THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF MATHEMATICS A RECURSIVE PRESENTATION OF THE GRAPHON SPACE

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1 THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF MATHEMATICS A RECURSIVE PRESENTATION OF THE GRAPHON SPACE PATRICK BLAIR NICODEMUS SPRING 2017 A thesis submitted in partial fulfillment of the requirements for baccalaureate degrees in Mathematics and Electrical Engineering with honors in Mathematics Reviewed and approved* by the following: Jan Reimann Assistant Professor of Mathematics Thesis Supervisor Nathanial Brown Professor of Mathematics Honors Adviser *Signatures are on file in the Schreyer Honors College.

2 i Abstract Graphons are objects that represent the limits of sequences of finite graphs - the completion of the set of graphs under the cut metric. The study of these continuous generalizations of graphs has been very fruitful, resulting in (among other things) elegant generalizations of the Szemeredi Regularity Lemma, advances in graph property testing, and applications to computer science, quantum computation, and statistical physics. This space is complete and separable, so it forms a Polish space. In this paper I show that, furthermore, it is an effective Polish space; that is, an algorithm exists which can decide in finite time whether the distance between members of a certain dense subset is greater than, less than, or equal to any given rational number. The space is naturally divided into equivalence classes of isomorphic graphons which can be transformed into each other through relabelling of nodes. The subject of Polish spaces partitioned into equivalence classes (called singular spaces ) is a topic which is an active research area in descriptive set theory, and so identifying the basic characteristics of the graphon space may be the first step to more interesting results obtained by studying this space through the lens of descriptive set theory and effective descriptive set theory.

3 ii Table of Contents Acknowledgements iii 1 Basic Concepts of Effective Descriptive Set Theory Polish Spaces Recursion Theory, Effective Polish Spaces Graph Spaces Graphs Homomorphisms The Sampling Distance Graphons Stepfunctions The Cut Distance Schatten p-norm Recursive Presentation Complete and Separable Distance Predicates Conclusion Bibliography 19

4 iii Acknowledgements My deepest thanks go to Jan Reimann for his continued support and tutelage throughout my undergraduate career.

5 1 Chapter 1 Basic Concepts of Effective Descriptive Set Theory

6 2 1.1 Polish Spaces The real numbers are an uncountable metric space. However, the real numbers are often defined as the completion of the rational numbers. Each real number can be represented as a Cauchy sequence of rational numbers, and the rational numbers are countable. In this way, we can think of the rational numbers as a countable backbone of the reals- the rationals are dense in the reals, and we can express every real number as a sequence from this countable backbone. We may fix an enumeration, A, of the rational numbers, for instance, A(0) = 1/1, A(1) = 1/2, A(2) = 2/1, etc., where we enumerate ordered pairs of natural numbers through a pairing function such as the well-known Cantor pairing function π(x, y) = (x+y)(x+y +1)/2+y, omitting repetitions of the same fraction so that each fraction can be represented by a single natural number. With A given, it is possible to establish a correspondence between sequences of natural numbers and real numbers. Given a sequence of natural numbers, n 1, n 2,..., then we can say that the sequence represents a real number x if the sequence of rational numbers A(n 1 ), A(n 2 ),... is Cauchy and converges to x. This concept can be generalized to other metric spaces. A Polish space is a complete metric space which is separable, meaning it contains a countable, dense subset. The real numbers are separable because the rational numbers are both countable and dense within the reals. A perfect space is one with no isolated points, so each point can be expressed as the limit of a sequence of other points. The real numbers are, again, perfect. In all perfect Polish spaces, the above method of representation is applicable. Each Polish space M has a countable, dense subset A. If we fix an enumeration of this subset, A = (A 1, A 2,...), then every point x in M can be represented as a sequence of natural numbers x 1, x 2,... such that the corresponding sequence A(x 1 ), A(x 2 ),... of points in M converges to x. The set of all sequences of natural numbers is called Baire space in set theory. It is common to think of Baire space as an infinite tree, where each node has N as its set of children; a sequence of natural numbers is a path through this tree. Baire space is equipped with the cylinder topology, where the basic open sets are the sets of all sequences of integers sharing a common prefix, so that for some n, every sequence in the set has the same beginning a 1 a 2 a 3...a n when truncated to the first n elements. It can be shown that for every Polish space M, there is a continuous mapping from Baire space to M.[1] 1.2 Recursion Theory, Effective Polish Spaces A function on the natural numbers f : N N that can be computed in an effective way by a formal algorithm or mechanistic process is called a recursive or computable function. The set of computable functions is formally defined as those functions which are computable by a Turing machine, or which are expressible in the lambda calculus of Alonzo Church; it is generally agreed that these formal definitions align with our intuitive beliefs about what kinds of functions are computable or can be evaluated by an algorithm. A Turing machine is a hypothetical device consisting of a long paper tape, stretching off to infinity in both directions, divided into square cells in which a single symbol can be printed - say, 0, 1, or the cell is left blank. A small moving head on the tape, about the same size as a square cell, drives up and down the tape, reading the symbols, erasing them, and writing new ones according to an algorithm hardwired into it. Suppose that we encode a natural number n onto the tape in the form of its binary representation, leave all other

