Functional Integration of Ecological Networks through Pathway Proliferation

Large-scale structural patterns commonly occur in network models of complex systems including a skewed node degree distribution and small-world topology. These patterns suggest common organizational constraints and similar functional consequences. Here, we investigate a structural pattern termed pathway proliferation. Previous research enumerating pathways that link species determined that as pathway length increases, the number of pathways tends to increase without bound. We hypothesize that this pathway proliferation influences the flow of energy, matter, and information in ecosystems. In this paper, we clarify the pathway proliferation concept, introduce a measure of the node--node proliferation rate, describe factors influencing the rate, and characterize it in 17 large empirical food-webs. During this investigation, we uncovered a modular organization within these systems. Over half of the food-webs were composed of one or more subgroups that were strongly connected internally, but weakly connected to the rest of the system. Further, these modules had distinct proliferation rates. We conclude that pathway proliferation in ecological networks reveals subgroups of species that will be functionally integrated through cyclic indirect effects.Comment: 29 pages, 2 figures, 3 tables, Submitted to Journal of Theoretical Biolog

Governing a Common-Pool Resource in a Directed Network

A local public-good game played on directed networks is analyzed. The model is motivated by one-way flows of hydrological influence between cities of a river basin that may shape the level of their contribution to the conservation of wetlands. It is shown that in many (but not all) directed networks, there exists an equilibrium, sometimes socially desirable, in which some stakeholders exert maximal effort and the others free ride. It is also shown that more directed links are not always better. Finally, the model is applied to the conservation of wetlands in the Gironde estuary (France).Common-pool Resource, Digraph, Cycle, Independent Set, Empirical Example

내차수가 0인 점을 갖는 유향 그래프의 m-step 경쟁 그래프인 수형도

학위논문(석사)--서울대학교 대학원 :사범대학 수학교육과,2019. 8. 김서령.Cohen [1] introduced the notion of competition graph while studying predator-prey concepts in ecological food webs. Among the variants of competition graphs, the notion of m-step competition graph to be studied in this thesis, was introduced by Cho et al. [2]. In 2000, Cho et al. [2] posed the following question: For which values of m and n is P_n an m-step competition graph? Helleloid [4] and Kuhl et al. [5] partially answered the question in 2005 and 2010, respectively. In 2011, Belmont [6] presented a complete characterization of paths that are m-step competition graphs. In this thesis, we study ``tree-inducing digraphs" with a source. We call a digraph D with at least three vertices an m-step tree-inducing digraph if the m-step competition graph of D is a tree for some integer m greater than or eqaul to 2. We say that a digraph is a tree-inducing digraph if it is an m-step tree-inducing digraph for some integer m greater than or eqaul to 2. We first completely characterize a tree-inducing digraph with a source. Interestingly, it turns out that if a tree is the m-step competition graph of a digraph with a source, then it is a star graph. We also compute the number of tree-inducing digraphs with a source.Cohen(1968)은 생태계의 먹이사슬에서 포식자-피식자 개념을 연구하면서 경쟁그래프의 개념을 고안하였다. Cho 외(2000)은 경쟁그래프의 많은 변형들 중의 하나로서 m-step 경쟁그래프라는 개념을 만들어 내었고 P_n이 m-step 경쟁그래프가 될 수 있는 m과 n에 대한 문제를 제기하였다. Helleloid(2005)와 Kuhl 외(2010)은 이 문제에 대한 부분적인 답을 제시하였다. Belmont(2011)는 m-step 경쟁그래프인 패스에 대하여 완벽하게 규명하였다. 이 논문에서는 내차수가 0인 점을 갖는 수형도 유발 유향그래프에 대하여 연구하였다. 점을 3개 이상을 갖는 유향그래프 D가 2이상의 어떤 정수 m에 대한 m-step 경쟁그래프가 수형도일 때, D를 m-step 수형도 유발 유향그래프라고 부른다. m-step 수형도 유발 유향그래프를 수형도 유발 유향그래프라고 부른다. 우선, m-step 경쟁그래프가 수형도인 내차수가 0인 점을 갖는 유향 그래프의 구조를 완전하게 규명하였다. 흥미롭게도, 내차수가 0인 점을 갖는 유향 그래프의 m-step 경쟁그래프가 수형도일 때는 항상 별 그래프임을 보였다. 최종적으로는 m-step 경쟁그래프가 수형도인 내차수가 0인 점을 갖는 유향 그래프의 개수를 구하였다.Abstract 1. Introduction. 1.1 Basic graph terminology - 1 -. 1.2 Competition graph and its variants - 3 -. 1.3 m-step competition graphs - 4 -. 2. Tree-inducing digraphs 2.1 Some properties of tree-inducing digraphs - 6 -. 2.2 An idle vertex of a tree-inducing digraph - 10 -. 3. Tree-inducing digraphs with a source 3.1 A characterization of tree-inducing digraphs with a source - 19 -. 3.2 The number of tree-inducing digraphs with a source - 24 -. Bibliography - 28 -. Abstract (in Korean) - 32 -. Acknowledgement (in Korean) - 33 -.Maste

