Asymptotics of the Number of Endpoints of a Random Walk on a Certain Class of Directed Metric Graphs
AAsymptotics of the Number of Endpoints of a Random Walk on aCertain Class of Directed Metric Graphs
V. L. Chernyshev, National Research University Higher School of Economics (HSE),Moscow, Russian Federation, [email protected]. A. Tolchennikov, A. Ishlinsky Institute for Problems in Mechanics RAS,M.V. Lomonosov Moscow State University,Moscow, Russian Federation, [email protected] 13, 2021
Abstract
A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpointsof a random walk at large times is found.Keywords: counting functions, directed graphs, dynamical systems, Barnes—Bernoulli polynomials
Let us consider a directed metric graph. Let us denote length of edge e k by t k and suppose that alllengths { t k } Ek =1 are linearly independent over the set of rational numbers Q .One could consider a random walk (see [3]) on a directed metric graph (see, for example, [2] for referenceson metric graphs). The main unlikeness with the often considered case (see, for example, [1]) is that theendpoint of a walk can be any point on an edge of a metric graph, and not only one of the vertices. Let onepoint start its move along the graph from a vertex (a source) at the initial moment of time. The passagetime for each individual edge is fixed. In each vertex, the point with some non-zero probability selects oneof the outgoing edges for further movement. Backward turns on the edges are prohibited in this model. Ouraim is to analyze an asymptotics of the number N ( T ) of possible endpoints of such random walk as time T increases. The only assumption about the probabilities of choosing an edge is that it is non-zero for alledges, i.e. a situation of a general position. Such random walk could naturally arise while studying thedynamical systems on various networks.The asymptotics for finite compact metric non-directed graphs was constructed in papers [5,6]. Moreover,in paper [4] the problem for wave propagation on singular manifolds was reduced to considering graphs witha finite number of vertices, but with infinite valences (but only a finite number of edges are involved at anyfinite time). So this article is the first to discuss the asymptotics of possible end-points on digraphs.We will consider a finite directed graph G = ( V, E ) of the following form: let there be an outgoing treewith vertices v and a root s , oriented in the direction from the root. The graph G is obtained from thistree by adding a finite number of edges leading from some vertices s to s . These graphs have an importantproperty: there is only one route (that does not return to the root) from the root to any vertex. Definition
We call a directed strongly connected metric graph a one-way Sperner graph (see [7] fordetails about Sperner graphs) if it consists of one-way tree started from the source vertex S and has abackward edges only leading to the source.Thus, in what follows we will consider only one-way Sperner graphs. N ( T ) Let us introduce several notations. 1 a r X i v : . [ m a t h . C O ] J a n igure 1: Example of strong digraph of discussed type.For any subgraph G (cid:48) of the graph G and the vertex v we denote by ρ in ( G (cid:48) , v ) and ρ out ( G (cid:48) , v ) the numberof incoming edges incoming to v and the number of outgoing from v edges of the subgraph G (cid:48) .For any route µ we denote by t ( µ ) the time of this route, i.e. the sum of the times of passage of theedges included in µ .In our graph G for any vertex v ∈ V there is a unique simple chain l v (i.e., a route in which all verticesare pairwise distinct) from s to v (in particular, l s = ∅ ).Let c , . . . , c k be the elementary cycles of the graph G (in our case k = ρ in ( G, s ) ). The set of allelementary cycles we denote by C = { c , . . . , c k } .In what follows we will assume that the passage times of all elementary cycles and all simple chains ofthe form l v in the aggregate are linearly independent over Q . This is a situation of general position.Let us consider a linear inequality of the form n a + . . . + n j a j ≤ T , where n , . . . , n j ∈ N . By { n a + . . . + n j a j ≤ T } we will denote the number of natural solutions of this inequality. Theorem 1. N ( T ) = (cid:88) v ∈ V (cid:88) I ⊂{ ,...,k } ( ρ out ( G, v ) − ρ in ( G (cid:48) , v )) (cid:40) t ( l v ) + (cid:88) i ∈ I n i t ( c i ) ≤ T (cid:41) , (2.1)where the subgraph G (cid:48) = G (cid:48) ( v, I ) of the graph G is formed by the union of the edges of the simple chain l v and elementary cycles { c i } i ∈ I . Proof.