7 3 squares on the tape blank, and start up the Turing machine with the head positioned above the leftmost binary digit of the number. The Turing machine will move up and down the tape, altering the symbols in the squares according to its program, and may eventually terminate. If it terminates while the head is positioned somewhere in the middle of a string of 1 s and 0 s, demarcated on each end by blank squares, we can interpret this binary string to be the output and interpret it as the natural number m that it encodes. (If the Turing machine stops while positioned above a blank square, we may interpret the output to be zero.) We say that the function φ computed by this Turing machine T φ maps n to m. It is possible that the Turing machine never halts when it is begun on a certain binary string, so strictly speaking, Turing machines compute partial functions rather than true functions. If the machine never terminates for a certain input, we say that the function is undefined on this input, or divergent. Not all functions on the natural numbers are computable: there are only countably many algorithms (for example, one could enumerate all grammatically correct programs in the C language), but there are uncountably many functions from N to N, so by a simple counting argument it must follow that there are uncomputable functions. A specific example of an uncomputable function, constructed by Turing, is the Halting function. This function takes as its argument the index of a Turing machine, and returns a 0 if the indexed Turing machine finishes its execution in a finite period of time, or 1 if the machine never terminates but continues its computation forever. It is possible to generalize this concept to speak of n-ary partial recursive functions. Instead of the input being a single binary string, we may have the input as n many binary strings, each one demarcated from its neighbors on both sides by a single blank square. An n-ary predicate P (x, y, z) on N can be thought of as computable if there is a computable n-ary function which returns 1 when the predicate evaluates to true or 0 when it evaluates to false. We can speak of computable functions on any countable sets B and C with reference to fixed enumerations B = (b 1, b 2,...) and C = (c 1, c 2,...). A function f : B C is computable if there is a computable function f : N N such that f (i) = j = f(b i ) = c j. In this way we can talk about the structure of the set, and functions on the set, indirectly through the natural numbers that encode them. A recursive presentation of a Polish space M consists of a countable, dense subset R (with a fixed enumeration r 0, r 1,...), together with two recursive (computable) predicates: P (i, j, m, k) d(r i, r j ) < m k + 1 Q(i, j, m, k) d(r i, r j ) m (1.2) k + 1 where d is the metric of M. This means that the subset R should have the property that an algorithm is able to decide in a finite number of steps that the distance between any two points r i and r j is less than, equal to, or greater than a given rational number. Such an algorithm would allow us to compute the distance between any two points of R to an arbitrary degree of precision - for instance, by means of a binary search algorithm. An effective Polish space is a Polish space that admits a recursive presentation. The task of this paper is to investigate a certain Polish space, the space of graphons, and show that it admits a recursive presentation. (1.1)

8 4 Chapter 2 Graph Spaces

9 5 2.1 Graphs A graph is a mathematical model of a network. A graph G consists of a set of nodes or vertices, V, which represent the points in the network, together with a binary relation E on V V, the set of edges or links of the network. Often E is symmetric and irreflexive. If a and b are two nodes in V, then aeb means that a and b are connected in the graph, or that an edge exists between them. An Internet social network could be modelled by a graph; we would say that the website s users are the nodes of the graph, and that two nodes are connected if the users are friends on the site. A subway map is also a form of graph. The stations are nodes, and two nodes are connected if the stations are exactly one hop apart. Computer networks are an extremely common type of graph - the machines of the network are nodes, and two machines are connected if they are directly linked by (say) an Ethernet switch. Graphs that can be described as above are called simple. It may be necessary to modify these definitions to account for more sophisticated types of networks. Some networks may involve selfloops, and the restriction of irreflexivity must be removed from E. We may also have a network where the description of connectedness is more complex than a binary relation, and so we must augment (or supplant) E with a binary function β : V V F which maps to some field F. (In this paper we will be working with graphs where F is R.) If the two nodes are completely disconnected, β(v 1, v 2 ) may return zero; otherwise β may return some useful edge-weight which reflects (for instance) the geographical distance between two subway stations, or the bandwidth of a network connection. A graph may also be directed, meaning the edges have an orientation and may flow in one or both directions, possibly with different weights in each direction (so the relation E and the function β are not symmetric). G may also include a node-weight function α : V F which assigns a value to nodes, perhaps representing the mass of a particle in a model of a physical system. n! In a graph with n nodes, there are possible pairs of nodes, and thus the graph may have up 2!(n 2)! to n(n 1)/2 edges. A graph where the number of edges is on the order of this maximum value is called dense ; if the number of edges is much smaller (say that n is large and the number of edges is on the order of magnitude of n rather than n 2 ), then the graph is called sparse. Sometimes the elements of V may be named or labelled ; for instance, V might be the set {1, 2,.., n}. (We will abbreviate this set as [n].) Two simple graphs F and G are said to be isomorphic if there is a bijection φ from the node set V F to V G for which v 1 E F v 2 φ(v 1 )E G φ(v 2 ); that is, the mapping preserves connectedness and disconnectedness. Often we are interested in these isomorphism classes rather than the labelled graphs, because the properties and structure of the set V are not of interest. We prefer to think of it as a homogeneous set where one node is not distinguished from another. Such an isomorphism class is called an unlabelled graph and are the more common objects of study. We are, in this paper, also interested in random graphs. A random graph X on the set of nodes [k] is a random variable which assumes the form of a simple k-node graph when sampled or instantiated. The distribution that governs X may result, for example, from choosing k nodes randomly and without replacement from a fixed large graph G, and connecting the nodes i and j in X if and only if the ith and jth nodes sampled are connected in G. Then X takes on the value of the resulting induced subgraph of G. Alternatively, the edges of X might be independent Bernoulli variables X ij, decided with probability p ij. We could then represent the simple, unweighted random variable