Population Dynamics on Complex Food Webs

In this work we analyse the topological and dynamical properties of a simple model of complex food webs, namely the niche model. In order to underline competition among species, we introduce "prey" and "predators" weighted overlap graphs derived from the niche model and compare synthetic food webs with real data. Doing so, we find new tests for the goodness of synthetic food web models and indicate a possible direction of improvement for existing ones. We then exploit the weighted overlap graphs to define a competition kernel for Lotka-Volterra population dynamics and find that for such a model the stability of food webs decreases with its ecological complexity.Comment: 11 Pages, 5 Figures, styles enclosed in the submissio

Network Characteristics and Dynamics: Reciprocity, Competition and Information Dissemination

Networks are commonly used to study complex systems. This often requires a good understanding of the structural characteristics and evolution dynamics of networks, and also their impacts on a variety of dynamic processes taking place on top of them. In this thesis, we study various aspects of networks characteristics and dynamics, with a focus on reciprocity, competition and information dissemination. We first formulate the maximum reciprocity problem and study its use in the interpretation of reciprocity in real networks. We propose to interpret reciprocity based on its comparison with the maximum possible reciprocity for a network exhibiting the same degrees. We show that the maximum reciprocity problem is NP-hard, and use an upper bound instead of the maximum. We find that this bound is surprisingly close to the empirical reciprocity in a wide range of real networks, and that there is a surprisingly strong linear relationship between the two. We also show that certain small suboptimal motifs called 3-paths are the major cause for suboptimality in real networks. Secondly, we analyze competition dynamics under cumulative advantage, where accumulated resource promotes gathering even more resource. We characterize the tail distributions of duration and intensity for pairwise competition. We show that duration always has a power-law tail irrespective of competitors\u27 fitness, while intensity has either a power-law tail or an exponential tail depending on whether the competitors are equally fit. We observe a struggle-of-the-fitness phenomenon, where a slight different in fitness results in an extremely heavy tail of duration distribution. Lastly, we study the efficiency of information dissemination in social networks with limited budget of attention. We quantify the efficiency of information dissemination for both cooperative and selfish user behaviors in various network topologies. We identify topologies where cooperation plays a critical role in efficient information propagation. We propose an incentive mechanism called plus-one to coax users into cooperation in such cases