Let us consider an arbitrary route µ from s to v . This route can be represented in the form of passingalong the edges of several elementary cycles and along the edges of a simple chain l v . In that case the passagetime t ( µ ) of the route µ has the form t ( l v ) + (cid:80) i ∈ I n i t ( c i ) , where n i ∈ N . Note that any time this form is thetime of passage of a route from the s to v . But, of course, different routes can have identical transit times.So, we have described the set M of passage times for routes starting at the vertex s . It has the form: M = (cid:116) v ∈ V (cid:116) I ⊂{ ,...,k } M v,I , where M v,I = { t ( l v ) + (cid:88) i ∈ I n i t ( c i ) | n i ∈ N ∀ i ∈ I } are the passage times for routes ending at the vertex v , and passing along all edges of the cycles c i ( i ∈ I ) .The condition of linear independence over Q of the passage times of elementary chains and elementary cyclesensure that the union is disjoint.Now note that the function N ( T ) is piecewise constant and jumps can occur only during the times M of passing routes. The jump occurs at time t ∈ M v,I is equal to ρ out ( G, v ) − ρ in ( G (cid:48) , v ) (where the subgraph G (cid:48) is formed by the edges of an elementary chain l v and cycles c i ( i ∈ I ) ), since for any t ∈ M v,I there is aroute µ with t ( µ ) = t , which ends along any given edge G (cid:48) , entering the vertex v .It remains to sum up the jumps over all passage times not exceeding T and obtain the value of thefunction N ( T ) . 2 Asymptotic formula for N ( T ) as t → ∞ Theorem 2.
Let G be a finite one-way Sperner metric graph. We consider a random walk on it with theinitial vertex s . Then for the number of possible endpoints at the time T has the following asymptotics: N ( T ) = T β − ( β − · (cid:80) e ∈ E t ( e ) β (cid:81) i =1 t ( c j ) (1 + o ( T β − )) , where β is a number of elementary cycles in Γ and T tends to infinity. Proof.
We know (see [6] for references on Barnes—Bernoilli polynomials) that the number of non-negativesolutions to the inequality n a ... + n m a m ≤ T grows as a polynomial of degree m . Accordingly, to find theleading coefficient, we need to take inequalities in the formula (2.1), in which either all cycles ( | I | = β ), areinvolved, or all cycles except one ( | I | = β − ).Let us consider the term N ( T ) in the formula (2.1) corresponding to | I | = β (then G (cid:48) = G ): N ( T ) = (cid:88) v ∈ V [ ρ out ( G, v ) − ρ in ( G, v )] (cid:40) t ( l v ) + β (cid:88) i =1 n i t ( c i ) ≤ T (cid:41) Let us write the two leading terms in the expansion (cid:110) t ( l v ) + (cid:80) βi =1 n i t ( c i ) ≤ T (cid:111) : β (cid:88) i =1 n i t ( c i ) ≤ T − t ( l v ) (cid:124) (cid:123)(cid:122) (cid:125) λ = 1 (cid:81) βi =1 t ( c i ) (cid:32) λ β β ! − β (cid:88) i =1 t ( c i ) λ β − ( β − O ( λ β − ) (cid:33) == 1 β ! (cid:81) βi =1 t ( c i ) (cid:34) T β − β (cid:32) t ( l v ) + 12 β (cid:88) i =1 t ( c i ) (cid:33) T β − + O ( T β − ) (cid:35) Note that the coefficient at T β does not depend on v , therefore [ T β ] N ( T ) = 0 due to the fact that (cid:80) v ∈ V [ ρ out ( G, v ) − ρ in ( G, v )] = 0 , i.e. by hand-shaking lemma. Thus: [ T β − ] N ( T ) = − β − (cid:81) βi =1 t ( c i ) (cid:88) v ∈ V [ ρ out ( G, v ) − ρ in ( G, v )] t ( l v ) Now consider the term N ( T ) in the formula (2.1), corresponding to I = { , . . . , β } \ { j } : N ( T ) = (cid:88) v ∈ V (cid:88) j =1 ,...,β ( ρ out ( G, v ) − ρ in ( G (cid:48) , v )) t ( l v ) + (cid:88) i (cid:54) = j n i t ( c i ) ≤ T , In this sum ρ in ( G (cid:48) , v ) = ρ in ( G, v ) = 1 for v (cid:54) = s and ρ in ( G (cid:48) , s ) = ρ in ( G, s ) − .We get that N ( T ) = N (cid:48) ( T ) + N (cid:48)(cid:48) ( T ) , where N (cid:48) ( T ) = (cid:88) v ∈ V ( ρ out ( G, v ) − ρ in ( G, v )) β (cid:88) j =1 t ( l v ) + (cid:88) i (cid:54) = j n i t ( c i ) ≤ T ,N (cid:48)(cid:48) ( T ) = β (cid:88) j =1 (cid:88) i (cid:54) = j n i t ( c i ) ≤ T We have: [ T β − ] N (cid:48) ( T ) = 0 , therefore [ T β − ] N ( T ) = (cid:80) βi =1 t ( c i )( β − (cid:81) βi =1 t ( c i )