10 6 X as an edge-weighted graph X where the weights of X ij are p ij. 2.2 Homomorphisms Broadly speaking, the study of graphons begins with the study of large, dense graphs, and measures of similarity between them. Given a large (labelled) graph, G, of size n nodes, and a smaller graph, F, of size k nodes, there are n k possible mappings of F into G, and ( n k) possible injective mappings. Of these mappings, some may preserve the edge relationships of F, such that whenever the pair of nodes (i, j) is present in the set of edges E(F ), that edge is preserved under the mapping φ : F G so that (i, j) E(F ) = (φ(i), φ(j)) E(G). A mapping that preserves the edge relationships of F is called a homomorphism. If we restrict our study to injective mappings in particular then we will refer to injective homomorphisms. An injective mapping that preserves not only connectedness, but also disconnectedness, so that (i, j) E(F ) (φ(i), φ(j)) E(G), is called an induced-subgraph embedding. hom(f, G): the number of homomorphisms of F into G. inj(f, G): the number of injective homomorphisms of F into G. ind(f, G): the number of induced-subgraph embeddings of F into G. This paper will involve homomorphisms into graphs with weighted nodes, edges and loops, so we extend the definitions of homomorphisms to include weighted graphs with self-loops. Let G be a simple graph with node weights α G (v), edge weights β G (u, v), and loops. For a map φ : F G, define the map weight α φ = α G (φ(u))) (2.1) and let hom φ (F, G) = u V (F ) (u,v) E(F ) β G (φ(u), φ(v)) (2.2) The definition of the homomorphism number can be extended by taking the sum of these weighted values over all n k possible mappings of F into G. hom(f, G) = α φ hom φ (F, G) (2.3) φ:v (F ) V (G) The injective homomorphism number can be extended the same way. inj(f, G) = α φ hom φ (F, G) (2.4) φ:v (F ) V (G) φ injective For simple, unweighted graphs, we can scale these numbers by the number of possible mappings of F into G to get the homomorphism density of F in G; this number represents the frequency with which F appears in G, or the probability that a random mapping of F into G will be a homomorphism.

11 7 t(f, G) = hom(f,g) n k, called the homomorphism density of F into G. t inj (F, G) = inj(f,g) : the injective homomorphism density of F into G. ( n k) t ind (F, G) = ind(f,g) : the induced-subgraph density of F in G. ( n k) Some comments on these definitions: If we choose k nodes at random from G without replacement, t ind (F, G) is the probability that the resulting k-node induced subgraph of G is F. These values are all between 0 and 1. These definitions are not meaningful for k > n, when F is larger than G. There is no obvious way to extend the definitions, but is necessary to define it for later, so we set each one equal to 0 whenever k > n. As G grows large for fixed F, t inj (F, G) and t(f, G) approach each other, because a random mapping of F into G is likely to be injective when G is much larger than F. A bound on their relationship is given by this inequality: t(f, G) t inj (F, G) 1 ( ) V (F ) (2.5) V (G) 2 t ind (F, G) t inj (F, G). This relationship does not, in general, become insignificant as G grows large. Let F be the graph with k nodes, but with no edges, and let G be the complete graph on n nodes. Every injective mapping of F into G is an injective homomorphism, but there are no induced-subgraph mappings of F into G. However, they are closely related as both deal with injections. Every induced-subgraph embedding of F is also a homomorphism of F, but so is the induced-subgraph embedding of any graph F which contains at least the same edges as F. Therefore they are related by: inj(f, G) = ind(f, G) (2.6) t inj (F, G) = E(F ) E(F ) E(F ) E(F ) t ind (F, G) (2.7) Likewise, ind(f, G) and t ind (F, G) can be determined explicitly by the values of inj and t inj in a slightly more complex way. 2.3 The Sampling Distance Suppose we choose k nodes randomly and without replacement from G - that is, sample a k- node graph from G - and study the resulting induced subgraph X. Then X is a random variable which takes on the value of a k-node graph when it is sampled, and P (X = F ) = t ind (F, G). The

12 8 probability distribution which governs X will be called σ G,k, so σ G,k (F ) = t ind (F, G). Define the distance between two distributions σ 1,σ 2 on the same (finite) sample space X to be d var (σ 1, σ 2 ) = 1 σ 1 (x) σ 2 (x) (2.8) 2 x X If d var (σ G1,k, σ G2,k) is small, then a random graph of size k drawn from G 1 or G 2 has similar chances of taking on the value of any given k-node graph. (Here, the coefficient of 1/2 is chosen so that the distance is always less than 1.) We would like to define a sampling metric between large graphs, δ samp, such that two graphs G 1 and G 2 are close under the metric if every small graph F occurs with similar frequency as an induced subgraph in both G 1 and G 2 - that is, for every small graph F, t ind (F, G 1 ) t ind (F, G 2 ). Equivalently, G 1 and G 2 should be close if σ G1,k σ G2,k for each k- the random distributions on k-node graphs induced by G 1 and G 2 should be similar. We can construct this as follows: δ samp (G 1, G 2 ) = k=1 1 2 k d var(σ G1,k, σ G2,k) (2.9) Or, equivalently, by combining equations 2.8 and 2.9, we can get: δ samp (G 1, G 2 ) = F simple, finite 1 2 V (F )+1 t ind(f, G 1 ) t ind (F, G 2 ) (2.10) Consider a sequence of graphs G 0, G 1,... whose size grows without bound as the index approaches infinity. A sequence of graphs is said to be convergent if for every finite graph F, the homomorphism density of F in G n approaches a stable limit; that is, lim n t(f, G n ) exists for each simple, finite graph F. By 2.7 and 2.5, this is equivalent to saying that lim n t ind (F, G n ) exists, or that for each k, the sequence (σ Gn,k) n N converges in distribution, which is in turn equivalent to saying that the sequence of graphs (G n ) is Cauchy under the metric δ samp. We seek to define an object which represents the limit of this sequence. Call it G. We should be able to extend the definitions of t(f, G) and t ind (F, G)in a meaningful way for each finite graph F, and G should have the property that lim n t(f, G n ) = t(f, G) and lim n t ind (F, G n ) = t ind (F, G)for each F. If this is possible, then we can generalize the metric δ samp using the new definition of t ind ; then lim n δ samp (G n, G) = 0. We say the sequence (G n ) converges to G. Graphons are these limit-objects. They represent the completion of the set of finite, simple graphs under the metric δ samp. 2.4 Graphons A simple finite graph can be represented in terms of its adjacency matrix : a symmetric matrix w of ones and zeros, where a 1 at index w ij means that an edge exists between nodes i and j, and a 0 represents the absence of an edge. We could generalize this method to represent an edgeweighted graph G w on the node set [n], by setting w ij = β Gw (i, j). As discussed in section 2.1, if we wanted to represent a random graph G w on the node set [n], where each edge (i, j) is present in G w with probability p ij, decided independently as a Bernoulli variable, we could encode it in