Hole의 관점에서 그래프와 유향그래프의 구조에 관한 연구

학위논문(박사)--서울대학교 대학원 :사범대학 수학교육과,2019. 8. 김서령.이 논문에서는 유향그래프와 그래프의 홀의 관점에서 계통발생 그래프와 그래프의 삼각화에 대하여 연구한다. 길이 4 이상인 유도된 싸이클을 홀이라 하고 홀이 없는 그래프를 삼각화된 그래프라 한다. 구체적으로, 싸이클을 갖지 않는 유향그래프의 계통발생 그래프가 삼각화된 그래프인지 판정하고, 주어진 그래프를 삼각화하여 클릭수가 크게 차이 나지 않는 그래프를 만드는 방법을 찾고자 한다. 이 논문은 연구 내용에 따라 두 부분으로 나뉜다. 먼저 $(1, i)$ 유향그래프와 $(i, 1)$ 유향그래프의 계통발생 그래프를 완전하게 특징화하고, $(2, j)$ 유향그래프 $D$의 모든 유향변에서 방향을 제거한 그래프가 삼각화된 그래프이면, $D$의 계통발생 그래프 역시 삼각화된 그래프임을 보였다. 또한 적은 수의 삼각형을 갖는 연결된 그래프의 계통발생수를 계산한 정리를 확장하여 많은 수의 삼각형을 포함한 연결된 그래프의 계통발생수를 계산하였다. 다른 한 편 그래프 $G$의 비삼각화 지수 $i(G)$에 대하여 $\omega(G^*)-\omega(G) \le i(G)$를 만족하는 $G$의 삼각화된 그래프 $G^*$가 존재함을 보였다. 그리고 이를 도구로 이용하여 NC property를 만족하는 그래프가 Hadwiger 추측과 Erd\H{o}s-Faber-Lov\'{a}sz 추측을 만족함을 증명하고, 비삼각화 지수가 유계인 그래프들이 linearly $\chi$-bounded임을 증명하였다.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts. In the first part, we completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we show that the phylogeny graph of a $(2,j)$ digraph $D$ is chordal if the underlying graph of $D$ is chordal for any positive integer $j$. In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles. In the second part, we present a minimal chordal supergraph $G^*$ of a graph $G$ satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly $\chi$-bounded.Contents Abstract i 1 Introduction 1 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Graph colorings and chordal completions . . . . . . . . 14 2 Phylogeny graphs 19 2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19 2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20 2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28 2.2 The phylogeny number and the triangles and the diamonds of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 A new minimal chordal completion 61 3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64 3.2 The Erd˝ os-Faber-Lovász Conjecture . . . . . . . . . . . . . . . 73 3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80 3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80 3.3.2 Making a local chordalization really local . . . . . . . . 89 3.4 New χ-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97 Abstract (in Korean) 107Docto

