) Compute the standard deviation of each column of M (considered as a set of integers) Let C , . . . , C m be the m columns with the largest standard deviation Return the n × m matrix (cid:98) V m T ( D ) given by concatenating the columns C · · · C m the entire microcircuit, we extracted spike trains only from L5TTPC1 neurons [10]. Thecode name stands for Layer 5 Thick Tufted Pyramidal Cell of Type 1 , and these neurons areone of the most prevalent morphological types in the neocortex and are assumed to be ofmajor significance in information processing in the brain. The structural data is publiclyavailable at [16].We consider five repetitions of each experiment, referred to as seed s , for s = 1 , . . . , .Thus we have distinct data sets, each consisting of a 250ms spike train, that is a list ofpairs ( t, g ) where t is a time (in resolution of 0.1ms) and g is a neuron (vertex) that spikedat time t . Each data set is then converted to a sequence of digraphs, using the transmission-response method introduced in [17]. The construction relies on three items of input: 1) theunderlying structural graph G , 2) a spike train data set D and 3) a pair of integers t and t .We then split the spike train into time bins of size t -ms, and construct a graph G r foreach time bin, where G r has the same vertices of G and the edge ( i, j ) if all of the followingconditions hold:(a) ( i, j ) is an edge in G ,(b) neuron i fired at time t in the r -th time bin, and(c) neuron j fired at time t < t ≤ t + t .See [17] for further background on transmission-response graphs.A typical collection D of spike train data sets consists of n distinct instantiations, denotedby D r , r = 1 , . . . , n , each of length L ms. Out of this collection we produce an n × m featurematrix (cid:98) V m T ( D ) , for some natural number m . The rows of this matrix are then fed into a k -means clustering algorithm. To obtain (cid:98) V m T ( D ) we apply Algorithm 2, which is a similarprocedure to Algorithm 1, but is adapted for use with functional data. As noted above, forthe collection D at our disposal we have n = 45 and L = 250 , and we set m = 6 so that thesize of our vectors is suitable compared to the size of the data set (see Remark 4.3).For the directed flag complex we compute (cid:98) V mβ ( D ) by applying Algorithm 3, which is verysimilar to Algorithm 2, but using the Betti numbers instead of persistent pairs. We only usethe first three Betti numbers ( d = 3 in the algorithm), as in this data set all higher Bettinumbers were zero. OMPLEXES OF TOURNAMENTS 21
Algorithm 3
Computing Betti number feature vector (cid:98) V mβ ( D ) for functional data Input:
Spike trains D = { D , . . . , D n } on a graph G and four positive integers { d, m, t , t } Procedure: for i ∈ { , . . . , n } do for j ∈ { , . . . , (cid:100) L/t (cid:101)} do Compute transmission-response graphs G ij def = G D i j , with parameters { t , t } Set M j as the n × m matrix with the i -th row as V dβ ( G ij ) Let M be the concatenation of the matrices M , M , . . . , M (cid:100) L/t (cid:101) Compute the standard deviation of each column of M Let C , . . . , C m be the m columns with the largest standard deviation Return the n × m matrix (cid:98) V mβ ( D ) given by concatenating the columns C · · · C m a a a a ab b b b bc c c c c51530 s1 s2 s3 s4 s5n g Tournaplexa a a a ab b b b bc c c c c51530 s1 s2 s3 s4 s5n g Directed Flag Complex Figure 10.
The cluster assignment of k -means clustering applied to (cid:98) V T ( D ) (top) and (cid:98) V β ( D ) (bottom), where D is a set of spike trains on the Blue Brainmicrocircuit reconstruction.Once we have computed both (cid:98) V β ( D ) and (cid:98) V T ( D ) , we apply k -means clustering to theirrows. The results of this are displayed in Figure 10. Here too one sees that the data issplit into three classes almost perfectly using k -means clustering on the corresponding flagtournaplexes filtered by local directionality, while applying the analogous clustering methodon the Betti numbers of the directed flag complexes yields poorer separation. This suggestsonce more that the tournaplex construction stores different, and possibly more detailedinformation about the network than the directed flag complex.On behalf of all authors, the corresponding author states that there is no conflict ofinterest. References [1] G. Alway. The distribution of the number of circular triads in paired comparisons.
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Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
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