13 9 a similar matrix with w ij = p ij. (The 0 1 valued matrix is only a special case of this kind of matrix; it represents the random graph which returns a specific graph with probability 1.) For such a random-graph matrix w of size n n, we could define sampling from w. To sample a graph F of size k < n, we may choosing k random nodes uniformly from [n] without replacement and deciding the edge (a, b) in F independently with probability w ab, where a, b are from the sampled node list [x 1,..x k ]. This is equivalent to instantiating the random graph G w of size n from w as described above, and then randomly sampling the k-node graph F from G w. Because it is possible to define sampling on these random graphs, it is then possible to define induced-subgraph densities of F in w (or into the random graph G w represented by w). t ind (F, w) is the probability that a k-node graph sampled from w is F ; t inj (F, w) is the probability that a k-node graph randomly sampled from w has at least the edges present in F, but possibly more. t(f, w) is the probability that a k-node graph sampled with replacement is F. We can define t(f, w) as being equal to t(f, G w ) where G w is the weighted graph encoded by w. A sequence (w n ) of these random-graph matrices (symmetric matrices taking on values in [0, 1]), whose size grows without bound as n, can be said to converge if lim n t(f, w n ) exists for each F. Every such random-graph matrix is equivalent to a symmetric function w : [1..n] [1..n] [0, 1], where w(i, j) = w ij. We construct our limit object by passing from a discrete node-set to a continuous one. A graphon - the limit of a convergent sequence of graphs - is a symmetric, measurable function on the set [0, 1] 2, so that W : [0, 1] [0, 1] [0, 1]. A graphon (the word is a contraction of graph-function) can be thought of as a random or edge-weighted graph on the set [0, 1]. Sampling a k-node graph from a graphon is performed by choosing a list of k real numbers uniformly from the interval [0, 1], where edges occur between the node pair (x i, x j ) independently with probability W (x i, x j ). Then t ind (F, W ) is the probability that a random k-node graph sampled from W is F, and t(f, W ) is the probability that the k-node graph sampled from W has at least the edges that F does. (We can here drop the subscript inj because this distinction is irrelevant in graphons: the probability of randomly sampling the same real number from [0, 1] twice in finitely many samples is zero and so a concrete difference between t and t inj does not exist.) How can we evaluate t and t ind? We have said that t ind (F, W ) is the probability that a random graph sampled from W is F. Suppose we sample a k-node graph from W. This graph is F if it has an edge wherever F has an edge, and does not have an edge where F does not. Begin by choosing randomly and uniformly a list (a 1,.., a k ). We decide each edge (i, j) independently with probability W (a i, a j ), so we have P (Sample(W, a) = F ) = (i,j) E(F ) W (a i, a j ) (i,j)/ E(F ) (1 W (a i, a j )) (2.11) Because sampling is done by choosing a random k-long vector a from [0, 1] we must integrate over the sample space of all possible k-long vectors - the k-dimensional cube where each side is [0, 1]. t ind (F, W ) = W (x i, x j ) (1 W (x i, x j ))d x (2.12) [0,1] k (i,j) E(F ) (i,j)/ E(F )

14 10 For t, we do not care about edges which are not present, so t(f, W ) = W (x i, x j )d x (2.13) [0,1] k (i,j) E(F ) Why can we say that these graphons are the objects we seek - the limits of sequences of simple graphs? This is justified by a theorem due to Lovasz and Szegedy: for any convergent sequence (G n ) of simple graphs, there exists a graphon W such that lim n t ind (F, G n ) = t ind (F, W ) and lim n t(f, G n ) = t(f, W ) for every finite F. In this case we say that (G n ) W, (G n ) converges to W. The converse is also true: for every graphon W there exists a sequence of graphs which converges to W.[2] Let G(k, W ) be the probability distribution on graphs of size k, where G(k, W )(F ) = t ind (F, W ) - the distribution that results from sampling a graph of size k from W. The metric δ samp can be adapted to a pair of graphons U, W as follows: δ samp (U, W ) = k=1 1 2 k d var(g(k, U), G(k, W )) (2.14) which again simplifies to δ samp (U, W ) = F 1 2 V (F )+1 t ind(f, U) t ind (F, W ) (2.15) Multiple graphons W, W may have the same homomorphism densities; it is possible that for each F, t(f, W ) = t(f, W ) and t ind (F, W ) = t ind (F, W ). In this case we say that W and W are weakly isomorphic. The set of all graphons is naturally partitioned into these weakisomorphism classes. Sequences of graphs that converge to W will also converge to W. With reference to 2.15, if two graphs have t ind (F, W ) = t ind (F, W ) for every F, then the sampling distance between two weakly isomorphic graphons is zero. Therefore, when applied to the space of graphons, δ samp is not a true metric but rather a pseudometric. (In order for it to be a proper metric, we would need to consider it as a function on weak-isomorphism classes of graphons rather than individual graphons as functions.) Because of the integral in the definition of t for graphons, two graphons whose difference is a set of measure zero will be weakly isomorphic. Weakly isomorphic graphons are also closely tied up with measure-preserving transformations on the interval [0, 1]. For instance, let W be any graphon, and let φ be a measure-preserving bijection from [0, 1] to itself. Let W φ (x, y) = W (φ(x), φ(y)); then W and W φ are weakly isomorphic. The application of φ to [0, 1] can be thought of as a relabeling of the graphon, which should be isomorphic to the original. One can also argue that it follows from the nature of sampling a graphon. A measurepreserving permutation does not affect the nature of homomorphism densities, as it does not affect random sampling. In sampling a k-node random graph from W, we begin by choosing k random real numbers from the interval [0, 1]; in sampling a k-node random graph from W φ, we first shuffle the deck with φ and then choose k real numbers again from this new shuffled interval. There is no difference.