생태계에서의 경쟁 관점으로 그래프와 유향그래프의 구조 연구

학위논문(박사) -- 서울대학교대학원 : 사범대학 수학교육과, 2023. 2. 김서령.In this thesis, we study m-step competition graphs, (1, 2)-step competition graphs, phylogeny graphs, and competition-common enemy graphs (CCE graphs), which are primary variants of competition graphs. Cohen [11] introduced the notion of competition graph while studying predator-prey concepts in ecological food webs.An ecosystem is a biological community of interacting species and their physical environment. For each species in an ecosystem, there can be m conditions of the good environment by regarding lower and upper bounds on numerous dimensions such as soil, climate, temperature, etc, which may be represented by an m-dimensional rectangle, so-called an ecological niche. An elemental ecological truth is that two species compete if and only if their ecological niches overlap. Biologists often describe competitive relations among species cohabiting in a community by a food web that is a digraph whose vertices are the species and an arc goes from a predator to a prey. In this context, Cohen [11] defined the competition graph of a digraph as follows. The competition graph C(D) of a digraph D is defined to be a simple graph whose vertex set is the same as V (D) and which has an edge joining two distinct vertices u and v if and only if there are arcs (u, w) and (v, w) for some vertex w in D. Since Cohen introduced this definition, its variants such as m-step competition graphs, (i, j)-step competition graphs, phylogeny graphs, CCE graphs, p-competition graphs, and niche graphs have been introduced and studied. As part of these studies, we show that the connected triangle-free m-step competition graph on n vertices is a tree and completely characterize the digraphs of order n whose m-step competition graphs are star graphs for positive integers 2 ≤ m < n. We completely identify (1,2)-step competition graphs C_{1,2}(D) of orientations D of a complete k-partite graph for some k ≥ 3 when each partite set of D forms a clique in C_{1,2}(D). In addition, we show that the diameter of each component of C_{1,2}(D) is at most three and provide a sharp upper bound on the domination number of C_{1,2}(D) and give a sufficient condition for C_{1,2}(D) being an interval graph. On the other hand, we study on phylogeny graphs and CCE graphs of degreebounded acyclic digraphs. An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i, j) digraph for some positive integers i and j. If each vertex of a (not necessarily acyclic) digraph D has indegree at most i and outdegree at most j, then D is called an hi, ji digraph. We give a sufficient condition on the size of hole of an underlying graph of an (i, 2) digraph D for the phylogeny graph of D being a chordal graph where D is an (i, 2) digraph. Moreover, we go further to completely characterize phylogeny graphs of (i, j) digraphs by listing the forbidden induced subgraphs. We completely identify the graphs with the least components among the CCE graphs of (2, 2) digraphs containing at most one cycle and exactly two isolated vertices, and their digraphs. Finally, we gives a sufficient condition for CCE graphs being interval graphs.이 논문에서 경쟁그래프의 주요 변이들 중 m-step 경쟁그래프, (1, 2)-step 경쟁 그래프, 계통 그래프, 경쟁공적그래프에 대한 연구 결과를 종합했다. Cohen [11]은 먹이사슬에서 포식자-피식자 개념을 연구하면서 경쟁그래프 개념을 고안했다. 생태계는 상호작용하는 종들과 그들의 물리적 환경의 생물학적 체계이다. 생태계의 각 종에 대해서, 토양, 기후, 온도 등과 같은 다양한 차원의 하계 및 상계를 고려하여 좋은 환경을 m개의 조건들로 나타낼 수 있는데 이를 생태적 지위(ecological niche)라고 한다. 생태학적 기본가정은 두 종이 생태적 지위가 겹치면 경쟁하고(compete), 경쟁하는 두 종은 생태적 지위가 겹친다는 것이다. 흔히 생물학자들은 한 체제에서 서식하는 종들의 경쟁적 관계를 각 종은 꼭짓점으로, 포식자에서 피식자에게는 유향변(arc)을 그어서 먹이사슬로 표현한다. 이러한 맥락에서 Cohen [11]은 다음과 같이 유향그래프의 경쟁 그래프를 정의했다. 