3t remains to prove that (cid:88) e ∈ E t ( e ) = β (cid:88) i =1 t ( c β ) − (cid:88) v ∈ V [ ρ out ( G, v ) − ρ in ( G, v )] t ( l v ) Let us show that by removing the edges, the expressions on the left and right decrease by the sameamount. First, we remove the edge e = ( v, s ) . The expression on the left decreased by t ( e ) , the expressionon the right decreased by t ( l ) + t ( e ) − t ( l ) , where the simple chain l leads from s to v . We continue until β becomes equal to zero, i.e. G will become a directed tree.We remove the hanging edge e : the expression on the left decreased by t ( e ) , and the expression on theright decreased by − t ( l ) + t ( l ) + t ( e ) , where l — is a simple chain from s to the beginning of the edge e .In the end, we get a tree with one edge and this equality is true for it. Earlier (see [4]) a formula was obtained for the leading asymptotic coefficient of N ( T ) in the case of anordinary (undirected) graph. Here is the formula: N ( T ) = T E − V − ( E − · E (cid:80) i =1 q jE (cid:81) i =1 q j (1 + o ( T E − )) , where E is the number of edges in the graph, V is the number of vertices, and q j is the lengths of the edgesof the undirected graph.There is no direct analogue of considered by us class of graphs in the undirected case. But we canconsider its special case, namely the class of graphs, which are oriented disjoint “ loops ” (cycles with twovertices), connected at the source. Such a graph corresponds to an undirected star graph. In this case, wecan assume that the sum of the lengths of the edges of the directed graph E (cid:80) i =1 t j is equal to E (cid:80) i =1 q j , and t ( c j ) = 2 q j . The number of vertices V is equal to E + 1 , then, substituting into the theorem proved in thisarticle, we get: N ( T ) = T E − ( E − · E (cid:80) i =1 q j E E (cid:81) i =1 q j (1 + o ( T E − )) , which, after shortening, gives the formula for the undirected case.We could notice that the presented formula for the leading term of N ( t ) is valid not only for graphsfrom the class of digraphs that we considered.Let us consider a graph in the form of a circle with two points on it and two directed chords.4he starting vertex is the vertex A = s . Let us find the times of possible routes.1. Times of routes from A to A : n ( t + f ) + n ( t + f + f ) + n ( f + f + f , n , n , n ≥ At these times N ( T ) increases by 2.2. Times of routes from A to B : n ( t + f ) + n ( t + f + f ) + n ( f + f + f ) − f , where not all n i are zero.The condition under which at these times the routes end at t and f : n > (cid:34) n > n > Here N ( T ) decreases by 1.3. Times of routes from A to C : n ( t + f ) + n ( t + f + f ) + n ( f + f + f ) − f − f , where n or n (cid:54) = 0 .The condition under which at these times the routes end at t and f : (cid:40) n > n > Thus, we get that N ( T ) is a quadratic function of T : N ( T ) = 2 { n ( t + f ) + n ( t + f + f ) + n ( f + f + f ) ≤ T, n , n , n ≥ } −− (cid:26) n ( t + f ) + n ( t + f + f ) + n ( f + f + f ) − f ≤ T, n > , (cid:20) n > n > (cid:27) −− { n ( t + f ) + n ( t + f + f ) + n ( f + f + f ) − f − f ≤ T, n ≥ , n > , n > } == T · f + f + f + t + t f + f + f )( f + t )( f + f + t ) + O ( T ) Conclusions
We found the asymptotics for the number of possible endpoints of a random walk at large times in thecase of the certain class of directed graph. Examples show that the formula for the leading coefficient stillholds for digraphs without the uniqueness of the path from source to every vertex. So to find a class ofmetric graph for which the derived formula could be correct could be the aim of further research.
Acknowledgments
The authors are grateful to V. E. Nazaikinskii, A. I. Shafarevich for support and useful discussions. Thework of A. Tolchennikov on Section 2 was supported by grant 16-11-10069 of the Russian Science Foundation.The work of V. Chernyshev on Section 3 was supported by the RFBR grant 20-07-01103 a.5 eferences [1] L. Lovasz,
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