15 Stepfunctions A stepfunction is a graphon W such that, for some partition of the interval [0, 1] into n measurable subsets S 1.. S n, the function W is constant everywhere on the set S i S j for any i, j. The partitions S i are called the steps of W. To every finite graph G we can assign an equivalent stepfunction W G - when we ascend from the set of graphs into the graphon space, G becomes W G. For a simple, unweighted graph with n nodes, divide the interval [0, 1] into n equal partitions of measure 1, so [0, 1] = S n 1.. S n corresponding to the node set [1..n]. For all ordered pairs (x, y) in S i S j, assign W G (x, y) = 1 if (i, j) E(G), and 0 otherwise. The graph of this function will look like the pixel picture that corresponds to the adjacency matrix of G. W G is strongly related to G in the sense that for every graph F, t(f, G) = t(f, W G )[2]. This allows us to naturally substitute W G for G in many equations which are defined with reference to t, easily translating theorems into the language of graphons. This can be generalized to graphs with weighted nodes and edges. For a graph on the node set [1..n] with positive node weights α 1..α n and edge weights β i,j (belonging to the interval [0, 1]), let α G = n i=1 α i. Divide the interval [0, 1] into n partitions S 1..S n such that the measure of S i is α i /α G. This way, we encode the node weights into the size of the partitions. The edge weights are likewise encoded into the function values: everywhere on the product set S i S j, let W (x, y) = β i,j. Similarly, every stepfunction W has an equivalent weighted graph, G W. If W has steps S 1,..S n then our graph is defined on the node set [1..n], where the node i takes on node weight λ(s i ) and the edge (i, j) takes on weight W (S i, S j ).[2] 2.6 The Cut Distance The sampling distance is an approach to comparing two large graphs that studies the subgraphs that appear in each; two graphs are similar if the same subgraph has a similar chance of being drawn from each one. Another method of comparing two graphs would be the edit distance, the number of edges that need to be added/deleted from G 1 to turn it into G 2. Formally, we can define the edit distance between two graphs G 1 and G 2 in terms of the distance between their corresponding adjacency matrices A 1 and A 2. The l 1 norm of a matrix is computed as A 1 = 1 n 2 n A ij (2.16) and the edit distance between two graphs is d 1 (G 1, G 2 ) = A 1 A 2 1 (usually without the scaling by n 2 ). The cut distance is similar to the edit distance in that we can define it in terms of a norm on the adjacency matrix. The norm being used here is the cut norm introduced by Frieze and Kannan [3]: A = 1 n max A 2 ij (2.17) S,T [n] The corresponding cut distance is i,j=1 i S,j T d (G 1, G 2 ) = A 1 A 2 (2.18)

16 12 To maximize the sum in the expression, one is trying to find two sets of nodes S and T such that in G 1, there are a great deal of edges connecting S to T, while in G 2 there are very few edges connecting those same subsets of nodes. This metric is useful because it directly compares two graphs without reference to other graphs; we are superimposing the nodes of one graph onto the other, and directly comparing the way edges in the first graph line up with edges in the second. The metric as presented here is highly sensitive to the labelling of the nodes, because the labelling of the nodes determines the way they are superimposed on top of each other. The same graph, relabelled, might not line up well at all with the original. If we want the metric to have the property that isomorphic graphs have distance zero from each other, we need to modify it. ˆδ (G 1, G 2 ) = min Ĝ 2 d(g 1, Ĝ2) (2.19) where Ĝ2 ranges over all possible relabellings of the nodes of G 2. Here, we find the best fit of G 1 on top of G 2. There is one more change required to fully define the cut distance. Given the graph G, let the blow-up graph G(k) be the graph in which every node of G is replaced with k child nodes. The new nodes are connected if and only if their parents were previously connected. In particular, none of the child nodes of a single node are connected to each other, unless that single node had a self-loop, in which case they are all connected to each other. This blow-up allows us to extend the metric to graphs of different sizes - if G 1 has size n 1, and G 2 has size n 2, then we may blow up each graph to the least common multiple of their individual sizes, or any common multiple, and compare them directly this way. We therefore define the cut distance as δ (G 1, G 2 ) = lim k ˆδ (G 1 (kn 2 ), G 2 (kn 1 )) (2.20) In the event that G 1 and G 2 are the same size, the new definition supersedes the previous one, and the distance is computed by taking the limit. (The two quantities can be different - how much different is an open question.) This definition can also be extended to graphons. Like the set of n n matrices, the set of measurable functions on [0, 1] 2 forms a vector space. At the core of the definition of the cut distance for graphs is the cut norm, so we proceed by generalizing this cut norm to graphons as W = sup S,T [0,1] S T W (x, y)dxdy (2.21) where S and T are measurable subsets. In this definition, the value of W (x, y) can be thought of as the edge weight between x and y. Alternatively, we can think as W as a sort of continuous generalization of a simple finite graph, where W (x, y) represents the edge density or connectivity between these two points. The supremum replaces the maximum in the previous definition. (Interestingly, this supremum is always attained, a consequence of the fact that the graphon space is compact.[2]) The cut distance between two labelled graphons is derived using this norm. ˆδ (U, W ) = U W (2.22) Again, in practice we often consider weak-isomorphism classes of graphons rather than individual graphons thought of as distinct entities. It is desirable to have a distance function which returns