유향그래프(digraph) D의 경쟁그래프(competition graph) C(D) 란 V (D)를 꼭짓점 집합으로 하고 두 꼭짓점 u, v를 양 끝점으로 갖는 변이 존재한다는 것과 꼭짓점 w가 존재하여 (u, w),(v, w)가 모두 D에서 유향변이 되는 것이 동치인 그래프를 의미한다. Cohen이 경쟁그래프의 정의를 도입한 이후로 그 변이들로 m-step 경쟁그래프(m-step competition graph), (i, j)-step 경쟁그래프((i, j)-step competition graph), 계통그래프(phylogeny graph), 경쟁공적그래프(competition-common enemy graph), p-경쟁그래프(p-competition graph), 그리고 지위그래프(niche graph)가 도입되었고 연구되고 있다. 이 논문의 연구 결과들의 일부는 다음과 같다. 삼각형이 없이 연결된 m-step 경쟁 그래프는 트리(tree)임을 보였으며 2 ≤ m < n을 만족하는 정수 m, n에 대하여 꼭짓점의 개수가 n개이고 m-step 경쟁그래프가 별그래프(star graph)가 되는 유향그래프를 완벽하게 특징화 하였다. k ≥ 3이고 방향지어진 완전 k-분할 그래프(oriented complete k-partite graph)의 (1, 2)-step 경쟁그래프 C_{1,2}(D)에서 각 분할이 완전 부분 그래프를 이룰 때, C_{1,2}(D)을 모두 특징화 하였다. 또한, C_{1,2}(D)의 각 성분(component)의 지름(diameter)의 길이가 최대 3이며 C_{1,2}(D)의 지배수(domination number)에 대한 상계와 최댓값을 구하고 구간그래프(interval graph)가 되기 위한 충분 조건을 구하였다. 차수가 제한된 유향회로를 갖지 않는 유향그래프(degree-bounded acyclic digraph)의 계통그래프와 경쟁공적그래프에 대해서도 연구하였다. 양의 정수들 i, j에 대하여 (i, j) 유향그래프란 각 꼭짓점의 내차수는 최대 i, 외차수는 최대 j인 유향회로 갖지 않는 유향그래프이다. 만약 유향그래프 D에 각 꼭짓점이 내차수가 최대 i, 외차수가 최대 j 인 경우에 D를 hi, ji 유향그래프라 한다. D가 (i, 2) 유향그래프일 때, D의 계통그래프가 현그래프(chordal graph)가 되기 위한 D의 방향을 고려하지 않고 얻어지는 그래프(underlying graph)에서 길이가 4이상인 회로(hole)의 길이에 대한 충분조건을 구하였다. 게다가 (i, j) 유향그래프의 계통그래프에서 나올 수 없는 생성 부분 그래프(forbidden induced subgraph)를 특징화 하였다. (2, 2) 유향그래프 D의 경쟁공적그래프 CCE(D)가 2개의 고립점(isolated vertex)과 최대 1개의 회로를 갖으면서 가장 적은 성분을 갖는 경우일 때의 구조를 규명했다. 마지막으로, CCE(D)가 구간그래프가 되기 위한 성분의 개수에 대한 충분조건을 구하였다.1 Introduction 1 1.1 Graph theory terminology and basic concepts 1 1.2 Competition graphs and its variants 6 1.2.1 A brief background of competition graphs 6 1.2.2 Variants of competition graphs 8 1.2.3 m-step competition graphs 10 1.2.4 (1, 2)-step competition graphs 13 1.2.5 Phylogeny graphs 14 1.2.6 CCE graphs 16 1.3 A preview of the thesis 17 2 Digraphs whose m-step competition graphs are trees 19 2.1 The triangle-free m-step competition graphs 23 2.2 Digraphs whose m-step competition graphs are trees 29 2.3 The digraphs whose m-step competition graphs are star graphs 38 3 On (1, 2)-step competition graphs of multipartite tournaments 47 3.1 Preliminaries 48 3.2 C1,2(D) with a non-clique partite set of D 51 3.3 C1,2(D) without a non-clique partite set of D 66 3.4 C1,2(D) as a complete graph 74 3.5 Diameters and domination numbers of C1,2(D) 79 3.6 Disconnected (1, 2)-step competition graphs 82 3.7 Interval (1, 2)-step competition graphs 84 4 The forbidden induced subgraphs of (i, j) phylogeny graphs 90 4.1 A necessary condition for an (i, 2) phylogeny graph being chordal 91 4.2 Forbidden subgraphs for phylogeny graphs of degree bounded digraphs 99 5 On CCE graphs of (2, 2) digraphs 122 5.1 CCE graphs of h2, 2i digraphs 128 5.2 CCE graphs of (2, 2) digraphs 134 Abstract (in Korean) 168 Acknowledgement (in Korean) 170박

On Opsut’s conjecture for hypercompetition numbers of hypergraphs

AbstractThe notion of the competition hypergraph was introduced as a variant of the notion of the competition graph by Sonntag and Teichert in 2004. They also introduced the notion of the hypercompetition number of a graph.In 1982, Opsut conjectured that for a locally cobipartite graph G, the competition number of G is less than or equal to 2 and the equality holds if and only if the vertex clique cover number of the neighborhood of v is exactly 2 for each vertex v of G. Despite the various attempts to settle the conjecture, it is still open. A hypergraph version of the Opsut’s conjecture can be stated as the assertion that for a hypergraph H, if the number of hyperedges containing v is at most 2 for each vertex v of H, then the hypercompetition number of H is less than or equal to 2 and the equality holds if and only if the number of hyperedges containing v is exactly 2 for each vertex v of H. In this paper, we show that this hypergraph version is true