17 13 zero if one graphon can be transformed into another by relabelling. In the definition for graphs, a relabelling was represented by a permutation on the set [n]. Here it is represented by measurepreserving transformations on the interval [0, 1]. δ (U, W ) = inf ˆδ (U, W φ ) (2.23) φ S [0,1] where S [0,1] is the set of measure-preserving transformations on that interval [3]. Note that this distance function is also a pseudometric rather than a true metric, as it returns a distance zero between any two graphons W and U if for some φ, the function W U φ has measure zero. We can turn it into a metric by defining it as a function on equivalence classes of graphons such that δ (W, U) = 0. The equivalence classes are exactly the weak-isomorphism classes.[2] The two different distance functions 2.22 and 2.23 can give rise to some confusion, because they give rise to very different topologies and have very different properties. I will refer to the first one as the distance function (or pseudometric) induced by the cut norm and the second one simply as the cut distance. The cut distance is extremely useful because, like the homomorphism density t, it also can be used interchangeably whether one is speaking of graphs or of graphons. δ (F, G) = δ (W F, W G ) (2.24) The sampling distance between two graphs does not have this property - it does not translate as neatly from the graph space to the graphon space, although as the graphs get larger and larger and approach their corresponding graphons more nearly, the difference approaches zero. Because of the small but nonzero difference in moving to the graphon space, it is less attractive than the cut distance for formulating the properties of graphons. However, as metrics on the set of weakisomorphism classes of graphons, they define the exact same convergent sequences of graphons, meaning that they induce the same topology on the set. 2.7 Schatten p-norm The cut norm on graphons is one of many operator norms which are topologically equivalent - i.e. they define the same convergent sequences of graphons. When p is an even integer, there is a corresponding norm on graphons W p = t(c p, W ) 1/p, where C p is the cycle on p nodes. We are specifically interested in the norm W 4 = t(c 4, W ) 1/4 because it has the useful property W t(c 4, W ) 1/4 4 W (2.25) (In source [2] this is referred to as a norm. Strictly speaking this should be considered a seminorm, because it assigns size zero to all functions whose value is only nonzero on a null set. Throughout the book the author does not distinguish between two graphons if they are almost everywhere equal. We will not adopt this convention and instead we will be careful to refer to it from here on out as a seminorm or pseudometric.) The distance function induced by this seminorm is thus topologically equivalent to the labelled cut distance which is induced by the cut seminorm. However, crucially, it is not the same as the unlabelled cut distance; in general, it does not assign distance zero to isomorphic graphons. We will use this metric in the next section to establish a recursive distance predicate on the set of (labelled) graphons, taking advantage of its simplicity.

18 14 Chapter 3 Recursive Presentation

19 Complete and Separable First, I establish a countable, dense subset of the graphon space. I will use the set of rationalvalued stepfunctions with rational boundaries. That is, the sets of the partition should be intervals, or unions of intervals, whose boundaries are rational; and the function assumes only rational values. Refer to this set as R. It is clear that these rational stepfunctions can approximate other stepfunctions to an arbitrary degree of precision in the sense that if U is any stepfunction, then inf U(x, y) U R (x, y) dxdy = 0 (3.1) U R R [0,1] 2 The integral above is the L 1 norm on the set of integrable functions on [0, 1] 2. Because graphons are measurable, it is possible to approximate any graphon under the L 1 norm arbitrarily closely using stepfunctions, and so it is possible to approximate them arbitrarily closely using rational stepfunctions as well (through 3.1). The bound W W 1 can be seen by referencing the definition of W in We also have, from 2.25, t(c 4, W ) 1/4 W. Therefore, if the L 1 norm can be made vanishingly small by choice of U R, so can t(c 4, U U R ) 1/4. Let us abbreviate t(c 4, W U) 1/4 as d 4 (W, U); then R is dense in the graphon space under the pseudometric d 4. Every stepfunction in R can be encoded as a q q matrix A U, where (A U ) (i,j) is a rational number representing the function s value on the square interval [(i 1)/q, i/q) [(j 1)/q, j/q). We can construct a practical Gödel numbering by assigning each matrix an integer code, from which the matrix can be effectively recovered by an algorithm. Let < x, y >= N N N be a bijective function which is computable in both directions, such as the Cantor pairing function described in Chapter 1. Let each matrix be represented by the ordered pair < q, c >, where q is the size of the square matrix A U. c, itself, is a list of all the elements of A U in order, from left to right and top to bottom, where c =< a 1,1, < a 1,2, < a 1,3,... >>> (3.2) Because the algorithm has q available after the first decoding operation, it knows to unpack c consecutively q 2 times before it has accessed all values. Lastly, each value a i,j is itself an ordered pair of the form < p, q >, representing the rational number p/q. This construction demonstrates that the elements ofr are enumerable and provides a method for an algorithm to perform computations on the stepfunction by performing operations on the natural numbers that encode them. This set of stepfunctions is inclusive enough to contain every set of rationally-valued stepfunctions arising from a partition on [0, 1]. 3.2 Distance Predicates It is here necessary to show that the predicates described in ( 1.1 ) and ( 1.2 ) are recursive; to prove that an algorithm exists to decide them. Our algorithm first considers the arguments i and j, the indices in R of r i and r j. The indices here are chosen based on the enumeration scheme described in the previous section; they are not arbitrary, but rather they encode the stepfunction directly. The algorithm unpacks i and j into the corresponding matrices A ri and A rj. Suppose that the dimension of the first is q i, and the second has dimension q j. The algorithm should blow up each matrix so that they are the same size

20 16 q i q j q i q j by replacing every scalar entry (A ri ) c,d with the q j q j matrix, full of qj 2 copies of entry (c, d). Then, the algorithm should subtract one from the other to create the difference matrix A U. This matrix encodes the stepfunction U which is the difference of the original two stepfunctions; our task is now to compute the C 4 seminorm of the encoded stepfunction U, t(c 4, U) 1/4. Fortunately, the fact that it is a stepfunction makes the integral definition of t computable. Recalling the definition of t and applying it here, we can write t(c 4, U) = U(x 1, x 2 )U(x 2, x 3 )U(x 3, x 4 )U(x 4, x 1 )dx 1 dx 2 dx 3 dx 4 (3.3) [0,1] 4 The product function in the integral above is itself a stepfunction. The hypercube [0, 1] 4 has within it (q i q j ) 4 hypercubes of equal size, arranged in a grid. Within each of these cells, the function has constant value, because the functions which are multiplied to create it are also constant. Everywhere on the interval [(x 1)/q i q J, x/q i q j ) [(y 1)/q i q J, y/q i q j ), the stepfunction U has value (A U ) (x,y). At each point < x 1, x 2, x 3, x 4 > in the hypercube [0, 1] 4, the ordered pairs (x 1, x 2 )...(x 4, x 1 ) belong to these stepfunctions. The integral then reduces to a simple sum, which can be evaluated by iterating over the values in A U. t(c 4, U) = 1 (q i q j ) 4 q i q j i,j,k,l=1 (A U ) (x1,x 2 )(A U ) (x2,x 3 )(A U ) (x3,x 4 )(A U ) (x4,x 1 ) (3.4) The algorithm should iterate over every ordered 4-tuple from < 1, 1, 1, 1 > to < q i q j, q i q j, q i q j, q i q j > and keep a running sum of the value in the above summation. When it finishes adding these up, it should multiply the sum by 1/(q i q j ) 4 to account for the fact that this is the volume of each hypercube incorporated into the sum. This final value is the value of the integral. The algorithm also takes the arguments m, k for the fraction m. It should compute k+1 m4 and (k + 1) 4, and directly compare the value t(c 4, U) to ( m k+1 )4. The results of this comparison allow it to decide predicates P and Q. Therefore the space of graphons, equipped with the pseudometric d 4, admits a recursive presentation and is thus an effective Polish space. 3.3 Conclusion The topology induced by this d 4 pseudometric is topologically equivalent to that induced by the cut seminorm, defining the same sequences of convergent graphons, although it is not an equivalent metric in the sense of being equivalent up to a constant factor [2]. We have here permitted ourselves to substitute the d 4 pseudometric for the pseudometric induced by the cut seminorm because it is the topology of the Polish space we are concerned with rather than the actual distance. It is important to keep in mind that these pseudometrics are not topologically equivalent to the sampling distance unless one alters their definition to include minimizing the quantity over measurepreserving transformations as described in The pseudometric chosen for this demonstration was defined on individual graphons, rather than isomorphism classes of graphons. I will here briefly describe the significance of this result and possible points of departure for future study. The set of graphons W is a Polish space because it is complete and separable. When studying it from a graph-theoretic perspective, we are generally not interested in studying each graphon

21 individually, but as unlabelled graphons or weak-isomorphism classes of graphons. That is, if we define the equivalence relation E to be U E W δ (U, W ) = 0, then our field of study is W/E, the quotient space of W partitioned into the equivalence classes of E. Quotient spaces of the form X/E, where X is a Polish space and E is an equivalence relation on that space, are an active area of research in descriptive set theory. Methods borrowed from topology and measure theory which have previously been applied to descriptive set theory are not directly applicable for their study. Thus they are often referred to as singular spaces [4]. From the perspective of descriptive set theory, we may ask questions such as: Is it possible to establish the cardinality of W/E as R by explicitly defining a bijection between the set of unlabelled graphons and R, rather than by a proof which relies on the axiom of choice? Is the equivalence relation E effective, or can it be decided with reference to an oracle? Does E belong to a well-known Borel equivalence class - for instance, is it Borel reducible to the Turing equivalence relation between sets of natural numbers E 0 or to the universal equivalence relation E [4]? Establishing the basic properties of W as a Polish space is the first step towards answering these questions. In applications of graph theory to computer science, we are often concerned with questions about the global structure of a graph, and its macroscopic form as a network. We are also interested in algorithms that can answer these questions. Any method of studying large graphs through homomorphism densities or random sampling of small subgraphs will be unsuitable for this study because these methods focus inherently on the microscopic properties of the graphs, so it is necessary to develop different approaches. One approach that might be useful in evaluating a graph s structure would be to describe the complexity of the graph in terms of its Kolmogorov complexity. Briefly, the Kolmogorov complexity of an object is the length in bits of the shortest computer program that can reproduce it. To define the Kolmogorov complexity of a simple finite graph, we could fix some encoding of graphs via natural numbers, analogous to the stepfunction encoding given in section (3.1). Then the Kolmogorov complexity of a graph G, encoded by the natural number n, would be the length in bits of the shortest binary string that, when given as the input to a universal Turing machine, would cause that Turing machine to print the binary encoding of n and terminate. Objects with low Kolmogorov complexity are simple and contain repetitive patterns that can be captured by an algorithmic description. Objects with high Kolmogorov complexity are basically random and contain no easily discernible patterns to speak of. Kolmogorov complexity is a concept which is applicable only to finitely long binary strings and objects that can be coded in such a way. However, it has an analogue for infinite binary strings, Martin-Löf randomness, which can be used to study the complexity or randomness of the elements of the Cantor space 2 N.[5] The Cantor space is equipped with the same cylinder topology for trees described in section (1.1), except that the prefixes are finite binary strings instead of finite sequences of integers. The theory of Martin-Löf randomness makes use of the framework of a sort of effective topology. It can be shown that every open set in Cantor space can be represented as the countable union of the basic open sets of the topology (each of which can be described by its finite binary prefix). An effectively open set is one which is the countable union of a recursively enumerable sequence of these basic open sets - that is, there exists an algorithm which can, one after the other, print out the finite binary prefixes which define the basic open sets of the sequence. An effectively null set is formed from the countable intersection of a recursively enumerable sequence U of effectively open sets U 1, U 2,... such that the sets of the sequence are nested (U i+1 U i ) and have measure that decreases with 2 i, so λ(u i ) 2 i [5]. The intersection of this sequence of effectively open sets is a G δ set whose measure is zero. These effectively null 17

22 sets basically include all infinite binary strings which deviate in an algorithmically describable way from the concept of randomness, and we say that an infinite binary string is Martin-Löf random if it does not belong to any effectively null set. Martin-Löf randomness may be relevant to the study of graphons. Simple, finite graphs can be described by their Kolmogorov complexity, and so it would be extremely useful to define an analogue of Martin-Löf randomness that applies to the graphon space, and to create a notion of effective topology which can be used to separate random graphons of high complexity from nonrandom graphons which display high algorithmic regularity, which may in turn be fruitful in the study of random and nonrandom graphs. Showing that the space of graphons is an effective Polish space is some small progress in developing such an effective topology. These questions provide the opportunity for further research in this direction and we leave them for future study. 18

23 19 Bibliography [1] Yiannis N. Moschovakis. Descriptive Set Theory, Second Edition. American Mathematical Society, Providence, Rhode Island, [2] László Lovász. Large Networks and Graph Limits. Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary, [3] C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós, and K. Vesztergombi. Convergent sequences of dense graphs i: Subgraph frequencies, metric properties and testing. Advances in Mathematics, (219): , August [4] Alexander S. Kechris. New directions in descriptive set theory. Bulletin of Symbolic Logic, (5): , June [5] Ming Li and Paul Vitányi. An Introduction to Kolmogorov Complexity and Its Applications, Third Edition. Springer Science+Business Media, New York, NY, 2008.

24 EDUCATION ACADEMIC VITA PATRICK B. NICODEMUS The Pennsylvania State University University Park, PA Schreyer Honors College Graduation: May 2017 Bachelor of Science in Mathematics Bachelor of Science in Electrical Engineering Dean's List ABOUT ME Undergraduate preparing for graduate study in mathematical logic. Primary interests: model theory, theory of computation, extremal graph theory, mathematical logic, cryptography, networking, category theory, machine learning. EXPERIENCE Moody's Analytics New York City, NY Data Science Intern June 2015 August 2015 Analyzed statistical data on network failures to evaluate the stability, resiliency and redundancy of networks Wrote R scripts to parse, analyze and summarize the contents of Neo4j graph databases Developed interactive data visualization applications in D3.js and Angular.js Undergraduate Research University Park, PA Studied Kolmogorov machines, a model of computation which operates on a graph December 2014-Present Coded simulations in C++ to analyze properties of random graphs Currently working on computability properties of graphons (generalizations of a graph to a continuous structure) SKILLS Competent programmer in a variety of languages. C++, Java, C, Bash, Javascript, Python, Octave/Matlab, etc. Some software development experience in a professional setting, knowledge of associated tools (e.g. Git) LEADERSHIP EXPERIENCE Penn State Collegiate Wind Competition Electrical Subsystem Lead Oct Present Designed and built a two-stage axial flux generator and an active rectifier for a 40W wind turbine Designed and coded the control system for the turbine, including a blade pitch controller and braking mechanism Boy Scouts of America Senior Patrol Leader Oct Jun 2013 Eagle Scout and member of the Order of the Arrow, Scouting's national honor society Organized meetings, resolved difficulties, assigning responsibilities and following up on their completion Served as Crew Chief at Philmont Scout Ranch, leading a 78 mile backpacking trek through the New Mexico desert and maintaining morale and unit cohesion through the 12 day journey. Moody's Mega Math Challenge March 2013 Placed 3rd in the competition, demonstrating ability to think critically, analyze and mathematically model Worked with teammates under enormous pressure and tight time constraints Demonstrated academic prowess and ability to critically reason through complex problems Awarded a $10,000 scholarship for placing in the competition